MATHEMATICAL ECON OM ICS

ECON6042

:':::;';,

emai I address : iosephwu(decon.hku. hk

Office hours: 6:15 - 6:45pm Thursdays

Room 904 KKL Building

Or by appointment Telephone: 2&50-6905

ECONOMICS 6042

M ATH EM ATICAL ECON OM ICS

DR. J. WU / Jan/2010

This course aims to show the interconnection between mathematics and economics.

The cornerstone of modem economic theory is general equilibrium theory based onoptimization. Static and dynamic optimal models will be presented for studyingmicroeconomic and macroeconomic topics. In particular classical welfare economictheorems will be covered within nonlinear programming framework in the Euclideanspace R". Such analysis will be extended to more abstract spaces as a lead-in to thestudy of topological foundation for static and dynamic equilibrium and optimizationmodels. Rehrning to the Euclidean space, application of these abstact concepts in theform of Ponfyagin optimal contol theory and recursive methods based on Bellmanstochastic dynamic programming will be used for analyzing the all-important macrotopics of economic growth and employmen! illustating the importance and relevance ofthese advanced theories.

Textbook: Lecture notes for the whole course will be available at my WebCT, HKUPortal

Reference textbooks:

Olq E.A., Real analysis with Economic Applications, Princeton Univ. Press , Princeton2007

Stokey, N.L. & Lucas, R.E. (with Prescott, E.C,), Recursive Methods in DynamicE conom ics, Ilarvard University Press, Canrbridge 1 989

Hoy, M., Livernois, J., McKenna, C., Rees, R., Stengos,T., Mathematics for Economics2ed. Cambridge, Massachusetts. The MT Press 2001

Ljungqvist, L, and Sargent, T.J., Recursive Macroeconomic, MIT Press, Cambridge 2000

Romer, D.,Advanced Macroeconomics 2ed. , McGraw-Hill, New York 2000

Takayama, A-, Mathematical Economics 2ed, Cambridge U. Press, Cambridge 1985

0

Reference Bibliography:

Akerlof, G. "The Ivlarket for 'Lemons': Quality Uncertainty and the Market Mechanism" , QuarterlyJournal of Economics 1970

Arrow, K. J., "The Role of Securities in the Optimal Allocations of Risk-Bearing," Reiew of EconomicStudies 3l (April l96a)

Bellman, kE, Dynamic Programming, Princeton University Press, Princeton 1957

Bellman, RE., Dreyfus,S.E.,Applied Dlmamic Programming,Princeton U. Press, Princeton N.J 1962

Border, K.,Fixed Point Theoremswth application to konomics and Game Theory, Cambridge UnivenityPress, Cambridge Reprinted 2003

Bouabdallah, K., Jellal, M., Wolff, F. "Unemployment and work sharing in an efiiciency wage model"Economics Bul/ear, Vo.' 0,No.3 pp.l -7, 20[,4

Browning, M., I{ansen, L.P. & Heckman, J.J., "Micro Data and General Equilibrium Models" in Tayloq J.& Woodford, M. (eds/ Handbook of Macroeconomics, Norlir Holland, Amsterdam 2000

Capinski, N., Kopp, P.,Measurc,Integral & Probability 2ed Springer-Verlag London 2004

Cass, D., Optimum Growth in an Aggregative Model of Capital Accumulation, Reiew of EconomicSndies 1965

Chiang, A., Wainwrigh\K, Fundamenal Methods ofMathematical Economics 4ed, McGraw HillInternational Edition 2005

Chow, G. C.,D)mamic fuonomics: Optimization by the Iagrange Merhd, Oxford U. Press, Oxford 1997

Debreu, G., Theory of Value,, Wiley, New York I 95 I

De La Croix, D., Mchel, P.A Theory of Economic Growth, Dynamics and Policy in OverlappingGenerations, Cambridge University Press, Cambridge 2002

Dixig A K.,Optimimtion in Economic Theory,2ed, Oxford, OxfordUnivenitypress 1990

Dreyfus, S.E., Law, A.M., The Art and Theory of Dynamic Programming,Academic Press, N.y. 1977

Farmer, R.E. A.,Macroeconomics, South-Westem College Publishing Cincinnati 1999

R E. Hall, "Tumover in the Labor Forcn". Brookings Papers on konomic Activity 3 ,1972

Heathfiel4 D (Ed ), Topics inApplied Macroeconomics. MacMillan Press Ltd., London 1976

Hillier, F.S., Lieberman, G.J., Introduction to Mathematical Progmmming,McGraw-Hill, N.Y. 1990

Hoy, M., Livemois, J., McKenna, C., Rees, R., Stengos, T.,Mathematicsfor Economics 2ed. The MITPress, Cambridge, IMassachusetts 200 I

Huang, T., Ilallam., 4., Arazem, P., Patemo, E. "Empirical Tests of Efficiency Wage Models", Economicavol.65, ppl25-43 1998

Intriligator, M.D., Mathematical Optimimtion & Economic Theory, Prentice Hall , New Jen ey l97l

Kamien, M and Schwartz, N., Dynamic Optimization 2ed Elsevier North Holland, Amsterdam 2003

Knight, F., Risks, Uncertainty &Proft,Houghton Mifflin, Boston l92l

Kubrusky , C., Measure Theory A First Course, Elsevier Academic Press lvfass, 2007Kusuoka" S., and Maruyama, T. @ds.) Advances in Mathemqtical konomics, Springer, Tokyo 1999

Layard, R, Nickell, S., Jackman R,(Jnemployment Macroeconomic Performance and the Inbour Market,2ed.,Oxford U. Press, Oxford 2005

Leslie, D, Advwnced Macroeoconomlcs, Beyond IS/LIvI, McC'raw-Hill Book Co. London 1993

Ljungqvist, L, and Sargent,T.J.,Recarsive Macroecanomic Theory, MIT Press, Cambridge 2000

Lucas, R. E. Jr.., *On the Mechanics of Economic Development ", J of Monetary Economics 22 (1988\

Luenberger, D.G., Microeconomic Theory, McGraw-Hill Inc., New York1997

I\{ankiq N .G., Macrceconomics 5ed, Worth Publishers, New York 2003

ilfinford, P, and Peel, D, Advanced Macroeconomics, A Primer,Edward Elgar, Chelterham,UK.2002

Muth, J. F., Rational Expectations and the Theory of Price Movements, Econometrica 29:315-335 (1961)

Negishi, T., General fuuilibium Theory & International Trade,North Holland Publishing Co.,Amsterdam 1972

Oh E.A., Real Analysis with Economic Applications, Princeton University Press, Princeton, 2007

Pissarides, C.A., Equilibrium Unemployment Theory,2d MIT Press 2000

Radner, R, Paths of Economic C'rowttr that are Optimal with regard only to Final States,.Review ofEconomic Sudies 196l

Ramsey, F.P., Al\4athematical Theory of Saving. Economic Joumal3S:543-559 (1928)

Romer, D ., Advanced Macraeconomics 2d, , McGraw-Hill, New York 2000

Romer, P.M., "Increasing Retums and Iong-Run Growth", Joumal of Political Economy 94 (Oct 1986)

Samuelson, P. A., Foundations of konomic Analysis, Harvard University Press, Cambridge, IVIass. 1947

Simon, C.P., and Blume, L.,Mathematicsfor Economislr, New York, W W. Norton 1994

Smith, W. T., *A Closed Form Solution to the Ramsey Model" Contributions to Macroeconomics: Yol.6:No. I Article 3, 2006 (http://www.bepress.com/bejm/contributions/vol6/issl /art3)

Silbe6erg, E. & Suen, Wing,The Structure of Economics, AMa*tematical Analysis,, Irwin McGraw-HillIntemational Edition, Boston 2001

Stiglitz, J.E. & Weiss, A "Credit Rationing in lzlarkets with Imperfect Information" I merican konomicReview l98l

Stokey, N.L. & Lucas, RE. (with Prescotl E.C.), Recursive Methods in Dynamic konomics,I{arvardUniversity Press, Cambridge 1989

Sun, W., Yuan, Y. Optimization Theory &Methods: Nonlinear Programming, Springer Science-BusinessMediaIIC, U.S.A.2006

Takayama, 4., Analytical Methods in Economics,, University ofMichigan Press, Ann Arbor 1993

Takayama, 4., Mathematical honomics 2ed , cambridge univenity Press, cambridge 1985

Tobin, James, (selected by) Landmark Papers in Macroeconombs,E dward Edgar Publishing Ltd.Northampton, Ndass. 2002

Tobin, James, "On the Efficiency of the Financial System", trloyds Bank Reiew July l9B4

Tinbergen, J. and Bos, H.C., ,Mathematical Models of Economic Growth,McGraw Hill , New York 1962

Uz"awa, H., Optiimal Growth in a Two-sector Model of Capital Accumulation, Review of Economic Swdies1964

Van, C.L., Dana, R, Dynamic Programming in Economics,Kluwer Academic Publishers, Dordrecht 2003

Vohrq k, Advanced Mathematical konomics,Routledge, Oxon 2005

Von Neumann, John, *Model of General Equilibrium', Review of Economic Studies 1945

Von Neumann,Joln,The Mathematical Foundation of QuanumMechanics, tanslated by R. Beyer,Princeton, Princeton Univenify Press 1955

Von Neumann, John, & Morgentstem,O., Theory of Games & Economic Behawor, Princeton, New York,1955

Wan, F.Y.M. ,Introduction to the Calcalus ofYariations and its Applications,Chapman & Ilall, New Yorkt995

Weinstock, k Calailus of Vaiations, Dover, NewYork 1974

Whinston, A, Moore, J., Wu, J. 'Resource Allocation in a Non-Convex Economy'', Review of EconomicSudies 1973

Chapter L A.

What is Economics? Tvpical definitions:

./\Study of resource allocation,production and distribution.

branch of social science.

-/ studies human behaviour.

Actually want

most efficie nt (Opti m al)way to allocate resources,to produce & to distribute.(if do not want to optimize, not necessary to study

IU

most efficient(Optimal)way to behave/do things.

econ)

: optimization, but scarce resources constaints(ifno scarcity, not necessary to study econ)

ut constrained optimization: max/min some goalssubject to some constaints

Prof. StevenN.S. Cheung FffitHtU ,\!llEArcadia Press 2000 page24-25(to paraphrase Prof. Cheung, only 2 basic principles in Econ.

l. constained optimization (by varying constaints, can predicVobserve behavioralchanges)

2. downward sloping demand function D.

Microeconomics:Study of individualProducers and consumers in markets

constrained optimization by economic agents

Macroeconomics:study of the economyas a whole

economy made up of economicagents' actions and reactions andthese economic agents (macro jargon:households and firms) all constainedoptimizing I micro foundation ofmacro

Remark:

But optimization as the main principle to study economics went through stages ofopposition and refinement.

Adam Smith's main theme of individuals pursuing seliinterest (optimaation) andinvisible hand (prices and markets) will lead the situation to a social optimality (result ofoptimization).

Social writers like Karl Man reiected this idea as economics manv times do notalleviate poverty.

J. M. Keynes wrote the General Theory because of unemployment during the GreatDepression in 1930s which ran contrary to the full employment predicted bymicroeconomics theory. The attention shifted from optimnationto aggregate concepts inthe economy.

All the while, classical economics, especially the Ghicago School, went throughrefinements and modification. In 1990s, the tide seems to shift back towardsoptimization by rational (self-interest) economic agents and the resultant marketequilibrium (G.E. models). Macroeconomics also was swept towards this generalequilibrium framework.

1. Static optimization is used to explain economic behavior at a given instance of time.2. Dynarnic optimization, dlmamic programming or Lagranglan methods are used to

explain economic behavior througb time.3. Stochastic features are added to include risks.

ECONOMIC -. OPTIMIZATION OR CONSTRAINED OPTIMIZATION

@

In Math, constrained optimization is called mathematical programming.

A) i) Static equilibrium models: A set of simultaneous math equations.

When 3 solution for this set of simultaneous equations, the P

model has an equilibrium point x*)

"*e.g. the familiar supply S@) and demand D(P) functions t

But what is behind S and D?S - derived from constrained profit-optimizing behavior andD - derived from utility-optimizing behavior under budget constraints.

In other words, there are optimal models underlying the equilibrium models.

ii) Static optimization models: Optimize (Max orMin) objective function f ,sometimes s.t. some constraint function g

e.g. Linear Programming LP- if objective & constraint fi,rnctions are all linear.

Nonlinear Programming NLP - when functions involved are not linear.e.g. concave programming: when objective F and consfaint G are concavefunctions over convex set - Max F (x,y,z) s.t. G (x, y, z) S K(Soln: (**, y*, ,*))

q*

B) i) Dynamic equilibrium models:

If time is discrete, math models described by difference equations.

If time is continuous, model uses differential equations.(solutions are family of functions or curves over time).

Underlyrng such dynamic equilibrium models are optimal dpamic models.

iD Dynamic optimization models: if objective function is an integral of differentialfunctions s. t. consfaints that are differential equations. (solutions are optimaltime paths f*(t))

Calculus of variations @uler's equation)

Ia,

More general theory

J,Pontryagin's Optimal Contol Theory

Find time sheam of Xnnol vble us.t. constaints on state variable x

-solution is opt control path u*(t)& opt state vble path x*(t)

Bellman Dynamic Pro gramming.,

basically a multi-stage decisionprocess based op recursive methods

--L/ \.\

stocl-rastic adaptive Bayesian

I O..irion TheoryI (dual contol)

Ii

I

certainty with risk risks can be J bylearning

(p d. fcn)

[C) Since this is a gadmath econ course, we will also cover general analysis, topologyand set theory to provide a math foundation of such consfiained optimization as well asmost general cases for optimization and existence of solutions. This is typical coverageof advanced grad math econ]

MATHEMATICS - OPTIMIZATION OR CONSTRAINED OPTIMIZATION

17ft century: pre calculus

18tr century:CONTINENT

Leibniz (1646 - 1714)

Developed calculus independently ofNewton,

ENGLAND

Newton (1642- 1727)

developed calculus to handlemotion (based on Fermat's idea)and used for Newtonian & celestialmechanics (laws of motion)

argued withNewton for calculus primacy.

LaGrange, Poisson, Johann & Jakob Bernoullitied to find general principle underlyingNewtonian mechanics so that calculus will onlvbe a special case ofthis general theory.

t0)=o ,*?IBased on Galileo 1630 discussion of brachistochroneproblem

(minimum time of sliding object from 0 ) p under Savlty g(which is a dyn opt problem)

in 1696 solved by Johann & Jakob Bernoulli, Leibniz and Newton

P. Mauperfuis, Euler noticed that many phenomena in Nature occur in the most efficientand most economical (no waste) way. Based on this metaphysical realization, Eulerestablished general principle to study all these simplifying, economizing phenomena inNature ) developed into a branch of math in1744 called Calculus of Variations.

E.g. In optics: Fermat's Principle I calculus of variations is aSoap bubbles I unifyrng theory for theseWater droplets I problemsGeodesics tHamilton's Principle IFields of elecbicity magnetism, relativity theory J

Nature economizing and optimizing) people subconsciously or conscious_ly mimicking nature and optimize?And since people optimizing before 17h century ) in-bred into human nature??Optimization is necessity for sustained Survival???

NATURE - OPTIMIZATION

INTERCONNECTED

MATH ECON

@

Chapter L B. Historical Development of Math Econ within general econ

'T*16m Century Mercantilism

-max Exportl

F *" precious metals (gold, silver)

I

-min Import.[

-Classical Economics

David Ricardo1817 Principles of Political Econ & Taxation

Econ activities by stateplanning & control

Adam Smith1776 nu rNeulRy tNTo rHE NATURE AND cAUSEs oF THE wEALTH oF NATIoNS

Invisible hand will bring

lffil*l(nade) I

social optimum& growth

"{.' ",

Thomas MalthusJohn Stuart Mill (state as "civilizer", forerunner of macro),Edgeworth, Pareto, Wicksell, Walras

-Neoclassical Econ

\ mparative Advantage@. Krugman 2008 trade pattems basod on econ ofscalg location. But still compamtive advantage?)

Karl Marx

Das Kapital

the

Rent

I

Alfred Marshall1890 Principles of Economics-Util max and profit fi max-Demand function D(P), +MU-Cambridge cash-balance D Theory- thouglr his writing mostly in plain English yet used math analysis

including phase diagram

///

A. C. Pisou I,/\

tvt*uritd"ti" *l\ I

of all i's util vs. L. Robbins I

(Pigou incorrect : as cannot'have interpersonal utilcomparison

J. M. Keynes

1936 General Theory of Employment,Interest & Monev

MACRO _

from whole economy's viewpoin!using aggregate variables like GNP,National Income, agg consumption C,

, agg Investrnent l, agg savings S,

I G (Gov expenditure), T (Tax overall)Pareto Optimality P.O.:

\situation where no one can be made better off \without somebody else beingmade worse off. \Thus avoiding problem of interpersonalutility comparison

MICRO _

view from each market and then addup to whole economy.

II

Micro DevelopmentI

1937 Von Neumann, J "Ueber Ein Oeknomisches Gleichungs-System und eineVerallgemeinerung des Brouwerschen Fixpunktsatzes" in Ergebnisse einesMathematischen Kolloquiums edited by K.Menger 1937franslated inl945-46 REStudies "A Model of General Equilibrium"

1930-40 Hicks, RGD Allen, A. Lerner, O. Lange, Samuelson using Calculus(marginal concepts like MU, MCrl\

r/\Theory of Consumer D Theory of Firm

Given: a set ofgivenprices (pr, pz, p:, ......., p$) =Pvectorconsumption bundle of consumer i (*t,, xa, xi3,withconsumptionmafixX: [xr, X2, X3, ......., xr]production bundle of produceri (yir, !iz, yit, yrN) : yj vectorwith productionmafix y: [yr, yz, yz, ......., yil

each consumer itvlax tI, (xi )

s.t. budget constaint Vi

s.t P**in(Incomeiconstained optimization

I

from ls order conditions of optimization, can derive solutions x io*(P), yj"*(P)

each producerj

Max rc1 = Max I pnyjo

s.t. production lonstaints Vj

good n j's supply fcns y;"*(P) of good n& demand firnctions of inputs interms of given P

which are

i's demand fcns x-*(P) ofin terms of price vector PLaw of D to assume I sloping D fcn

Define competition = all consumers and producers take prices as given.

(Price taking behavior using atomism as rationale: all economic agents are toosmall to affect prices and everyone will take prices as given).

I

I

If can find price vector P* : (pr*, pz*, p3*, pN-) s.t. in each nft marketTotal Demand D" (P*) =I p,,* xin * : Total Supply S" [P*) = I pr,*y;,,*

rithen P* is the market clearing price AND we note

[P*, x*, y*] solves Max Ui s.t. budget constaint Vi

Max ri s.t. production sst Vj

and P* clears nft market V nft markets

Then [P*, x*, y*] is called a General Equilibrium GE under Perfect CompetitionOR a Competitive Equilibrium C. E.

I

Fundamental Theorem of Welfare Economics (: Theorem of InvisibleHand):

Under perfect competition assumptions :

a) perfect information (no information cost, no risks)b) perfect exchange (no tansaction cost TC)c) no externalities (a11 econ agents consumption/production are independent)d) perfect knowledge (no risks)

[P*, x*,y*] is C.E. t [x*, y*] is P.O.(i.e. Equilibrium t Efficiency)

1ra 1'6f-0

Policy implication: explains why laissez faire (each agent left to pursuehis own constrained maximization) leads to social optimality; whydecentalized (as opposed centralized planned economy) is "good"because it is Pareto optimal.

|4ko20

Pure Price Theory(Welfare Economics)

Applied Price Theory(Chicago School)

1930-60 Keynesian Macro popular but U. ofChicago maintained Micro discipline with itspolicy implication of laissez faire/market econ

IViner, Hotelling, Simon (1978;no full information,decision based on bounded rationality), Douglas,

I

Frank Knight (risks & uncertainty)

if market rrotp.reri

Coase (1991) G. Stigler (1982)

no perfect exchange no perfect information:) Transaction cost in real life

Coase Thm: if TC:0 :> information cos!CE € t P.O. search cost to setProperfy rights to informationclarifu externality problems

F.Oshom (2009) common proptyRb can be managed by groupO. Williamson (2009)firm asstructure to rcsolve conllicts

if no atomism : not price taking behavior

M. FriedmanA. Director, Coaseproperfy rights toclarifu 1) incentive touse Price system(laissez faire &decenfralized market)2) externality (Coase

Theorem)

e.g. duopoly (2 sellers) oligopsony (few buyers)oligopoly (few sellers) monopsony (1 buyer)monopoly (single seller)

facing J sloping D facing fsloping S not P-taking

?

PX

b.Aa..6

6^df ir^/l*t U44f\

e,

sc

,/

I P-searching (Stigler's search cost)

rent-seeking behavior (Buchanan I 986)consumer surplus exfraction

\--

real life competition(Industial Organization topics)

incl. design of econinstitution: Hurwicz,E. Maskin, R.Myerson

il rtft.A.0

non-pricing behavior

- packaging- sales promotion

uGeneralized Neoclassical Micro Model (incorporating real life features)

How about risks?Via information cost; incur information costJ risksuntil at the margin, additional info cost: benefitfrom additional risk reduction. (But unsatisfactory)

Pure Price Theory (main model GE: C.E. t P.O.)Lemer and Lange proved converse in 1930-40

P.O.+C.E.

But calculus proof I small neighbourhood ('.' derivatives use limit concepts)I results only local and not global

1950 K. Arrow (1972) and G. Debreu (1983) (teacher M. Allais) independentlyextended to global results, using mathematical programming over sets (convexproduction set and concave utility functions);Hurwicz (2007), Uzawa ; Takayama, Whinston

) Given perfect competition (perfect information, exchange, no externalityand price-takers) and certainty (no risks) assumptions:

C.E. €) P. O. globally

General analysis & set theory in math used to extend results from Euclideanspace R o to more general topological spaces & provide proof of solutionexistence.

pncmge.g. cut prices

to compete

t4 6cQo

\

\

How about risks?/_ =_\_-*_ %

Game Theory Decision Theory(esp. B-School)

@ayesian, adaptiveModel )

Von Neumann-Morgenstern VN-M

Expected Utility

VN-M 1947 Theoryof Games &Economic Behavior(M. Allais 1988)

VonNeumann 1928Zur Theorie derGesellschaflspiele

I

action <+ reaction

IRadner, Shubik, Scart ShapleyNash (1994, Nash EquilibriumHarsanyi (1994), Selten (1994)

IGenerafrzed reaction function &Best response fi.rnctionsstrategic pricing and market models- core, coalition,- cooperative & noncooperative g:rmes

- bargaining, agency problems- stochastic games

R. Aumann, T. Schelling (2005)

D. KahnemnQ0A2)V. Smith Q002)

Arrow-Debreu State-contingentCommodities Model (Full Insurance Model)based on VltlM expected utility theorem

proved: given convexif5r assumptions andwith risk-free assumption relaxed, if I a fullinsurance market (or full state-of-nature)

then C.E. still €+ P. O.even under risky situations.

I

Qualifications : asymmetrical informationStiglitz, Spence, Akerlof(all 2001)

in fact, insurance market and other financial product markets(e.g. swaps, derivatives, options, stocks ... ): all markets for efficient risk re-allocation.Note: risks cannot be eliminated, it can be reduced by

informational activities until at the margin, benefit = cost;and then the remaining risks are re-allocated efficiently,again resulting in a P.O.

@

other a comPuter. If the qJres-

tioner cannot determine bY

the responses to queries Posedto theri which is the humanand which the comPuter, thenthe computer can be said to

be "thinicing" as weU as

the human.Ttrring remains a hero to

DroDonents of afificial intelli-gence ln part because ofhisEtithe aszumption of a rosy fu-hrre: "One day ladies will taketheir computers for walks inthe oark and tell each other,'tvty little comprrter said such a

funny thing this morning!'"Unfor' tunatelY,. realitY

causht up with Turing well be-fore-his vision would, if ever, be

realized. tn Manchester, he toldpolice investigating a robbery

at his house that he was having"an affair" with a man whowas probablyloown to the

burdar. AluraYs frank abouthis iexual orientation, lirringthis time got himself into real

houble. Homosexual relations

were still a felonY ia Britain,and Thring was tried and

convicted-of "gross indecenqlin 1959. He was sPared Prisonbut subjected to injectionsof female bormones intendedto dampen his lust 'fmorow1rriq,breasts!" TLrring told a

f,iend. bn June 7, 1954, he

committed zuicide bY

eating an apPle laced withcyanide. He was 4I.

Ttue setuior uniter ?utl GraY

arites on o Turing manhinn

irtually all comPuterstodaY, from$l0million supercomputers to the tinychips that Power cell Phones and

Furbies. have one thing in'common:they are all "Von Neumann.maehines,"variations on the basic comPuterarchitecture that John von Neumann' - ..

building on the work of Alaii Turing, laid outin the 1940s: Men have become famous tor-''less. But-in'the lifetima of.tbislHungarian- ' -'

born mathematician whorhad;.his handin: "

everything from quantum physics to U'S'policy during the cold waq the Vonfu"uriunn m-achine was almost the least of

By NATIIAN ilfYHR\rOtD the University oi aud"pesi ai'ttre age -ot

zei''After. immigrating to the U'S. in 1933'

von'tieum"nn ias hired, along vi'ith Albert6nstein" by the newly formed'lnstitute'for ;-' '

Advanced StudY in Princeton, NJ" a

nonprofit research institute set up by the

Barirbergei famity.with profits'lrom theirdeoartmlnt'stores. The l-Ls- proved to be

the -oerfect intellectual playgtrould for'Von - . :

Neumannts.boundless genius:-He;threw - ;

i,irit"i? "iiin-a"{trusiast

into"bneiintractabie*oroblem after another;ranging-'fror* the'' :

abstractrnathematics:ofiguanturn.. ) .

mechanics to. the practical problems ofweatherprediction' hydrology and'the'

iliilldiiis, such as numbers ortext, also

held the stepby-step instructions that would

allow the machine to be programmed tooerform any task Von Neumann persuaded'ih.

t.n s.'t Jomewhat skeptical board oftrustees to allocate $1O0,OOO-quite a sum in

1945-to buitd the trnNnc, the first in a series'of'earlv Von Neumann machines thatinctua6a the JoHNNIAc (at Argonne National

' Laboratory) and the IBM 70L one of theprogenitors of IBM's enormously profitable'

i' mainft ame lihes.. ' . I recently visited the i.A-s. archives

'ndpaged throu!h Von Neumann's handwritteniaf,notebooks describing the constructionand testing of his primitive computersvstems. Interspersed with technical data

a?e,comments such as; "5 a.m.: l've been at

this all night, and I still can't find theproblem.J'm disgusted and I'm going tobedl"-a sentiment anY computer

. programmer will recognize..Von'Neumannbian't just design the stored'programcomouter; he was the first hacker.

hs rivatry with the Soviet Union heated

uo, Von Neumann became a strategic adviser

on defense policy. He was appointed by

President Dwight O. Eisenhower to theAtomic Energy Commissioni which oversaw

the oostwar LuitOup ot the U.S. nucleararsenal. Von Neurnann's game theory became

a tool to analyze the unthinkable-glob^alnuclear war-and led t6 the doctrine of;mutu"tty assured destruction'" which would

shape U.S. strategy for the next two decades'

Von Neumann also became an icon of the

"oiO *"t. Oit"Uled with pancreatic cancer' he

stoicallv continued to attend AEc meetings,ntii nii death in 1957. The wheelchair:b'oundscientist with the Hungarian accent who

mathematically analyzed doomsday is

said to have bien a model for StanleyKubrick's Dr. Strangelove. r

Nothan Muhnold., chief technologg ofrcttJor Micrositft, colbcts old ntpercornputers

his accomplishments.

.: a!. economic's and.'evolutionarytheorv: The 1997 Nobel Prize in '

I Economics was awarded to'eame theorists, the seventhiobel Prize that grew out of VonNeumann'b ideas:.' With theonset of WorldWar ll, Von Neumann was

:!E5 recruited tor the Manhattan:l-\ Project and PlaYed a role in

ffiffi j: *' :l**'d:'fllfJ',i:;f *HT"'"",JI:ONE OF SEVERAL SEMINAL"VON NEUTJANN:MACru!-'IES* bution was supervising the vastiL'Euni lr'iAE tNsrtrurE FoR ADVANcED sruoY ind comptex mathematical

BorntoprosperousJewishparentsincalcu|ations_donefirstbyhandand|aterbyBudaoest in 1g03, vun N"u."ni w"i a primitive electronic computers-required to

ittiiJiroaiey who could divide eight-digit design,the bombs'

nur"Ulri in'[,is f,eaa by age iix,-lEarned' After the war, he returned to the l^.s.

;;i;;ilt-;; "LL "igt

t jndamuiea nis and became obsessed with computing' von

parents' friends by glanclng at a phone Neumann's vision for the machines went

book and reciting wnote pales verbatim. beyond the rote arithmetic tasks for which

Mathematics quickly oecanie the {ocus of they were o1iginally d::g!:9:.L}t idealized

his studies, culminating in a Ph.D. from computer, thJsaml memory units that held

I

HtS COLOSSUS CRUNCHED NUMBERS FOR BRITAIN IN WORLD WAR II

104 TIlvlE, lvlARCH 29' 1999

Samuelson (1 970), Robinson Stone (1984)

MACRO

Agg concepts C,I, G, T

parallel development

GNP as measurement ofnational income, measurementof C, I, G, Y, L, M etc

ECONOMETRICS(Econ + statistics)

Tinberger (19 69), R.Frisch( I 969)Klein ( I 980),Kuznets( I 97 I )

Haavelmo(l 989 Identification Problem)

Samuelson' s Keynesian Crgss

(,r,Q t lz

+*I

t 1 ,EW u-{4*r*(;€

Y.Tac;{

if replace G by net gciv expenditure g: G - T

get lhcal policy AG and AT

letg:0 )equilibriumYe:C + It Y- C : I

defineS=Y-C butY-C:I

I I: S get IS curve

money supply: M exogenous (e.g. cenfal bank conrol)get monetary policy AM

demand: liquidity preference (prefer to hold money) Lvarious theories: 1) Quantity Theory of Money

2) Marshall Cambridge equation3) Keynesian monetary theory (using consols)4) Tobin (1981) LP as Behavior towards Risks

REStudieslgsS - (laid partial foundation ofmodem Finance)

5) Baumol's Inventory Approach6) Friedman (1976) New Quantity Theory ofMoney

+5U Lts+kl

ctt th

But how about monev:

get LM curve

u ws1950-60

IS-LMfremework

r970-94

1930-60

I Aggregate D

But how about labour, production

(i.e. supply side, e.g. Cobb-Douglasaggregateprod funotion, Solow (1937) i Aggregate S

education for labour force; Becker(1992)human resources for productivity

{}IMarschak-Brownlee Aggregate S - Aggregate D Macro model(analogous but different from GE built up from individual econ agents)

u

1960 Growth Theory (=tper capita Y &lor Yfrr".ploym*t bytechnological advances and labour productivity increase -education/fraining)Inflation, unemployment issues (esp. Phillips Curve and laterFriedman-Phelps (2006) expectations-augmented Phillips Curve)Friedman Permanent Income Hypothesis (expectation)

U

Rational Expectation TheoryI\{arschak (1953 Economeffic Measurements for Policy & Prediction)Robert Lucas (1995)Micro level - individuals max obj

s.t. various consfraints ANDs.t. rational expectation of prices (both present & future like

infl ation/deflation - macro), of intertemporal consumptionpattern and of actions of other economic agents, thuslinking macro with micro foundation.

Equilibrium now just steady state in decision rules space

fiust like Micro ArrowDebreu State-contingent GE model)

So far most models are static models (one period instantaneous models)

Dynamic l\dodels (with time dimension) for both macro (e.g. growthmodels) and micro intertemporal models, mostly using difference &differential equations and calculus of variations.

@

tTIff TYALI }I'ITUS'I firu$T[AL\ I$focn

OPIMON DECEMBER 3,2008

Economists Have Abandoned Principle

Twelve months ago nobody could have imagined government

interventions we now take for granted'

By OLIVER HART and LIIIGI ZINGALES

This year wili be remembered not just for one of ttre worst financiai crises in American history, but also as

the moment when economists abandoned. their principles. There used to be a consensus that selective

intervention in the economy was bad. In the iast 12 months ttris belief has been shattered.

hacticaliy every day the government launches a massively expensive new initiative to solve the problems

that the last dav s initiativi.did not It is hard to discern any pnncrples behind these actions. The lack of a

coherent strategy has increased uncertainty and undenained the public's perception of the govemment's

coropetence and trustworthiness.

The obama administation, with its highly able team of economists, has a golden opportu:rity to put the

counfry on a better path. We believe that the way forward is for the government to adopt two key

principles. The first is that it shouid intervene only when there is a clearly identified market failure. The

second is that government intervention should be carried out at minimum cost to ta,tpayers.

How do these principles apply to the present crisis? First, ttre market economy provides mechanisms for

dealing with difficult fi11195: Take bankruptcy. It is often viewed as a kind of death, but this is misleadine.

Bankuptcy is an oppornr:rity for a company (or individlal) to make a fresh start. A company in financial

distess faces the aungrr thai creditotr *itt ffy to seize its assets. Bani<ruptcy gives it some respite. It also

provides * oppormrlty for claimants to figure out whetherthe company's financiai trouble was the result

of bad luck or bad management, and to decide what should be done. Short-cuuing this process ttrrough a

govemment baiiout is digerour. Do.r the government really know whettrer a company should be saved?

As an exampie of an effective bankruptcy mechanism, one need look no further than the FDiC procedure

for banls. when a bank gets into touble the EDiC puts it into receivership and ties to flnd a buyer. Every

time this procedure has bien invoked the,depositors were paid in futt and had access to their money at all

times. The system works we1i.

From this penpective, one must ask what would have been so bad about letting Bear Stearns, AIG and

Citigroup (and in tire iuture, General Motors) go into receivership or Chapter 11 banlcruptcy? One

71u2009

Lia4

Page 2 of2

',, .

argument often made is that these institutions had huge numbers of complicated ciaims, and that the

bankruptcy of any one of them would have 1ed to contagion and systemic failure, causing scores of further

banh-uptcies. 41G had to be saved, the argument goes, because it had rillions of dollars of credit default

swaps wrth J.p. Morgan. These credit default swaps acted as hedges for tillions of dollars of credit default

swaps ttrat J.p. Morgan had wi*r other parties. If AIG had gone bankrupt, J.P. Morgan would have found

itself unhedged, putting its sAbiliry and that of others at risk.

This argument has some validity, but it suggests that the best way to proceed is to help third parties rather

than the disftessed company itself. In other words, instead of bailing out AIG and its creditors, it would

have been better for the goyernment to guarantee AIG's obligations to J.P. Morgan and those who bought

insurance frorn AIG. Such an action would have nipped the contagion in the bud, probably at much

smaller cost to ta:cpayers than tire cost of bailing out the whole of AIG. It would also have saved ttre

govemment from having to tnke a position on AIG's viabiliW as a business, which could have been left to

i Uuot *prcy court Filally,'it would have minimized concerns about moral hazwd. AIG may be

responsible for its financial problems, but the culpabiliW of those who do business with AIG is less clear,

and so hetpile them out does not reward bad behavior.

Similar principies apply !c the housing market. It appean that many people thought that house prices

would never fali nationaily, and made financial decisions based on this premise. The adjusunent tc the

new reality is painful. But past mistakes do not constitute a market failure. Thus it makes tro sense for the

govemment to support house prices, aS Some economists have suggested.

Where there is arguably a market tailure is in mortgage renegotiations. Many mortgages are secwitized,

and the lenders are dispersed and cannot easily alter ttre terms of the mortgage. It is unlikely ttrat the

prcsent situation was anticipated when the loan contacts were written. Government initiatives at

facilitatine renegotiation therefore make a lot of sense.

Our desire for a principled approach to this crisis does not arise from an academic need for intellectuai

coherence. Without principles, policy maken ineviably make mistakes and succumb to lobbying pressue.

This is what haBpened with the Bush administation. The Obama adminisfation can do better.

Mr. Hat is a professor of economics at Harvard. Mr. Zingales is a professo: of finance at tle Chicago

Booth School of Business.

Hanlon's Razor: Never attribute to nalice that which can be adequately explained by stupidity.dscott,s corollary: The line between malice and stupidity is called depraved indifference'

r ?6J'- ----4'

- /r ,^nnn

Dynamic econ - optimize multiperiod objective function (of variables)in different time periodss.t. resource confraints of these variables over time

1960- presentDynamic optimization under uncertainty (explain econ behaviorthrough time, under risky situations)

Remark: most analysis relies on function differentiability, hence opt are local results;with present-day powerful computers, computational iterations &/or approximationalgorithms are employed to estimate functions, whether differentiable or not.

Pontryagin Optimal Confrol Theory(for both continuous & discrete cases,mostly m4thematical econ)-l

Dynamic opt + stochastic assumptions

-LaGrangian method(continuous optimization under risks)

Discrete Models (mostly Macro, esp. growth models)F. Kydland (2004) E.?rescott (2004)

IBellman's dynamic programming

{Dynamic programming is a way to approach certain types of dynamic problems andmany time solutions limited by dimensionality)

In particular, graduate Macroecon theory course: study dynamic macro models basedon GE and using dynamic programming

under ce@inty stochastic models

Principle of optimality + Markovian process to add risks

tlllrecursive models Markov process: measure theory & integration

IBasicdlly sequence of probty distribution functionsconverging to an invariant distribution = stochasticequivalent of detenninistic steady state.

So now not just frnd steady state dynamic system (equilibrium) but theoretical modelswith equil described by stochastic processes over different periods fiust like real lifeeconomic behavior). But many times, such problems are analytically insoluble orhave no evident closed form solutions, or can only be solved by common sense, or nostandard solving procedure, &lor can only be approximated by some algorithmsometimes.

Summarv:

RISKS???

Game theory

Hist development of Econ

MercantilismI

Adam Smith Invisible HandI

MarshallPigout -Keyn"s

MICROTheory of Cons. D Theory of Firm

GE model-r

compet and perfect mkt asstrmption

Thm: Inv Hand: CE €t PO locallv

Pure Micro P TheoryCE €+ PO globally

MACRO

C+I+G

(usual undergrad)

+AggS

Applied P Theoryesp. Chicago School

realJife features by relaxingcompet and perf mkt assumptions

Generalized Neoclassical Micro Model

Stochastic modelsDecision Theory Arrow-Debreu full insurance

(VNM Exp Util Thm)

+LM

Agg D

| 1960 growth, risks???Dynamic systems - Simple descriptive dynamic systems

1 (difference:tions: Harrod Growth Model; stochastic 2-dim

I difference +ions: companies growdr, Hall Employment model )I to*l

Rational ExpTowards GE model (Op$

Dyna mic optimizationrnodels

,/\Discrete time case continuous time case

I non*urlo opt control theoryBelknan dyn progand multi-stage opt

I

RISKS??? Stochasticdynamicoptimization

Chapter I. C. Unconstained and Constrained Optimization

REF: Silberberg & Suen (2001) Chapter 4Takayama (1985) Chapler ITakayama (1993) Part2

This section and optimization in general, in skeletal form:

Optimization

2 indepvbles

Y: (xI, x2)

n indepvbles

y : f(x1. x2,

1 equalityconstaint

2 indep vbles 4 vblesmax y: f(x1, xz)

s.t. g(x1, x2): k(without matrix)(with mafrix)

max y : f(xr, X2, ... , .,)

S.t. jg(xr,xz, "', x,1;2kixi20

i:1,2,j:1,2,

(The most general case will be covered in the nonlinearprogramming section)

4Unconstrained

-/ \-'-{functionwith Iindep variable

Iy: (x)

I

max y: f(x)II

max y : (xr, x2) max y: f(x1, x2,

(without matrix)(with mahix)

multivariate function

-/\_/\

consfrained,1,

multivariate function with n indepvariables

----4\ m inequalifyconstramts

max y : (xt, x2, '.. , xn)

s.t. g(x1, x2, "', xo) : k(with mafix)

i) Unconstained optimization

How to optimize (max or min)??

Assumption: In this section, we assume functions are differentiable.Since differentiation deals with small neighborhood around a point, the max or min we obtainwill be called local or relative max (or min) and not necessarily global max (or min).

Case a) unconstrained optimization of function with 1 independent variable

Optimizef(x) x € R ) solution x*

l$ order condition (necessary condition):f ' 1x*; : 0 (critical point)

2d order condition (sufficient condition):if f "1x*; < 0, then x* is a relative maxif f "1x*; > 0, then x* is a relative min

In case f " 1x*; : 0 at critical point x*, then we need to have successive derivative test,We must keep taking derivative at x* until we get a nonzero higher-order derivative.

even'numbered (4ft, o*, g*, ...)

II

and

-&-<0 / -\" >0

llx*isa x*isa

6

lf such nonzero higher-order derivative is

--\t'\odd-numbered (3t, 5", 7*, ... .)

I

then x* is an inflection point.

relative max relative min

RMK: For a rigorous derivation of the l't and 2od order condition using Taylor's series,please see Silberberg & Suen or Takayama [985].

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Economic application (unconsfrained opt of univariate ftrnction):

A firm wishes to max profit n(Q) defined as Revenue R(O - Cost C(0, & givenR(Q) : 407Q - Q' ; C(Q) :Q' -lQ'- 50 where Q: outputquantrtyofthe firm.

Maxn(Q):407Q -Qt - (Qt-3Q2-s0)

: 4o7Q +2Q2 -Q' + 5o)

1o order condition to get critical point Q*:

setdn(Q/dQ:407 +4Q-3Q' : 0

+ 3Q'- 4Q-407 : 0

+ Q* _:

,.:i): { l6-4(3x-407)l t 2(3) : [4 r .l+roo1r0

or - 66 / 6 (we rule out this soln '.' assune output not -ve)

Can check critical point condition at Q*:

da(Q) / d Q :407 +4 Q-3Q' : 407 + 4(37/3)-3(378)2 :0

2od order condition at Q*:

Tt" (Q):4*6Q- 4-6(3713):-70<0 ) localmaximum

trliOn!)tr

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An example of unconstrained optimization of functions with 2 independent variables:

z(x,y) :x3 + y3 -18x2 + 6y' +81 x -36y + 23

1" order condition: set

zx :3*-36x+81 :0 t *-tzx+27:0 [x:{+12 +'{Wnt l2(l): {12+ 6} 12: 3 or 9l

t x*:3 or9

zy:3t'+l2y-36:0 t y'+4y-12:0 [y:{-4+{ 16 - 4(rX-12)| t2(1): {-4+ 8\ / 2: 2 or -61

t 4 critical points (3,2) (3, -6) (9,2) (9, -6)

2"d order condition

z,u<:6x-36

tz **(3,2) : 6 (3) - 36: -18 < 0

z *" (3,-6) : 6 (3) - 36: -18 < 0

z**(9,2) :6(9)-36: 18> 0

z**(9,-6) :6(9)-36:18> 0

zw : 6y+12

z n (3,2): 6 (2) * 12: 24

z r, (3,-6) :6 (-6) + 12: - 24

zn(9,2):6(2)* 12: 24 >0

zn(9,-6):6 ('6) r 12: -24 <0

z *y: z y* : 0 for all critical points t (" *)' : 0

for (3, -6)

z*" (3,-6)<0 zr, (3,-6)<0 and z xxzw: (-1SX-24): +432> 0 : ("r)'I relative maximum

for (9,2)

z,*(9,2)>0 zn(9,2)>0 and zxxzw :(1SX24):+432>0 :(z*y)'I relative minimum

for (3, 2)

z*"(3,2)<A zo (3,2)> 0 (different signs) &2"*zw:(-1SX24) : -432<0 : ("*)'

@)

+ saddle point

for ( 9, -6)

z *" (9,-6) > 0 z n (9,-6) < 0 (different signs) & z **z w: (18X-24) : -432 ( 0 : (r r)'I saddle point

Economic application example :

Afirm's revenue function depends on its two products a and b and is given as:

z(a,b):32a-a2 +2ab -2bz + 16b -9

Find a*, b* to max the firm's revenue function.

l't order condition to find critical point:

za :32-2a+2b :0t a*:40 b*:24

26 :2a -4b+16:0

2d order condition at (a*, b*):

zaa : -2

zab : 26 : 2 t ("ua )' :4

zM zbb:G2)(-4): 8 > 4: (zuo )'

Hence revenue is maximized at a* : 40 , b *: 24 and

revenue z (40,24): 32(40) - 402 + 2(40)(24) -2Q4\ + 16(24) -9 :823

@

Case c -1) unconstrained optimization of multivariate functions with p2 independent variables,without using mafix algebra (generalae from n:2 independent variables case):

Assumption: below multivariate function f has continuous partial derivatives to any order.

We can see with the many combinations of i, j when n gets large, the terms becomes veryunwieldy. One way to overcome this is to use linear (matrix) algebra.

Remark: A formal derivation of the l" and 2d order condition can be found in Silberbere &Suen [2001] and Chiang & Wainwright [2005].

o

Optimize f(x1(a), xz(a), ... , xi(a), ... , xn(a)) ) at solution (x1*(a), x2*(a), ... , xi*(a), ... , xn*(a))

ls order condition: set following : 0 to get critical point

fi (xr (a),x2(a),... , xi(a), ... , xo(a)) : 0 V i

2nd order condition: (using Taylor series and sign definiteness of quadratic forms)

n n dxi d":IlfiitFl:ot-r r-r da da

if [F] < 0 then negative definite (f stictly concave) and we obtain local maximum.and ltr] > 0 then positive definite (f strictly convex) and we obtain local minimum.

Case c-2) (alternative method) unconsfrained optimization of multivafiate functions withn independent variables.

optimize f(xr,xr,...,xn) ) atsolution f* =(x,,*1,...*l)

1$ order condition:

vector x' = (xi,*!,...*l) is such that

.f,(7')=0 i=1,2,...,n critical point x.

2od order condition:

a'f a'f azfax? &r&, &,&oazf a2f

matix H: *,7'a2f

&r.&"

a2f..--;Ax;&n&,

Remark Note H is symmetrical

Hessian lHl alternates in signs starting lH,l .0,lHrl ) 0,...

=+ lHl is negative definite (itrf is stictly concave) and

2d order condition for x'being a local maximum is satisfied

Hessian fHl with principal minors lH,lt O V, = 1,...n

* lHl is positive definite (itrf is srrictly convex) and

2oo order condition forx'being a local minimum is satisfied

Chapter I. C. ii) Constrained Optimization

We mentioned at the start of the course that resources in the economy are scarce. We alsocannot do anything that is technologically impossible &lor we are bound by existing sociefy'sproperty right structure. So when we optimize we will be constrained by resource scarcity &/ortecbnology &/or some other conditions (like private property rights consfaints whereby youcannot just take other people's resources without making proper payment).

i constrained optimization, called mathematical programming (;rcomputer programming).

In particular, if optimize linear functions subject to (s.t.) linear constaints, the mathprogramming is oalled Linear Programming (LP). The most general case is NonlinearProgramming where both objective and consfaint ftinctions are nonlinear.

Familiar examples in Econ:

(a) Aconsumermaximizeshisutility U(x) x € Rn p € Rns.t. his budget constraint p . x < his income I

(b) A firm maximizes profit zc(y) y € R n

s.t. production is feasible within the existing technological set.

(c) A flrm minimizes cost C(y) y € ft n

s.t. technological level and inputs/labor availability.

How to do constrained optimization???When we are given a consfained optimization problem, we just convert it into an unconstrainedone (using a Lagrangian equation), because we know how to do unconstrained optimization.

For instance, given a function f(*, y) with 2 independent variables and I equality constraint:(univariate functions is a special case of the following, just set y :0)

Max f(x, y) f is called objective function

s.t. g(x, y) : k called constraint, this case is an equality consfiaintcan write alternatively as k - g(x,y) : 0

[Remark: to minimize an objective function h(x, y) is just the same as max - h(x, y) so we willonly show constrained maximization.]

We set up a new objective function Q called Lagrangian, consisting of both f and g:

Original objective firnction f constraint (k- g)

o (x, y, rl: (r,lv) * t t k - Jf*, ,ll l. called Lagrangian multipliert

IAA) tk - Bl :0 I f [k - g] :0 so optimizing Lagrangian = opt original fBB) The Lagrangian is unconsfained t we can apply the unconstained

maximization which we covered previouslyCC) constraint will automatically be satisfied when we are solving 1s order cond]:

Max O (x, y, I) : (x, y) + l, I k - g(x, y)]

ls order condition: to get critical point with 3 variables x, y and l,

@" :0@v :0@r :0 (= k-g(t,y):.0 whichisjusttheconstraint laftersolvingforx,y,l,fromthese

3 simultaneous 1o order condition equations, the consfaint is automatically satisfied).

Interpretation of the Lagrangian multiplier l. : at the optimal level, l.* shows the approximatechange in the objective function when the constraint is changed by I unit.

We can derive the 2"d order condition for oonsfained optimization but this again involves manydifferent combinations of partiats and cross derivatives like (Dii , (Dtj as well as solving forsimultaneous equations in the 1$ order optimization condition, all of which can be handled moreeasily with mafix algebra.

Usual econ examples:

Max utility function U (x, y) x , y commoditiess.t, prx+pzy: t Ot:priceofx; p2:priceofy; I:income

If U(x,y):x2 + 5xy + y2 and (pr, p z ): Q,1) and I:100

1o order condition: (critical point)

Q*:2x+5y-21,:0

@r : 5x + 2y - l" -0

@l -100-Zx-y:0 (rfiiohisjusttheoonstraintlaftersolvingforx,y, l,fromthesimultaneousl$ordercondition equations, the constraint is automatically satisfied at the critical point)

) x*: 10 Y*:80 l* :5x+2y:210 atthecriticalpoint(x*,y*):(10,80)

{J*:(10)2 + 5(10X80) + (80)2 : 10,500s.t. 2(10) + (80) : 100 and the constraint is satisfied

2d order conditions can be checked to see this is indeed a maximum.

Now if we change the constraint by a small unit, say 1 unit(should be arbitrarily small unit but this is just for ease of calculation)

) constraint 2x+y = 100 changedto 2 x+ y : 191

from l" order condition, we get y: 8xand(D1 :101 -2x - y :0 t 101 -2x- 8 x:0 t 101 :10x

t x* : 10.1t Y*:80'8

) U* now: (10.1)2 + 5(10,1X80.8) + (80.8)2 : 10,711.05

compared with the previous U*, the difference is (10,711.05 - 10,500) :211.05 = 210 : I ,F

Optimal Lagrangian multiplier /' x tells us if we relax the consffaint by one unit, the optimal valueof the objective function will change by L x units.

For programming (constained optimization) problems, we often times use a value function V (x, o)where x is the endogenous variable and o the parameter. Value firnction is defined as the optimalvalue of the objective function given the parameter. Hence in the above example, O is theobjective function, x,y are the endogenous variables and consfiaint amount 100 is the parameter.Then

V(xn, Y*, 100) : 10,500

V(x*, Y*, 101) : 10,7I1.05

In this context, the Lagrangian multiplier 2 * is the imputed value of the consfiaint.It is the maximum amount that the optimizsr is willing to pay to change one unit of constraint (i.e.the shadowprice of the consfaint).

ln the above example, we used equality consfraints (in the budget conshaint case, flris meansthe consumer will use up all his income). We will extend the case to inequality consfraints.

Alternatively,

Constrained optimization of 2-independent-variable functions with I equality constaint,in terms of linear algebra.

Optimize f(x,,xr)s,t. $(xr,xr) = k k: constant

we form the Lagrangian

O(x,, x2, X) = f(x, , X, ) * l,(k - gix,, xr;)\

note = 0

ls order condition:

set Or=0 -)

Oz = 0 | to get critical point (xi,x])@r=0 fNote:O^=0 )

+g("rr,xr)-k=0=+ constraint is satisfied when solving simultaneous

equations]

2n order condition:

andcheckat (xi,xi)

if llt-rJ=1"-l " o =ltrl is positive definite + (xi,xl; is a locat min.

\(-O uS3+ 2Q ngg z - A ,rS? )

if lFrl>O *lHl isnegativedefinite =+(xi,xi) is alocatmax.

if 2"d order not met, need firther test.

setupborderedHessian 1o1=13,, 7,: i;l "' ll fl, ;'*lg, Ez o I le, (D,, Qn

t )l \\ J() /

.-odHoo6.oadlo6

Ec.eI6E3€&- b;- bo 'Dr0 e' .9

_b6 a.*oo R

,>,trEq ho=#E EEh, r tx EHi 6== Ege -6d 9!6 n.:-^e nS!';E6 "lp EFE ;XF EEh ^ JJ; 8E

r* 6lio & ,$, '-' Al > Fr{ ;Y-; ? ?? qtg>ee \\rl e--,^ p:

€ rR*s t s$e T€-? c.8lPa.O ,8*r-. -E."3 '-lt r- =F'6 :,S: : - a - - aa H;E ed"edd ;;; 'Et

c.5Ebo,Og

.:'sts 6 s.. s

E-d- A a.'...]t/u

c>6So

^ 60bo\\

:*-q- ry< d ,. ,. bo v{a

;d" e *+

II

e,e

|t'lJ"

::ge\:t lJ

'e*€rtsi R.H;ti EEtr .9 " E: o

# 3 E rEf5H#.E.EFg €€€s E

g € H5€€ &j s€ iEs IE €Eg€€ iE!gH:E : "

;E*H€,q { €

EgEEEi : ^gsflsse-$ * JS9lPPFoo: (}E

EE$TsE ; rg- m\ > -Jl tr.8.6;€PE ,: ! 3

s.EE*,a'f , 3.-EFFi{€: { SrE

*€$qf!" *: $EE EE;#:* s E #E E'q.ei€E .R "E RH E6;EEB 'H 5 5i gcEs€F E iEP

0z

o1i

x

.ta

..9 6$o'Fx6o5.o= n rl

E= €'rj:?tr:ro..

EF s'o T

dd dd 6a

q q dll t IxA g ,Og 6

tl

rf;

ll

;

'' ^" J<

ls

Jo

We extend to constained optimization of multivariate function with one equalityconsfrain! using linear algebra:

optimize f(x,, xr,...,xo )

form Lagrangian (D= f(xr

l3::Bordered HessianlHi= | :

lo"tlt,

s.t. g(xr ,x2," ',Xr ) = k k: constant

, X2 2..-, X1) + l"(t - g(x,, xr,..., xo ))

-12

8z

orn

orn

o-o|'n

or=

8n

orn

@rn

o*

lols,ls,l'tolon

9r

8z

g9n

0

8r Ez

@,, @rt

Q^

Ort ":

lo order condition:

set O, -Oz =@r...@o -@r. =0 togetcriticalpoint x. =(xi,xl,...,x;)

2"o order condition: check at x.

if lH-,1,1H,1,...1tr,1<0 =lHi nositivedefinite + tocatmin

\order of principal minor

if ltrrl t O,lFrl .0,ltrrl > 0... alternating in sign+ lHl negative definite =local max

if 2"d order condition not met, we need further test.

c)

*{ uzt$*M+refe {*u Jf' wl

u4, f(*,rfIr-'-'r-i)(;l', iX C*, ,*r, "', x.) e ft't

({ * +g (x,, --,*"))g *n f,{r,,-*, -- ,y^}+

ral :

gp (3

o1,tl

b o '-o

l.r\

Lxli=t v

't,'to"*-"'1,^'8, *0, - -- '$"

t lt+

iE'*f,"---.$n'g,"X, --ry, ![,, [,o -- fl,.'?" ',-'1 9" '_':

\

'x^ 'e^ -'h* i*, *'--- {o^l-rt:T I tu I

Lt,**t I .,It-,^+"\ -- -

?ffr I

frlh;4*t l"-"tr4'

>d ur{t- tia*t{

\ f, \ gr-*. (r'.t*-g *3 k, f * 4;\ l"tn^*r \ !\

" #'{lc."r&

Gl,"s a.I*at<-- tS b*oi=4 'f -- 1" Kt* Lrs-{'af

6"^'. I .d?t.ag-q &nefr* u-1,-<-,

; 5*- * ,t,i-u'&6rt

Econornic application :

Derivation of Marshallian Demand function D for economic agent fu\A.

fuAu\ MaxutilityfunctionU (x, y) : 5 xy s.t. budgetconstraint g(x, y): I

whereg(x,y) : prx + pzy and I:constant- A\rA\r{'sincome.

Formlagrangian@andmax: Max@: Max 5xy + l,[I-(p1x + pzy)]

1$ order condition (to get critical point):

O":5y -Ipr :0l,:5ylpt:Sxlpz * y: x(p1lp2)

@v:5x -l"pr:6 \

@r:I-(pr" +pzy):O t I-prx -pzx(prlpz):0(tlrisisjusttheconstaint) ) I -2prX

) x: I / 2pt - Dfunction forx dependentonpricepl andhis income I

check 2"d order condition using bordered Hessian: (note partia derivatives I r = pr , s: = p: )

l4,l:lu-l: lorr en er | : lo r o,

l@r, Qzz Ez I 15 0 pz

| *' Ez o I lp' pz o

Laplace expansion along l$ row

|u" u" I: u

"l u,, u,, l* ( -1) arz

dzt urrl I ua zzz

l+ at' Idtt ulll lu,t dtz

: 0 + Cl)55Pt

Pr0

- PtPz

+Pr

:5prpz + 5prpz : l0 prpz >0

t lFl negative definite I local maximum.

50

Pr Pz

ll

5Pz

'.' prices Pr>o pz>o

@

Chapter I. C. iii) Constrained optimization with inequality constraints.

So far we used equaliry in the constraint. For example in the budget constraint case, equalitymeans consumer will use up all his income. How about the case when he doesn't spend all hisincome and save some of it? Then we need to look at inequality constraints. How can we handleinequality constraints? Answer: convert the inequality constraint into an equality constaint('.' we know howto solve opt s.t. equality constraint):

4- inequality consfaint

x>0Note we can convert inequality x > 0 into equality "

- s' tr equalitv constraint

where s : slack variable, squared to get a +ve si€n; s e R

x> 0 meansx)0 or x:0

Max f(x) s.t.

X,x: 0 case

when s:0

Noteifs#0 * x : s' > 0 * interior solution (not corner soln)w)

)turtv.&*) t€e)4ah

u4 Arfr:afr'^xr MM fC*>\ (!, >(..7_9

xtso

) Max Lagrangian O : Max f + ?, [r - r' ]

l't order condition (to get critical point):

u!\Ny {b0

Noteifs:0)Subcase I

x>0casewhens * 0

{e>X-

*x7Ox : st : 0 t corner solution and constraint is bindine.

subcase 2

O": f' +]" - 0

(Dr: -2)"s :0

ol,: x-s2 -o

........n u

.........12.21

[3.3.]

@"* O*, g^l

?: ?: %'l

*(0-4r')-o+1(2

-4 s2 fx*+27, >o

must > 0+lf**0

I

r)

(Nwd4,.f(*) #xAo

4 (interiorsolution. *

Casel) if s l0 from 12.2) + tt @:f+1,[x-s2]:f+0[x_s,]:f

2"o order condition (bordered Hessian)

i@z : lo I

0

-2 ?,,

-2s

tf

.. .. t4.41

Case 2) if s : 0 ) corner solution x* : 0 ('.' from [3.3])and , 2?,, >0 ('.' from[a.a])

) l. >0 t f'< 0 ('.'fromll.l] f'--1.)I function f is falling and not critical point (i.e. f ' + 0 )

) x*:0 is the maximum.p'*{u\ *x. . {8}

: 0 I constraint not binding

f

(if we want amaximum)

-rr€ x* ;c(:o*

:+ we get the following conditions:

we can generalize to multivariate function

max (x) s.t. x ) 0

generalized,l'1 order condition: f'(*) <0 and if f,<0:+ x = 0 max

Note above principle was derived using the 2nd order condition + ws are saying if we wantthe max (i.e. set (H)t o ) then the principle is hue, but this does not tell us how to find the max.

Similarly for

min f(x) s.t. x > 0

generalized l'tordercondition: f'(x) Z 0; if f'>0+ x=0 min

max f(x,,...,xo) s.t.

generalized 1" order condition:

1) For those variables x, ) 0+ fi <0 andif f, <0 then Xi =0 ismax

2) For those variables x, not restricted to > 0

*fi=0

(-)

max f(xl, X2 ,. . ., X1 ) any n, m

s.t. tg(*,,'"x,,)>0

'g(xr,...x")>0 jg(*,,".,Xn)=fi.rnctions

I,(*,,'"x")>o

xi >0 i=l,"',n

we form the Lagrangian

o = r( )+ir,,iej=l

1$ order condition:

(also known as O, = fi *t f, j g' s 0j=t

KUHN-TUCKER and if (D, <0, = Xi:0

condition)

(D^ =jg 2 0

andif *r,t0, = fu,=0

For min f( ) s.t. 'g( )< 0 x, ) 0

1$ order condition 0, = f; *if, jg, ) 0j=l

and if >0, then x;: 0

0r, =rg 2 o

and if *r, t O,then I: = 0

Analogously, c&r add slack variables and do the same derivation to get optimum of

Remark: again these conditions only describe max/min but cannot be used to find themax/mn.

Show Kuhn-Tucker K-T condition for constained maximization can be rewritten as:

(B) :[Qi: fi+ I I: jg, . 0 and x' Or:0]

] at tfre optimal point x* L*

(Bl):[@rj:'g] 0 ; l.i @i1 :ol

Pf: K-T condition is stated as frst part (A) and second part (Al) below:

K-Tconditionfirstpart: (A)[@1:f1+ I f i'g, = 0 andif (Di<0flrenxi:0]

(A):[@r:fi+ f l:'gt'0 and if @i<0then xi =0]t [xi (Di:0&ofcoursestillOi:fi+ I Lj'gt < 0]:(B)

(B):[@i:fr+ I l.i 'gt'0 andxl @i:o]t [if (Di< 0 then x1 must: 0 to make xi (Di : 0 & of course

still (Dt: f i+ f L: t g, s 0l : (A)

Hence (A) € + (B) or (A) = (B)

Similarly for second part of K-T condition:(A1): [@r::'g ] 0 ; I: ] 0 & if Oij >Othen l; :01

€ t (Bt;:[@rj:'g ] 0; I: > 0 & li @ri =01:(81)And (A1) : (81)

+ K-T condition which is (A) + (Al) is = (B) + (Bl)

Economic Application of Kuhn-Tucker (K-T) condition:

John minimizes cost C(a, b) under the following conditions:

Min C: Pu a + Pu b Pu: price of apples aP6 : price ofbread b

s.t. his utility function U(a,b) > K a,b

assuming goods are not free so Pu, Pu > 0

Question: in minimizing his cost, what is John's utility level?

We note Min C = Max - C

+ above minimization problem becomes ,

Max-Pua - Pub s.t.U(a,b)-K> 0

[Let U(a*,b*) be denoted by U*]

)Lagrangian@: -Pua - Prb +1.[U(a,b)-K] ]"> 0

and K-T condition:

- Pa + 1* 16U*/ D a) S 0 and a* (-Pu + ),* (AU*i d a)):0 [K-T-l]

-Pr + 1r*(?U*lab)< 0 and b*GPo + L*(AU*/db)):0 [K-T-2]

u*_K> 0 I*> 0 I*(U*_K):O [K_T_3]

If we assume John wants to consume both a and b, then a, b > 0I from [K-T-l] (-Pu + ?u* (AF*/d a) :9 ot l* (aF*/ d a): Pu > 0

t l.* ;e 0 and since l,* > 0 ) l.* > 0

i from [K-T-3] ( U* - K):0

'U*:KEconomic interpretation: to min cost, John is at the same time "maximizing" his utility levelby consuming up to his constraint level K.

il4

Chapter I. C. iv) Nonlinear Programming

The most general optirnization problem is when multivariate objective function f and

conshaints rg are nonlinear functions.(again we look at maimizattonbeoause minirnization is just maximizing the negative f)

Max f (x1, x2,X3, ... Xn) I :(xr, xz, xr, ... xr,) e XcR"

s.t. jg(x1. xz,x... .. "") ) 0 j --1,2,... ,J

all functions are nonlinear.

We notice tle following points:

1) Our analysis so far are characterization of optimum using calculus, there is no knownalgorithm for finding the optimum (except in Linear Programming LP - we have thesimplex algorithm based on extreme point theorem.)Usual method for nonlinear case is to find feasible points (points that satisfy the

constraints) and check Af when A x i ; and keep changing x 1 to hopefully max/min fThere are many approximation algorithms using computers but these are not closedform solutions. This brings up the question of

2) Existence ofa feasible point that is optimal? and

3) Can we characterize constrained optimum for general firnction case?

ln fact, in math eoon, questions 2) & 3) occupy the bulk ofdiscussion.For example, concave programming with saddle point characterization.

Given f, . jg are concave functions then the nonlinear programming problems of

Max f(x )

s.t. jg(x) > 0 V j ( constraints)

is called concave programming.

Remark: f and jg need not be differentiable; hence the analysis is very general.

We show we can characterize the optimality of this concave programming with saddle pointand tle optimal result is global.

Defu: Given any two points x1 and x2 in a set X: o x1 + (1 - d,) x2 is called the linearcombination of points x1 , x2 and is the equation of the straight line joining these two points.fNote: xl , x2 can be vectors.]

Defn: in particular if 0 < a < l,thencr xl + (l -o)x2 is called a convex combination ofxl ,x2

Definition: asetCisconvexif x1, x2 e C then convex combination cr x,+ (1 -cr)x2 e C(for any 0< cr < l).

[i.e. ifxl and x2e C) line joining these 2 points alsoe C, the C is convex][hence relationship: convex functions requires convex set for its domain]My teacher the late Prof. Takayama used the Chinese character for convex

a set is convex if A (protruding) everywhere )

Below set S also not convex

Concave function: f is a concave function over a convex set CifV x,ye Cand0< cr < If(ct x1 + (1 -cr)xr) > a f(x1 ) + (1 - c.) (x, )[Geometrically, line joining any 2 points on flies on/below the function f l

c

If strict inequaiity f(c x1 +(1 -c)x2)> af(x1 )0< o <1 x + y

strictly concave lirnction(no shaight lin€segmetrts allowed) $,t;U tn't a'* '(a,.'

{e;also concave(but not strictlyconcave) finction

*, :"o,,!,{-{)xt

L

Definition: fis (strictly ) convex function if - f is (shictly) concave.

convex function

/r L'l L,UNot strictly convex function strictly convex function

Generalized defi nition: f is concave over a convex set X g R.' iff Vintegers rn > I

f(cr1 x1 + 6s2 x2 +

with cr1 > 0 and

We first state a theorem in linear programming (which is a special case of concaveprogramming because linear functions are both concave and convex functions but notconversely. i.e. concave (convex) functions are not necessarily linear).

Defii: Given a function O(x,l.) = f G) + Itrg(x)1, (x *, l*; is a saddle point Sp of O (x, ),)if@ (x,l"t) S @(x *,2,*) S @(x *,2,) V_x V& in the domain.

[Remark: SP means if viewed fiom one cross section, the fiinction is a relative maximum ancif viewed from another cross section, it's a relative minimuml

* cra,Xn )> cr1 f(x1 )+crzf(xz)+.... + o.f (x.)

m

0j

Theorem (Goldman - Tucker): If i'g ,Vj:1,2,... J, are linear functions defrned over aconvex setc cRl , then

x * is a global maximum over the domain C solving

Max f(l ) s.t.rg(x) >0 j:1,2,....,J(i.e. x * is a solution for the linear programming problem)

iff

3 Lagrangian multiplier l"* > 0 s.t. (* *, l*) is a saddle point ofthe Lagrangiano(x,D:f(x ) + l['g(x )] with ]":(],r. ),r. l"r)

[Tfus tleorem actually points out the important primal-dual result in Linear Programming. Forproof, please see Takayama [985]1.

Before proceeding to more general concave programming theorems, we will look at Kuhn-Tucker Theorem. As a simple illustration, we will only prove the theorem for the case of 2-variable differentiable concave functions and one inequality constraint. We will also state themore general case theorem for n-variable concave firnctions (not necessarily differentiable) andj inequality constraints and its proofcan be found in Takayama [1985].

Lemma:For concave firnctions f (x1. x2) and g(xl, x2) with xl. xz 2 0; (xr. x2 ) e C c R2

thera do€s not existJ

(A) If / (x1#, x2#) e C s.t. f(x1#. x2#) > 0 and g(x1#. x2#) > 0(i.e. no points in the domain for f and g to be simultaneously positive),

then 3 nonnegative weights po. , pr- ) 0, not both zero,

s.t. p6- f (x1, x2)+pr- g(x1, x2) < 0 V (xr, xz)e C gR2

@) firthermore if I (xrO, x20) e C s.t.g(x10. x20) >0 [Slater's condition]

then p6- > 0Proof:(A) WLOG, let f be +ve then g must be -ve so can choose pr. very largo to make weighted

negative p1. g to at least offset the weighted positive p6- f;and hence po. f (xr, x2)+pr. g(xr. x2) < 0.

(B) Ifp6- not> 0, then p6- :0 ('.' po. > 0)) (0[(x1. x2)+pt- g(x1. x2) < 0 V (x1. x2)e C gR2

) g(*r, *r) . 0 V (x1, x2)e C -cRz

contradicting Slater's conditionHence ps* > 0.

Theorem: (Kuhn-Tucker)Given f (x1. x2) and g(xr, x2) are real-valued differentiable concave functions defined over aconvex set C c Fi and

If f (xr0. xz0)e C s.t.g(xr0, x20) >0 [ Slater's condition]

then (x1*, x2* ) is a solution for the concave programming problem:

Max f (x1, x2)s.t. g(x1, x2)) 0 xr,xzZ0

iff

I Lagrangian multipliers ),* : pr- / po. ) 0 for the Lagangian

o (x1. x2, l") : f(xr. x2 ) + l" I g (xr. xz ) ]

satisfu.ing following Kuhn-Tucker conditions:

0@/0x;_ : f i(xl*, x2*) + l,* gi(xr*. x2*) < 0 xr*. x2* > 0

xi+ (d@/dx-i-) =Q i:1,2

A@/A),.: g(xr*.xz*) 2 0 l"*>0

l,* (ao/a).) -0Proof: If(x1*, x2* ) is a solution for the concave programmirrg,

* f (x1*. x2*) ) f (x1. x2) with g(x1. *r) > 0 V (x1, x2)e C

I / any (xr, x2)e C s.t.

f(x1*. x2*)<f(x1, x2) withg(xi, x2) - 0including g (x1, xr) >0

meaning / any (x1, x2)e C s.t.

f(xr. xz)- f(x1*, x2*) >0 with g(x1. x2) >0It,(concave - numoer, sl t concave concave

then by above lemma:

3 nonnegative weights po., pr. i 0, notboth zero.

s.t. p0.[f(x1, x2)- f(x1*, xz*)]+ pr- g(x1, x2) < 0 V (xr. x2)e C

i ps- f (x1. xz)- po. f(x1*. x2*) + pr. g(xr. x2) s 0

) po- f("r. x2)+ p1" g(xr, x2) < po, f(x1*, x2*) tl-L-gj

In particular, ifwe insert (x1*. x2* ) into [I-L-9]we get pr. g(xr*. *r*) s 0 [-L-10]

but pr- 2 0 and g(x1*, xz*) Z 0 ) pr g(xr *, x2*) > 0 [-L-11]

[-L-10] and [I-L-l1] t pr. g(x1*. x2*)=0

) from [I-L-9] /,=o

po. f (xr, x2)+ p1- g(xl, x2) S po- f(xr*. x2*) * p1* g(x1*. x2*) [I-L_12]

Furthermore, for any pz = O t O, g(x1*. x2*) - 0

/r=o .z>oI p6. f(x1*. x2*) * p1* g(xr*. x2* ) < p6- f(x1*. xz* ) + pz g(x1*, x2* ) [I-L-13]

iI-L-121 [I-L-13] combined give the follow Saddle poinr Sp:

p0* f (xr,x2)+pr" g(xr.xz) ( po. f(x1*,x2* )rpy. g(x,*, xr*f pn. f(x1*,x2*)+prg(x,*.x2*) [Sp]

and by above lemma and slater's condition, po- > 0, so we can divide throughout [Sp] by p6.getting following for all p2, x1, x2 ) Q

f (x1, x, )+(p,., p6, )g(x1. x2 )3f(x1*. x2*)+(p1.7 po. )g(x, *,x2*)< {xy +. xr*)+(prpn)g(x1*. xr* )[Sp1]

and if we define l.: fu27p6) and given l.* :pr. /po- > 0 tsplj becomes

f (x1. x2)+ l"tg(x1, x2 )<f(x1*. x2*)+l.*g(xr *.xz*)3 (xr *. x2*)+l.g(xr *, xrr XSp2l

[SP2] means (x1i. x2*) maxim izes Lagrangian (D (x1. x2, L) while l+ mifimizes the sameLagrangian (D. And since f and g are differentiable functions then l't and 2nd order conditionfor max and min are the same as those depicted by the Kuhn-Tucker conditions (i.e. thev bothstate: (xr *, x2*) maxirnizes and l,* minimizes Lagrangian @).

I (*r \\7

Conversely, if(x1*. x2*) , l"* satisfy Kuhn-Tucker conditiolst = [SP2]* f(x1, x2) + l.* g(xr. x: ) < f(x1*, x2*) + Ir g(x1*,x2*) Vxr.xz _ 0 tI-L_141

Case i) ifl"* >0 and from Kuhn-Tucker condition ),* (6 @ I 6l,) ( which : l.* g(x1*. x2*)):0 [r-L-15]

) g(x1*, x2+) : 6

[-L-l4l+ f (x1.x2)+1"*g(x1.x2)< f(x'*, x2*) vxr,xz ) 0t l*g(x1,x2)S f(x1*. x2*)- f(x1.x2)

But l"t ) 0 and when s.t. constraint g(xl, x2)>0&Vx1,x2) 0) 0< f(xr *, xz*)- f(xr,xz) s.t. constraint & vxr.x:) 0

= f(x1*, x2*)2 f(xl,x2) s.t. constraint & vxr,xz ) 0

= solves concave progrzrmming problem

Case ii) ifg(x1*. x2*)>0thenby[I-L-15] ,,* g(xi *, x2*):0 t ],* -0[-L-14]t (xr*, xz*) > f(x1, x2) Vxr,xz 2 0 V g (x1. x2) ) 0

There are two more general and useful theorems covering conoave programmmg.

Theorem (Kuhn-Tucker-Uzawa): Given t rg ,Vj:1,2,... I,are real-valued ooncavefunctions defined over a convex set C g Rn

;1* is a solution for the concave programming problem:Max f(x ) s.t. jg(x) > 0 Vj:t,2,....,1

i f coefficients p0,, pr*, p2.. ..., pr. all > 0 s.t.

po- (x) + pr' tg(x) +pr"g(x )+ .. *pr- jg(x) spo- ({*)

and p1. tg(X*) +p2.2g(x* )+ *pr. ig(d):0

Proof of this theorem uses Minkowski separation Theorem and the details can be found inTakayama [985].

The following theorem requires Slater,s condition assumption:

f x0 e C s.t. ig(x0 )> 0 j:1,2,....,J [Slater's condition]

Theorem: if f(a ), jg(x ) j:1,2,....,J are all concave functions defined over aconvex set C c R.' and suppose Slater's condition is satisfied, then

x * is a global solution (global maximum) for the concave programming problem:Max f(x ) s.t.rg(x ) > 0 v j:1,2,....,J

iff

: )"*> 0 s.t. (x *, i,*) is a saddle point of the Lagrangian @ (x,l) : f(x ) + lt je(x )lwith I:(?q. 1'2,. .l.i):

O(x,1.*) 5 @(x*, l*) < O(x*, l) Vxe C V),> 0

Proof of the above treorem follows from the above Kuhn-Tucker-Uzawa Theorem and can befound in Takayama [1985]

The above theorems form the basis for showing existence of competitive equilibrium, paretooptimality and other issues in welfare economic theory which occupied thJ main part of matheoon development in 1950-1980.

Above analysis is also extended to more absfact level using fourdations of math (set Theoryand math logic, general analysis) to provide foundation of math econ.

We will study all these topics in the following chapters.

Chapter II. A. Main Micro Econ Issues within Math Framework

Microeconomics: principal theory is price Theory and main theme is Adam Smith'sInvisible Hand Theorem (= Qlassical Welfare Theorems)

Adam Smith's Invisible Hand rheorem Iinks equiribrium and efficiencv concepts.

Competitive Equil C E.

untler PerGct CompetitionEquilibriun

i) Importance of market vs. planned economy - significance ofprope(y rights.

Given: scarce resources (if resources are not scarce, no need to study economics)

) two questions A1, A2 :

-.---

€l Pareto Opt P.O. globally under weakest assumptions.I

efficient in

Pareio Optimal sonse

A1. resource allocationwe need

mechanism to allocate resources

A 2. income/wealth disaibution

mechanism to distribute income

Planned (: s6p6-6;econ ICentral Planning Board CPBdecides resource (including human resources)allocation, production, distribution.

1) To properly gauge D, S of everything:Transaction cost TC (incl contracting, actual transaction& enforcement cost) and information cost incredibly high-- resources utilized for nonproductive activities

like monitoring reporting, enforcing.

2) if CPB gauging incorrect, further wastage ofresources -unwanted goods (S > D), or shortage (D >S) means longqueues for products (waiting time : opportunity cost), blackmarkets, bribery - all employing resources, but for non-productive activities, i.e. not for production of goods & services.

Market econ (: decentralizedprice system)

allocation bydecenfr alized price system(Micro hice Theory)

A&

\m slmplest sense

ifS>Dforsomething ) pJ* supplier will supply less and demander will

demand more) Pf and S & D change until at certain P*,

S(P* ) : D(P*) ) P* called equilibrium price

if D> S for something t pf* supplier will supply more and demander will

demand less) P I and S & D change until at certain p*,

S(P* ) - np*; ) P* called equilibrium price

decision (for allocation) deoentralized and nobody hasto gauge what other people want. All they need to dois to optimize according to the changing prices. (e.g.

max utility s.t. budget constraint or max rr s.t.production). TC for gauging limited to own selfwhich much less than gauging by CpB.

We can show such price system at equilibrium isefficient in the sense of Pareto Ootimalitv p.O.

We now proceed to studv this.

Adam Smith (1776):

malntleme

ii) Classical Welfare Econ Thms within NLp framework in Euclidean Spare R":

Economists for the next 200 years formulated his concepts oflaissez faire, invisible hands,sooial optimality etc., culminating in the classical welfare Theorems with two imDortantmodern day concepts:

Competitive Equilibrium C.E.: General Equilibrium under perfect competition

AND

Invisible hand will brins

propensity to work

\]v

Pareto Optimality P.O.: an economic state (equilibrium or allocation) is p.O. if no one canbe made better off without somebody else being made worse off. (This is to avoidinterpersonal utility comparison which is impossible)

Remarks on P. O.:1 . P. O. is a static efficiency conoept for question A1. Welfare Econ looks at the

equilibrium to see whether it is socially desirable in terms of efficiency.P. O. is not concerned with the (other) question 42. of income/wealth distribution(which depends on subjective value judgment). e.g. a dictator may own 98% of thewealth of his country while the people only own 2%, this situation may still be p.O.because any re-distribution will make the dictator worse off -- objection to thisinvolves value judgment.

42. income/wealth distribution

involves subjective value judgmente.g. in pure communism and socialism,theoretioally there should be absolute equalifin income/wealth distribution and all capitalgoods are owned by everybody (commonproperty rights).Under private property rights systerns,income/wealth distribution depends onendowments and capability of individuals.

Which distribution is better depends onsubjective value judgment which is outsideour realm.

Main point on .A2.: property right stucture

is if I private property rights, then peoplehave materialistic incentive to produce; whilecommon properry rights system channelseffort towards non-production endeavors likeclimbing the hierarchical ladder of the rulingparty

More importantly, private property right isessential for people to have incentrve to useprice system./market econ.Without private property rights, allocation isby CPB or CPB-controlled price system.

fiL

2.

Back to C.E. and P. O. in question A1. .

The historical development starts from

\,

47,

I I I . CPB bureaucrats are not disciplined norrewarded by changes in the marketvalue ofresources they control) no incentive to adjust priceaccording to market value changes.

222. matket players cannot compete forresources by paying higher prices whenoptimizing) no incentive to use price system.

333. Also no incentive to preserve resourceswithout private property rights

(these are Industrial Organization topics).

i-I) Theory of consumer D: Marginal Utility MU, Marginal Rate of Substitution MRS& equilibrium occurs when ratio of MRS for any two goods ofany two persons are equal.

Then use Edgeworth box of two persons' jndifference curve to prove P.O.D functions from 1$ order conditions of utility maximization subject to budget constraint.

iII) Theory of firm: equilibrium will occur when ratios of Marginal Rate ofTransformation MRT for any two inputs of any two producers will be equal. Thisequilibrium can be shown to be P.O.S frrnctions from l" order conditions of maximization ofprofit subject to production.

i-III) Theory of General Equilibrium : Combine theory of consumer D and theory of firmand D & S ofeach market and find prioe vector p* such that all D: S.

This type of coverage is common in all micro price theory course (e.g.: Luenbergertextbook [1997] )

We will cover the above subjects ftom the viewpoint of the Classical Welfare theorems inthe Euclidean space R n (R n is a vector space)

Assume perfect competition :

1 . perfect defined by perfect information (l no risks, no uncertainty), perfectexchange (no TC), no externality, property rights perfectly defined.

2. competition defined as all econ agents are price-takers.

fRationale is atomism which states that all econ agents are too small to affectmarket prices so just take market prices as given when optimizing.l

i-I) Theory of consumer D: each individual i maximizes his utilif' U; fti) s.t.budget constraint: L li S Wi. : his incorne * endowment

[we assume his utility depends on his consumption ofgoods bundle represented byvectorl i: (xir, xi:, xi:, ...,xiv) : say (apples, ioranges, 1clothes, 1watch,.....,ipen)

vector p - market price vector (p1, p2. ...,pv),assumeditakesthisgivengandputinto his budget constraint when he is optimizing;in our example (price of apples, price of oranges, price of clothes, ... , price ofpen).

Since i can only afford bundles x I within his budget, we call these feasible bundles.l

Max utility s.t. budget constraint is constrained optimization, so we set up Lagrangian (D

) each individual i Max Oi:Max Ui(x;r. xiz, xi:, .xiv)+l(Wi - ,- p* xi,,,)

and from this consfained maximization problem we can get demand function of theindividual, in terms ofprice vector.

x1* (pr. Pz, . . . ,pv ) -- i's demand for good 1

}t-ftt, pr, ,Pv) - i's demand for good2

x l"r. Gr. pz. ...,pu)- i's demand for goodM

I i's demand depends on tle given prices and his optimization behavior; and x* will befeasible (i.e. within his budget constraint)

We defrre individual i's preference ordering Fi between two goods bundles x(represented by veotor 1) and y (represented by vector y)

.

x >i y stands for: i prefers goods bundle x over goods bundle yx -i y stands for: i is indifferent between x and y.

ifbodr above two cases, we use the syrnbol x L 1 y

fNote: b; represents preference reflecting emotionaVpsychological factors and isdifferent from > which compares real numbers.

We assume that i's preference ordering can be represented by his real-value utilityfirnction U; (hence can use >) . We will study the necessary conditions for existence ofsuch utility function in real space and in metric spaoel.

We assume i's preference ordering can be represented by his utility function as follows:x tsi y if & only if U, (l) > U, (y)

r -i )L if & only if U, (x) : U, (y)

x li y if&onlyif UrG) >U,(y)

$Er

iF EE

5$ i$iFF

c !.Fi'

lit-

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irl'ilrii[it il !irIft' il

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iiifi!iili riiiiir, r!i!l,r,5rlitillrlI

After i solved the above constrained maximization problem given price p the solution

bundle x* (feasible) will be preferred over all other feasible bundles x (i.e. x* L; x )

i-II) Theory of firm: Forj's production vector yi: (y1t, yj2, ...,y;rvr), positive entrystands for his ouQu(s), negative entry stands for his input(s) and enfy 0 means he is notproducing that good nor using that good as input.

e.g. j's production vector maybe(apples, .ioranges, ;clothes, . . . ,1bee{ .;hides, .;homs, . , . .,lcattle feed, 11and ,1labor,.. ., lpen ):( -3003, -60 6 , -36 , -.. ,9 ton,3 ton, .1 ton, ... , -18tons ,-3 acres, -18, ...,-5)

Given price p, each produoer j maximizes profit: p (yjr. y.i2....,y:r,,a)inj,sproductionset Y . We denote solution of this math programming problem by y1

[Since price p> 0, when we multiply the two vectors p and y!, 1le +ve ]jn (outputs)multiplied by p- will be the revenue and the -ve yt* (inputs) multiplied by p. will be thecost and we automatically get the profit function.l

i-trI) Theory of General Equilibrium under perfect competition C.E.

We combine the above Theory of consumer D and Theory of fum by first confiningtotal consumption by all i in good m must be < total production by all j + initialendowment of all i and j in that good rn, for all goods m : 1,2, ...,M.i.e. feasible consumption s.t. x < y+xo, where x u

= initial endowment goods vector.

Market is cleared at certain price vector p! means: g!( y + x0 - I ) : 0,

[i.e. at price pl , total supply = total demand for each and all goods.]

We can frame above in terms of nonlinear programming:

Assume no externalitiesgoods m: m:1,2,...,Mconsumers i with consumption vector xj:(xil, xo, x$, . .,xir) i:1,2,...,1producers j with production vector yi: (y;1, yJ2. . . ..yjM), j:1,2, ...,J

initial endowment vector held by i orj : xl or nf

utility firnction for i : U1 (x11, xi2 , xB , . .x \a ) continuous and real-valued fimction

letx = Lxi y= IJi "o= Ixoi+dii=l j-l i.j

Definition: An array ofconsumption vectors { 1 } is feasible if3 array ofproductionvector{yi} s.t. y+ X0 ) x 1e consumption setX g RM

Definition: given U(1) = [rU (x), rult;, .. .rU(X) ]is a vector-valued function

and r g(x), ,g(x) , MgG) are real-valued functions; 4 e consumption set X c RM

vector X* e X is called a vector maximum ofU(a) s.t. *g(D>0 m:1,2,...,Mif(l)

'" g(l*))o vm

(2) / a s.t. rU(x)2 rU(x*) k= 1,2, ...,Kwrth 1U(d> ru(><|) at least one kand '.g(dZ0 m:1,2,...,M x e X

n particular if u(:r) represents utility vector for all economic agents and constraints arethe feasibility consfiaints, then this vector maximum definition is actually equivalent tothe following P.O. definition.

Definition: Feasible { 1 *} is Pmeto Optimal if / feasible { I .} s.t.

Ur (l ') > Ui (X_i *) for all i, widr strict inequality for at least one i .

Remark:

Vector Max -ve definition

U(x):[rU(x),zU($.,rU(x)]

constraints ,,g (x) > 0

4* vector max ofU s.t. g(a)> 0

,vfeasible 1

IT

l) 1+ feasible (i.e. g(x*)) 0 )

P.O. -ve definition

U(x): [lU(5),:U(x). . ,ru(x)],7

2nd person util firnctiol

consfaint g(l) = y + ><o -21 >0

X* yi vector max of U s.t. g(g $ ) 0

,vfeasible I y

1) x* y* feasible (i.e. g(x*. y*) )0)

2) cannot furd any feasible xr s.t.

hu (ro) , ru(xr), .*U (x#) I

2) cannot find any feasible 1t y# s.t.

LU(x),rUG#r. ...U(xu)l

with sbict inequalityfor at l€ast one i

11, 2,

lr, 2,

3.01, 4l

3, 4)

hu(x.),zu(r"). ..xu(x')l /\ [ru(x').zU(x-), .ru(x-)]i.e. atleastsoms,uftl ' ruG)

"/ | \ ,.". ",,*"tsode iu6') > iuG)

I x+ y*] is P.O.means when feasible x* sav :[rU (x-) , :u (x*.1. 3u 6* y." .g 1*r';1:1r,2,3,41

cannot furd feasible some xt s.t.

LU (xt , rg1t*;, ,U<xnl. rUtllt:1r,2,3.0r,41

('.' if can find such 5", then x* not p.O.)

Definition: Array ofvectors [p*,{ xi *},{ yi*}] withp!>Q lJLi *e X, = i',consumptlon set, yi+e Yj: j's production set V i, j, is aCompetitive Equilibrium ifi

(CEl) U, (x.l*) >Ui(xi ) with p*_x_i Sp*x;* V1i e X; Vi(cE2) p*Ij* > p*& v_y, e Y; vj(CE3) l*S(y++Ie)and p*(y.+ xo-x,*1 -6Econ interpretation:(CEl) means each i will max his utility at x* according to p* s.t. his budget consfaint.(CE2) means eachj will max his profit at y* according to p* s.t. his proJuction set(CE3) means feasibility ofconsumption and market clearing for each market.

e)

Theorem (Vector Maximum): ru(d, zU($. ....rU(x); rg($, zg(x),...,Mg(x) areconcave real-valued functions defined on a convex set X c RMand that Slater's condition holds i.e. f x' s.t. rg(X,)>0 Vj

ff 4* is a vector maximum for U(x) s.t. 19 (a) > 0 V jand define (D( x , &) = Q U(s) +l"g(x) , then

f 9>0 but+q and?,*>0 such that ( x*, i"*) is Saddle point of@(1,1")

[i.e. O( 1, ],*) S O( x* , !*) S O( x* , l") for all 1e X and i" ] QJ

Proof: Please see Takayama [1985].

Now we are ready to proceed to prove P. O. ) C.E. globally

We note the following:equivalant definition

l)PO ::) vector rnax

I 2; * 8t*n slat"t" "-a

"*T-.,_,*^:*)*_l Q>8but+ 0; l,*>0 s.t. (s*, l"+) is Spof O(5, ),) = Q U(1) +l,g(x)

I 3)SP 1*r sr"t ',si €l x* is global max for following concave programming problem:

Max Q U(a) s.t. g(x) > 0 [ConProg 16]

[by NLP theorem in Chapter I. L. which states given concave functions over aconvex set and Slater's condition, SP ( x* ,l,*) of Lagrangian O( l,l,) is: global max for conoave prograrnming problem]

) 4) If above U(x) = utility vector functionand g(a) to be the feasibility oonsraintf + x0 - x I QThen [ConProg 16] becomes following concave prog problem;x*, y! is the global max for [ConProg 16-1]:

Max Q U(a) s.t. feasibility C(x, y, r0) > Q [16-A]

) 5) same x*, y! is the global max for the following concave prog problem:Max U(x) s.t. feasibility g(x, y, x') > Q [Conprog 16-16]

[Pf : Assume d,l is NOT &e global max I : i ..t. U1i 1t U(!*) s.t. feasibility.

Butg>g,q+0 t CU(i) > 0 UG1) s.t feasibility + contradicting [16-A']i d, y! is the global Irra\imrn for [CorProg 16-16]l

I 6) By Kuhn-Tucker-Uzawa theorem in Chapter I, x*, y! is a solution of theconcave programming ) 3po*,p1 :(pr*, pz*. . . .,py* ) >0 s.t.

po* U(x) 1plfu + ro - d s po* U(d) anp

-p10i+ r'- rIJ=0 s.t.feasibility(yl+ x"- x'))0 117-17)

) 7) from[17-17] pe* U(g +pj(y+ x0- dS ps* U(f) &we justadda zero temr : pl e(i!) to RHS of above inequality:) pq*U(g+ Ll (y + r0 - x) s ps* U(f) + pl (y1 + 40 - rI) [18-18]

In addition, if we have following assunptions, we can show [P.O. t C.E] :

(A-1) lSUefs conaftionl I x# € X;y# eYs.t.y#rao-1# >g(if s0 )some i may not survive atr#)

(A-2) Y convex set

(A-3) U,(x_r ) oontinuous and concave function V i

(A-4) [Cheaper Point Assumption] given 41 *, p*: 1 4_; # a X; s.t. p*Li *> p*x3 # , V i[without this assumption, i may only be minimizing his expenditure but notnecessarily maximizing his utility s.t. budget constraint].

Theorem [P.O. ) C.E]: Under assumptions (A-1) + (A-4)

if [{rli*}, y_*] isP.O. thenl p* >0 and{yr*}s.t.[p*,{t*},{yi*}] isaC.E.

Proof: We know [{xt*}, y*] isP.O. andby above steps 1)-7)wehaveAA) from |7-l7l1po* , tr*. : (pr*. pz*. ...,pl,a*) 20 s.t.

po* U(x) + pl fu + x" - E) S ps* U(d) and

pI(yl+ r0- r:):0 s.t.feasibility(yl+ru-xI)>0 t (CE3)

and BB) from [18-18] pq*U(r+ pj (y + :0 - x) 5 ps* U(rj) + pj (y! + r0 - E)we letx:* andy:y1 except some arbifary producer q then [18-18]becomes pl (yn)s pi(yn1 )

since q arbitary from j : 1,2, ... , J

tp1(vi)<p1(vi1) vj t(cE2)

similarly we let x: xi &V:C except some arbifary consumer h then [18-18]becomes

Bixul-pl*u s ps*uu(xf) - ps*uu(xh) vxr',e Xi anvi=1,2,...,r [19-191

2 cases: ifpq*>0, then [19-19] means

Uu(xrJ) >UlGr.) with.Blu* )plxa Vhfromi:1,2,...1+ (CEt)

ifps*:0, then [19-19] means pl xb )plxl* Vxr contradicting (A_4) cheaperpoint, )po* must > 0 (and above p6* > 0 case works) t (CEl).

This also brings out the following lemma_

Definition: consumer i is satiated at4-;# ifu,(4#) > U,(xr ) Vx-.; e X;

Lemma: If I one consumer i who is not satiated at 6 * then p* + Q

Proof: assumep*:0 then from [19-19] p6* Q(d < ps* Ui (Xt)

t i is satiated at x_; * @eductio ad absurdum - proof by contradiction)_

[C.E. t P.O. Theorem is almost trivial]

Theorem: [p*, { :r i *}, { yr-*}] is a C.E. ) I{ x_, *}, {g*}l is p.O.

Proof: from (CEl)Ur (x..i * ) >Ur(I, ) with p*3 Sp*xi* Vx3e X; Vi) cri Ui (l * ) ) oi Ui (x_i ) with feasibiliry oi > 0

) x o; Ui (x i * ) >: o U1(3_1 ) with feasibility 120-201ll

now if [{ 1*}, {1i*}] notp.O.

)3 feasiblefu')s.t. U, (X,')>IJi (a_1*) for all i, with strict inequaliry for at least one i

)X cr,1 U1 (l') > ! o.; Ui(1 * ) with feasibility conhadicting [20_20].ri

t [{ x-; *}, {;vi*}] must be P.o.

iii) Existence of C.E. using Brouwer's and Kakutani,s Fixed point theorems:

[193040 Hicks, Samuelson, studying C.E. ) p.O.and Oskar Lange and A. Lerner (Ref: Econ of Control Ny MacMillan 1944) werestudying the converse P.O. ) f p* s.t. C.E.l

A1. Allocation problem: From the above welfare thms we know ! an equilibrium pricevector at which supply equals demand in each market. But then a cpB can also achievethis by rationing demand & supply of everything. what is different is this perfect marketprice allocation is PO efficient as shown by the welfare theorem.

A 2. Income/wealth distribution problem: As we mentioned, the above perfect marketequilibrium allocation may be Pareto but it may not be "equitable', by some subjectivevalue judgment. In fact, within a static model, any distribution of benefit is paretol

One possibility is to distribute aggregaie income p*(y* + x0 ) by giving p*lr., * to each iand then we also get C.E. and P.O. But this is artificial, contrived and controlled.

We want to show the "real" existence of C. E. with math.

Existence ofCE using Fixed Point Theorem

J

Definition: Ii(p* , y*) :p* 1;u + max { 0, X,0ti !* $*}

where 03 i is i's share of profit from j

e.g. income distribution as

X 01i - I ('s profit completely distributed to a1l i)

(CEl) now becomes Ui (ti*) > U, (X_, ) V-x.i e X1 Vi with p*1 < 11

Need additional assumptions:(A"5) Z: {(xr. xz. ,xr,y): x S y*+ x0 y_e y 4ie X; Vi} iscompact {inreal spaoe = closed (including all boundary points) and bounded (does not go to infinite))

(A-6) Nonsatiation: Vlae X; V i [{ I }, y] e Z

l3 xi'e X1 s.t. xr' bi &_ Vi

(A-7) Inaotion allowed: 0eYi Vj (togerp)0)

(48) Suwival: 3 x1m e X1 s.t. &oo. *,0 Vr

(oonesponding to cheaper point. -- to guarantee i's dernand function upper semicontinuous. Also implication xf is an interior point of Y + X0 in a C.E. and not bebelow subsistence level. We will study the concept of semicontinuity in the followingsections)

(^

[Existence of C.E.]Theorem Ee: Under assumptions (A-1) + (A-8)

I p* >0 butpg*+Q s.t. [p*,{x_r*},{g*}] isaC.E.

Proof use Kuhn-Tucker Theorem and Kakutani's Fixed Point FP Theorem. Please seeTakayama [1985]page289. (Kakutani's FP Thm is normally used when utility functioris concave. In case of strictly concave utility functions, we can use Brouwer,s FixedPoint Thm.)

In the above analysis, we are concentrating on the theory and making all kinds ofassurnptions necessary for the theory to work. e.g.. we assume I utility function; swvivaland semi-continuiry in above theorem etc.

In order to explain the rationale behind all these assumptions, go into details of some ofour proofs and extend our results, we need proper math econ foundation. Ald we alsoneed various FP theorems. The next section on general math analysis and topology istypical grad school coverage for such purpose.

lRemark: For students who want to see a preview of the next section :

A) Brouwer's FP Thms nonemp4" convex, compact c R" and continuous f: s ) S

) - fixed point xx € S s.t. f(x*) : x*.

=Every continuous function f from closed unit ball B " ) B o

has at least one fixedpomt.

[closed unit ball B o & closed unit simplexes A o are homeomorphic].

Application of Brouwer's FP Thm: Proof of C.E. existence in an exchange economywhere utili|., functions are strictly concave.

Sketch ofproof: Assumptions 1) existence ofa continuous, strictly concave and locallyinsatiable utility function u;(xi): non-negative quadrants R" + )R V i

and 2) feasible set closed and bdd (i.e. compact).

Since budget constaints remain unchanged when price vector p is scaled by any constanlwe can normalize p by scaling to get p € A ".Construot use excess demand funotion E(p) to construct a mapping f (p) : p + E(p) .

Such that f: A' ) A' and satisfying all Brouwer's Thm assumptionsi by Brouwer FP Thm 3 fixed point p* s.t. f (p*): p* but f(p*) ulro: p*+E(p*)I p* : p* + E(p*)) I p* s.t. E(p*) = 0 [i.e. p* will make aggregate excess demand function:01I - p* s.t. supply = demand resulting in C.E.

w

B) Kakutani's FP ThmX nonempty compact, convex set e R.'and conespondence | :X 3 X is upper semicontinuous (u.s.c.) s.t. | (x) is convex_valued, then I fixed point x* € S with x* € | (x*).

Note above f in Brouwer's FP Thm is a point-to-point function. Kakutani's Fp Thmgeneralizes Brouwer's FP Thm in a slightly different direction. We stay in Euclideanspace R " , + compagtness + convexrry, but now look at set-valued (-mu1ti-valued :point-to-set = multifunction) functions f called correspondences, in particular semi-and hemi-continuous oorrespondences

Application: Proof of existence ofNash Equitibrium in Game Theory and aboveTakayama's proof of C.E. existence. We shall cover all these topics in detail.]

Chapter II. B. Generalization of math econ concepts using general math analysis

i) Set theory and short review ofreal analysis

In previous seotion, we have all these assumptions on preference (emotional aspect),nonsatiation, compactness of feasible sets Z, semi-continuity of functions etc. whichbasically are assumptions necessary to do the math parts. To understand this we need tostudy the axiomatic foundation of math.

General topology is used in virtually all branches of modem matl. Together with settheory topology provide the axiomatic foundation of math and hence foundation of mathecon.

For undergrad math econ courses we mainly do problems in Euclidean space R.'. @leasesee footnote below). But not all econ problems are Euclidean. Something asfundamental as preference is not really Euclidean. Hence in grad level math econ, wemove up to a more abstact and more basic level to work in general topological spaces (ofwhich R' is a special case) and theory of functions in general (not just real-valuedfunctions). For instance, we work in a space of bundles of goods (not real numbers) inwhich cons'rmers will have a preference ordering (psychological, emotional rather thanreal numbers).

Topological spaces .ne math structues drat allow the formalization of concepts such asconvergence, connectedness and continuity. The three main structures we are interestedin are1. order structure (ordering things so we can talk about finding higher/highest (or

lower/lowest) order for maximization (or minimization)).2. metic struohre (concept of "distance" for measurement purpose, i.e. how much

higher or how much lower, how near. We also need metric spaces with arbitrary set Xto show preference ordering can be represented by real-valued functions).

3. linear structure (to add vectors and do scalar multiplication. This is necessary to haveproper consumption and production sets consisting of vectors.)

And of course, when we move from our abstract level back down to the Euclidean level,we get the usual undergrad defrnition of convergence, continuity of real sequences/seriesand real analysis.

Footnote:e.g. typical undergrad courses use Edgewortl.r Box in RP space to show PO. and CE.

br'C--,a / EIn advanced undergrad and grad courses, we use vector maximum in R:'space for PO. [x*, y*], CE [x*, y+,p*] concept.In proper grad coursies, we can start widl any space like a commodities space - not Rl- and preferenceordering. Then modi$ the space (make it compact etc) so that preference can be represented by utilityfunction etc., making the analysis both realistic and general.

6ncttt^l

(;-' ''-'

Terminology and basic notions.

Given arbitary sets X and Yboth * O; set of natural numbers N: {0, 1,2,....,n, .....}

FtINCfiON fis a binary relation € X x Y Cartesian product:1)Vx€X,3yeYs.t.xE/2)V y,z €Y, xS and xfz ) y:"

Notation: f : Xdomail codomaio

ffiNotation: yx:1f :X ) y)

X is COUNTABLY INFINITI, if 3 bijection (1-i conespondence) f : X ) N

X is countable if it is 1) finite or 2) countably infinite.

Cardinality of a set : number of elements in the set.

[example : N is countable but set of real numbers R is not countableCountable set has cadinality : cardinaiity of some subset of N]

[X, Y countable ) Cartesian product X x Y a]so countable.l

fintuitively if (x, y) € Cartesiaa product X x Ymeans we caa find a way(namely, a tunction g: {X,Y}) U{X,Y} s.t. g(D €Xandg(Y) €Y VX,Ye{X,Y})to pick one element x fron X and one element of y from Y. For finite and countablyinfi:ritely many Xx\xZ ..,., we can find such a function gJ

For Cartesian products of infinite number of nonempty sets Xx YxZ ........, we needAxiom of Choice:

V nonempty class Y ={Y Y is a set}, 3 a function g: Y ) uY s.t. g(Yl € Y VY € Y

[ntuitively: we just rely on this axiom that we can pick one element from each of theinffnitely rnany sets in the Cartesian product. This is equivalent to :

1. every n-nary relation conteins a function.2. Zorn's Lemma which we will cover after introducing order structurel

Given set X * Q, binmy operations + and . on X ; x, y, z€X.The list(X, +,. ) is a FIELD F:

if 1) commutative: x+y:y+x ; x.y:y.x2) associative: (x+y)+z:x+(y+z) ; (x. y) .z:x.(y.z)3) distributive: x(y+z):x.y * x.z

4) 3 identity elements 0 and 1: 0+x:x+0:x ; 1.x:x.1=x5) 3 inverse elements -x and.x-r : x+ -x : -x +x :0,

Vx€K{0} x .x-t : x-r .x = I

[Intuitively: A field is an algebraic structure for doing arithmetic. x € X is called a scalarExample: set ofreal numbers R; set of complex numbers Z]

VECTOR SPACE V overthe freld F:

A set V with two binary operafions:Vu,v,w€V; x,y€F

i) veotor addition v + w: Vx !-+Vii) scalar multiplication xv: V ---+ V

satisfuing axioms below:

l. associative: u + (v + w): (u + v) + w. .t1o

2. commutative: v + w=w + v.

3. 3 identity element zero vector 0 € V: v+0:v

4. I inverse elements -v €Vwith v+-y:$.

5. Distributive: x (v + w) : 1 v I * to.

6. Distributive: (x + y) v : x v + y v.

7. x (y v): (xy) v.

8. Soalar multiplication identity element 1: 1 v : v

flntuitively: Vector space is a set of objeots called vectors that can be scaled (scalarmultiplication) and added (vector addition)Examples: real vector spaces R ' , R ', ... ; in particular R n

is n-dim real vector space]

Ordered BASIS B for vector space V : { br , bz , .... , bn } s.t. for any v € V ) 3 1!linear combination a1 b1 * azbz+ .... * aoln : y

a'lU,

I: I Coorcmate:la.l.J

L>l

A,

1-/T

v

lvls:

[e.g.forR3: b1 : (1,0,0) br:(0,1,0) b3 -(0,0,1)and all vjn rlat vector space can beexpressed as linear combinations of (1,0,0) (0,1,0) (0,0,1)

Now we add order structure into a field to set ordered field:

Order: Defrne a binary relation E : X) X as:

(Vx,y,z€X)1. Transitive (T) ifx E yandy E z )xD z

2. Reflexive (R) if x E x

3. Complete (C) if either x E y and y E x

4. Symmetical (S) if x E y I y E *5. Antisymmetrical (A) if xE y& yE xl x=y

P called preorder ifTR andthe list (X, E ) ca[ed preordered set

E called partial order if TRA and the list (X, E ) called poset

E called linear order if TRAC and the list (X, E ) cailed loset

Boundedness (Bddness): S c poset P is bdd from above if I a € p s.t. a P p Vp € p; and

bbd frombelow if 3 b € Ps.t. p E b Vp € p.

If S is bdd both from above and below then it's called bounded fbdd).[intuitively, it means S is contained in an intervall

[intuitively, E i. u generalization of all "greater or equal,' Spe ordering, ilcludingsubset ordering I in math, preference ordering ) in econ, 5 ) 4 in real number Rl

Application to Theory of choice in Econ: preference relation ) on a set x of arbitraryobjects is usually assumed to be TR, so list (X, )) is a preordered set. A preordered set

(X, P ) is just a generalization of (X, >). From tlere we can extend to utility functionexpressing such preference ordering.

o

Szpilrajn Theorem: Every poset can be extended to a loset

[krtuitively: we can extend the TRA to TRAC with a more general E I

Define an equivalence relationship = if it is TRS

Define equivalence class of x with respect to : : {y € X ; y = x} denoted by [x]=

Application to Theory of Choice in Eoon: [x]= is just a generalization of indifferencecurve going though the point x.

[f we move back down to the Euclidean space R.', we get the familiar indifference curve]

Given preordered set (X, E ), @ +Y c X and y* € Y

Y* E-maximalif/y€Ys.t. y > y*

y* E-minimalifiy€Ys.t. y* > y

ify* Ey V ye Y I y* called E-maximum of Y.

if yEy* V y€Y ) y* called E-minimum of Y

Gven a poset (X, E ), ** is an upper bound for poset if x* E x V x € X

Given a poset (X, E ), ** is an lower bound for poset if x E x* V x € X

The supremum of X, denoted by sup(X), is the least upper bound. i.e. sup(X) is the E-minimum of {upper bounds xi of X}lnfimum of X, inf(X), is the geatest lower bound: the biggest of all lower bounds of X.

A sequence in a nonempty set X, denoted as {x.}, is an ordered array of elements ofX,(xt,xz, .... ,x.,....) x. €XSo basically it's a function f: N) Xwithf(m):x. m€NGiven seq(x1 ,x2,.... , x-, ... .), take the seq m1 (rrlz (m: <.. < m. <.... and the seq{x * "}

is a subsequence of {x,n}

Z! space which has a dlstance measure

[Example: a Cauchy sequence in metric space - its elements becomes closer and closertogether as it progresses. i.e. after n terms of the sequence, the maxirnum ilistancebetween any two elements becomes smaller than an arbitrary e > 0.

e.g. {xJ = 0, v,, 1/3,....1&, 1/m,...) and l1& - l/ml < l1&l + l1/ml ;

as k, m ) co Iim l1&l + ll/ml = 0l tlistance between l/k & l/m squeezed) no "holes". But since we are talking about distance, we can do this only in metricspaces which we will cover.

Later on, we will look at comptrete metric space M (where all such Cauchy sequencesconverge to a limit, i.e. evely Cauchy seq of points in M has limit also il M). And we canalways complete the space by fflling in all the "holes". E.g. given Q which is incompletebecause there are "holes" of irrational numbers like .f,2 and z ..., we can make itcomplete by filling in all the irrational numbers.

Metric space l1{ js sorn.pact ifr M is complete and totally bdd (versus S C R " is compactiff S is closed and bdd so we see compactness in metric spaces works like the abstractgeneralization of finiteness due to the bdilness.l

Zorn's Lemma: If all loset in a given poset has an upper bound ) poset has a maximalelgment-

[Zorn's Lemma is equivalent to Axiom of Choice]Hausdorff maximal principle: f a f -maximal loset in all posets.

ordering defmed in terms of subsa (inclusion) relation.

ORDERED FIELD (X, + , . , 2) is a freld with partial ordering on Xs.t.x 2 y) x+z2y+"aod if z2 0thenxz2yz

Notation: X. = {x } 0l X*n: {x > 0}

X ={x <0} X--: {x<0}

Note in any ordered freld (X , + , . , ]), the triangular inequality holds:

llx+vll s llxlr - lyll t";a.pl.' f.]r". 'utr*riXii>rx-g l{,,tl+llYrl-11xtJlltt1tt/ \$/ €t '"oZ

..n> -,K #;+# r

[example: rational numbers Q form an ordered field, R is an ordered field but thldifference is the following completeness axiom:

Every nonempty, bounded from above S c R has a sup@) : a real number r*.

Intuitively: R is complete (no "holes" between its elements) while Q has "holes"(irrational numbers slots between its elements)]

ii) Topological spaces

In modern math foundation, the concept oftopological space is essential.We study topological spaces to see the foundation structure of math econ. These include1. order structure, part of whioh is covered in the above analysis, for optimization2. mefric sfucture used for measurement and optimization3. linear structure for vector operation.

We start with the concept of space.In any discussion, we have a universe ofdiscourse (the universal set).

We call this a space: a collection ofobjects (= points).

Deft: Given an arbitrary nonempty set X (x can be anrthing, e.g. X can = {oonsumption bundles forJohn) );

let r : a collection of sets Oi c X s.t.

3.

1. @er, Xer

Oi e t i: 1,2, ..., m )Ooer aeA )

fli=r Oie t

Uo66 Ooe t

2.

then the pair ofobjects (X, t) is a topological space with topology r.A member 01 of t is called an open set.Collection t is called the topology on the underlying set X.

ln addition, if the topological space (X, r) has a distance (= metric) frrnction d for itspoints, then it's called metric space, denoted by (X, d).

Defn: (X, dx) is a metric space if X is any nonempty set and has a distance firnctiond: XxX) Rs.t.l) d1 (x,Y)=A |ff1:t x,Y e X2) dx (x,y) + d(v, z) > d(z,x) tiangular inequality3) d;(x,y) Z 0

a) d1 (x,y): d(y,x) slmmetry

flntuitively, with the distance firncfion, we can talk about how near (the distancebetween) any point xo is to another point x in the space; hence we can have notions ofl. length between 2 points - used to define open/closed intervals ; and

\ t\ |

.'.?Lo , ''-n--lt,

2. neighborhood of xe - used for function continuity ( A function f(x) is continuous at x6if f maps points nearby to x0 to points that are nearby to f(x o)) I[n analysis, we study metic spaces for connectedness, separability, compactness andcompleteness concepts.e.g. Econ Application: connededness + separability + Richter-Peleg ) utility )Rader's Utility Tluu ) Debreu's Utility Thml

The real line R is a topological space @, t), associated with the metric space (R, d) ofreal number system with a distance function (called Euclidean mefic) d(x1, x):lxr*xzl(i.e. length ofthe interval (x1 ,x2)) between any two points x1 .x2 eR

If interval does not include endpoints x and y, then it's an open interval. If intervalincludes the endpoints x and y, then it's a closed intenal.

o?6t l'^je*lh|-. CIASt+ tilfe&,A,

On the real line, neighborhood is in terms ofopen intervals, namely,given any xs,r e R" r>0,the set B"(x6) = {x eR:d(xs,x)< r}: {open intervals around about xo with radius r}.Open sets in this metric space can then be characterized as a union of open intervals.

l-dimensional @, d) metric space

flI p.ta',co I

P.

nedr*il^;L 6"-(xn)We can extend to Cartesian products ofR. namely, R " space:

@ , r) topological space associated with the metric space(R.' , d) with a distance firnction called

-

Euclidean metric d(x,y) : .,/l (.r, - y, )'?I i=1

tu -+F tu

and open intervals are conespondingly generalized to n-dim open ballsB. (x o) : {x e Ro: d(xo, x) < r}

Example:

2-dim (R'z, d; ig.a metric space Euclidean metric d (*,y):{G:;:Gfldenoted also by ll . ll . Not" open interval generalized to an open circle (2-dim ball)

,rc6.tx'\={tt f:a{'";)arJ

(.

3-dim (R', d) metric space

u&,>4."t ebbrc)<ltJ

R-

K';'l--" *ut,v\l'''"

[We note in the (Rl, d) space, points can be expressed in terms ofvectors.

VECTOR: Ordered n-tuples ofreal numbers is called a vector, denoted by

x.= (x1,x2,....,xi, ...,xo) xi e set of real numbersl

Analogously, in a general metric space (X, d), an Open BalI around a point x 0 e (X, d) :B,(xo): { xeX: d(x,xs) <r} and xois called the cenfe and r the radius.

Defit: nonempty S c (X, dx) is an open set if for any x e S, I +ve real number rs.t. B' (x ) c $.

[example: all open balls are open sets, collection of all open sets in X is open. Note openball are nonempty because it at least contains the oenhe]

[Intuitively, since we have d in metic space (X, dx), we can define open balls and hencedefine open sets. We let t be the collection of open sets defrned in terms of open balls,then we get a topological space (X, t )l

Since a metric space have these open (and closed set) set properties, we can leverage onthese to move to even more abstract topological spaces. @lease see below)

In addition, in metic space, we have idea of one element being "close to" anotlerelement in terms of distance metric d in an abstract space. Using this "close-to-ness" wecan talk about convergence of seq in metric space because convergence intuitively is justelements in the seq getting arbitrarily close to the limit.

Given mehic space (X, d1), x 6 e X andS C X, a sequence {x.}convergestoxoifforany real number e> 0 I n* e N++ (+ve natural number) s.t.

Vne N++ 2 n* ) dx(x6,xsr)<e

We write lim {x-}: x o or {x-}) x 6

If x e € S, then seq {xo'} is a convergent sequence in S

iii) Generalization ofecon ideas in abstract topological spaces

'YDefir: Given topological spaoe (X, t ),S c Xis a closed set if its complement S " is an

open set. In other words, S " €T

[examples: in R, a, b are finite rea] numbers: intervals [f,A], [a, .), (-*, a] are closedsets. (a, bl and [a,b) are neither closed nor open sets.]

Given topologioal space fi, T ), sequence {x,o} in X is convergent in X if I x o € X s.t.

for any open set Y containing xe 3 n* € N++ where Vn€ N*+ 2 n* I xn € Y.

In an arbitrary topological space, such a limit x o need not be 1 t

In particular, for real seq {x,n} € R, it is bdd from above if sup {x. : m € N} < co

And bdd ftom below if inf{x* : m € N} > - co

And bdd if bbd from both above and from below

[ntuitively, bdd from above means all the terms x'o in the seq ( some real number B]

Every convergent real seq is bdd and we have

Bolzano-Weierstrass ThmEvery bdd real seq has a convergent subsequence.

Defo: (X, fi) and S c X is closed iff V sequences in S convergent in X converges to apoint s € S.

Sisbddinxif I e >0s.t. Se neighborhood N.,1(x)t

{y € X: fi(x,y)< e }

Continuity of single-valued function f.

Defir: Given metic spaces (X, dx) (Y dv) and function f: (X, dy) ) (! dv),fis continuous at point x0 if for any real number e > 0, I real number 6 s.t. foranyx €X , d1( x, ro). 6 ) dv( f(x), (xo)) < e . Iffis continuous at all points ofX, then fis continuous in X.

flntuitively, continuity means for points x close to x6, (x) will be close to f(x6) in thespace.l

Since R is also a metric space with the usual Euclidean metric ll . ll , we can move back

down to this R spaoe, then f: R ) R and we get the familiar Bolzano-Cauchy epsilon €-delta 6 definition for continuity:

Areal function f; XqR) YcRis continuous at a e Xifve )0, =6>0s.t.vx€X lx-al < 6 t I{6d-(d1.,This is the neighborhood idea we mentioned previously. f is continuous means: iJ wewant to restrict flJ values in any arbitrarily small r -neighborhood, we only need tochoose small enough neighborhood ofx values around a. In other words, if we can dothis regarclless ofhow small the neighborhood offld then fis continuous.

Alternatively, we can also define continuiry in terms of seq. Letx € V c R, I V ) R;if X seq (x-) in V\{x} withx-) x or if ! such a seq & lim u-* f(v): (x), then f iscontinuous at point x.

Third way is to define continuity by open sets. This approach is most general and easiestto gxtend to more abstract sDac€s:

-. .- ^-1^:*f is continuous rn X ifff -'(W) is open in X whenever W is open in Y) continuity definition is extended to topological spaces as:

f: (X, Tx )) G Tv ) is continuous if frflID € T* whenever W € T'

We know continuity is a local concept because we define continuity of f at a point. lf f iscontinuous at each point on an iatervayset, we say fis continuous on an intervaVset. Butws still have the "each point" local concept.

To extend this local into a global concept, we need to extend continuif based onuniformity:

Given f: (X, dx) ) (l dv), f is uniformly continuousif for every real number e > 0, I6>0 such that for all x, xo € X with dx( x, xo)< 6,we have that dr( f(x), f(a)) < e

fintuitively, funiformly continuous means fnstly srnall changes in x I small change inf(x) (continuif part); secondly the size of change in f(x) depends solely on size of changein x but not on the porn I x itself (uniformity part).

In other words, uniform continuity is a global concept.l

s9

Note:l. Uniformly continuous functions are continuous but not conversely. (e.g. (x) : 1 I x, x€ R*' is continuous. But as x ) 0, changes in f(x) become unbounded, hence notuniformly continuous).2. A function f that is continuous on a closed bdd interval is also uniformlv continuouson that interval.

[f we move back to real space R2 with Euclidean distance function ll . ll , then denr ofuniformly continuity becomes the familiar: if for any e > 0, f 6 > 0 such that for all x, xo€Rwith llx- xo) ll l< O I ll (") -("o) ll< ' t

[Concept of uniformly continuif can be generalized into topological vector space.]

Econ application: Ordinal utility theory

Deftr: Given set g c preordered set (X, >) (TR). If Vx,y€ Xwithx>y, !z€ Ss,!x| z 2 y, then S is Z-dense in X.

kr particular, let ) represent preference relation as we know it in Econ; and X as

{ outcomes} or {conunodities} or {alternatives} ) econ agent views altemative x to be

at least as good as alternative y ifx 2 y --- including strict preference relation ) and

indifference relation ! on X.

We let preference relation on X to mean preordered set (X, >)

Reflexivity: x ) x is easily acceptable as rational behaviorTransitivity: is usually accepted as a rationality hypothesis. i.e. if you prefer x over y andy over z, then you will prefer x over z.

Note for Completeness: you must know x )y or y ) 1 Vx,y€ X. There will beobjections that this be included as rational behavior since many times we don't know.For instance, trying to decide whether to go to Chinese U to study Chinese literature or toHKU to study madr. Unless there is complete ordering in such n-dim choice set, we maybe indecisive.

Defir: Given preordered set (X, )) where ) is the above preference reln. V x € Xweak upper --contour sets U.(x) ofx is defined as:

U.(x):{yeX:y)x}

Analogous defo for stict upper )-contour sets U, (x) ofx:U,(x):{y€X: y>x}

And weak/stict lower )-contour sets L.(x) / L, (x) of x:L.(x):{y€X: x)y}I-,(x):{y€X:x>y}

In order, to do utility theory in Econ, we need to find the class of preference relations thatcan be described by a utility function and then maximize the utility function.

Defir: Given 0 * S c any arbitrary set X and preference relation ) on X.Function u: S) R represents ) onSif

x.y itr u(x) - u(y) Vx,y€S.

If I such function u, then u is called a utility function and preference relation ) isrepresentable.

Note such a utility function is not 1! As any strictly increasing self-map f(u): u(X) ) Rcan also be a utility function. .Hence this is an ordinal (order only, noi for calculations,not cardinal utility function) utility concept. U ^r-p^.-t--* -

€ithei x > y or y>x

T\nt: 0 # X is a countable set and ) isa complete preference relation onX I ) isrepresentable.

BirkhoffTlvn:0 * X is a set and ) is a complete preference relation on X. If f acountable l-dense S c X, then ) representable by a utility function u: X ) t0,11.

[A lexicographic preference reln on R2 is not representable by a utility function.]

Hg) -t'l/'7L

I

A preorder which is not complete cannot be representable. In 1960-70 people (M.Richter, B. Peleg) were trying to relax the completeness assumption. They modified theutility function u: X ) Rto:

x>y t u(x) >u(y) AND x=y i u(x) = u(y)

called a fuchter-Peleg utility function.

Lemma [Richter]: Given 0 * X and ] is a preference relation on X.X contains a countable )-dense subset I 3 a Richter-Peleg utility ftnction.

fRichter-Peleg utility firnction: domain X is not completely ordered ) range R iscompletely ordered. Hence there will be cases of indecisiveness or agent cannot comparethe altematives.

Shortcoming: (information contained in Richter-Peleg u) is < (information contained in>\

Because uG) > u(y) only tells us y is not strictly prefened over x but does not say econagent actually prefer x over y.

One possible way out is to use a set of real-valued functions as utility index to representpreference:

SetU: X * R represents ) iff u(x) > u(y) Vu€ U Vx,y€ X

[e.g.: givenR",n ] 2.

We represent the partial ordering 2 by a setof utility frrnctions u i(x) =xi V x€ R"Thus x > y itr ui(x) 2 ui(v) i:1,2,,....,n

We now move the above analysis up to the metric space level.

Defu: Metic space (X, dx) and ) is a preference relation on X. ) is uppersenicontinuous if strict lower contour set L> (*) :{ V € X: x ) y}is an open subset of X;and is lower semicontinuous if U, (x): {y€ X: y> x} is an opel subset of X, Vx €X; and is continuous if ) is both upper and lower semicontinuous.]

[Intuitively, for upper semicontinuous preference relation, if altemative x is preferredover y, then x is also prefened over alternative z which is very close to y.

Alternatively upper semicontinuous preference reln means:Seq{y''}inX)y t:B€R s.t. x)y.VmZB.Thus linking emotion ) with math metric ) through semicontinuity.l

closue of S wrt to X is tlre smallest closed set in X contai[ing S

Defu: (X,dx)andS c X. if cl x(S) = X, the S is dense in X (or S is a dense subset ofx).[Intuitively, S is dense in Xf. if any point in X can be "approximated" by points in S,

2. any nghd ofx N .,x(*) contains at least l point from S.l4

{y €X: dx(x,y)<. }

Defu: (X, d1) is a separable metric space if X contails a countable dense set.

fExample: R is separable metric space '.' Q is a countable dense subset in R]try9*uct 19e9l9ey on {functions: R ) R} is a separable Hausdorff Space with cardinalityof 2' (called beth-two, with c as cardinality of R) i separable spaces can be ..not

small".But in general, when we talk about separable space, we think of it as not very .,large,'

because there's a countable set in this space almost the size of the space itself.AIso intuitively, separability means the metric space has few open sets.]

We wish to have metric spaces which are "nice". Separability is .hice". (please seebelow Rader's Thn.). Also all open sets can be described as countable sets oi open sets) know all open subsets of (X, dx) ) know all its closed &/or connected set i knowall seq convergent in that space ) know about upper and lower semicon and continuity.other properties making a metric space "nice" are compaotness and connectednes;.@lease see below after Rader's Thm).

Thm: Rader's Utility Representation(X, d1) is a separable metic space and ) is a complete preference relation on X.) is upper semicontinuous - it can be represented by a utility firnction u: X ) tO,ll

Before covering compact sets in topological spaces, we need to remark on class of sets.Gven a set X, its power set denoted by 2x :{S: S c X} ('.' if I n elements in Xt f 2'elements in 2x e.g.-X: {a, b, c} with 3 elts, then 2x has lA, fu\, lb}, {c}, {a, b}, {a, c},{b, c}, {a, b, c}} 23 : 8 elts. I cardinality of2x > cardinalig, of X. S

[Cantor's Paradox]: X: {S: S isa set} cannotbe a set.Assume @: {S: S is a set} is a set) [VeltSin2@:> S g @ andisasett S€@jmeans2@c @I cardinality of 2@ < cardinality of contradicting s. So @ = {S: s is a set} cannotbe considered as a set. we can just call it a (proper) class which is a collection oi sets.

Defo: (X, dy) and S c X:Cantor's Paradox (Also Russ€ll,s peadox a set ofan sets.)

tAclassC of subsets of X is said to cover S if S c u C.If all such subsets in C are open in X, then C is an open cover of S.

lAlternatively: class c of open sets {c 1 } with i € I (an index set). c is a open cover ofS if each point in S belongs to at least one C 1l

Defu: (X, d1) is compact if every open cover of X has a finite subset also coverins X.S e X is a compact subset of X if every open cover of S has a finite subit atsocovering S.

[Lrtuitively, compactness provides a finite strucf,re for the infinite concepts we aredealing with.compactness helps to extend-results {iom metric spaces into more general topologicalspace settings Examples are function spaces that are not metric spaoe;. For ourpurfose,it is very important for optimization.l

In oase of R o space, we have the following thm.

Thm [Heine-Borel]: Every subset S G R n is compact iff S is closed and bdd.

Note this is only valid for R o space and not for any metric space because there are closed& bdd nenic spacas that are not compact. In other *o.ds,

"o,npactness is a broader

concept than closed & bddness. In fact, compactness iff closed & bdd + equicontinuous.6 collection.of functions is equicontinuous if ar the functions are continuous and haveequal vairation over a given neighborhood.)

And the important thm:

Thm [Weierstrass]: (X, T ) and continuous funotion f: X ) R.Compact S g X t f achieves a maximum and a minimum in S

Thm A closed subset ofa compact metric space (X, dJ is compact.

Econ application:

(x, dx) with x = {choices of econ agent} with subset Fc X where F :{feasibre choices

of econ agents) assumed * Z

Define optimal choice setC(F): { x€F: /y€Fs.t. y > x}

If C(F) + @ thenwe have a optimal choice solution x* s.t. feasibilitv.

We will show that C(ry + O and it is compact.

case 1) ifF finite and since we have transitivity of) , we can always order all elts x and

findthex*€Fs.t. ly€Fwithy > x ) C(g + O

case 2) if F is Nor finite (e.g. infinite-dimensional commodities space)BUT compactAND ) is upper semicontinuous:) is upper semioontinuous t L, (x):{ y € X: x ) y}is an open subset ofX for Vx in Xt Alleltsx€F c X also € u[open subsets L, (x)j )u {L, (x): x€F} is anopencover of FAt the same time we know C(F) - {x€F: Xy€Fs.t.y ) x}= n{F\!(x*) :x€F}Suppose C(F): n {F\L, (x*) :xeF} -At F:F\Z:F\ n{F\L,(x*): x€F}:u {F\ F\ L, (x*) : x € F}: u{ L, (x*) :x€F}is an open cover ofF.Since F compact) f finite subsetFF c F (F is a cover) ) Since FF finite and by transitivity of>) I )-maxim:rl elt x* in FF and note x* € F\ L, (x*) (..' x* ) ;x)I x* must € L, (x+) for some x+ e FF\{x*} (defn ofcover) ) means x* is in strict

lower contour set in FF\{x*} with x+ ) x+, contradicting x* being the )-maximal elt in

FF) C(D+ o

To show C(ry + A is compact:

Pf: L, (x) is open set in F ) F\ L, (x) is closed subset ofF V x € F) n {F\ L, (x) : x e F} is a closed subset ofF which is compactand [r {F\ L, (x) : x € F}: C(F)t C(F) compact. (closed subset of a compact metric space is compact,t

[Above means C(D + O> ! opt (most preferred )-maximal) elt in feasible set; andexplains why we needed semicont and compact set as ass'mptions for Thrn Ee. To fullyprove that thm and also its counterpart thm in more abstaot spaces, we need to coverfixed point theorems below.

Completeness properly of space

Deft: A Cauchy sequence is a sequence in a mefic space (X, d*) if for any e > 0,

lB€Rs.t. d" (x-, xn) <e forallm,n )B

[Intuitively: the later terms beoome arbitrarily close to eaoh other]

Thm. A seq in metric space is convergent t it is Cauchy.

Thm. If a Cauchy seq has a subseq that converges in X, then the seq itself converges inX

Deft: (X, da) is complete if every Cauchy seq in X converges to a point in X[Intuitively, completeness means there are no points missing either in the interior or theboundary.

Q is not complete (points like irrationals are missing) but R is complete.l

Thm. Metric subspace S of complete (X, d1) is complete iffS is closed in X.

Compactness I completeness but converse not true (e.g. R is complete but not compactbecause not bdd).(X, d1) is compact iff it's complete and totally bdd.

\S c CK, dx) totaily bdd if any e > 0, I fuite T c S

s.t. Sc u {neighborhood N .,;(x): x € T}

[e.g. hfrnite discrete space is bdd but not totally]

Reason we want completeness is because contactive self-map (contraction) on acomplete metric space has fixed point.

Defii: a self map d: X) X on a metric space (X, d") is a contraction (= contractionmapping) if 3 real number 0 < r < 1 s.t. d,.(d(x), 0 (V)) < r d"(x,y) V ay€X.[Intuitively, g shortens the distance between two points. S is uniformly continuous.]

K sn..,e X_

.1,--

rd'"t)

r

d[e),4@)

| -- -aQ&)t,@,4611

-- -cg,

La(x-,t) > az(\eYdtas) \8'7 )

Gven a self-map f :X ) X, f(x) = x is a fixed point x€f

EO1{

Fixed Poitrt FP Thm [Banach]:Given complete (X, gs), contaction d : X )x*ex.

In a complete (X, dv) a contraction mapping has a Mxed point. such a fixed point isthe limit of the convergent (cauchy) sequence defined by the recursive equation x ,*, :6(xJ, n=0, 1,2, ....... This is formulated in the thm below:

&

4()i.|

L

X'/

l1! fixed point d(x*)=**

4en

4@t't)

dtrorlD))

+r,ft

l*t{ r. t) Qk) x-1

[The Banach FP Thm does not hold for a metic space that is not complete. The Thm canalso be used to prove existence ofordinary differantial equation solutions]

Defo: Gven any nonempry setX and self map d: X) X, define 6r= 6 21d 6n+r=0(O'),n:1,2,3,.... The self map0n is called the nth iteration of d.

Thm: Given complete (X, dx), d' (nth iteration of self map O ) is a contaction I dhas a fixed point d (x*): **,

"* a

".[Intuitively, if the iteration results in a smaller and smaller interval (confaction), then itconverges to a fixed point. ]

flteration is the starting point of recursive methods, difference equation and its phasediagrams.lDerivation ofgenera.l solution formula for l$-order linear autonomous difference equation:

xt = bxt.r * a a, b constants

We start iteration for t = 0, l, 2, ....,t

xr= bxo + a

x:= bxr + a = b(bxo + d + a=b2xo+ba+a = [2 1s+a(b+f)

xe= bxz + a = b(bsxo + a(b +1))+a= b3xo +bza +ba +a

a:a,

But geometric progress of t terms of sum

+ a( bz + b+ 1)

a(bfr + bt2 + ... + b+1)

- bsx

= bixo +

1-bt ifb l1

1- b

(bt.r +bt.s + ... + b+ t) {

) General solution (when no initial condition is specified and we get family of solutionsby varying initial condition x o ):

{xo - [a/(1-b)]]b'+ ia /(1 -b)] b* 1 [6-6]

{xr = txo + at

ifb=l

l,- r

Ifinitial condition is specified I get specific solution.

Phase diagram of nonlinear difference equation: y t - (y t_r)2

tt

1r+-*tt

.)

[Importance of FP theorems in Math Econ:

If we wish to specifi conditions for existence of a solution x* ro a sysremof simultaneous equations, where the solution is in the form off(x*) : 0.ifwe can express d(x)=(x)*xwhere d: X) Xis aself-mapand introduce the same conditions required by some Fp theorem on O (x)into the systemthere will exist some fixed point x* by this FP ThmFP means d (xx) = x,rf(xi) : d (xx) - x* : 0 and we have accomplished what we set out to do.

E.g. Given an exchange economy with consumer i's demand D; (p) and his initialendowment Wi :

we define an excess demand function f(p) : ), @, (p) - W; ) (when f(p) : 0 meansequil in tle market '.' demand : supply).

we define o (p) = (p) + p and impose some conditions (like some continuity conditionson f(p) and on concave utility functions, compact convex feasible set) to ensure bvBrouwer's EP Thm I a fxed point p* (: price vector) where d(p*): p*+ d (p*) = f(p*) + p* = p*t f(p*) = 0

This is the FP part ofthe proof of Thm Es

2. For game theory, if S _ I is a mixed strategy for all n players except for player i.

Define a best response correspondence g 1(S_i )toS_1 : {all prob distrib overopponent players profiles ) to { i' s strategies}.

Define @ (S): O, (S_r)x O2(S_2)t ..., d"(S_")

If3 a fixedpoint S* s.t. S* e O (S*) = I strat€gy set which is a best response to itself= no player can I by deviating from that strategy

= Nash equilibrium

) set up the conditions for Kakutani Fp Thm so that d (S) has a fixed point and we getexistence of Nash Eouil.

t)t

@

[It is said that when in 1949 Johr Nash presented his (-Nobel-prize-winning) theory tovon Neumann. Von Neumann's response: "That's trivial, you know. Thafs just a fixedpoint theorem".]

3. Suppose we are given an:tio1; ;: (x) + c where fis a vector and c is a real numberIf f is a contraction, then we can define self map g(y): (y) + ct g(y) also a contraction) by Banach FP Thm f a I ! fixed point y* s.t. y* : g(y*)I y*:f(y*)+cI y* is a solution for x: f(x) + c.l

In general, any topologioal space in which any continuous self-mapping must have afixed point to said to possess a fixed point property. Note not all topological spaces havethis _!xed

point property, e.g. a continuous self-map rotating the annulus of i dougbnut(or disc) has no fixed point. If annulus is filled in, then the centre will be the fixed point.

since Banach FP Tlm provides the theoretic foundation of Bellman DvnamicProgramming DP, we will cover this subject first before rehrning to various other Fptheorems.

ffi'#ifi;3,Hf*i"ffffi.i,I

Lemma: S a nonempty set and 0 +X c B(S) meansfext f+a€Xclosed under addition by +ve constant function r'

oeaosf ,g€Xandf -g) O(f) > O(g)Gven increasing self map @-. X ) X I

and 0<5<l s.t. @(f+a) <O(f)+6a

| @ is a contaction

Given following functional equation (note fon both sides of equation):

o (("): max { d (x, y) + 6 f(v)} o<x,y < 1 0<6<1

with d (x, y) being any continuous mappilg: [0,1] x [0,1]) [0,1J x [0,1]

Note:1) d (x, V) obviously bdd real function (hence complete).2) 0 (x, V) is closed under +ve constant function addition'.'@ (x, y) continuous t O (x,y)+a with a> 0 also continuous

Note: rD: C[0,1] t {[0,1] x [0,1]]\set of continuous real functions on [0,1]

And O maps a continuous function to a continuous function I (D is a self map on C[0,1]

Iff(y) -g(y) t d (4v)+6(y) > d (x,y)+s g(y)) max{ O 1",y;+6f(v)} ) max {O (x,y)+Og1y;;t O(D > (D (g) t @ is an increasing self map

Furthermore@(f(x)+a):max { O (x, y)+ 6 ((y)+ a)}

: max{ g (x,y)+d6(f(y)}+6a< O (f ) + 6 a

Then all hypothesis of above lemma is satisfied and @(f) is a contraction.

) l ll fixedpoint O(f*)=f* f€C[0,1]

=f * solves the functional equatior f(x) : max{ g (x, y) + S f(y)} with 0 <x, y< I0<6<1

which is called Bellman's functional equation.

One other property of metric space is comectedness.Deft: (X, dx) is connected if / 2 nonempty and disjoint open subsets X:, X:c X s.t.X1u X2 :X

[tntuitively, metric spaoe is connected if cannot be written as union of two or moredisjoint open sets; so pizza is connected brt two separate pizzas each representing a space!s not connected. As an aside, dou€ftnut is not path-connectedl

We shall return to our discussion on Bellman Dynamic Programming later. First let us gothrough the other fixed point FP theorems so that we can wrap up our discussion onclassical welfare tleorems and game theory.

Observation: Connected sets 6 a: {all intervals on R}.

Lemma: Given metric spaces (X, d,.), (Y, dy) and continuous function f: X) Y.

Xconnected ) fOO cormected c Y.

Pf: Assume f(X) not connected I 3 2 nonempty & disjoint open subsets Or , Oz of f(X)s't' Or U Oz= fCXl)'

Foraayxe f -t(Or), f(x) € Or whichis open ) - e >0s.t.N .,y(f(x)) cO1

Furthermore fcontatx + f 6>0s.t.f(N 0,1(x)) cN .,y(f(x)) butN .,v(f(x)) eOr) f(N a,1(x)) c Or. Alternatively, N o.;1x) cf-r(Or).Since x is any arbitrary x e f -r(O1)

, t f -t(Or) nonempty open qX.^. ,, . ^-t-^,Smllarly 1'-'(O2) nonempty open cX.

These 2 nonempty open f 't(Or) , f -t(Or) must be disjoint (if they are not disjoint ) ! x

€ Or anp 02 with (x) € Or and (x) € 02 contradicting Or and 02 are disjoint,hence f -'(O1)

, f -'(Oz)

must be disjoint.)Moreover f -1(O1) U f -t(Or):f -r(Or U Or):X, contradicring X connected ) ffimust be connected.

Actually the simplest FP thm is the lntermediate Value Thm IVT. (= l-dimensionalBrouwer's FP Thm)

Version I of Intermediate Value Thm IVT(K, d") connected and continuous function f: X) R.(x) <c < (y) anyx,y € X) latleastonez € X s.t. f(z):c.

Pf: From above lemma (X) connected c R t (X) interval in R and on this interval,every point is : to some f(x).) any c in the interval (x)<cSf(y) must: to some f(z) , z €X.K llol

f cont & X, f(X) connected) no break in fgraph and [x, y]) f going from f(x) to (y) mustgo tlrough e : (z) and cannot skipover it.

tf,TlJI)lsl

fG)="

!e>

q

*3-o^

fq-)

x-c*__

v

x*, J&*)

ftt< t

IVT CorollaryCase 3) f(0) > 0, f(i) < 1

fcont ) from (0) to (1)must cut 45o line'.'no break in f graph andCannot skip over 45o line

Real life example inciudes: continuous water evaporation over time, temperature changeor fever over time:

Or lost 30 lbs weight over 15 day ) sometime withil the 15 days must have lose 9 lbs.

l.a*

t5 )4

-w3olh l--

lo t+

l0

Version 2 of IVT (Bolzano)

[a,b] nonempty, compaot, convex QR, f continuous: [a,b] ) R and f(a).(b) < 0) 3xx € [a,b] s.t. f(x*):0.

Corollary: f continuous : [0,1]) [0,1]

) 3fxed point xx € [0,1] s.t. (x*) = y*.

Pl Case 1) (0):0 ) done.Case 2) f(1): I ) done.

Case 3) f(0) + 0t f(0)>0; f(1) + 1t f(1) < 0 @lease see diag on previous page)Define F(x) = f(x) - x. Note F : [0,1] ) R and continuousAnd F(0) = f(0) - 0 = f(0) > 0 and F(1) = (1) - I < 0 t F(0).F(1) < 0 and by above IVT

) 3x* € [0,1] s.t. F(x*):g also = f(x*)-x* I (x*;:t* frxed point.

Converse of IVT does not hold. E.g. f reai-valued function on interval I. Any a, b € Iand Vu € ((a), (b), 3c € (a, b) s.t. f(c) =u i fcont?No! e.g. (x) = sin llx x *O and f(0) = 0. f has FP but not cont.

Above corollary is a special case of a more general FP thm called Brouwer's FP Thm.Where cont f: S ) S and S no longer: [0, 1].

Retraction: (S, ds) metric spaces c metric space (X, d*)

continuous function f: X) S is a retraction if f(x) = x Vx€ S and (S, ds) is a retract of(x, d,.).

[Intuitively, every point of the codomain is a fixed point of the function f. Note R can becondensed continuously by the foliowing cont f into [0,1] in a way to leave each point in[0,1] intact (x) : 0 for all x < 0 and f(x) = 1 for all x > 1; for 0 < x < 1, f (x) :

" gr"

identify map.Note [0,1] cannot be condensed into (0,1).1

Metric space (X, dJ has FP property (every continuous self-map f on X has a fxedpoint) does not guarantee its mekic subspace also has FP property. E.g. [0,1] has FPproperty but (0,1) does not.

Lemma @orsuk) If (S, ds) is a retract of iX, d"). (X, dJ has FP property t (S, ds)also has FP properly.

Brouwer's FP Thm: S nonempty, convex, compact c Rn aad continuous f: S ) S

I - fxed Point x* € S s.t. (x*) : 1*.

a0

Pf: Please see Border (uses Sperner's Lemma in combinatorial topology).

[sketch of an altemative pf:a unit sphere So in a normed vector space V = {v € V: ll v ll = 1} a closed unit ballBn in a normed vector space y= {v € V: ll vll < 1} and in case of Ro, 5 n

=1112 +x22+... + x"'?= 11 andBD:{x12+ xz2 + ...+xn2< 1}. Note the borurdary of Bois asphere S

o'1 (e.g. projecting a ball in 3-dim space onto 2-dim spaoe we get a circle).'.' B n is nonempty, convex, compact G Rn , we can rc-state:

Brouwer's FP Thm: Every continuous function f from closed unit ball B " ) B n has atleast one fxed point.Lemma @orsuk) fl any retraction fiom B n onto S

n't n = 1,2,3, .....[Intuitively, there is no oontinuous mapping from all the interior poinls of a disk onto the circleforming the boundary ofthe disk.l

Suppose X FP x* : f(x*). Take any x in B I and f(x) #x I caa draw a straight linejoining x -- f(x) and extend the straight line to cut S "

-r (at the boundary) at g(x). g is acontinuous function: B o ) S'-1 and is a retraction, violating above Lemma @orsuk)) must 3 FP x* : f(x*). l

fNotes:1. Generalization of Brouwer's FP Thm into infinite dimension by extending unit ball ina Euclidean space to one in a Hilbert space is not true '.' uait ball in infrnite-dim Hilbertspace is not compact. So such generalization require additional compactless and manyli-nres convexity assumptions on the space.2. Above Thm does not hold for open balls.3. Brouwer's FP Thm is used to show existence of solution to differential =.tions,existence of CE and existence of equilibrium in Game Theory.]

Examples in real life application of FP Thm:Directory map of HKU campus, airport, subway etc and point "you are here" is a fxedpoint. Map of the world on the floor--some pt in the map lying directly over the point itrepresents. But if you are haveling in a space ship, drop world map on space ship floorwill not result in a FP because you are outside &e set S.

Application of Brouwer's FP Thm: to prove existence of a C.E. in an exchange economy(no producti on).

Consumers i: 1,2, .....,I commodities j : 1,2, ..., J

i's consumption bundle vector x; =( x1 , xiz, ...,x;:) xi € rt'* n

& his initial endowment c; : ( c;r, c;2, . . .. c.ts)

allocation x = (x1, x2, x3, . . ., x1)pricevectorp: (pr, pz, p:, . . ., pr)

Ary *' e R.'*)3 yi € Rn* inneighborhood of x;

.s.t. u ;(y;) > u i(x1)

Assuming existence of a continuous, strictly concave and locally insatiable utilityfrnction u i(xi): non-negative quadrants R: o ) R Vi. And assume X q < Q,making the feasible set closed and bdd (i.e.compact).So we have a set of utility functions {ui(x)}

[f utility function is concave but not strictly concave, we need Kakutani's FP Thm toprove.l

For a function f: X ) Y, argument of the maxargmax(x)={x€ X: f(x)}(y) Vy € X}

Define C.E. (x*, p*) as

l x;* € arg max{u r(xi): s.t. px; <pc;(budget)&0<x<Q(feasibility)Vi}2. lxi *: Ii ci

Define function difu): arg max{u r(x): s.t. px; -< pc;(budget) & Ixi S Ici < e(feasibility) Vx€ R.'* )This is the most preferred bundle in the feasible budget set : demand by i (analogous toderivation of demand function ftom the l'r order condition in utility max).Since u(x) continuous, stictly concave ) d 1(p) well defrned and single-valued V p.

ITerminology:the standard (= 6it) n-simplex denoted by N: {(r,, Xz, X:, .. ., *t) € R.': x1 } 6 66Ii xt-1) ( Intuitively it is the n-dimension analogue of a triangle)0-simplex is a point1-simplex is a line segment plus all interior points.2-simplex is a triangle plus all interior points.

3-simplex is a py'ramid tetrahedron plus all interior points. (viewed from above X )4-simplexGiven a regular (n-1)-simplex, pick a new vertex and join it to all original vertices by acommon length edge to form a regular n-simplex.In topology, simpiexes are the simplest convex sets.Note: all balls and simplexes of the same dimension are homeomoryhic,(homeomorphism = topological isomorphism = isomorphism between topological spaceswhich preserve topological properties. Hence homeomorphic spaces can be consideredas same topological spaces) ) balls and simplexes are same in topological spaces.]

l) For any a> 0,px; < pci E apxi < apci which means dro) = d(a p) because thebudget constraints remain unchanged in the optimization problem when p is scaled by a) in our optimizatlon, we can scale any price vectorp'by letting a: | /\ p;' s.t.p;.)newpricep:'/E pi' and summation of all these new prices ! (pj' / Ij p; ' ): 1 (i.e.normalize the price vector) F-9) for the following analysis, we can just use normalized price vector which means ourprice vector p € {p € R" * and I p: =1 } t p € unit simplex N in Euclidean space.

2)'.'dr(p)generatesmaxutilityinfeasiblebudgetsetsoforanyui(x')>ui(di(p))thensuch \'must be outside the feasible budget region with px i ' > p d ;(p)

3) if x;* solves max{u 1(x): s.t. px;3 pc; (budget) & 0S x s Q (feasibility) V i} aadsuppose px1* < pci) by local insatiability assumption 3x;'in neighborhood of x1* s.t.px 1' < pci and u i(xi')> u 1(x;*) conhadicting x* optimal ) pX i* = pci conshaintbinding.

Lemma (Walras Law) p€ Ri*and p. [}(dr(p) - ci)l =0

Pf: by3)px1*:pcr ) pxi* -pci :0* p(x;* -ci )=0 V i) summing over all iwe Betp.[Ii (d;(p) - q)] :0

Lemma: lim sequence { p, } in A'=p, then d;(p) ) d;(p;t) oo

Pf: Please see Vohra.

We defme an aggregate excess demand fimction B(p) : XrI (d iO, - ci ) so Walras Lawsays p.E(p):0

Standard procedure to prove existence as we mentioned is to let function p + E(p) to havea frxed point i.e. let -p* : p* + E(p*) ) E(p*) = 0 or p* will make aggregate excessdemand function to be zero and supply : demand for the equilibrium.

@

We use a variant of the E(p) function, call it g(p), for the proof:

Defrne g (p): (gr (p), crb), ...,g1 fu) ) on An

CjO) = p;+max{O, L(dt(p) - crj)} i (Lr(p.+max{0, |1(di*b) - ci.)}):p;+max{0, |i(dii(p) - c,.;)} / (l+I'r max{0, Ii (di,(p) - ci.)})

Note this is a variant of normalization of price F-9, modified to have nonnegative excessdemand of all p.

t g (p): (er (p), crfu), ...,sr (p) ) is a continuous function by above lemma and also

l. g,(p) = 1 so g: ^^"

) ^^"

By Brouwer's FP Thm f fixed point p € Ao s.t. p= C O) t p.i = g: b) t(unit balls and unit simplexes are homeomorphic).

pj:pj+max{0, }(d,;(p) - c,t)} / (1+I",'max{0, Ii(di-1p) - "-)}))p: (1lX-'max{0"L(dr.6) - ci"')}): p;+max{0, }(d,:(p) - c,:)}lp;€,' max{0, Ii(di.b) - ci,)}):max{O , l(d,:(p) - c;1)} [ff-10]

We know that max {0, tr (d rib) - c i)} > 0; and we only need to look at cases wherepi > 0 (we restricted ourselves topl > 0andifp; = 0, commodity j is afree good and allconsumers will consume j up to tl-re constraint t } (d ri(O) - c11 ):0 and we havesupply ofj = demand ofj in market j).

case 1) max {0, } (d,:(n) - c ,,:)} = 0 and p; > 0

) (:,'max{0, !; (di.G) - "r)}):0) each max {0, } (d ^(p)

- ci-)}:0) X, (d;-(p) - c;') < 0

Again we only need o look atp,n > 0 ( if p,n:01m is free commodity and supply =demand in market m):Subcase l-1) ifXr (di,(p) - ci.) : 0 we have supply = demand in market m-Subcase 1-2) if Ii (d m(p) - c i-) < 0 and p- > 0)p* Oi(d^G) - ci.)<0 Vm) Lp. Oi (di.(p) - ci.) )<0 violating Walras Law Lemma) we can only have Ii (di,(p) - cin) = 0 and supply = demand in market m

case 2) max {0, } (d,:(p) - ci.i)} > O

[F-10] and for p; ,p. >0) (tr max {0, } (d",(p) - ci-)}) > 0

€e

I some m s.t. l; (d *(p) - c;-) > 0* p

^(Xi (d'-(p) - c,-)) > o

t t* p'" {x; (d l"b) - c *) ) > 0 contradicting Wal ras Law Lemma) can only have Ii (d,-(p) - ci-):0 and supply = demand in market m

So in ;'all cases, we have sunnlv: dernand in eech msr!.et and the f;xed poini p is.uhcequilibrium price guaranteed by Brouwer's FP Thm.

Kakutani's FP Thm generalizes Blouwer's FP Thm in a slightly different direction. Westay in Euclidean space R n

, * compactness + convexity, but now look at set-valued(:multi-valued : point-to-set : multifunction) functions called correspondences, inparticular semi- and hemi-continuous conespondences.

Terminology:

Correspondence | : eachp point ofa setX 3 subset ofthe setX.(use notation 3 for multi-valued functions). So now f(x) is a set c X.

l:x32r/{a} = x€X = f (x)c YIf f (X) c X then it's a self correspondence on X.

For correspondence | (x) , a fixed point x* means xt € | (x*).

Correspondence l(x)is: 1) convex-valued if I.(x) is a convex set, Vx€X.2) closed-valued if | {x) is a closed subset c Y

For both firnctions (point-point) f X > Y and conespondences (poinfset) I-: X 3 !the graph, denoted by Gr is:

Grf:{(,{,y) €Xx Y y=f(x)} Gr | :{(x,y) €Xx Y y€ | (x)}

Since now we are dealing with correspondences, we wish to have the concep ofcontinuity to be consistent with continuity in the single-valued function case: i.e.if we collapse a multlvalued correspondence continuity to that of a single-valuedcorrespondence, we should get the ordinary continuity notion. [Intuitively, ordinaryfrrnction continuity is a graph showing no breakl. We can do this either by uppersemi/hemi continuity or by lower semi.&emi continuity of correspondence.

r .?('jeuwa.r +&,t tl.a!\

(*vt-<{-o..J.a^<t

Ytr-t

Y

*D

\a^ f $.@q)

L

And we know that to understand Kakutani's Thm properly, we need to clarify theanalogous concepts of continuif for both single-valued functions and multi-valuedconespondences.

For single-valued function i.-.. f Topqlggrg{l qpace

Defir: Given G, dx): f: X) R is upper semicontinuous (u.s.c.) atx* €Xifforanye>0 3 6 > 0 s.t. d1(x*,y)< 6 )f(y)<(x*)+ e Vy€d;(x*,y)

f is u.s.c. if it is u.s.c. Vx € X.

[Intuitively, fis u.s.c. at x* means points ppp nearby x* )1. (ppp) not > f(x*) by too much but can be much lower; or2. f(ppp) very close to or < f(x+)l

Example:

R

l.>

I

0,5 ,F{

x' foro<x<1f(x): { 15 forx:1 is not continuous but is u.s.c. atx- 1

1-x for I <x

Ofct)$ ,r-s.c. aE x!= 1

Xz5=t J.'

tGiven (X dx): f: X) R is lower si:micontinuous (l.s.c.) at x* € X

ifforany+g f 6>0s.t.dx(x*,y)<6t(V) l (**)-€ Vy€dx(xn,y)

f is l.s.c. if it is I.s.c. Vx € X.

flntuitively: (ppp) very close to or > f(x*)]

fc*: t;,n*v,s.c .Y\q

L.s.c.F,Of uJ. C, '.'f cert) t tcrit)J

8

t)

(a

In real space:

Altemative defir: f: X c R') R is u.s.c. on Xif Vr€R {x€X: (x) 2 r} isclosedinX.= {x € X: f(x) ( r} is open in X.

Alternative defu: f X c R.') R is l.s.c. on Xif Vr € R {x € X: f(x) ( r} is closed in X.= {x € X: f(x) > r) is open in X.

u.s.c. and l.s.c. is a weaker notion of continuity.Above f(x) is continuous on X iff f(x) is both upper and lower semicontinuous.

The reason we split the definition of continuity into u.s.c. and l.s.c.:

Most students lmow that real-valued function over a compact set X assumes its max/minin X if f is continuous by the following theorem:

Weirstrass Thm: (X, da) compact and continuous f: X ) Rt J x, y € X s.t. (x): sup fCX) aad (y): inf fCX)

,v7supremum = least upper bound infimum: greatest lower bound

foraselS g & if =

a bouod s.t. set S isbdd fiom above. then tie smallest oftheseupper bounds is sup

But actually f only needs to be u.s.c. to get the max and l.s.c. to get the min:

Lemma @aire) (X, d1) compact and continuous f: X ) R1)fu.s.c. ) I x€X s.t. (x):sup f(X)2)fl.s.c. ) I x€X s.t.(x):inf f(X)

Application: we only need u. s. continuity for preference to be representable by utilityfunction.

Rader's Utility Representation Thm II(X, d1) separable aad ) complete preference relation on X. Then) u.s.c. ) ) representable by a u.s.c. utility frrnction u: X) R.

Debreu's Utility Representation Thm([ d;) separable and ) complete preference relation on X. Then) continuous I ) representable by a continuous utility function u: X) R.

In fact I continuous (/u.s.c.) utility representation for> iff ) is continuous (/u.s.c.)

We now look at meaning ofcontinuity ofa multi-valued correspondence | .

Terminology:

[CAUTION: In the literature, especially in earlier textbooks (e.g. Tatayama [1985j),semicontinuity (s.c.) is sometimes used both for single-valued functions and for multi-valued corespondence. More recent publications us-e semicontinuity for frrnctions andhemicontinuity (h.c.)for correspondences. However, we can usually distinguish betweenthe two by looking at the function/conespondence in question.

Confusion will oniy arise if we are discussing real-valued single-valued correspondencebecause a semicontinuous real-valued frmction f (single'valued function) is not ahemicontinuous cortespondence (multi-valued function) unless fis continuous.]

Defn: given (X, dx) , (Y dy)correspondence f:X 3 Yis compact-valued if l(x) c Y is compact V x€X.(i.e. if image is compact subset of Y).

[We leamed the following 2 definitions already. Here we just state them again but thistime in terms of metric spaces:

Deftr: given (X, dx) , (l dv), correspondence f :X 3 Yis closed-valued if f (x)cY is closed.

AND Defn: ifY c R' and l(x)convex V x€X, fis convex-valuedl

[CAUTION: Defn of h.c. is not standard and is defined differently by different writers.E.g. Berge required compact-valuedness in his defu of upper henicontinuity (u.h.c.)The general defrr ofu.h.c. is dfferent from Berge's definition ifimage set Y is notcompact. When compact-valuedness is absent, Berge definition becomes the definitionof closedness of I at x. However in most econ writings and applications, and also inour notes, compactness of image set Y is usually assumed so this normally will notpresent any problems. Specifically, when closed-valued I is mapped into a compact

space, Berge's and the general definition of u.h.c. coincide.Observations: Y is compact and I is closed I f u.h.cAND I is u.h.c. & closed-valued ) f is closed.l

Thm: Topological spaces X and X correspondence l: X 3l I is u.s.c- inXiff |closed in X and Y is compact.

To extend continuity concept from single-valued fimction to multi-valuedcorrespondence, we define hemicontinuity.

General Deftr:

Correspondence I isu.h.c.atx ifwhenever x is in the upper inverse ofan open set* a neighborhood ofx also in the upper inverse of that open set.

= I. -'(o)- { xexs.L F(x) co} voc Y

Conespondence |:X 3Y isu.h.c,atx iff upper inverse image f-l(O)isopeninXV open O c Y

Given (X, dx) , (Y dv)correspondence f: X 3 Yis u.h.c. at x € Xif Vopen subset Or c Y with f (x) c6r, I 6 > 0 s.t. f O{o,x(x)) qOr

=Alternative Defn:| : X 3 Y is u.h.c. at x € X if any two sequences seq{x "} from X and seq{y

"} from I

Y compact, with:x,,)x; y"€f(x,) Vn; and y,) y€Y) y€ f (x) (i.e. seq {y"} € f (x") ) some end point y, if endpointy€ l(x)

then u.h.c. at x)1.

Alternative Deftr: Correspondence | : X e Rl 3 X (compact) is u.h.c. ifGr | ={(x,y) €Xx Y y€ | (x)}isclosed.

Defn: If u.h.c. V x € X then I is u.h.c. on X

[When u.h.c. (and similarly for l.h.c.) concept is reduced from set-valued conespondenceto single-valued correspondence (essentially like a single-valued function), thensingle-valued correspondence fis u.h.c. itr fis a continuous function) Every u.h.c. (similarly for l.h.c.) single-valued correspondence is a continuousfrrnction and conversely.AND a semicont real-valued function is NOT a hemi-cont correspondence unless it isalso continuous.]

[Intuitively,

u.h.c. at x means small change in x # image set | (x) to suddenly becomes larger.

Alternatively,l) if move from point x to nearby point x' there will not be any point in | (x') that is notnearby to some point € | (x).

2) as we move from x to nearby x', | (x) will not suddenly contain new points.

3) image set | (x) can implode but not explode given small change in xl

Numerical example:Conespondence: f :R3Rx<0 f (x) : {y: x-3<y<x-1}x:0 I-(x):{y: ,3<y<3}x>0 I.(x): {y: x+1<y<x+3}fNote f is convex-valued Vx: I- (x) is a convex set]

Vx* € R I is u.h.c. at x*

Check:any sequence seq{x

"} from R

and seq{y,} from R with:xo)x*; y" € f(x") Vn;andy") limit point y*€ R

Is limit point y* € | (x *) ?

Ifyes, then I u.h.c. at x*)

WLOG let x* < 0andseqxn)x*ifseq {y"}€ r(x")note y o) intewal (x*-3), (x* ,1)li limit point y* must be

x*-3<y*<x*-1) y* € | (x*)) I- u.h.c. at xt

Similarly for x* ) 0

can show seq {y "}€ | (x,)) some end point y*,and y* € | (x *) then u.h.c. at x*

Double-check for the case x*:0:I(0) contains all limit points of f(x)in neighborhood of x*-0 so okay.

flntuitively;I is u.h.c. at x* means when x is approaching x*in t]le domain with y in | (x) approaching its limitpointy+, I will findy* in f (x*) set.l

( u.A... ix *e R,

Y u.at l-.A.c d' f=o-: *t *1fx"\ * o

t c1 f1-3 ,^.,r 3.el-e<".)

rf 1n-+ o ere)

Numerical example:Correspondence: f :R3Rx<0 f (x) : {y: x-3Sy<x- 1}x:0 r(x) : {0 }x>0 l(x) = {y: x+ 1<ysx+ 3 }Note I] ris sfill cnnrrev-',ol"o.l

Vx*€R x* * 0

I is u.h.c. at x*'.' same as above example.

But at x* : 0 not u.h.c.:

WLOG, any seq{x ,}> 0 from Rand seq{y"} from R with:xn)0 ; yn e f (xn) V.,;*d yo) y* >2so y* 4 | (0)I I not u.h.c. at x* = 0

Y -h... il z*1ont u,&.c, aJ z'x =o

f v,c+ L,l..c. or r-x=o

R-

e

,: qt &- o H 4f 1,3,,'-v €rF,,>r$. g. -+ o (f@>

I'ell

xlhct u.t.c.ax *t

Ye>

T,^./'"

f at4o L.las.wf u,Le,

lttI

6 zrz u,j,-c

ttat t,t,c.

fu>

tn.4, e : hsd,* t.h. c"

a.-.* L.A.e.

tr- t-.t .c "t xg3

x-

u+"/,2

al

tlt

I

vl x.> /,,

fu'.] el1 r.)tf,,

g

'4|

4x- 7t 1- X'1,a '+ >rRt-s.,- I"rffi-{ v€Et

- I [!^1e tTxn\ sf, t" * S :ft-3eGc**) rt ,a.-+t

t''+

v.\

Defn: correspondence I is l-h.c. at x if whenever x is in the lower inverse of an openset ) a neighborhood ofx also in the lower inverse ofthat open ser.

r -(O)= {x€Xs.t. f (x) nOl u } VOc y

f :X 3Yisl.h.c.atx€X iff lowerinverseimagef_1{O)isopeninX VcpenO G y

Given ()i dx), (y, dv)correspondence f: X :i Y is l.h.c. at x € X ifVopensubsetOr c Ywith f (x) nor I @, I6>0s.t. l(x,)OO1 I oVx'€N6.11x;.

[.h.c. means any element in f (x) can be approached from all directions]

Altemative Defn: f :X 3 yis l.h.c. atx€X ifany 2 sequences {x,} inX andyiscompact, with:x")x andanyy€ f (x)I - seq {y " }

in Y such thaty")yand y"e t(x") Vn

If l.h.c. V x € X then I is l.h.c. on X.

Deftr: given (X, dx), (X dv)correspondence f: X 3 yis continuous at x € X ifcorrespondence is both u.h.c. andr.n.c. at x-

It is continuous if it is continuous on whole set X

[Intuitiyely; I is continuous at x means small change in x # image set | (x) tosuddenly becomes larger upward or downward.]

]hile we are on the subject of correspondence continuity, we cover the following BergeMaximum Thm.

@

Maximum Thm @ergg Debreu)

Given (l dv), CX, dx) andspace ofcontinuous real maps on X x y

Vy€Y,'C\,.)er

compact-valued correspondence B: y 3 X; u€C(XxY).

Define d(v): ar-g max{u(ay): x €and u*(y) = rn*{u(x,y): x e e(V)}

IfB continuous at some y € y t1) d: Y 3X compact-yalued, u.h.c. and closed aty2) u*: Y ) R is continuous at y

S nonempty convex set q R", function f: S) Risstricdy qu|si.concrve if(ct x + (l_a)y) > min{f(x), f(y)) Vx,y€S ,0<cr<l

(quasi.concave if f((l, x + (l_a)y) =

min{f(x), (y)}),v

[ifu strictly quasi-concave * d is continuous]

fPlease note l.s.c. of u is necessary for Maximum Thm]

B(y))V

<-Y 4ha.,',r...,k J

[when optimizing continuous real-valued firnction over compact set, and the compact setis varying continuously with some parafteter vector, then opimal *iution..tir ri ufp".hemicontinuous correspondence with compact values

= optimal solution changes upper hemicontinuously when constraint set chaneescontinuously.

knportance of Max Thm is that it tens us whenever constraint sets (rike budget set) andmaxrmands are continuous at x*, then the set of maximizers is u *"tt-u"t uuJa u.t .'".

^-

correspdence at x*. Thus letting us prove various important math "cono-i"

tlreoi"ms.

From Berge Maxm Thm can ) to Michaer Serection Thm for fhe suffcient condition forselecting continuous ftrnction from the set of frrnctions in tlte correspondence )Brouwer's FP Thm ) Kakutani Fp Thm.

Berge MaxmThm also provides foundation for one-sector optimal growh model arrdstationary Bellman DP and we will retum to this topic later.f

Application:

ln the above Maxm Thm:Let price vector p € R*n (strictly positive Euclidean space); w : wealth/income € R.,*Since budget constraint p xS w depends on both parameters p € R++n and w €R*andw9 can lef Y: {(n w)} and q in the above theorem becomes a budget conespondenceB(p,w): Y3X = P*'*i 311

ln the above Thm, d(p, w) will then be the demand correspondence, is well-defined (bvWeierstrass Thm), dependent on parameters (p, w)and d: R*n*l j R

ur in the Thm can be utility function and is assumed to be confinuous with u,n: X x y)R with u(x, (p,w)) ) R.

Maxm Thm I B compact-valued & continuous correspondence at some (p, w) € y:1) demand correspondence d(p, w): R.,*"+1 3 R;

compact-valued, u.h.c. & closed at (p, w)2) u* : {(p, w)}) R is continuous at (p, w)

t If X x Y is compact; and B, u are as in above Thm, then u* is continuous and bvWeierstrass Thm, we can find (x*, (p, w)) to max u

{ow we are ready to look at fixed point theorem involving concept of hemi-cont. wefirstly state different versions of Kakutani's Fp Thm as preview:

Kakutani's FP Thm: X nonempty compact, convex set c R.' and correspondence| :X 3 X is upper semicontinuous (u.s.c.) s.t. | (x) is convex-valued,

then 3 fixed point x* € S with xr € | (x*).

(here semicont means hemicont as I is a correspondence.)

Altemative Statement:Kakutani's FP ThmLet S be a nonempty, compact and convex subset of Euclidean space. Let | : S 3 2s bean upp€r hemicontinuous set-valued function on s with the property that | (x) isnonempty, closed and convex V x € S. The I has a fixed point.

Pf: Please see Border [1985 reprinted 2003]

chapter tr. c. Kakutani's fixed point theorem and application to Nash Equilibrium mGameTheorv.

Kakutani's Fixed Point Theorem:Graph of conespondence G(b):. b:y+X={r;,, r)e yxXs,t.xe b(;)}

Grft) is closed in product metric space y x X = b has a closed graph Cn@)

X nonempty, closed, bdd, convex set G Ro.If b convex-valued self correspondence: X) X having a closed graph Gr(b)) 3 fixed point x* € b(x*).

fNote importance of convex-valued.

+50

CotL*4-rn-lu*l*

#

e,"t 3 F** ?E

W$. c, cev'rS**a--'c"e*

**t Lm*gc'

#F

l( tt>)

Application:

Game Theory: strategic interaction (action €) reaction/response) of a group of pplayers.

Action space ofplayer i Xi: {actions feasible to player i} V i=1,2,...'PWith game outcomesx=(xy, x2,......, xp) €Xr x Xz x....... xXp:XAnd payoff function of i: r;: X) Rs.t. n i (x) > r; (y) means i better offwithx:(xr, x2,......, xp)thany=(y1 , y2.......,yp)Action space of player j other than i: X-1 : { (w,, w, ...,wp_1 )s.t.w1 €X.;V j<iandwl-r €Xj V j>i) fori=1,2,...,P(xi, x -J: i's action xl given a1l otherj I i players action x _i €X-1e.g. i: 5 , P*7 thenX-s: { (wt, w2. w3, w4, wo, w7 )

P-person sfiategic game G: { (Xi, ri) i=r.2....,p }

Defn: Given game G as above, outcome x * is a Nash Equilibrium NE if

x* € argnax { ri (*r, x-1*)s.t.x1€X1 } V i:1,2,....,Pand

NE(G) : { Nash Equilibria of a game G}

If X i nonempty, compact c R.' , then G called a compact Euclidean Gane. If payofffunction zi is continuous real function: X) R V i= 1,2,....,P then G is aoontinuous and compact Euclidean Game. If each X i is convex and compact and each zi(x;, x-i) is quasiconcave for any given x-i € X-; , then G is called a convex andcompact Euclidean Game. A compact Euclidean Game that is compact and continuous iscalled a regular Euclidean Game.

Thm (NasQ For a regular Euclidean Game G: { (X1 , n i) i=t,2,....p }, NE(G) + O.Pf: Definebi:X-t 3 Xi and b: X 3 X is a self-conespondence with

bi (x-i): argmax { ri (xi, x-;*) s.t. x; €Xi }b(x) =6r(x t) * b z (x- z)*....... x bp (x-p)

Weierstrass Thm I b is well defined

V x€b(x) ) x1 € b; (x-1) V i

t x € NE(G).

Each Xi compact and convex ) X cornpact and convex (by thm).

For x€X and 0< cr < 1, andy,z €b(x) best responsest ni(y;, x-i): ni(zi, x-i)

Since ,ri quasiconcave tz1(oy; +(l-a)zi, 11) > min{ [ z i(yi ,x -j):nt(zi,x _; )] , zr ; (z;,x-; ) ] > r;(wi, x_ 1)(all other strategies) V wi €Xi

I linear combination ay+(l-cr)z €b(x)) b convex-valued.

By Berge Maxm Thm b has closed graph (and is self-correspondence: X ) )Q

i b satisfies all condition ofKukutani FP Thm) bhasaFPmeans lx* €b(x*): br(x-r*) x b2(x* 2*)x....... xbp (x_p*)

i.e. x i * is best response action given all other players best response action x _ 1 *

I no incentive to change and reached a NE.

Direction of generalization of Kakutani's FP Thm:

Kakutani-Glicksberg-Fan FP Thm: generalize Kakutani's Fp rhm into infrnite-dimtopological space:

Given S nonempty, compact and convex subset of a locally convex topolog.ical linearspace. Let f :X) 2x be a Kakuteni map (i.e. X, y topological spu""s; y coorre*;f: X ) 2Yis upper hemicontinuous and I (x) nonempty, compact and convex V xex.)) f has a fixed point.

Lefschetz FP Thm: applies to almost all arbinary compact topological spaces. Giveconditions in terms of singular homology (set of topological invariant of i topologicalspace X) that guarantees : a fixed point.

chapter III Continuous Time Dynamic Equ'ibrium and optimal economic moders.

A. Review ofdifferential equations in dynamic analysis.

Differential equations depict functions over continuous time (any instant of time) versusdifference equation over discrete time (points i" tin," lirie iast rrour, this hour, next hour)

Review

f differential don f difference :tiont

indefinite

I

differential

\ J(x)dx

integral

----^------,_definite

,ll" r19ax

l -variable

Iy = f(x)

1't order derivative

dy/dx: f '(x)

l

mulfivarialeI

Y : (xr, Xz, x:, ..., Xn)." --""--

r"- order partial derivative

I

af af af

dxr dxz 6xo

I2od order partial derivative

a2f a2f azf

a xt2 a x22 o xo2

.l2* order cross partials

a2f a2f

totalderivatives\(show 2-variable which easilv'".t..a"u[

t" "-"ffi"?*il \af af2fr order denvative

I

d(dy/dx) / dx

: d2y / d.*: f .. (x)

d (xl .x2) = ----- dxr + ____-4",

dxr oxz

iip

dy0x2 dxrdx:

nth order derivatived\/ dx": f \x)

a2f ^)^o-r

6xzdxr b xz? xz

nth order partial derivativesa"f

dxi"

nth order cross partials

6"f

dx1 8x; 6xr.......

Cl, C2 oonstants of integration

_r1l\ :.-bt a cl-cz"-J

wherec=ecl-c2 [tr-D-10]

tdy dx called, -)differentials

(df(t)/d0e.c. (0: where (t) is a function & b is a constant [f-D-9]

$olution of4 differential equation is a function (or an equation) without anyderivative or differential terms, defined over an interval and satisfies the differentialequation for values in that interval.

e.g. solution for above differential equation is: (reananging terms of [II-D-9])

(df(t)/d0 / f(t) = -b and integrating both sides of equation

-b

i -t at

-bt+Cl

) f(t): exp ( - bt + Cl

tf(t)=ce-bt(is a function without ary derivative or differential terms.)

We can always check the solurio! by differentiating the solved f(t) irr lll-D-l0l:dfit)/dr = d(ce-"' ) / d r = c e - o , ( - b) = - b

" " h'= -b f(l)

and get back the original differential equatior [II-D-9].

(O

when solutions contain unsoecified varue constant c, the solutron is ca'ed a generarsolution (i.e' a famr.-ry of functions by varyin! c, .rtiffit ffi..entiar equation).

initial conditionWhen c is specified by or solution called parti culzu or deftnite.

boundary conditions

+t;^14g

speci6, f(t) = 0 boundary cond itionthen AI' defrnit€ solution

For the above differential eouatign tII_-D-gl, f(t) can be radioactrve erement at time t,then the differentiar equation is d".".iuiog il, ;onti";;;;"uy process of radioactiveelement f(t) over time'rn phvsics:

Initially decay occurs faster because there is more (radioactive) material, then slows*111*"u1.: kss and tess (radioactive) material.

'So rate of decay ifi(rfi;G-"proportlonal to amount of material f (t)

i.e.f(t)+ : rate of decay (d f (t) / d0 J continuously over time. In other words

df (t) / dt =-bf (t) b = constant showing proportional change.(a differcrrtjal e{uaro! rvjft firr)ction (t) and ib derivatilie dtru / diJ

e g ln econ, suppose labour force L(t) grows cnntinuously at 9% at any instant of time t.(i.e. at any instant of time percenfage growh of labou, fo.l"j(d L(t) / dt) / L(t) : 0.09 a dtfferential equation containing L &its derivative.

We get solution by the same method as above:

Solufion: L(t) .ce 000, c rs any constant.

(a! equadon wih I{t) but no d€rivativ€ydiffeftotials & such thar. \\rhen we dilfej:€ntiate the solution we get bsck theorigiDEl difier€ntiat €quation dl(t) / dt = 0.09 L (t) )

check: dLldt=d(ce 0.0e,1/dt: c e 0.0et(0.09): 0.09 c e o.ont : 0.09 119

The solution is a general solution.Ifc is specified e.g. c: 5, then we get a particular solution L(t): 5 e o.oet

e.g. sp€ci& P as initial conditiollhen AB defurite solution

( rt11

Yjlf.l{ft1.:rtation, solving differential equations are not always simpte andsrrargnuorward l hat is why we.have different categorization ofiifferential equations,each of which has different sorution methods. ot"titi.o, r"r"tions are not knorm.

Remark: f can be a l-independent variable function or a multivariate f'nction.

Following are some terminology for categorization.

f(x) l-independent variable

Differenti al equation (oategorization )

f(x1, x2, x:, . ., x,r) multivariate

called pArti4l differential equatrons(some total differential equations)

i.e. diflerential equation is an equationrelating function fto its partial/totalderivatives and/or its partiaVtotal differentials

e.g. 4+ (y-t)dy + (4 l+40 dt= 0

ote g. F(y, t) = --- dy + dt (partial

dy differentialequation)

e.g. dF(x, t) : (aF/ax) dx + (dFldt) dt(total differentialequation)

e.g. in particular, if d F (x, t) : 0 thencalled exact differential ecuationinfactdF=0 ) FnoAg F constantrn dynamics, F (x, t) called a stea<lv state

called ordinar,v differential equations

i.e. differential equation is anequation relating function fto its derivatives or differentials

e.g. above radioactive decaydf (t)/ dt =-bf(t)

and

Labour force gro*th functions(dL(t)i dt)/L(t):0.0e

Terminology for firther categorization:

Order of differential equatior: order of tbe highest derivative in the equation

e.g. dy(t)/dt:ke et l$ order (ordinary) differential equation

6 2f(x1, xr)

a xl2

e.g. n >9

aof aef+ __-_____ * xt :5

d xrn 6 *un

d4f(x)---------- + x7 :5ox

7-x :)

'.' highest is I $ order derivative and l-independent variable

2no order (ordinary) differential equation

'.' highest is 2nd order derivative and l-independent variable

+ & :5 2N order partial difterenttal equation

'.' highest is 2nd order derivative and multivariate f

e.g. d'?r1x;

d, x2

dy

dx

e.g.

Degree of differential equations:

highest power to which the highest order of derivative is raised.

e.g dy(0i dt=ke e1 1"t degree ls order ordinary differential equation

nth order partial differential equation

l " degree 46 ord er ordirnry differential equation

3d degree 4h order ordinary differential equation

^tr , -rhv qegree )- order ordrna.ry differential equation

d5r1x; ^ a4r1xy{--:-}'r {-------- }t' r *'= 5dx' dxo

Notation: when time dimension. is.involved as a differential, we adopt Newton,s notationf and not Leibniz dy / dt t." !, : a j i'at -- t: ;;;;;t""

Definition: Autonomous (= time-independent) differential equations are differentialequations not specifically involving time.

e.C. i(0 =9 y+7 notrerm

ji (t) =ke5

Definition : Non-autonomous (= time-dependent) differentiar equations are differentialequations specifically involving time.

e.g j,(t):9y2'

ii(t):kes(t+l)

Differential,equatio:r (categorization based on order,leading to Cauchy_peano Theorem )

f(x) I -independent variable f (x1, x2 , x: . .. xn) multivariate- partial differential eauations

1$ order (only I't order derivative/partials)

e.s. 4+6i -r)dy +@f +4t)dt:A

2od order

a2 FF(y, t): -:- dy + dt

af

:

nth order

:hit/+\2nd ordcr ordinary differ-rirr" */*\o'"

- ordinary differential equations

1$ order (only I "t order derivatrves,l

e.g. above radioactive decaydf(0/dt:_b f(r)

2d ordere.g. Newtonian mechanics

m:massofyy(t): A position ofy when A tji (t) : A_(A position of y when A t)

when A t= acceleration a

| ( v(t)) : ro.ce acting on yLaw of Motion F(y(t!: n a

(highest order is nth derivative) nth order

11119 4:t" Thegrem (= Fundamental Theorem for differential equations)

U)lease see any math textbook for proof)

A system ofn ls order differential equations with following 4 assumptions * )3 function g (t) which is a solution of ihe .y"".. lf o io ,atrsfies the iniriar condition,then solution is unique.

tf assumprions for a system [ (t) = f tl, u (r) , y(t) ](A-l) frurction f is conrinuous

!e-]) nartials 6fi/Oui exist and are continuous(e-:) V(t) at least piece-wise continuous(A-4) (uo, to ) grven

Application ofThm: nth order differential equations can be converted to n x 1", orderdifferential equations and then 3 solution by Cauchy_peano Thm.

e.g. given2M order D : flt,u(t),f (t) I n(f : du/dt n@= 6za6rzconvert to hvo 1$ order y :f i t, u(t),y(t) I and y(t)=f (r)

Tenninology for more categorization:

Differential equations are linear if 1$ degree and no product ofy and je.g.Y :cr y + c2 (1$ order linear)

e.c. ji (t) + 9iO + l9y(t):81 ( 2od order linear)

Nonlinear differential equation are >l"r degree and,or with products ofy and derivativesof y like jt.

e.c. Iy (0 l2 : c y(t) + k

e c. bi (t)l 3 :e1i, 1tly1tl1s

Remark: often times use linear-d-ifferential.equalions (/system), which are easier to solve.ro approxlmate nonrinear differential equations iisystem) and information from -- "-"-'

linearization of the nonlinear syste- in tf," n"ighUo.hoo?JrL ,r.uay.ore. (e.g. fornonlinear I linear tangent line is an uppro*i*ui.onj. '

@

)1w that wg have categorized trre various differential equations, we can start solvingthem according to their catego(es.

The first step is always to check to see I'f the differential equahon can be solved byrntegration alone.

0. Check if can solve by integration or successive intesration

e.g. d2 y (t)---------- - 9

d t2

seeifcanJ[d2y(t)/dt2 ldt

d. y(t)/dt 9t +c,

t try integrate again (successive ntegratron)

itav(t)lat lat : j(st+cr)dt

: ,/Zl- + Crt * C: is then the solution (noderivative/partials)

check solution by differentiating:

dy(t)/dr :9De)t + cr= 9r +

d2y (t) / d,t2 :d(gt + cr)/dr : 9

l"tdegree 2nd order ordinary linear differential equation

= i sat

which is the original differential equation.

Application:

Simple Endogenous Growth Model (Let y : nationai producvmcome; K : capital)

Prior to mid 1980s, almost alr gxowh moders (including Solow tho*th Moder) assruned1) technological changes are exodgenous, and2) production functions exhibit diminishing returns to scale.

(i-e- the production with respect to each of the inputs are concave firnctions).This is lnown as Law of diminishing marginal productivity = Law ofdiminishing returns. It s.ays if we keep o"-" irrput

"ooat nt, say l0 workers, and Imachinery @apitar); initia y production f with each ex*a machine but after more andmore machines are added, such I w l J This is because tle sarne 10 wJers ";;ioperate so many machines. Similarly if we hold machinery constant and increase

w-orkers, after a while, crowding and administration of wtikers will J marginal - -

proouctlvltv.

In mid 1980s, economists started relaxing these two assumptions which give rise to

Endogenous Growth TheoryRef Romer, P. "Increasurg Returns and Long_Run Crrowth,, JpE Oct/19g6

Endog Growth Theory assumes technological progress is not exogenous.In fact capital K should include knowledg". I"".;riog.";s to knowledge is plausibreas knowledge is cumulative. Ifwe use knowledge witt increasrng return to ofrbetdiminishing returns to other inputs, then it mightie a"".ptuoi" ro u.r,r.e constant returnto scale (or even increasing return to scale if linowleage i ;;fficiently;.

If we assume I rate of capital will be same as f rate ofproduot y:

dK/K : d Y/Y (interpretation, given % change inK, y same % change, in whichcase there is constant returnsJ

above is a l$ order differential equation and variabres are separabre, (different variableson different side ofequation). wi can sorve for v:(integratJtoth sides of the quation)

ttt+

iarlr = jaylylnK+sr: lnY+c,2 we let ca: h_bca: lnY-lnK

e"3 =A:y/K note e c3 =[ >9

t Y = AK is the production frnction note: dY/dK = A > 0 and dyrldk = 0 not <u

This is ofcourse the production equation for the AK Moder for Endogenous Growth.

Furthermore if we assume investment : s y and K is depreciated at rate of 6,then dK : sY-6K

t dKi K :sY /K -AK/K

= sA - 3

butdK/K=dY/Y (=sA - 6)

) as long as sA > 6t dy/y >0 which means national product will growpersistently and there is no need for exogenous tech change assumptlon.

There are many forms of Endogenous Growth model with varying degree ofsophistication. Please see for example Romer [2000].

Another macro econ application:2-sector income determination model

Given: time t (instant of time, not a period)aggregate consumption c(t)aggregate Investnent I(t)National Income Y (t)and respective equilibrium levels C" I. %and respective deviations from respective equilibriurn levels C 1 I a y 6

t c(t):c" +c^I(0: Ie + I^Y(t): Y. + Y^

Assume Cr: c Yr c: mmgnal propensiS' to consume

16: i Y6

dY^ /dt : k(Co+Ia -ya) excess aggregate demandY grows ocly to excess aggregate demand

0< i,c,k < 1

) dY^/dt : k(cya +iya -ya): k(c+i_I)y6

d Ya+ -:--- : k(c+i- l)dt isa ld order linear differential equafion

Y,1

dYat I----- : Jt(c+i-1)dt) In Yr : kt(c+i-1) +cl cr:constantfrom integration

t YA : ekt(c+i-l)+cl

att:0 Yr:ecrJ

Also: Y(0)-Y"

Since Y (t): Y" + y^

: ye + e kt(c+i-l) e c1

: ye+ ekt(c+i-t) (y(O)-%)

i ast+"o Y(t) will -+ Y" only if ekt(c+i-r) * 0 notek,t>0

i.e. only if (c+i-1)<0i.e. only if (c+i)<1

So stability condition of2-sector income determination model is ( c + i ) < 1

That is, marginal propensity to consume and to invest must sum to less than l.Altematively (c + i) Y < Y.

So far we only looked at 1'r order linear differential in the form of

df (t) /dt:-bf(t) = df(t)/dt +b f (r) :0 ftnown as homogeneous form)

w*hich_ can be solv-ed by integration. we now proceed to cover the more general form of| "' order linear differential equalion:

df (t) /dt + b(0 : a (note a, b can be a(0, b(t) and a not necessarily:0)

as well as other forms of differential equations.

The derivation of solutions can be found in nrany math econ books (e.g. Hoy, Livernoiset al). Here we will just state the solutions as rules based on a few categorizaiion. thecategotizafion is by no means exluustive. we remark again that often times, solutions ofdifferential equations are not known.

l. I s order (highest derivative is 1

$ order) linear (no product of f, derivatives,differentials nor f > power of l,like f2)

dy(t) /dt +by(t)-a (note a, b can be ao, b(t))

Rule: (must memorize)

General solution: y(0= e -Jbdt (t-<a

---oalled complementaryfunction (arbitrary constant c)= deviation ftom equilibrium

e -lbdt ( Jua jlat dt) is called the particular integral: rntertemporal equilibrium level of y (t)

StabiliS, condition for this rype of d ifferential equation:

if lirn "

-ftat " = 0 theny(t)dynamicallystable.

t ---t a

e.g.i+4y=16

tben solution by above formula

y(t) : e -J+dt (6+116e j4dt dt)

: e -4r (c +J 16 e 4t dt)

-ja at: -4t+crwe lump cl to constant c so herewe just ignore it for the time berng

there is constanl c2 of integrationwhich we lump to constant c as well

: e -4t(c+ (16/4)e4t)

- ^ ^ -4t r- c e -+ 4 ts the solution

check solution by differentiafing dy (I) / dt : iOt(t) =d(ce-4t+ 4)ldt: ce-4t(-4)+(0) :_4c e.4t

: 16-4ce -ot -16

_ - 16-4(ce -o'+ 4) = 16-4y(t)t i(t) +4y(t):16 which is t}le origiaai differential equation

and lim e -Ibdt c: tm e -l4dt c: lim e -ar c: lim c /(eat) : 0t --) co t --+ oo t ___+ co t -__+ co) dynamically stable

-lbat (c+ Ia" Ioot dt;

2d order bnear dif,ferential oquatrons

j(t) + br j7 (t) + b2 y(t) : a a, 6 ,bz constants

Particular integral

f: b,*oI oot'

v, = jfit u, =o rr, *o

l;* b'1=b2=o

complemertary function

!o=Jt+Yz

Yr = Are"t

Yu = Azet"

complex roots

_h___:! < 0

2

U

convergent

A1 4,2 constants

(and if O12-4b2)<0+ complex root)

-A.1 eft +,{2 t e(

- u, t.,,6 j -16.rr,12 : ----i=----_'

br2 + 4b"

if b12 : 4 b2 then nsolution y(t): yn + y"

Stability Condition for 2nd order linear differential equations

both 11, r2<0

convergent,. - il( . e--'--_+ U aS t__- oo)

ri>0U

not oonvergent

2 parl

with distinct orrepeated real roots

Econ application:Suppose market price P(t) of computers at time t is adjusted by hvo factors:

/\1) excess Demand : D(t) - S(t) 2) unsold inventory for period [0,1]

with excess D(r) 1+ r19t

Given s(t):E +G P(t) ; D(t): H + J P(t)

E, G, H, J € R ; assume G-J * 0

Find the price path P(t).

We nore: S(t1= 6P1r1o"46Q): JP()and we can write price adjustment differential equation as:

p(r)=atD(r)-s(t)l- pllsk)- D(t)ldt o, B >0 [rrr-D_32]

We differentiate [II]-D-321 wrt t:p(r) = alo(r) - s(r)l - Bts(r) - D(r)l

+ P@ = zUPO - GP@) - BIE + Gp(t) - H - Jp(t)l

with unsold inventory | at time t t p(rlunsold inventory = total accumulatedpast exc€ss supply : S(t) - D(t) represented by(assume excess S(t) > 0),tr.^.

.

)15\r)- D(tlldt "As

0

[Thm: function f continuous on closed

interval[u,v]; V te [u,v],

F(t): ltfQptl t F '(t): f(t)l

a in solution formula

2

Assume (-a(G-.I))t > lpyC-4> real & disrinct rr, rz

+ Aze z A1 , 4,2 arbifary constants

plc-Jt tG-Jr\

\ .r\ Pc:Ar e'\,7t,.\'.

-.\, \)\ ,i

+ P{0 + gg_!y{t) + p(G - J )p(t) = p(H - E) nore B(G - J )+ 0br b:

) pp -- PIH - El :lH -El - alc - Jlx 1!(-a(G - J)), _ 4BlG _ Jl

t P(0: Pp +Pc

3. Exact l"order (1"'order derir.ative, multivariate) differential equations[Remark: most economic books are confined to 2-variable difierential equations

e.g. ii(t) : fl t, u(t), ri(t)l

Given: d F (y, t): (0F/0y) dy + GF/A| & - 0 Find F(y, t).

ls steo: check ifexact differential <ion by checkinga2F _ a2F

dtdy dydt

2"o steo: if exact, go to 3'd step. otherwise maybe able to _make exact by multipryineby an integrating factor (a multiplier) then go to 3d step.

3d step: find solution by successively partial integrating

e.g. given differential equation

dF(y,0 :4::_dv +$f +qtJ at :o [3_3_t]

2"d.: multiply whole equation by integrating factor t * *

[3-3-l] becomes 4ytz dy +@f t+4t ) dt:0

I\-av^{a fr{+s,t')=ts,t fr{tfr)=ttt = exact

3d: re-arrangeterm" ff=+f > OF=4ryz4

instead ofand partially integrate wrt y constant c

+ lar = r = !+vtay =lr',' *of,l some t terms lefl

After getting F, tale derivative of F wrt to t to e"t ffdF-' =4v't+G'(t\ot

which can now be equated with the given ff = +tt + +rt

=G'(t)=4( + Jc'(t):c(r)=J+t' at= lt' +c

tF(y,t) = 2y2f +4Bf +c

cheok 4=4vt'? 9l- =4w' +4t'fu'at

ar : $ay * $a t = t|Gfi)dy + (4y' + +t;dt]= oav' asarne as [3-3-1]

** If3 integrating factor: Rule for finding integrating factor for 1$ order nonlinearmultivariate differential equation. *r--r

f"/ar) "rdF))/Ar ir I llql-1E11=rru) aoo.,h.o .I',,,0,(A)'lfaF)l-t---ry

| ' isintefuatine

(ait.)[ ^rart ^(ar't)

^, ,. , I

ol;i _ "ltr r

'-, - aFl ry a a

l=sttl alone then tJst'xt t integratrngTactor

avl )

r, llc2 J

. .. l- . ''" .:

i,l' t ++{ aompreviouspage

:, '',,

t . - /1.,| | .\ | -+a____:__ | r+q _ 6* l= u

4y-* 4* L (- s ''r t,,.L.,, L'c *tr t-.-

chcck (A), raL%t _at*Ji/L.

jJ'-A li-' ra fimction not of y alonebut ofy, ti.e. {y,t), not {y)

rf)tr1-ut -t

I'- ad ) I

)+l(

qtr J

It, I tl t)- *t | = .=4-

0 r rr.Xt

-a function of t alone=l{ -tr i.e. onlY involve t

Ivlc (t) = ----

tien integrating factor = ^ .] ?ttt'lzC

lJ- )i^ lt *

^ l.^t

t

4 . seoaration of variables : for solving I $ order I "t degree nonlinear differential equation

If equation is separable s.t. M(x)dx + N(t)dt = 0 then can just J

dxe8 -;=x-tot

*$=,0, - [{ =f,arxo J xo

11 f-;;*"' =tt",

t2 l^+t+ Sx5

+c, -cr =u solullon

d(t' IcnecK Dy orllerentratrng

* 17 *

5; *.,

I fdx)-+ t ----;1 - 1=gx.1.drldx

-_+ - = x"tdt

5. Bernoullinonlineardifferential equations.

y +p(t)y=q(t)y"

manipulation: set s(t) : t V(t) I t-

) ds(t) dy(l-n)y i;-nr-t ----- = (t-n)y-' [qy' _py] : (t_n) [q_ps]

d t dt ls order linear difftial don, use formula

Solution: s(J) : e -J(r-')pdi[c+ i(t-o) q . Jo-n)pdt dt]ll

lylt;1t'-";

in particular, y = y (a- by) a, b > 0 called logistic or Verhulst or S-curve growth

-.,.]=,*J**(a)=e') 5 \drl

1.e. f /y = a decreasing functionSolution: y(t): a/ [b+ke-"1 k oonstant

Before moving onto Solow Growth model, it will be helpful to have aRecapitulation of Differential Equations:

I -dimensional (single equation)differential equations

-.../.--.-\1 -independent variable: ordinary

step 0:

linear,

(no product linearterms, no power

I> l) e.g. Solowe.g. most simple Growth

radioactivedecay Modela: dy/dt + by

General solution v (t) -e'Ibd( (c+ ja; Io?' dt)

+ stability conditions

2nd order

linear5i1t)+brf(t)2 -part soln

a

b2

IAt'=lu '4c1

nonlinearI

separation ofvariablesmethod

I

if use Cobb-Douglasin Solow, solve byBernoulli nonlinear

!o=!t+Yz

Yr = Are"'

J z = Aze"'

1't order

nonlinear

a. check if exactdifferential

b. solution bysucoessive integration

c. if not exact, make exactby integrating factor

multivariate: partial/total

try integration or sucoessive integration firstif unable to do, proceed to next step

e.g. Simple Endogenous Growth rnodel.

step l: solve according to different categorization ofthe differential equations

1"' order (l$ derivative)

+ b2 y(t) : a

Y(t):Yp + y"

br*0

b, =0 br +0

b, =f, =Q

+ stability conditions

II.

nth order nth order

Cauchy Peano Theorem converting nth order differential eqution to n x 1$ orderdifferential equations.

m-dimensional Differential Eouations

[m >l simultaneous differential equations ) if we wish to have solutions, then we musthave m variables I always multivariate;mostly study dynamic systems with steady states (stability conditions)l

2-dimensional(2-simultaneous differential equations)

m-dimensional

-----\linear nonlinear

2d order linear - trendand momenfum modei+ stability

Istability usingeigenvalues

I

stability usingeigenvalues& Jacobian

Remark: We repeat that unlike differentiation, solving differential equations are notsimple and not straightforward. That is why we have different categorization ofdifferential equations, each of which has different solution methods. Often times,solutions zue not kno$'n even.

I ri(1)

Chapter III. B. Neoclassical Gro*th Model

Ref: Solow, R.M. "A Contribution to the Theory of Eoonomic Growth" eJE Feb/1956

Simple Solow Growth Model (1"'order 1" degree linear differential equation.)

Given Y1 : F(Kt, Lt ) where Yt : aggegate output at time t [3-3-3]F is the aggregate production functionK t : capital in economy at time t > 0L 1 : labour in eoonomy at time t > 0

Assume F homogeneous ofdegree I, twice differentiable and

AF AF A2F A2F----- > 0

AKt OLt AK,, 6L,,

Kr:(:dKt/dt):sY, 0<s-savingsrate<1 [3-34]

Lt:Loe" L o : labour at time 0 and is given t3-3-51r : labour growth rate

ietkt: Kt/Lt capital: labour ratio

How to find the equilibrium growth path??

Implicit assumption of this model: since K1 and L l are the same in all model equations) all Kt and L l are fully utilized) full employment of both capital and labor are implicitly assumed.

[Standard manipulation of macro growth models: transfer the production function F withtwo inputs in to production function f with k t: Kr / Lt ratio:

Define(kt)- [1 lLtl F(Kt, L1)

we multiply both sides of equation [3-3-3] by I / L I

+ Yt lLt: F!Lt: F(Kr lLt, Lt/Lt):F(k,,1) since F homogeneous ofalso ll degree I

f ft 1) by definition of f

t Yr :L1 f (ki) :Lo e" f(kt)

alsoK, : kt Lr : k. L6 e'I

differentiate wrt t and get

)dKt/dt: kt Loe't(r; + dki dt Loe'talso ll ) ) ,k,* dk/dt =s[r(k,)]sYr = sLoe" f(kt)

t dk/dt:s[fft1)] - rk1 ) k1 time path moves according to s, r rates [3_3_6]is l* order ln degree linear diffarentiar cquaron

[Given F is conoave, how about f?Following two lemma show if F is concave, then f is also concave.]

Lemma: f ' (k, ) = Marginal Physical product of Capital : MppK

Proof:Knowl/L1[F@t,L)J =fGr) t F:Lt(k)

aF al.tf(k) a(Kt,Li)MPPK:---- : Ltf'(kt)-----------=

dK. 6K, 6K,

A?F

-ir,rto ifandonlYiff "(kt) <o

a( K')L,f'G,)(1/L,) --- = f'(k,)

dK,

Lemma: Kt Lt > 0 t

Prgof < o -> diminishing ftrum ro taboura'F a aF a all4f(k,)l a a(K,tL,)---. : --G--, : ---G------) : -:-[f(kt)+r,r f '(kr)-------------lau, aLt aLt 6Lt 6L, ar4 ua3-,

A (K. lln )= f '(k,) lLtf '(k1X- Kt /Lt2)l

f'(k,X- K' /rnz) + ---- tf'G,X- k, )la14

f'(k,X- K,/L,2 1" If'(k()(+K, /Lr2 )f"(kt)(- KttI.:x-k,)f"(k,)(- Kr t\' )(-K, /r-,)f"(kr)(k,2XliI4) k,'> o

+: .0 ifandontyif f "(kt)<0dL,"

a

6L,

a

/^-4<--=-J

:

t

+ f"(ktx- Kr lLtz X-k,)l

(1/14)> 0

We see that [3-3-6] the equilibrium growth path d h / d t or( :s [f(k1)] - rks a lsr order difer€ntirl equation

To solve this differential equations:

if f(k1)notknown if weknowf (():or we don't want tosolvef(\):use phase diagram use various rules according toto show the path and category offmovements withoutactually solving thedilferential equation a particular example

whenF(K1. Lt ): L, to K,o 0 <q, < Iwhich is a Cobb-Douglas funotion, homog ofdegree 1

t f(k,):1/LtlF(Kt, Lt )l :1/LrlL,lo K," l: Lt o Kt": Kt" / L,*: k,o

and [3-3-6] becomes

kt :slkt"l -rkt 13-3-jl

a nonlinear ( .' product k1 " ) differential equation -Bernoulli differential eouationcan be solved by setting m(t) : k1 t -"

cL + 0,1

+ dn(t)ldt : (1-o)\ tr-c''r-t dk,i dt

: ( 1-o)\-" k,

(1-a)(s Ikr"] - r k.)

1.dr\t

: (1-ct)[s -rk,{r-"1 1

: ( l-o)[s -r m(t)]

) hansformed into a l"torder linear differential equationin the form a : dy/dt + by

--J--(1-cr,)s: d m(t)/dt + r(1-o) rn(t)

solution by formula: m(t):e -Jrrr-u)dt [c+ it f-a)se Ir(r-d)dt dt j

f ,rr )

f(kt )

f'>f"<

n');J'

f(kr)

s f(kt)

04s4I

concave functi on

aa. <oLc|.J Aga.rll-l FldD€,L-

K equilibriumwhererkr=sf(kt)when i. = o

7tfor thase d i agram

/(@9e- .L.^+t'*

&.L

('*D

:e -lr(r-{r)dt [. + j( t, cr; s e .(1*)t

dt1

:e -Jr(r-c')dt [c + {(1-o,) s / (1-o) r} ( e .(ro)t +c2)]

_ e -r(r{r)tc+ (e -L(tcltr, s/r(e r(r*)t, + c2 (l-o)s)

- . -r{t-c)t"+ {s/r} + e

- .(1-")t{ c(l-cr)s)

: e - r(r-".)t (c+ (l-cr)sc2)+ {sir}

let oonstant ca : (c + (l-ct)s c2 )

) m(t) =" - ro-c,)t ca+ {s/r} t3-3-Sl

in particular at t = 0 m(0) :(1)ca+ {Vr}

I ca: m(0) - {s/r}

+ t3-3-Sl m(t)=e-'('-a)t (m(0)- {si4) + {sh}

since m(t): k t-"

:;' 1r r -ct - e '0-)t 1mi0;-1s/r)) +{s/r}is the solution (no derivative/partials)

note 0 < cr. < I 0 < 1-ct andgivenr>O (population growth rate)

>0 >0

and iim e-'(r-,,)t - 0

t-+co

--> as t-+ co, i.e. over time, k t-" -+ 0+ {Vr}

= k -+ l-"rl s/r= (s/r1 r/(t-"1

Econ interpretation: Equilibrium gro*th path when capital: labour ratio-+ (s / r; t r<t - " ,

Bquilibrium level dependent on s savings rate (related to investment rate) and rpopulation growth rate.l

Above is an equilibrium dynamic growth model. To extend it to an optimal growthmodel where some objectives are optimized, we need to cover Calculus of Variations orPontryagin Optimal Control Theory or Bellman Dytamic programming and add in morevariables/equations to t}re above simple Solow Growth Model.

Chapter IIL C. Optimal Growth Models - Neoclassical Grow.th Model using Calculus ofVariations

in 18s Century, Newton and Leibniz were vying for calculus primacy.

Newtonian mechanics model:Newton wanted new type of math for dealing with motion and forces (represented byvectors) operating in physical system.Approach: equations of motions (represented by differential equations). Simple casesare solvable but typically coupled (i.e 2-dimensional) vector differential equations arevery formidable.

On the Continent,Leibniz, Bernoulli (farnily), Lagrange, Euler, Poisson, Jacobi (French,Italian, Swiss, German mathematicians) tried to develop a more general theory such thatNewtonian mechanics becomes only a special case.

Such development actually stemmed from Galileo discussion in 1630 to find theminimum sliding time of a particle moving from 0 -+ pby gravity. The problem (calledbrachistocbrone problem) was solved by Joham & Jakob Bernoulli and also by Newtonand l€ibniz in the early 18* Century.

Euler further deveioped this formulation into a branch of math in 1744 called Calculusof Variations (to find extremals and form basis for Newtonian mechanics).

Euler and his fe1low mathematicians on the Continent noticed that Nature has certaingeneral properties that conform to principle ofeconomy or simplicity.

e.g. in optics: Fermat's Principle which as stated by Fermat was incomplete but basicallysays that actual path between 2 points taken by a beam of light is one which is traversedin the least time. (minimization)

e.g. geodesics : -- moving bodies seeking the shortest distance between two points on asurface. (geodesics are actually cuwes in which distance between two points on a surfaceis a local minimum -.- like Great Circle route when traveling the globe. Einstein used itin his general theory ofrelativity where his curved space-time has geodesics --curves inwhich the distance between 2 events is a local minimum).

e.g. soap bubbles: deformable thin film objects will take the shape that minimizessurface area.

P&e,tr)

e'g in electromagnetic and erectrodynamic theory: electromagnetic phenomena alsofollow similar minimum principle.

e'g' -statistical

mechanics: particles system organize themselves into equilibriumconfiguration that minimizes their energy lHaiitton's rrincipre of reast action: motionof a system of particles explained by J i - u T = kinetic energy u : potential energy.)

e'g' in thermodynamics, trre dissipative processes maximizes the dissipation rate.

The- above approach is called I.agrangian dlmamics and later Hamilton (rishmathematician) focused on slightly different emphasis calied Hamirtonian dynamicsestablishing the most general formulation of mechanics.

Basic idea: look at the movement of a particle in a trajectory from point A to poinf B.We perturb the trajectory by a very smill amount ut "ll".y

poirrt of time between A & Band then minimize the perturbed trajectory to get th. ..rtubi" o. stationary,, trajectorl,.

Note above all extremum principles (least, min, shortest etc) and can be formulated by

optimize J ) called integral principle or variational principle

i..;,ft":: simplicity principles expressed some stationary path of line integral (correctpath, stable path)) arbitrary variation about this correct path will lvaiue of rine integral and will not bethe minimum.

variational calculus - rook at relationship between corect path and all arbitrarypath in neighborhood ofconect path in order to get the coriect path solution.

c - coraect path

c' arbitrary path inneighborhood of c

i.e.c'(x)=c(x)+ s c(x)I \'a"tl\+

variation pathvariationparameter

c' continuously differentiable

o(x) goes to zero at end points pl p2

) many c' possible

idea is extremizing arbitrary integraltr

I = I Ftt,x(t),i(t)1dtt6

x (t): (xr(t) , x2 (t), x3(t) , . . . ,x.(t) )

llLetl(e) = J p(t,xG), x(e)) dt

g0

the stationary path has the property d I/d e l_ =0tomakeX(e) -yFor I to have an extremum, its f'st differential dI must : 0, solve and we get Euler,sEquation

AF- --- t

oxr dt

We state this as a theorem:

AF

) :0 i = r,2, ... ,m

a 2nd order differential equation.

AF

dxi

trTheorem:.In order that x(t) maximizes integral f F [t, x(r), ,i 6y1 A t, the necessarycondition is Euler's equation ta

daF)=0 i=1,2,...,m

6xi c t Oxi

Proof: Please see Takayama [1993]Altematively we can prove it by using the operator 6called variation ofl s.t. (dI/del]rd. = At

Given F tt, x(f, i1t)1 let

similar to change, A, d, d ). 6

rtt),(tJ + rl1t(t)

+l t* F - F t-t, x(tF mg(t),;(t)+mc(t)l

. .F g x(t), i(,) rn ariitrary constant, g(t; aruitrary

function but g(t6) : g(t r) :0 s.t. x1q:49 + mg(;) at to t r . By Taylor eifansion

x(t): x(t6 ) + i (to ) ( t - to) + '"'(to ) (t - to)t /2! + ....

F [t, x(t)+ ms(t), io+mg@] = F [t, x(t), i(t)] * a r ra * (mg) + d F /d i (.g) * ....

: OF lA x(mg) + a F ia i (-!., - ....t r'[1, x(t)+ mg(t). x0)+mg(t)j -F[t, x(t), x(t)]t,/

l\t, define o1 uu

I l+>)

IfF=x

IfF=;

6x =dx13x (mg) +

I

6i:ailax 1mg; +

A x /A1. @g) : mg (definition of 5x)

0

a. .oxldx(mg) =mg

01) 6 F=(6Fldx) (5x ) + (aFlai)(6i )

Necessary condition for findipage 4l4in Takayama

'lggsf:

""o"tul in dynamic optimization (please see Lemma on

t1 tlu l.t,x(r),i(t)l dt:0 r larp,"1t1,i611 ot:o"o"+., 1o) Jl(aF/dx)6x + (dF/Ai )6il dr =0

+ JftbFra$me + (aFlai)nll dt:0 = I(6Flax)s + (aFlai)El dtLlno: {(aF

laisl dt + [116p 16i; !1 at

ll- t ( inteention bv earts lpn' -prr-ltp')

= J (a F /a x)gldt + @ F /a\) sf o

- f", to rr, 1 (a F /d i )l ) dr : 0

but g( t6)=C(tr) :0tr- i,{tdrlaxl - d/dr[ @F/ai)l] e dr =04n

il0... g arbitrary and need not =0

t (dF /0x) - d,/dtt (aFiai) : 0 Euler's Equation

3]i-!y]"t.r :n.ation which gives.necessary condition for dynamic optimization isanatogous to first order condition in math piogramming problems.

The analogous sufficient t'2nd order) conditions are known as Legendre, weierstrass,Jacobi etc conditions. e.e. Leqendre condition for suflicienf: for minimum F *, *, 2 0 ;formaximum F- ,. <o iun"r"*l='J;ii";il'.'-''Most math econ coruses iust cover the.necessary condition (Euler,s equation) and assumethe 2no order conditions are checked._ ett"-utiv"ty, iir ir-a'irr"r"rrtruule and concave inx(t) and x (t), x(t) twice or*l:r]i9|":-"

"lor"d #;ii;bl *h";;i;t=ffi;i;;= b' then Euler's equation becomes the necessary and sulficient condition for x*(t) to

M* j f (t, x(9, i1t.1y dt s.t. boundary conditions x(t6):aandx(11):S

(9

We employ Euler's equation to dynamic optimization involves maximizing orminimizing an integral which define an area under the curve F, F being a-function of(time t, function x(t) and derivative function * (t) for the time period [t 6, t 1]):{r

Max (ormin) It fr, *ft), i ttll a r s.t. boundaryconditionsx(re):aandx(r r ) =bto

lince 1ve are. dealing in space of functions, JF is called a functional. The solution ofdyramic optimization is*a funclion x*(t) without any derivativevpaniars and which

opumrzes the mtegraI J F (t, x(t), i(t)) dt s.t. boundary conditions. The funcrion x*(t_;is caf fed an extremal. to ' ,,

It now becomes very easy because we know if we wish to max J Fftt, x(t). i(t)l dtwe.just substitute integrand F into Euler,s equarion form, solve,ff" F "ii"iliiitr"*r"rEuler's equation and the solution is the answlr.

Hence:static optimization

find point (x1, xz, x:, ...,x^)

which ma,x objective function

f(xy, x2, x:, ...,x,n)

dynamic optimizationt;

frnd function x(t) max I F 1t, x, i I O t.rrd

just plug F into Euler,s equation to getpossible extremal solution x*(t) (a function)('.' Euler's equation only necessary butnot sufficient condition, unless F concave)

Math example of dynamic optimization to lustrate above discussion:

How to calculate length L of any curve C joining points a : x (t 6 ) and b : x( t r)

Pythagoras theorem dl z : d. t2 + d x, .

+dL2 /d,t2 =l+dxz/d(

) dL / dl = V! d;./ dtT= {r +l?tr

add all d L with t -+ o (i.e. smaller & smaller intervals)... there,s no dr (width) in integ.alJ, J dL Ike adding up

all points on C and not adding up all area urder the curve.

cet f'dL : L<o

tr: I r7T *;ioit

t

>cct)

= lenglh of any Iine joining a and b

what is the shortest distance between a and b? (i.e. find a line joining a and b that has theshortest length). We iust solve' .tt7-min L : min f "rn " ;,,7- d t -minimize lengrh of line joining a and btswe recognize this is a calculus ofvariations problem and now know how to solve:step 1. Just plug the integrand as F(t, x(t), x1i)) into Euler,s equation:

same math example as above: F: {l + i (t )2, integrand

AF

dx

d '6Fr,l-t---lat \ail

(as there are no x(t) terms, only i terms)

d:--1"% (l+ i'z) % zi1 1"hain rule;

dt.,.............--

= d/dtfk /^,1 (t+ *')lt Euler's equation 0: d/dt[il./(f+ ir)]

Step 2: solve Euler's equation which is a 2nd order differential equation.

t i o at : J alat ti t.l tr* ilt at

) ct : \ t ,,1 1t+ ir, 1 el constant orintegration

-cr2: 12 t1t+ *21 I cr2 +cr2Vt = '*z

lcr' = |t-cr';2=11 -c,2;i2

) c12 /11- cr2) =i2 ) c = ilet ll

^2

) i c dt = j i at >, c2 constant of intesration

) c t + c:-= x (t ) is a linear equation.representing a straight line )shortest distance between 2

points is a straight line.

So far all growth models we studied are equilibrium models without explicitgolimintion. (optimization being the cornerstone of modem econ). ts.g. sorow GrowthModel describes continuous equilibrium growh path based on ,uuirrgrlo.

"oo.o-ption;.we now study models with growth objeciive optimization, i.e. optimal growth -odels. '

= d Jt'J-;?;-ar

, trf

Neoclassical Growth Model

Ref: solow' R'M' " A con*ibution to the Theory of Economic Growfl,, eJE Feb/r956

(Using preliminaries we had in Chapter III C. we look at a more elaborate solow,s Modeland then introduce optimization ofgrowth objective.)Assume at tlme t' economv orgluces a composite product calred nationar product y, with2 factors ofproduction labour 1: populati on) L 1 and capital K,

AF AF----- , ----- > 0dK, 0L,

Given aggregate production function:Yt: F(Lt,Kt) where Lt = labour in economy at time t > 0

K 1 : capital in economy at time t > 0

:Jffi ;i""$: A,x:ffif,1ffi#T'o t'

= R2, homogeneous ordegree 1 (constant

a2F a2F

; i; '

-;-;,-;- '

0 1ai'nini'hing."tm lvith 'espect

to each factor)

(Remark: an example of a fun;tjo_n th3t satisfu all these assunptions is Cobb_DouglasfunctionF(L,,Kr): A L,1-" K," i, . " . fJ"'let aggregate consumption at time t: Xr ; aggegate investment at time t = I tEquilibrium condition: yt = Xt * I t (i.e. aggregate S: aggregate D) t3D-21

let p = constart a depreciation of existing K 1

tz gross tr = Kr + UK, [3D-3]

[3D-1]

[3D-4]

Xt : (1 -s)(Yt -pKt)average propensity to consume

new net inveAent in K: *, o, \ .eplenish depreciated srock

assume L1 : Loe't Lo = labour at time 0 and is sivenr : labour growth rate

o f I :: (oc savrns behaviour) t3D-sl

This model has 5 variables L s, K , I, , Xt, y, and 5 equations.

I implicit assumption: full e-ll!4nent of all factors, since use same K1 L 1 inthe equilibrium conditions and other equations.

forll >0 define kr= K, /L, capital: labour ratio yt= yt / Lt

t l/ Lr {F(L,,Kt)} = F(1, Kt/ Lt)o"on" = f(k,)

t F(Lt,Kt)=Lt f(kr)

We recall the following 2 lemma in Chapter III. B.

Lemma: f ' (k j ) : Marginal physical product of Capital : Mppr

a2FLemma: Kt Lt > 0 )

AL:

We can also easily show following 2 lemma:

Lemma: MPPL : AF / ALr = f (k1) + f.(kt) (_k,

Proof: F (Lr, K, )=L, f(k,):>aF /aLt: f(k,) +L, f,(k,)(_Kt/Lt2):(k,)+ f,(kt)(_kr)

a2FLemma: K, Lr > 0 1

6K"Proof: 62Fl6K,2 = o ff,(k,)l/dK,: f.,(k,)1r1il,; Lt > 0

Lemma: L1/L1 =r

Proof: L.=dL,/dt= Lee'tr =L1 r

NowYl = Xt * Ir : X1 +(Ks + pK,) t3D_6]

Assume L 1 ) 0 (atways need labo*); we define X t = X t / L t (per capita consumption;and(r/Lr ) [3D.6] get

aaf(k'): x, + Kt/Lt + pkt note k1 : d(Kt/Lt)/dt :Kl(l/Lt)-Kr(LrrlLt 2)

.a: Kt(1/L, -Lt(Kr/Lt2)

,1A _:=lL

-.'- & Lt'/'"=:>Kt = Lt(kt +Lt(kr /Lt ))

:+(kr): xt + Lt(kt +Ltkt /Lt))i L1+ pk1a

->kr:f(k1)-(r+p)k,-x, kr, xt > 0 [3D_7]

called the neoclassical aggregate feasible growth path.

Once initial condition ko , x 0 specified, equations [3D-7] will bethe attainable path_for (L1, Kt, It, Xt , yt ) .

Note only used 4 equations and3 variables kt (Kt, L t ) x1 I many paths possible.We add the other equations and variables.

from[3D-5] (l/Lt)JXtl = (1/Lr)l(1 .-s)(y1 -pK1)l

-; xr : (1 - s) [f(k1) - p k1 ] substituting into [3D_7]

-1 kt: f(k,)- (r+p)k1 - (1 -s)[f(kt)-p kt]: sf(kt)- rk1 -trrk,+pkr -sp kt

: sf(kt)- (r +sp)k,

If we i.ntoduce the following assumprions:

(A-1) f'(k,)> 0 f..(k,)<0 v kt> 0(A-2) f(0):0(A-3) f '(0)> (r +sp)/s altematively (A_3') f ,(0):co(A- ) f '(o )< (r +sp)/s altematively (A _a,) f .(oo):0(A-5) 0<s <l

Theorem (Solow) Under assumptions (A - I ) .-- (A 5 )

I feasible and unique growth path (k sobw, XSolow) constants k s6bv', xs.q6w ) 0

s..t. any attainable gowth path with oc saving behaviour will converge monotonically to(k s"r"* , x sorow )[i.e,. k1-+ k solo*, Xt -) X sotow ast-+oo regardless of initial k0 > 0]and(k soro*, xsor*) determined by sf(kso6*): (r +sp)*ks"6* where kt:0

and xs.l.* :( l - s)[f(k s.lo*) - p ksolo,u] from [3D-5]

such a (k sorow, x sorow) path is called Solow's path.

The following diagram depicts the proofofthe above theorem. C**tffan -

Note Kt / LtI K1 and L1

I Yt and Xr

z4.b dg.-(et-=o): K sot"* rs constant

both grow at constant rate I ('. : L, grow)ng attate r)

also both grow at constant rate r (. Yri Lr= f(k soro*): constantandXt/ Lt: (1 - s) [f(k soro* ) - [i ksoro*] : constant)

)

tI r aiso glow at constant rate r ('.'It:yt- X, , both y r,X r grow at constant rate r)

called abalanced growth path (L,, &,Ir, Xr, yr )

If applied to a Cobb-Douglas production function F(L 1, K1 ): L,t-o K," 0< cr < 1

f(k1)= l/L1 {F(Lr,K,)} = 7/Lt Q-t1'" K,") = (Li" K,") : k,"

I Solow's path s f(k solo*): (r +sp)ksor"*

I s k s.6*," : (r +sI)ks.,.*

t s/(r +sp): k.o,o*t-*

t(kc)A;6.4)r<

.r t3-lJ ':ttt

V- cooslanr , (analogous resull in Ch Ilt.B.. using Bemoullr diffrial lion )

) ksoroo= ''-''! s/(r -sF) L,>0.

Econ interpretation: we see from above equation, an economy's savings rate s and laborgrowth rate determine capital size and thence affect the production level.Higher s &/or lower r ) higher output level and highei per capita oueut.

Solow type growth model was used to explain why postwar Japan and Germanyexperienced rapid growth (due to high savings and investment rates). It can also be usedto explain why high population growth rate countries like India and china have lowerpercapital output growth. we caution that grouth determinants are more complicated thanas proposed by the Solow model and there are many other considerations.

We note above k soto* : constant - K1 ,{L I) K,: k soro* Lt -- ray fiom origin (0,0) with slope d K1i dL1 : k soto*

SoLod f*-q

Lt

[Ast-+"o kr-+ksoro* but **> (Lv,K1) )kso6* ray. In faot ca'show it cannotConvefge tOk Solow fOY. lRefr Deardof, A. v. "Growth Path h rhe sotowN€oclassical C,ro$rh Model eJE leb/t9ro]

Another econ interpretation: Solow type model is based on X t consumption: constant

i ol. Y t which means average propenslty to save ts constant regardless of inoomedistribution between L land K I owners.

Shortcomings: no money/finance, no tech progress, no intemational trade ... in the model.

In addition, solow fpe growth model cannot explain the sihution why most countries,after reaching the steady state oueuf level, can siill grow persistently. To do so, we needto have exogenous technological progress to launch growth onto a niw equilibrium paflr.

(*t't- \*r;;

(+oF.>

4

x*t

But in 1950-60 much discussion on growth models and more and more jargons.For example:Definition: Golden Age Path - is a ne oclassical feasible path ft i, x 1 ) with k 1, x I bothconstant over time t .

) set ofall balanced grou.th paths of(L r, Kt,I,, Xt, Y1 ) satisfying [3D-l][3D-2] [3D-31 [3D-4].: set of golden age paths.Sincekl= constant I kt: dkt/dt:0 on golden age path) Xsoto* = fG s"r"* ) - (r + p) k s.1o* from [3D-7]

Growth theory was firrther extended to

Optimal Growth Theory (some growth objective is optimized)

Note from assumption f '> 0 f"< 0) f t slowing whiie (r + p)kl tconstantlyI x; function bonding downi can find k* which max x t globally.

find k* from 1't order condition:dlf(kr)- (r+p)k1l / dk1 s"t:0+ f'(k ): (r+ p) tosolve fork*

2nd order conditionfconcavel fft1) - (r + p) k 1 also concave and 2od order condition satisfied. We haveboth necessary and sufficient condition for k* global max. Hence we get following:

Defn: Golden Rule Path: path maximizing per capita consumption xi at every t .

k+\

Theorem: Under assumptions (A - l), (A - 2), (A - 3 ,), (A _ 4 ' )

3 unique golden rule growth path (k*, x*)

Econ interpretation: Golden Rule patl to max per capita consumption x, ) marginalproductivity ofcapital f' (k*) : labour growth rate + capital depreciation rate.

Problem: initial L,2, K,1 maynotbe on golden rule path of ft 1, x1) ) needadditional assumption s.t. k e : L 6 / K 6 converges to goiden rule path as t _+ oo .

Policy implication: Note the appealing nature of the above various golden growthpath to a politician or social planner. For instance, he oan try to manipulate labourgrowth rate and capital depreciation rate to equate them to marginal pioductivity ofcapital and tbus guide the econ to grow along an optimal path thaf maximizeseverybody's consumption. But note the limitations and the unrea.listic assumotionsof the_ model. More modern growth theory concenfiate on technological progr...,including knowledge and labow productivity.

Macro optimal growth model using calculus of variations.

fGrowth models with time dimension are most suitable for dynamic optimizationapplioation.l

Question: what is the necessary consumption X 1 to maximize certain tarset likemaximizing (stream of future consumption)

Xo

X.JX2

where f'(k*) :1+ p andxr:fft*)-(r+p;k*

X n *.n-

\

l.-feasibility

obviously making choioe between X t and I t (intertemporal resource allocation) )consume more (less) now = invest less (more) now = consume less (more) in thefuture (unless technological/labour ohanges to increase (decrease; pioauciivity;.

PV(X,)

:

PV(X,)

s.t. oondition and given boundary condition.

L^TgR

,lad-

tv1

xt

&rt

1..$c'

aeqs s4 71-q

We now start to build optimal gro*th theory based on previous model. Assume:

It=r(r,,K) ; L,>0 Kt>0 [3D-l-llF homogeneous of degree I & ooncave

{''.uK, = Yr - X, (=I,) t3D-l-21Lr / Lr : r [3D_l_3]

From previous discussion we get the feasibilif condition as summarized by theneoclassical aggregate feasible growth path [3D_7] :

kt: fG,)- (r+p)k, - x1 kt , Xr > 0

Question: Find optimal time path of per capita (divided by population) consumption x 1 =X I / L 1 that satisfies the target ofT

max J x, d t s.t. feasibility [3D-7] and boundary conditions k 6 ,k

1

{max total sum ofper capita consumption over planning period [0, T] s.t. feasibility andboundary condition.l

(We rccognize this as a d).oamic optimization problem)

Problematic points:1) planning time horizon [0, T] is crucial because if the two cwves in above Figure

converges over certain time nterval, the height difference between the 2 curieshave different econ interpretation. So who decides which plan, S_year, tO_year,50-year plan?

2) should population exhaust everything on or before T, if not, how much K1 Iashould remain at T.

3) if r very far ahead, we wi have technorogical changes(computers, more efficientproduction methods etc.), labour changes (better training aod education; trr"i

--^'

affect the factors productivity; also soiio_political changes (from common toprivate property rights, from planned scon to decentraliied market econ), as wellas X 1 pattern may change (taste, customs changes).

In the.literatwe, always igaore all these points and assume T_+ oo . paliooule being alleligible paths closely approach fixed balanced growth paths ) all paths root uuo,it ,u-"for very long T horizon. obviously math is moie simpiified. Ifnew technology, til"" ."-do problern with new tecbnology incorporated in model and let T+ oo again.

Broaden flre x 1by u(x 1) where.u is society's utility function, maybe sum of all typicalindividuals (e.g. all clones) utilib, functions to avoid interpersonai utility comparison,which are-very stong and unrealisfic assurnptions. But then whole

"r.u"." ri o" ""it,strong ard unrealistic assrrmptions - rike a "national product", aggregate production

function ... ..

Objective now becomes dynamic optimization problem:

max ir u(x,; g-ot dt e- pt : djscounting teru and p discount rate.

where u defined over [0, co ) strictiy concave and twice differentiable.st to feasib iry [3D-7] Lr= f(k,)- c.k1 -x1 ; cr: (r+ p) and boundary conditions

Econ interpretation: p > 0 ) discounting utility ofour children compared to utility ofpresent generation ) ethical and interpersonal oomparison issues involved.p:0 no problem. In the literature just assume p > 0

So dynamic optimization problem for Optimal Grovth Theory now looks like:

rnax Iru(x,1s'ot dt p>oo

s.t. i,: fG,)- crkl -x1 kr ,xt z 0 [3D-1-8]

boundary condition k6>0 kr >0

* max I u [f(k,)- ak,-krJ s-ot dt p>0[3D_1_e]

k;,x1)0 ko>0 kr>0

How about convergence: ifassume satiation, then f bound from above and if+assumption *rat econ productivity 1 then bounded from bel0w ) convergence ofpaths.

{e se.e t}at [3D-l-9] is a dynamic optimization problem which we can solve by calc'rusof variations.

To apply Euler's equation, the integrand O: u [f(k1) - gk1 _ir 1 s- ot

aodao) set --- ( --r- )6kr dt 6k,

a@

'[f '(k,)- 0.]e-nt0kt

d ao d

;* ( - - -;- ) : --- [ u ' ( - 1 ) e" et ] : [-u" i, s-0,+ u.(-l) e-o'(-p)]dt 6k, dt

: e-pt [_u..i, + u.pJ

Euler's equation ) u ' [f' (k,) - cr] e-

) [u'f '(k,) - u'cr - u. p] / -u,, :

)i, : (u'/ -o,,)lf.G,)- (a + p)l

nf - ^t -'' -e '[-u"x,-u'p.l

Xt

+(&+)

Sinoe O strictly concave ( ... u shictly concave on finite T)

) Euler's equation is both necessary and sufficient condition for x, optimum.

solve for kr xt from system of differential equation, i., ,i, at k6 , k1to get the solution path (kt,xt ) - called feasible

-Euler's path.

Assumptions (A-1) u'>0 u,.<0(A-2)f.(k,)>0 f..(kt)<0 v k,>0(A-3) f'(0): * f'(-):0 f(0)=0(A-4) lim u(x)--+ -o osXt*)0 xq)0 (this is to guarantee

interior solution but note tlre extreme disutilitvof low consumption period as x 1 -+ 0 )

eQt

({&+\

.&"t

f11maryis: Euler's path is (k t, x t ) satisfuing following system of 2 simultaneousollrerenuat eouatrons:

akr: f(kr)- ok1 -x1 [3D-r-lt]

x1 : -(u'/u")[f '(k,)- (o + p)] [3D_1-12]

If form offis given, we might be able to solve above simurtaneous differential equations.Ifno! or if we don't want to solve, we can use a technique called phase diaer.am io .". -how the two solution curves behave, without actually solving for k , , & .

&-A

Phase diagram 4ethod:1) Set differential k , = g 1o get and plot steady-state line.2) Determine which side of steady-state line f, >0 lmeansk, tastt)andi, <0

(meanskl Iastt;.3) Do the same for i 1 : 0 steady-state line and determine sign of i 1 on eaoh side of

steady-state line.4) combine various directions to get behaviour of the 2 differentiar eouations.) for our above two differential equations:

A. Form the k 1- x 1 space. Get the two steady-state (when differential: 0) lines below.

B- From^[3D-l-l1] set i , : O g from above figure, as t -+ "o, f 1 but tslowing whilecrks t constantly ) xt = tf(kt)-cr k1l has the following shape in figure anddividesthekl-xl Spooe into two regions. Letk. bes.t. f(k") =61L,

C. -From [3D-1-12] set i, : g ) and let k "1 s.t. f .(k"1 ):(cr + p). Note kt: k.ris a vertical line in the k t - x 1 space in {igure and divides the k , _ x ,- ipace into tworegions. 0<k,r S k. by(A-2)(A-3).

D. Now determine the signs of the differentiat, i, i, on either side of steady-state lines.In figure, fork,2 < k.1 t f(k.2 ) < f(k", ) (...f .>0

)and f '(ke ) > f '(k.r) ('.'f',<0)

)f'(k*)-(a+p) > f .(k.r)-(a+p)

{<I : -Jy-:gltt' (k e ) - (o +p)l > - (u, /u.. ) If

. G

", ) - (c[ + p)J : i, I : 0

K sz >o k.t

<0 for k":) k"r

lx,

t;, 1 >o and +k"z

xtl^i3

Similarly we can determine the signs of k 1

e.g. af k5a >k" f(k"a)-ak"q: x"r <0

k,l=f Gg)- crk5a 'x1 <0k"+

on either side of k 1: 0 graph.

and - x1 <0 ('. x, >0 )t)forkl<k" kr>0

ra

tt: t"r(L; o srl',4 sIr€r-)

&tto*t7o

&r*

IJo

I+ne6"3"cl

J

f-tt &,

!_L,,<a

i.

{(tr). *(tal(i.:o)+ xr:e \56,-c *4 i,."QX 7+.ro

otl ,*1 Lbo

f')o

'*

<---;nt/ I"est

f, x1l- {::--f,a (.o :::.

&L

(<. =oi\r.,lr!

,/., ,,,

V,,

&,r

Depending on initial k o and boundary condition- .k r ,' &e arrows will trace out the

possible p-aths which describe shape oipath satisfying the simultaneous differentjal

equations [3D-l-l l] and [3D-l-12]

i, = O intersecting k1= 0 steady states = equilibrium

,, ^\€

12\ ,

iF;i|

&,r

Chapter III. D. Growth model with nonlinear production and ininite horizon: Ramsey-Cass-Koopmans Model.

Ref: narnsey, f. P., "A Mathematical Theory ofSavings" Econ Journal D€c/1928Cass, D., "Optimum Grorth in an Aggregate Model of Capital Accumulation: A Tumpike Theorem',

Ecorcmetrica Octl966Koopmans, T.C., "On the concept of Optimal Economic Growth" in The Econometric Apprcach to

Deveiopment Planning, PASSV, Amsterdam, North-Holland 1965Smith, W. T., "A Closed Form Solution to the Ramsey Model", Contributions to Macroeconomics: Vol.6:

No. I Article 3. 2006

The optimal savings rate in the previous section was originally studied by Cambridgephilosopher Frank Ramsey [1928] who did not solve the problem completely. In 1950sthe problem was revived when gro\4.th topic became popular in macro. ln 1960s Cassand Koopmans formulated and solved the problem with nonlinear production functionand infinite time horizon. (Cass used Pontryagin Optimal Control Theory.) Basically wewish to find optimal savings rate path s+ r that max per capita consumption (altemativelysociety's utility function) till eternity; noting however that more consumption (: less s 1 )now I less capital now ) less consumption in future, & vice versa.

Let us restate the problem with all the assumptions:

max J* u(",) e-Ptdt

s.t. feasibility and with following assumptions:

[A-1] u'(x,)>0 & u"(xt)<0 Vxr)0 (strictly monotonic & strictly concave frurction)

[A-2] f' (k,)>0& f"(k1)< 0 V k, ] 0 (saictly concave tuncrion)

tA-31 f(0):0, f'(0):co, and f'(o ):0lA-41 lim u(x) )-co x>0

x-0

lA-51 xt>0 k,>0 Vt

To solve the problem, we start with the finite horizon problem that we solved in theprevious section, namely:

max J'u(x1)e-ot dt0

s.t. feasibility

e- pt: discounti ng term, p discount rate. [3-E-l]

e- pt : discounting term and p discount rate.

[3-E-1-A]

I and we get feasible Euler path solution from previous section as:

aaxt: (u'/ -u") [f '(k,) - (o + p)] with feasibility kt : f(kt) - ok1 -x1

together with the relative phas e diagam.We define: f '(k*(p )) :(o + p) and from IA-21, tA-31 0< k*(p)< k# where k# isk rintercept and x*(p) is s.t. x * = f(k*(p)) - o k* ( p)

@)

u, fconcave by assumptions tA-UtA -21) integrand for Euler equation [3-D-1-9] inprevious section will be concave)l a global maximum & a 1! optimal savings path.

We now extend this finite analysis to infinite horizon by letting T--+ co in [3-E-t-A]:) get improper integral [3-E-1]. We need to check if this integral1coJ u(xr) e- P' dt converges (i.e. not explode to infinity which will be meaningless)

0

In fact, when p = 0, the integral does not converge.

To overcome this divergence problem, Ramsey (and later Koopmans) used the followingingenious method:A bliss point is where economic agents ale satiated and society's utility function starts tobe non-increasing. € p=oAssume ! same bliss level utility for all generations, denoted by u(x*(O))) due to satiation, u(x*(0) = max utility achievable for generation t Vt :

utt&) uLro)

s (x*(e))

uLro)

i -/" f@), ,i@) .,. , Jo' lj-L-l-bl Y_- ,l\\

i Ramsey then defined the following integral with a bliss point: -\--\ . , I di ' l:ou'alel' tr"*'\ t* i@ [u(x t)-u(x*(O))] d t L3-E-21 otDeve'

\ .*. r L*\..,/-...-(u)udt LJ-E:21

\ .... rco - ....'"''\ which is equivalent to Min I [u(x*(O)- u(x t)] d t [3-E-2-A]\. 0

-...-"\ " -.,,-/--'*-*----..*-=,? Dt =--- ---

uptor)

aO

To ensure integral convergence (i.e. bounded from both above & below for optimal path):

1. We assume fiom some initial k s> 0, consumption path x, is always strictly positivei u (x ) f shictly monotonically ) both integrals [3-E-1] & [3-E-2] are bounded frombelow for the optimal path. LL

lx+ +F-+I

u(xt)f * V'e.>- 'i>lot)+

dt2. A.lf p > 0, [3-E-1] integral will converge as the integrand will tend to zero as t ---+ oo

( since e - Pt * 0 as t ---

"o I bounded fiom above)

) f l! feasible Euler's path (given any positive initial k6) tending monotonically tok*(p ) and x*(p ) as T --- o.

(ln terms ofdiffer€rtial equations, the point [k*(p), x*(p, is a saddle point SP. We can show by lineadzirg the fwo

Euler paths differential equations around SP, eigenvalues ofthe coefficient matrix will be real and of opposite signs.)

B. If p = 0 the integral does not converge. We use above Ramsey's idea byconstructing a reference utility path u (x*(0)) and converting problem to [3-E-2]

Mar J

*1u1x ,;-u(x*(O))l dt which = Ma* J-[,r(x, )-,r(x*(0))]e

- ptdt withp= 000ll

[3-E-2-Blwhere u(x*(O)) is the utility level ofper capita satiated consumption for individuals.

From the above 2 diagrams [3-E-1-B] with bliss point, the integral l3-E-z-Alis obviously bounded from above for both two possible cases (MU constant or falling).Hence integral l3-E-21 is also bounded from above and convergent.

""Tfl'We note [3-E-1] integrand is [u(x)e-

pt] while [3-E-2] integrand is [u(x s)-u(x*(0))]e -pt

withp:0(plssee[3-E-2-B]))[u(x*(0))]e-pr n P-E-21will drop out when we aredoing differentiation in Euler equation ) Euler equation for both [3-E-1] & [3-E-2] arethe same.

Now we can analyze the Ramsey-Cass-Koopmans model:

When we solve [3-E-1]

M* j- u(x1) e-ptdt with above assumptions0

lThere are four types of Euler's paths in the phase diagrams [3-E-3] [3-E-4]:

Type l: if xo=0and xt:0 V tThis is not eligible as optimal path because assumption [A-4] means utility will be -"o.

Similarly forko:0 and kt:0 V ttf(0):0) xt:0 V t means utility willbe -co; so again not eligible as optimal path.

Type 2: if k, >k*(p) V t > T , given certain T > 0We note such type path will eventually enter zone of kt > k*(p ) andxl < x*(p ) andreach say kr2 at time T. We note from t> T, we can improve by consuming k I(disinvesting k 1 and consuming it as additional x t ) ) Ik, towards k*(p)and fx1towards x*(p ) and we will be moving towards the target of max it u( x,; e

- pt d t .

And after we reached k*(p ) and x*(p ), we just maintain the path there and this will be abetter path than type 2. So type 2 is not eligible as optimal path.

Type 3: if k, < k*(p) V t> T, given certainT>0) x 1'[ and people will keep I x1 tomaxu(x1)) such type path will eventually move to k 1< 0 violating our assumption k t > 0

and hence is not eligible for optimal path.

Type 4: if k1---+k*(p)and x1-'x*(p) V t---+ co This is the optimal pathmaximizing either one of the integrals [3-E-1] or [3-E-2] s.t. feasibility and our otherassumptions.

Above can be summarized by the following Theorem:

Theorem: (Ramsey-Cass-Koopmans)

Given assumptions [A-l], IA-21, [A-3], [A- ] and [A-5], from given any initial k o , if

p > 0: 3 1! eligible and feasible optimal path converging monotonically to path

[k*(p ) x*(p )]. Such optimal path will max integral [3-E-1] and this integral will beconvergcnt for this optimal path.

p:0: 3 I ! eligible and feasible optimal path converging monotonically to path[k*(0 ), x*(0 )]. Such optimal path will max integral [3-E-2] and this integral will beconvergent for this optimal path. {Rmk: note this path is just the golden rule path}

Rmk: if ko: k*(p ), then feasible optimal path is just [k*(p), x*(p)] path V t > 0.

Following is a phase diagram [3-E-5] for when p < 0 which has a funny interpretation ofa negative discount rate. Aoyway, for this case, / feasible, eligible Euler's path for theinfinite time horizon.

We note parenthetically that recently gro*th models use u (x 1) - ln x 1 which is aspecial case of the following CRRA (Constant Relative Risk Aversion) utility function:

x, t-o 0l l and>0u(xr) =

1-e

: lnxt 0=1

And 1/ 0 = intertemporal substitution elasticity between xl and x111: measure of willingness households substitute consumption over time periods.

The larger I / e t more willing to substitute.

Ramsey model has also been modified to use above CRRA utility functions and thatfirms are included to hire labor and rent capital from identical households. Yetassumptions placed on firms make t}rem merely passive agents and not intertemporaloptimizing agents. e.g. firms maximize only instantaneous profit: at every instance oftime, frms employ production factors (labor and capital) by paying them respectivemarginal products and then sell resulting output, thus generating zero profit. Hence eventhough the modified model looks more like a G.E. structure, yet households still are theonly intertemporal optimizing agents.

{'9

rt).q)TY?E L

dB€L- T' "+#

,*;l;9-o&Q?

|

[u-*'.]

[s -e-+1

A

fieP

,t)

+{.f>

z''

xt" fdc) -"'kt

\_.-+ +\\*\

\t

{p>

t l-e-sl

chapter III. E. Alternative exposition of Neoclassical Growth Model withponhyagin,s Optimal Control Theory

In 1960s, Pontryagin's Optimal Control Theory replaced Calculus ofVariations as a tool for dynamic optimization.

Terminology:Functional: an integral I {*(t), u(t), tl dt whose numerical value isdetermined by each x(t) in the family of functions of x(t).

For optimal control theory, solution are contror path u*(t) and state path x*(t)which optimize the functional s.t. constraints on state variable x(t).

[calculus of variations is.a-speciar case of optimar control rheory, seekingonly the optimal state variable path x*(t).1

AAA) We first look at univariate x(t) and u(t) functions;

First Case: Continuous finite time horizon and.fixed end point T

Given: t = time ; T = fixed constant (finite time horizon)x(t) real-valued function called the state vmiableu(t) real-valued frmction calted the control variable

Find opt path of control function u(t) to max functional (or some other target)s.t. differential equation constraint on state variable x(t):

Max objective functional | fl*(tl, u(l). tJ dt (or target I c ix itf ) ts.t. l't order differentiat equation

i = I

r-eal_valued firnction (+partial derivative)xi:fi [x(t),u(t),t] i : 1,2,...,; p_tzl

where x(t):[xr(t), x:(t), . ..,x"(t)][3-13]u(t): Iu1(t), u2(t), .., u,(r)]

s.t. boundaryconditions x1(t6)=x16 x1(T)=x;1 i: 1.2.....n

Solution: Optimal time path u*(t) for control variable and simultaneouslyoptimal path x*(t; xe, t 6) which solves differential equation(s) (containingcontrol variable u(t)), depicted bV l3-l2l and satisf,ing condition [3-13].

Note on optimal decision path x*(t), it is optimal at each time point t, notjust at initial time t = 0 or t 6 and ending time t : T.

Pontryagin formulated necessary condition for dynamic optimization calledPontryagin Maximum Principle, involving a function called Hamiltonian Hwhich is analogous to Lagrangian function in math programming.

H is defined to be the integrand ofobjection function + a multiplier calledcostate variable times the constraints.

i.e. H: f[x(t), u(t), t] + Ioci(t) f i [x(t), u(t), t]

For sufficiency conditions, we need both the objective and constraintfunctions to be differentiable and jointly concave in x and u; and costate

variable oc(t): 0 ifconstraint is nonlinear in x or u, otherwise oc(t) can takeon any sign.

Let us start with the objective function being a functional l-tl*t,1, u(t), tl dtwith fixed endpoints; x(t) and u(t) are l-dim, , and only I c'Bnstraint on x(t):

.TMax J" f[x(t), u(t), t] dt

s.t. i : g [x(t), u(t), t] {derivative of state variable x(t) determines thedynamics of the system)

s.t. boundary conditions x (t s) : x6 x(T): x 1

We define Hamiltonian H : f[x(t), u(t), t] + oc(t) g[x(t), u(t), t]

oc(t) is the costate variable (analogous to the Lagrangian multiplier) =marginal value of state variable x(t) with respect to the objective function.

If H is differentiable in u(t) and strictly concave, then f interior (not comer)solution and Pontryagin's necessary conditions for max:

A) aH-0

0u

0oc AH Pontryagin's

MaximumPrinciple

AH

0x.

C) boundary conditions x (t 6) : x6 x(T): x 1

Remark: If above f, gare differentiable and jointly concave in x(t) and u(t)

with oc(t) ) 0 when constraints are nonlinear in x(t) and u(t), then abovenecessary conditions also are sufficient.

Let us look at an example:

t* {t x*[u(t)]'zdt IEll

s.t. i(t) :10u(r) tE2l

s.t. x(0):2 x(l): 5 tE3l

We set up the Hamiltonian H: x - u ' + oc ( l0 u)

A) setO: a H I 0 u :-2u+10oc ) u:5oc [E4]

B)seti=-0H/Ox:-l tE5l_ BY IE4l

seti: 0 H / O x - constraint:rcu1ton tE6l

B)

0t

dx

dt

\ t bt )

[gs]+ a(t) - 1*at:J(-1)dt - -t +kr kr:constantof integration

tE6lt x(t) = "f iot: i C50t+s0kr)dt= -(50/2) 12 +50kr t+ k2 k2:constantofintegration

From [E3] x(0):2 : {-5012 (0)' + 50 kr (0) + k2 ) h : 2

From[E3] x(1):5:-25Q)2 +50 kr (1)+2 ) kl:28150:14125

) optimal path x*(t) : -25 t 2 + 28 L+ 2

I optimal control u*(t) : - 5t+ 2.8 with u(0) : 2.8 andu(l) : -2.2

Second Case: continuous finite time case but with free endpoint:

T

tvtax I f[x19, u(t), t] dt

s.t. i : g [x(t), u(t), t] {derivative of state variable x(t) determines thedynamics of the system)

s.t. boundary conditions x (t6) : xe x(T) free

if I interior solution, then the maximum conditions remain the same exceptfor the boundary condition, namely:

A) aH :00u

B) Ooc AH

0t

0x

at Doc

boundaryconditions x (to) : x6

0x

AH

Pontryagin's

Maximum

Principle

c) oc (T) :0

oc (T) = 0 is called transversality condition for the free point. Since the

condition means value ofx at T is free. the relevant constraints is

nonbinding so the costate oc must be : 0.

In general, transversality condition means when there are various solutionsof optimal control problems (e.g. when solving 2no order differentialequations, there will be 2 constants of integration generating family ofsolutions), we need endpoint transversality conditions to specif whichdecision path should be our specific answer.

Example:4

rrlal [ax-5 u21dt

s.t. i:l0ux(0):5 x(4) free

We setupHamiltonianH: H:4 x-5 u2 + oc(t; 10u

A) set0: a H l0 u : -10u +10oc ) u:oc [E4-1]

B) set& = - d H / 6 x -- -4 tE5-11n1[E+-11

seti= 0 HIO x=constraint:tOri-= tOo. tE6-11

[E5-1])oc(t)- J;dt :JG4) dt -- 4t + kr k1 : constant of integration

lE6-11t x(t) = .i iat - i l0udt: "f to o. dt:J 10(-4t +kr)

-(40/2) t2 + 10 k1 t + k2 k2: constant ofintegration

From [E3-1] x(0):5 =. {-4012 (0)' + l0 kr (0) + k2 i k2 : 5

From transversality condition oc (T):0 ) oc (4): 0: -4(4) + kr)k' :16)optimalpath x*(t):-20t" +1601+ 5

) optimal control u*(t;:6. (T): - 4t + 16

Third Case: problems with endpoints s.t. inequality constraints

T

Max J flx(t), u(t), tl dtt6

s.t. i =g [x(t), u(t), t]

s.t.boundaryconditions x(ts) : x6 x(T)>x1

Note if x*(T) > x r ) constraint nonbinding and the problem becomes an

ordinary free endpoint problem with oc (T):0 when x*(T) > x 1

if x*(T) < x 1 then constraint is binding and problem becomes an ordinary

fixed endpoint problem with o. (T) > 0 when x*(T) : x 1

Thus when we have endpoints s.t. inequality constraints:

step 1: solve it like a free endpoint problemstep 2: if x*(T) Z x t , then we are done.Step 2a: ifx*(T)<x1 setx(T)= xr andsolve as afixedendpointproblem.

Example:4

Maxi t4x-5u21dt

s.t. i :10 ux(0):5 x(4)> 225

We solve as if it is a free endpoint problem, from above, we know)optimalpath x*(t):-20t" +1601+ 5

) optimal control u*(t) -oc (T): - 4t + 16

) x*(4): -2og)2 + 160(4) + 5=325 >225) constraint nonbinding and solution is the right solution.

BBB) In cases where x and u are multivariate functions (> 1-dimensional):x(t) - (xr(t), xd|, x3(t), ..., x,(t)) u(t) : (u1(t), u2(t), ..., u,(t)).

Optimize objective by finding x(t) n-dim real-valued function, time t, with u(t) r-dimvector-valued firnction

n x 1" order differential equations

i'(t)=r'tx(t),u(t),tl i=1,...,n (3-F-12)

7\,/

real-valued function

where x(t):(xl(t),xr(t),x3(t), ...,x(t)) Iu(t) = (ul(t), ur(t), ..., u(t)) | tr-r-tr)

)boundaryconditions x'(t")=19 i:1,...,n

Again Pontryagin optimal control is used to find:

solution path x(t;x", t" ) satisSing path depicted by (3-F-12) and condition (3-F-13)

Remark: CAUCHy PEANO Theorem provides sufticient condition for existence oflocal, unique solution x.

u(t) is called a control= [ur(t), u2(t), , . . ., u(0] u1(t) called Control variable

If u(t) e U : set of admissible controls

range../

u(t)domain + U ------+ U range U called control resion

x(t) path depends on control u(t) which is not specifred a priori, we just pick u(t) fromthe set of u(t) that max a target.

objective function

nt-e.g.Max S=l qx1 (T) T:fixedconstant

i:lu(t) e U

s.t. i i(0: r' 1*6;, "1t1,

tl i: 1, . . . , n

and x 1(t o): x 1u i:1,.

with u(t) deiined on t o < t

Once found u*(t), can find state variables x*(t) (solution of differential equationssatis$ing boundary condition)

x i(t) assumed continuous fuaction on time tf; assumed continuous in each x, t, u and continuously differentiable.

(x', t" ) s.t. x' e X x(t" ) = x'

Theorem 1: If 3 continuous vector-valued function P(t) = [Pr(t), Pdt), ..., P(t)lvanishing simultaneously for each t such that

1) H is maximized with respect to u(t)

i.e. H[x*(t), u*(t), t, P(t)] > Hlx*(t), u(t), t, P(t)l Vu(t) e Ufiom conslraints i, - f;

where H[x(t), u(t), t, PC) = I P, (t)f, tx(t), u(t), t]i=l

2) P(t) and x+(t), u*(t) solve II.AMILTONIAN system:

i'-:aH*/ap, P;:- dH*/dx; i:1,2,

.,n

<T

not

aHl 4laPJ*.,u. *-/

3) TRANSVERSALITY Condition

P(r) : q i: t,2, ... , n

From objective functions ) c; x1(T)

4) x11t)=*o i:1,...,n

Pontryagin MaximumPrinciple

RMK: .' I constants ofintegr t get familyof solutions, so need trarsve.sality conditionto determine sDecific solution

then x*(t), u*(t) solve

^+max S=)c,x,(T) T: hxed constant

u(0€u i=t

s.t. xi(t)=fi [x(t), u(t), t] i=1,...,n

xi(to):x o i--l,...,n

Above is necessary condition, furthermore iffl are all concave functions in x and u, thenabove also suflicient condition for u* to be global optimum.

Analogy: H & Lagangian Pi & Lagangian multiplier:

4./COSTATE variables

Rewrite (3-F-12) as

i,-(t):f;[x*(t),u*(t),t] i,*:- oH* loxi

:-lPr (afl* / 6 x )+ P2(0f2* I 0 x )lz tsf.

=I-P, --'; '&,

:+ 2n equations (lst order differential equations)

ii:... i Pi:.... i:1,.,.,n2n variables

xi, Pi i:1,...,n2n boundary conditions

xi(to):x o PiG):ci i:1,...,n

transversality condition provides additional boundary condition

SolutionT* called optimal control, x* optunal trajectory

//e.g. speed, direcfion ofthrult for rocket min time missile intercepts a plane

Above includes a special case

Ifwedefinex0(t) by io(t) :fo [x(t),u(t),t]u

with x"(t')=o x" = If" dt

k'then t = J'f. Ix(t), u(t), t]dt = x. (t)Ji 5

x.(t):0: x"(T)

so maxl:.l*j]tt I dt = max x.(T)

s.t. io(t) : f6 [x(t), u(t), t]xi(t):Iixi (t') = x: =0

= define new Hamiltonian A[x(t), u(t), t, P. , P(t)]

= {{[x,u,t] + lP,{ [x,u, t]i=l

and conditions 1) 2) 3) a) ofTheorem 1 can be written as

i ; f4**,u*,t,n,p(t)l> Hlx*,u,t,p.,p(t)l vu (3-F-1s)

y'drop all other P; termsP.(T):1 P(T):0 i :0,1,...,n (3-F-16)

2 ) P(t), x *(t), u * (t) solve rhe Hamilronian H

ii:afi*m,

3 ): J)

b:q)

Pi:-AH */axi i:0,1,2,...,n(3-F-14)

/r7f)

Since P.(T): 1, by (3-F-I4) Po: -dA */dxo :0)N.

>P"(t)=l Vt

:+ A witten as '.' Po:0io:afi*/aPo:o

tr = 1t;q1*1t;, o1t),9 + tr,!(x,u,q ,/is funotion of x, u, t, p ,/=Htx(t).u(t),t,P(t)l ,/t/

Since P.(t) = 1 then AH */A Po : 0 '.'no P6 term in H

so can omit i : 0 case and rewrite as

1') H[x+(t), u*(0, t, PC)] > H[x*(t), u(t), t, P(t)] Vz(r)

no need 0

I IJ *

2'\ x' =""3P,

^ 6H*oxi

i=1,2,...,n

3) q(7)=0 i=1,2,...,n

necessary condition for max I : max I q1*1t;,o1t1,tpt

ueU u e U

s.t. xt(t)=f,[x(t),u(t),t] i = 1,..., n (3-F-17)

xi (t') = x: i=1,...,n (3-F-18)

Note

AS

5 = I.c,x, (T) can be converted to target as J & roic" rr"rru

-,t" -t "

s = J' I Ic;x;(t) pt +)c,x,tt" ),,--z r '=,

r .Ttrtllc,x,(t) |L i=l Jr"

nan= !c,x, (T) - lcixi (t") + Icixi(t")

i=l i=l i=l

/Cancelled out so no need to includein max process

+max S s.t. (3-F-17) (3-F-18)

= .no I r"l ]dt where f, s.t. (3-F-17) (3-F-18)

Note above problem T is fixed a prioribut x(T) not fixed a prioriftom i equations we get i(t) and

once fi(t) obtained we get i(t)

Various other cases:

Fixed T with fxed end points * need rnodifred theoremFinal T open with fixed end points - need modified theoremFixed T with variable RH end-point - need modified theorem

depending on x(T) a priori fixed for some coordinates ofx(T)final time T a priori fxed

e.g. Final T not a priori fixed(i) RH end-point x(T) partially fixed

s.t. xr (T) = xJ i=1,2,"',m(n(frxed)x,(T) not fixedfor i:m+1,"',nthen condition 3) transversality condition A to 3')P'(T)=c' +Ii i=1,"',m

P, (T) = c, i=m+1,'..,n1,, unknown variables constant over time

e.g. Fixed T with fxed end-point problem

max f f" tx(t),u(t), tldt (3-F-1e)

s.r. i,(t): f' [x1t1. u1t1. 1] i=1,...,nt'=0x, (0) = xi x,(T) = x]

\\-,2\\T frxed and fixed

=I",f,

i=1,.",n

J]D

Define x.(t) by x 0 (t) =f 0 [x(t), u(t), t]

TT)x6fi): I ro at =J xo dt -00 xo(T)

so (3-F-19) =max xo(T) s.t.xi:f i x0(t)=0 i:1,2,...,n

Now we apply Pontryagin Maximum to get following modified Ponfryagin Theorem 2solving above (3-F-19) problem

Theorem 2 [.ftl, tAil optimal for above (3-F-19) problem

if I nonzero (n+l) vector-valued continuous function P(t)=[P.C),P,(t),Pr(t),.r\,P,(tnwhich has piecewise continuous derivatives s.t.

i"O, i(t),0@ and P(t) solve

Hamiltonian fr : A = Hti( t), n(r), t, p(r)l = t pt (t)fr ti(t), rD(t), 4 with

(1) Hti, t, t, PI > H[i, u, t, P] vu e u

(z') ii: aH rcPi

Pi: - aAlaxi i:l,2,...,n

(3') P"(t) = c6m16r1 2g Vt 0<t<T

(4') x'(t")=xi xi(T)=xI i=l,A,n

We have conesponding corollary theorems when T open and not fixed but we will notcover as we only need above Pontryagin Theorem 2 for optimal growth theory.

0:0xo(

rvxol =

0

Econ application: optimal gro*th theory again, using optimal control theory.

like fo

maxJ u(x(t))e-e'dt

like i,/ -_'rir"r

ike uIas.t.k(t):ftk(t) -ok(t) -x(t) k(0):k0>0

k(T) = ft' ;' px(t) > 0

p>0Assumptions:(A-1) u'(x)>0 u"(x)<0 Vx>0

(A-2) f(k)>O (k)<- f(k)>O Vk>0

(A-3) f(0) > 0 f(0) = oo f(o) = 6

Here per capita consumption x, control variable

k, state variable

control region f) S R nonnegative > closed set

Assume T final time fixed k(T) fixed

=+ apply Pontryagin Theorem 2

/uno'H [k (t ), x(t), t, P @] = P.u[x (t )]e

- P t + P @V Ik (t )l - G k (t ) - x Ol

Po: constant ) 0 can show Po(0) > 0 and WLOG can assume Po:l

= H[k, x, t, P] = u[x(t)]e-p' + P(t)[f[k] - crk - xl

from (1') u[i]e-at + pIi - ak -;l> u[x]e-ct * Ort' - ^ - r,

= u[i]e-ot - u[x]e-oi > Pti - cri - xl - f1i - o[ - i1= -P(x) - P(-i)=P(i)-Px Vx(t)>0 0<t<T

assume liq u(x) - -oo

a

^n(from dH /APi =) k(t)

=i(t)=0 = u(x)= -oo(boundary) lJ

not optimal

=*(r)>0 Vt 0<t<T(i.e. interior solution)

- /\ A=r(k(t)) -ck(t) -x(t)

>0

(from - aA/a kO :) i t = - p(t)tf 'G(O)- o l

I since x is max in condition (1') for H

afr- ^ =0 critical point

dx

afr+: =0 =u'(ile-o' +P(lX-l)=0dx

- P(t.) = u'[i(t)] e-er > 0>u >o >o

@

tdP(t)i dt:itt) :t"ie-p'+f,' e-p' (-p)

rittl: (Qt" -pt') ""p' uuti(t) = -p(t)trit) -ol/1

= -t' 1*1e*'gr'6y * a1

)(it" -pt') : -0'rtrlth -qr

t ?: -G'rirrrGt -ot-pt')rt-/\.UA

t :- -^-(f '(k) -(a+p)

same as in previous secton using calculus of variations

- same phase diagram for x 1 and feasibility k t

G)

Now show calculus ofvariations, in particular Euler's equation, is a special case

of optimal control theory:

I = J" rtx(t), x(t),tldt

final T time fixed, end points fixed

^ ' +--Hamiltonian: H = P.f + /P,u,ftl

,=l

. ,arr

=x, =-=u,' ,p.

6:-drl:-pld I

ox, \ ox, I

x(t) = {x, (t), x, (t),..., x" (r)}

x(a)=g x(b)= P

Vi

('.' u(t) no x,term) (3-F-20)

i=1,...,n

/ Calculus of variations: choose x(t) to max I s.t. x(a)=c {b)=pII like f, in theorem

\ if let ;t(t) = u(f, then same as Pontryagin optimal control

\o'oJ'{*to,u(t),tldt

aH _ af -set=max crifical point i- = P. :+l = 0ou, ou, r---\

i=1,".,n

note \ =constant>0; suppose Po =0+ P; =0vanishing simultaneously.

at P .d(af\ P= r- > u =

_ = ____ and _l _,1_ l= ___" aui P. dt[aa, / P.

. af draf)=)lf ers eouauon' ax, dt [ai(r)/

\aui

Vi contradicting all P1 not

( ar\-plal(3-F-20) '.\ a, I af

- - \ '/ =-LP. 0t,

chapter IV. Discrete time dynamic equilibrium and optimal economic moders:

A. Discrete dynamic macro equil models:

i) Review of difference equations (dfce :tions)

Most undergrad models are static models with same-period instantaneous adjustment andwithout any time dimension. E.g. demand function ai present, suppry firnctiin as ofnow,equilibrium prices found instantaneously and hold for ihis moment.

In real life, we have intertemporal (between time periods, e.g. last month-this month-nextmonth, today-tomorrow etc. ) situations and adjustment processes. To handie suchsituations, we need difference and differential equations.when we are dealing with discrete trme change, we use difference equation and withcontinuous time change, differential equations.

Difference equations are functions between time-lagged independent variables atdifferent time intervals and dependent variables.

Terminology: Order of difference equation : greatest number oftime period lagged.e.g. 1 period lagged ) l'lordere.g. n periods lagged ) nth-order

Symbol in difference equations:A to measure discrete (not continuous) change over time period t.

AxAx, = -------- - xr -t - Xr

At I A x 1 : $(10 - 9) trillion: $1 trillion )

e.g. (last year GDP this vear GDp: $10 trillion =$ 9 trillion

solution of a difference equation yields values of x for every trme t. It is therefore aftrnctionl

Further categorization:

Linear (with respect to x t ) difference equation: one with x l only raised to l"r power (soproduct terms like x, y 1 or-exponentiar terms like x 1

- , m any real number, noi attowldy.e.g.xr:bx1-1 + a" a, b constants (note term araisedto nth power okayj

Otherwise we have nonlinear difference equation.e.g. xr :b(x,-r )'* a a, b constants

Autonomous difference equation: one independent of time. i.e. without any time t termsin the function. Otherwise it is non-autonomous which is dependent on timi.e.g. autonomous dfce *tion: X, = bx,_t - ae.g. non autonomous dfce -tion: xt : bXt_t * at

In the previous chapter, we derived the general formula for l tt-order linear autonomous

difference equation as follows:

xt:bXt-r * & d,b constants

General solution (when no initial condition is specified):

We start iteration for t = 0, 7, 2, . . .., t

Xr: bxo + a

x:=bxr+a=b(bxo+a)+a=b2xo+ba+a : b2xo+ a(b +l)

x:: bxz + a: b(b2xo I a(b +l))+a= b'xo +b'a *ba *a

: b3xo + a(b2 +b+ l)

X1 = iiiiiiii ::::::::::::: : btxo + a(bt-l + bt-2 + ... + b+ l)

But geometric progress of t terms of sum

: i__1_- ifbrll- b

: t ifb=l

) General solution (when no initial condition is specified and we get family ofsolutions by varying initial condition x 6 ):

Ix s-(a/(l-b))] bt + [a/(1 _b)] b+ 1 tz_rlfXt: 1

xs f !t b=l

If initial condition is specified + get specific solution.

(St-1 ',r-6t-z +... + b+l) {

We can re-write [2- 1] as

x1: Abj +

?"complementa.yfirnction =deviation ftom equilibrium

.Cr where A=xo - (a/(l-b))\particular solution= intertemporal equilibrium level ofx

C = a/(l *b) t22l

If deviation A bt + 0 ast-+ co then solutionx,* = C

C = intertemporal equilibriumalso : steady state of the system (= rest point, stationary value).

C is said to be dvnamicallv stable i stability depends on b (e.g. b = I/lrl , N a naturalnumber I as t -) oo, A b' -+ 0 )

Acfually Steady State x * means: as t-t oo, xt - xr+t: Xt+2 -- ...... =somex *and we will retum to this subject later in our discussion of dynamics of the system.

Remark: We have to be very specific with the categorization of both difference and laterdifferential equations because there are different approaches and different solutions foreach categorization. Above system is single difference equation, i.e. one-dimensional.Later we will look at simultaneous system of m difference equations i.e. m-dimensional.

Finance application as illusEation: deriving formula for Future Value FV usingdifference equation:

Given present value P 1 for period t, interest rate per period: i

) P,: (1 +i)P, 1 in l't-order linear difference equation form

and i >0 t(1+i) >1;6 1

so apply solution formula

P,= {Po - 0/[1 *(l+t]](1 +i)' + 0/tl -(t+i)i: Po (l + i)t

which is the formula for FV.

f,.(+i)(,.L ^- Q+t) ro

( I rt

(v= ?o 7, =Qti,)f" Fv',= Pr,= [+e f Po

SIMPLE RI,CURSION:

we note in difference equations, x t is a function ofprevious period x 1_ 1 (for all t) ,which means the problem is recursive. For linear reiursive problems, we use iteration toderive the solution, as we have done for the linear difference equation solution formula.

In particular, if the linear difference equation is of the simple form as in the above FVproblem:

Ps+1 =a P1 (interpretation: principal P in t + 1 period = a times p in previousperiod t ) deposit accumulating)

I solution by iteration is much simpler & we can solve without using solution formula:

whent:0 Po*r=Pr=a Po

t:l Pr*r=P2:a P 1 =a(aPe): a2p6

t-*2 Pz*r = P: = aPz = a(a2po)= a.po

:

t=n-1 Pn =a Pn-1:a(ao-tP6 )= anP6

and this can be an alternative way to calculate FV:We Iet P6 : present value ofdeposit and interest rate is i, then a-1+i wi bethe

principal plus interest eaming per $ per period and

Pn: a"Po = ( 1+i)'po wili be the future value after n periods and is thesame formula we derived above.

The reverse process ofcompounding is discounting.

i.e. given FVn and discount rate i, what is the pV p 6?

From above equation we get P o : p,' /( l +i)'

) PV=FV"/(1+i)"

ln particular, we let 1/1+ i: B, then pV: B 'FV"A formulation we will see in macro growth model lften.E.g. social planner maximizing discounted society's utility stream Max I tS

tU( c, ).

OBSERVATIONS AS PREVIEW FOR RECURSIVE DYNAMIC PROGRAMMING:

l.

2.

J.

4.

From above iteration, we get inkling ofusing recursive method to solve adynamic problem with sequential relationship. We break up the dynamic probleminto sequence ofproblems, solve one ofthes- ,"qu"n"", *i th"n use solution inthe sequential relationship to solve the whole dynamic problem.Similarly, for dyramic programming where we are doing multi_stage dynamicoptimization. We break up the problem into sequence oiproblems, solve onestage and then use this solution in sequbntial relationship io solve the entiredyramic problem.To inhoduce risk, we use stochasric models which basically consist of sequenceof probability distribution (p.d.) function for the sequentiairelationshio.Regarding system equilibrium :

A) For deterministic models, when the sequential solutions converge to arecurrent state, then we get an equilibrium.

B) _For

stochastic models, it's the sequence ofp.d. converging to an invariantdistribution which is the equilibrium.

Dynamics of a discrete system:

A discrete dynamic system reaches a steady state when x 1 : X t _ r V t > some n.

aFor b =+ve fraction 0<b<1It is a monotonic path convergence

A linear autonomous l't-order dfce =tion xs = bxt_r + a has the properfy ofhavinga steady state iff lbl < 1. Thismeansthatthetimepathxt,t:0, 1,2,.... traced outfrom a difference equation x1 = bxt -r +a will

"onu".g" to a steady state iff lbl< t.

This is obvious ftom [2-2].

For b = -ve fiaction -l<b<0It is an oscillating path convergence

Harrod Gro*th Model - a descripti.,r" growh .oa"tlGg aiff".err"" "q*ti_-_

-t-.

we. will be st'dying gronth moders with dynamic optimization. As preparation, we startyith the simple deterministic macro Harrod growth moder using difference .dil;, ;;look at the dynamics and intertemporal equilibrium.

marginal propensit, to save

Given aggregate savingsaggregate investrnent I ,

St = sYt= a(Yt - Yt-r

5-9U<s< l

) a>I (this is AccelerationPrinciple in Macro whichsays aggfegate Investmentis proportional to rate of

AY i.e. rate of ly t ) I 1)marginal capital: output ratio

equilibrium ) St: Ir

) sY1 = a(Y,- Y,-r )

t (a-s) Y1 =oYtr

) t, : (a/ (a-s))Y, 1

solution by formula

) Y, = 1Yo-1 (a / (a-s.1)

(IS curve over time)

+ 0 is in l'r-order difference equation formX1 :bXj-t I a

0

) (ai (a-s))t+ ( ---------- )1-(ai (a-s))

: Ys {a/(a-s) }t

But a>_1 s< 1) (a-s) > 0 t a/(a-s) > 0 which means equilibriumnon-oscillating.

Furthermore since 0<s<1anda>l ) a/(a-s) >1 (see below)@.c. a: 1.8, s-0.3 t 1.8 /(1.s-0.3) : 1.2)

) Y 1 (non oscillating), explodes without bound and not convergent to y e path.

Ya

uvt,

Note in Harrod Model , in case

a / (a-s) <1

non oscillating

fboth these cases violate Harrod assumption of s > 0 ]) a<a-s ) a:als

<1

) s<0 ) s:0'a (a-s) *ve fraction Y,: yo{1}, = yo ast___} ooi (+ve fraction) t*0

as t--- oo

) convergent ) conversent

One more case is when a / (a-s) :0 [violates Harrod's assumption]thenYt:Yo{0}':0

so depending on where Y6 is, yt : 0 maybe below or: yo andnote the econ implication of a zero national income over time.

a1=Lo

tg,

convergent

,>

Corollary to Harrod Model: Y 1 in some cases explodes so change question to whatgrowth rate g will wanant St = I1 for each t -- warranted growth rate problem.

We know Y r =Yo { a /(a-s)}t [Not€: using solution of lst sequence to get equil for whole recursiv€

probleml

and definition of growth rate g = ( Y1- Y6)/ Y6

) g = ( Y6 {a / (a-s)}'- Yo)/ Yo= a/(a-s) - 1

: (a-a+s)i (a-s) : s/(a-s)

Econ policy implication: Suppose policy maker knows a = 115%o; s: l5%oand his objective is to ensure savings : investment at all time, then must try to getgrowhatrate g = s/ (a-s; : 15% / (115-15)% : 15o/o / 100%o: 15%to ensure S t = I t foreacht.

chapter I[. A. ii) Review of I't-order nonlinear difference equation with phase diagram(discrete growth model with nonlinear production fu.;il;.we see how above l tt-order rinear autonomous difference equations can be sorved. Infact, linear 1"-order and 2nd-order dfce

=ion, *" uf*uf, explicitly solvable.For nonlinear difference equations, *. .un rot frJ roiuiioo'. ,or, or the time. However,we can use phase diagram to study the characteristics ofthe sorutions.

A dfcd +-ion phase diagram depicrs y , as a firnction of y ,_1.:t axes lor phase diagram will be y

,_1 - y ,

?, l,t-]:4n9" where Y, = Y 1-1 will occur when the difference tuncrion crosses the..o ,. ,,__A -- ure uurer(,l4) rlne (+)- trne equates the vertical with the horizontal axis).

For example, given nonlinear 1.t-order difference equation:

Yr: (Yr-r)"'

It

Yc-l

Find steady state: Yt =ytr :y:

y=(y)" ) y(y3-l)=o )

) difference equation will cross 450axis) at Y = 0 and 1

y:0 or y = I are steady states

line (where vertical axis Y 1 = y 1-1 horizontal

Note:

check

dY,

d Yt-r

d, Y,

Y r-r :0 ) Y, = 0 so the origin (0,0) is on dfce equation graph in phase diag.

l"t and 2nd derivatives ofthe difference equation:

: % (Ya)-% >0(=slope)

= (-3/4) (1/4) (y i ) -1-3la < 0

tfrl,raaaa,

dYu,'

,, -9L.-)

Y4

Tt-l

)function is concave & we can draw phase diagram:and from some arbitrary points likeYr-r : 0.1 and 1.5,the graph converges to Y1-1 = yt:1

t(l,1) is a locally stable steady state.

Lo,

4o.t=Io

6s7

The divergent cases are where ] r" order derivative at steady state point l> 1. Hence wehave the following test for local stability for 1't-order auton6mous nonlinear dfce =1ion:

given steady state y

If d Y, < 0 then oscillation

> 0 then non oscillating

< I then y is locally stable

> I then y is locally unstable

If

d Y,-r

Again using the example of y t =aty:9

dY,------------__l

I ury-o : %8 ,, ) t'ol

dY,-r

(y .r ) ". [steady states (0,0) and (1,1)'

ary-o ---+ +oo ) non oscillatine

l*,

l-rl

dY,-----_-___l

I aty=0d Y"l

.---+ * co ) locally unstable

atY=1

dY,-----*-llatY=ldYur

dYt"-----*-l

I atY= Id Yt-r

= r/a )0 ) non oscillating

: %<1 ) locally stable

We can check similarly for the other three cases.

An economic application would be the following typical macro growth model:

Given: Yt national product andkt capital (assuming unit labor) both at time U6 capital depreciation rate; s savings rate as "%

ofy,Aggregate production: yt = kto

Investment function: kt*r = kt - 6kt +syt

kt >0 0< o <lt e N set ofnatural numbers6 < 1 0< s <l

) kt+r = kt dkt +skt"

= (1 -6)kt +slt" Ilgr.ord€r nonlinear difi.erence equariotrI

steadJ states from solvi-ng following equation:l( =ft'5k +sk o

t k[6-sk"-1 ]-0t k=0 or !=o-r.i(o I s)

Econ interpretation: equ'ibrium of the growth morlel depends on depreciation rate 6and savings rate s Equi.librium olr*. *h"o ""0r1",

i" the (o_ t)th root of(depreciation/savings) ratio.

let order derivative:

d ktu---.'.-...-..1latvdkt

= (1 'o) + cr sv >0 >0

k," - t > o>0 >0

2"4 order derivai;ive:

d2kt+r.........,.-......1

latrd kt 2

(o-1) q s kr"-2e <0 >0 >0 >O

<0

) -concave

function and graphano converges m onotonically:

ft,t* r

as follows and the non-zero steady state is locally stable

+50

ll rr-t=-

-ar

arD

Remark: Some nonlinear autonomous dfce :tions like xt : bx1_1 (1 _xt_r )hasparabolic-shape graphs - its 1 't derivative changes sign - in the phase diagram where x 1converges monotonically at first but then oscillates as it nears the steady itate. This leadsto chaos theorv..l

,.L

Summary: tb41r7;E-

I!//

/iautonomous

II

l" order I

*, = t1",., n u

Solution by fomula(recursive method)

2nd order

bx, l+at xl = blnxr-l+a,/

some functionsxl - rxr-r(l- xl_ r)can lead to Chaos Theory

xr = bxr-l *axr-u *cSolution by formula& eigenvalue/roots

Difercnce equations

--t \Linear (with respect to y r )- always explicitly solvable

nonautonomous

b, a are b, a i- or bothxl = br xr-l+ar_Solution by recursive method

autonomous

nonlinear- in general no explicit solutions- use phase diagram to analyze

Solution+ stability ofsieady states

nonautonomous

(@

Chapter IV. A. iii) Briefreview of eigenvalue and eigenvectors for square matrices.Stochastic descriptive dynamic model with Markov transition probability matrix:fsurvival of firms)

In above two sections we studied single difference nions. How about n-dimensiondifference equation models consisting ofn simultaneous difference equations?To handle such n-dim difference models we need EIGENVAIUE in linear algebra.

We recall to solve a set of n simultaneous linear equations with n variables, we set it upin matrix form A x : K , where A is a square matrix, x and K are vectors:

E.g. Given linear system:

311 x1* apx2 : k1

O2yXr* A22x2 : kr

t LHS can be wriften as A X

ltRHS K: IKI ILl

rrJ

:[:;r ;i[:;]

We then solve A x : K using various methods in linear algebra.

For a square matrix A, we can calculate a number called determinant (to see if J 1 !

solution for the linear system),

For eigenvalues, we are solving A x: cr x (not K) where A is a square matrix.Also we can calculate another number called an eigenvalue ftom that square matrix.

ct, crx: crIx

a Ixl

null and given scalar

f -lr -!

" 11 0ll*, 1:| 0 1l I x) |L JL-J

+0

[for eigenvalue, we first note if x is note.g. for 2x2 case. r I f ..1

ax:lc x' l= la 0lP,'l-L" *'l Lo "JL]'J

To derive the eigenvalue number:given vector x : (xr, xz, x:. . . ., x, )

and square matrix A [-a ', o,, a ,:

lu:' a22 a23

I

l:I

l'1.' u" u "

i_i9.;

If we can find reaVcomplex scalar cr

s.t. A x : cr x then a is called eigenvalue and x is called eigenvector

Note if a is an eigenvalue t A l: s x : crl x) (A- crl)x:0: null matrix (A- crI)called

characteristic matrixLemma: for square matrix B, if Bx : 0 but x + 0 ) determinant lBl:0Pf: Assume lBl + 0, then I B - I

given . B x : 0 we can multiply both sides of equation by B-1+ B-rBx - B-roil * il-tIx:0 il- [-

txt lBl:0

0 confradicting x-+ 0

So by above lemma, deteminant l(A - a I )l :0

so we see that eigenvalue o is s.t. determinant l(A - o I )l : 0

and this suggests the method to calculate eigenvalue.In fact $(cr) = l(A - o I )l :0 is oalled the eigen equation (or characteristic equation).

Remark: we can write Q(cr) as a polynomial

:crn+aroo-l + arc "'2 + a:ct"-3 + ... + a,,-lctl+an :0

where a 1 are lirnctions of a;i

Then by Fundamental Theorem ofAlgebra, this polynomial has n (not necessarilydistinct, not necessarily real) roots and these are the eigenvalues.

Since (A - aI)issingularI matrix (A - cr I ) is linearly dependent) vector x can assume an infinite number of solutions.ln order to get a unique vector solution, we normalize by I x i2 : 1 .

Eigenvalues analogy :

Mafix : Person eigenvalue : person's ID cardo. is not a mafix, but a number (analogous to ID card not a person but characterizing aperson).

To normalize means we choose the unique veotor L= ( x 1 , x2 ) which has a length of 1.This implies that x12 * x22: l. Geometrically in Euclidjan space R2, u."Io. * c-be represented by an arrow fiom the origin to the point ( x 1 , x2 ) and has a unit length.

Numerical example to show how to calculate eigenvalues:

"= [-s :l[: -e)

we know for eigenvalue ct I

butlA-"tt: [n^-*L'

A-o, Il=0

', "]

=81 + 18 (t +ct' -9:72 + 18ct +ct2 :0

) ct : 1- rat./1tt'-+1t11tzy1y t z1t1) cr: {-tst.[TFZEE]l7z

: (-9-cr;2-f1:;

) *= {- ax,,[*G"11r zu

) cr, = {-9t3} : -12 or-6 both eigenvalues (roots) are negative

We can use any one eigenvalue to calculate the eigenvector 1 , say use o : - 6

substitutinginro (A- ct)-r =[r-*-u, , I [-,.| :[, ;lf; II r ., (_6lJ L.,J L3 :|l,.il

t -3 xt +3xz=0 and

3 xr - 3x2 = 6

t x 1 : x2 and to normalize x 12 + 1r2 = 1

t xt2 +xr2 =1

.) 2 x12 :1

t xr ={-072 + x2 '- f -r= xr ={(l/2) andx =lfat72l

I

lwz'lL

: f'lLOJ

Similarly using the other eigenvalue = -12, we get the conesponding eigenvector as:

['- ''' ,-',,,1 lt,] =[l

lfll =fsl

t 3xr +3xz:0 and3 xr + 3xz : 0

) xi = -x2 and to normalize x1

t x12 +1-xy;2 :l 9, 2 xt2

, + *r, =l_I

t *' ={@) t Xz=-Xr : -./ ltlZ; andx :

Another numerical example for illustration:

given square 2x2 (hence 2 eigenvalues) matrix A =

to frrd eigenvalues o sel 0 - det lA - crtl : l-6 - oIrI

= 6 s, + az -7

[1.':?,,1

llu 7

Ll ro

o-"1

",41, l=,

check I & -7 are eisenvalues

There are many usages for eigenvalues:1. to test for sign definiteness2. to test for matrix singularity

_

3. for obtaining solutions of 2no order difference equations4. for determining the stability of dynamic systems5. for derivation of general solution for n-dimensional dynamic system etc.

In addition, if square matrix A is also symmetrical (i.e. A = Ar ) then we have thefollowing convenient properties regarding eigenvalues:

r-texample of symmetrical A lt t : l: et

lz 6 4ll: 4 5J

We note symmehical mahi* rri.t U. square because a ij - a ji in symmetrical matrix.

S)'rnmetrical matrices have1) only real eigenvalues (no complex roots).2) enough independent eigenvectors to diagonalize and not necessary to work with

generalized eigenvectors.3) eigenvectors that are orthogonal to each other

l"i' ,.,1:,

+ economists many time assume matrix symmetry when modeling for optimizationor statistical (econometric) problems.

Numerical example: (let e ; denote eigenvalues)givensymmetricalmaiixa f-s 21

12{set lA-eI | =9

->(_5__€)2- 4: 2 (r)

(still discrete & intertemporaln , n+l period )

l;-0:>

all e < 0:> A negative definile and all e are real numbers.'.. A_symmehicalcan furd eigenvectors x-, J./tal from e, - -7 ", ar_ {'lltfl" f.o.

", = -3Lr'4 hi,oj

After frnding the eigenvectors x ! x I fiom the respective eigenvalues el e2 we cangroup t}em into a Transformation MatrixT= [ lr xz- ]

We now apply n-dim difference equation model to economics.

1-dimensional difference equation with single difference equation enables us to study e.g.how price of apple in period 1 will affect price and consumption of apple in period 2 etcl

In real life, we require more complicated dynamic systems than those described by Iequation. We need dynamic systems describing economic phenomena and theirintertemporal relationship as well as their intenelationship. This necessitates a system ofn simu^ltaneous difference equations. e.g. this month's price of apples and oranges will{fect ltue months'price and consumption ofapples, oranges, banana .... as describedby difference equations; or present period R&D (research and development) will affectfuture technological level ) affect future production ) affect futuri supply, prices,demand etc.....

we illustrate with 2-dimensional first-order linear difference equations which can beeasily extended to n-dimensional first-order linear difference systems.

2-dimensional l"-order linear difference equation system (= fwo I st-order lineardifference equation forming a dynamic system):

Xn+1

Yn+lott Xnilz t Xn

* av Yn+ a, v^

-^il". *t+lI

{-10 r !1100 - 4(1)(2r)l} /(- 10+ 116) /2 : -7 or -3

We can rewrite in matrix form

znr|n I n+l

I14',- )

[-"-,.l = [",, ",,"l [.".|tllLY",,J Lul urr) LtJ

UUUZ n+r = A Zn

.lThen T_'A T =le1 0 0

[ 0 er 0

I'It:lo oL

[' :,]

T-rA T :T-r T

A solution to the system means Z n = ( x, , y n ) that solves all the difference equationsin the system simultaneously.To derive a solution, we need to assume that A has 2 distinct real eigenvalues(characteristic roots) e1 , e2 and corresponding eigenvectors 11 , v 2 .

Review: 1) get eigenvalues by solving lA - e; I | = 62) get corresponding eigenvectors by solving (A -e I )y 1 :0 null vector3) form transformation matrix T as matrix [1 11: ]

Following properties ofT are needed for the solution derivation.

Theorem: A is a m x m square matrix (not necessarily symmetrical) with m distinct realeigenvalues el ,e2, ....,em and corresponding eigenvectors v r , 1L2,..., v-!q.T is the transformation matrix: I v1 y_2 ... v ,n ] and if T -r exists,

ol:l9rn I

Proof: for sake of brevity, we prove the 2 x 2 case. The m x m case proof followsexactly the same steps but with i extended fiom 2, ..., to m.

By defurition T : I v I r:] ) AT: [A11 Av2]

But Av; : eillj by definition ofeigenvalues and eigenvectors

) AT = [e ryJ ezv z I : r,r ..,, 1r,

0",]

:T

t'ler

LoiJ 'f' :J

: F' :l

Qet

Corollary: if A is in addition a symmetrical matrix, then T -r: J r

Given a 2-dimensional linear dynamic system in matrix form:

I.". 'l : [, ', ",,.I [.".lttlL'' 'J : L"'' ^-J L'"J

UUZn+t AZ"

and A is a square matrix with 2 distinct real eigenvalues and 2 conespondingeigenvectors y1 L2.

I't step: useT:Ivr v z ] to transform Z:

LetZ i=T Q t

UU[t ,,,,1 [0,,,,.l

[z rrrzJ

[er;nJ) q= T-t Z for allz including

m panlcular Q nrt = I LD*t =

i.e. Zn :TQ.

Zn+r =TQ o*r

Qn = T-rZn

Q o*1 = T-r Zo*t

T-r (Az^)

r-' A (rQ.)

(T-' A T)e"

[", ol q"lo e!

F, ol [0,",,-lLo .,J p.,4

: l-l' '*',,.*"]: [:;n]

= een

t q(n+l)t = er g(n)rq (n + r) 2 : e2 q (n)2 t2-31 are two I -dimensional simultaneous

difference equations

Recall from our discussion on finance application using iteration method to solvedifference equations in the recursive form ofQ n+t: eQ n the solution is

Q n = eo Qs where Q 6 is value of Q at period 0

f r f Iwelet Qo= lCrorrl:1c11[o

rorz-l L.

,l where c I c 2 are constants

' t2.3r

1x';:',lN :

[l ;;{ I |;$ =[:::;llJ ::i: :inow transform Q o backtoZ"

>2, :re" :rfo,",,l:rF,:.,1Lo,",,J [:,".'J

=[rr "r,[:;::J

)Zn :cl ern IL-! +c2 e2n y_2

: general solution form fora 2-dimensional dynamic model Zn+r : AZ"

and the solution consists of constants c 1 c2 , eigenvalues el o e2

n and eigenvectors

The above can easily be extended to m-dimensional (i.e. m simultaneous I "t-order lineardifference equations) system:

l':Ct,'

Zn+t = A

[i::lt;* jTheorem: LetAbe the m x m matrix with m distinct real eigenvalues el e2 ... enwith corresponding eigenvectors v r v 2 ... va :

) general solution form for a m-dimensional system ofm l't-order linear differenceequations Zn+t = A Zn :

zn

z 6yrz 612

:

z6y^

7 -^

where c;

er' yt +c2 e2" L2 + *cm emn llm

: constant i= 1,2, ...,m

Economic application of m-dimensional difference equation system (discrete m-dimensional d1'namic model) :

We construct a descripive dynamic model for growth of firms. To add more realism, werntroduce risks into the model. This can be done through Markov process ) a stochasticdynamic model with Markov transition probability matrix. A model for firms growth canbe stretched to imply econ growth with policy implications.

Some preliminaries:Markov matrix M is a mahix whose entries are > 0 and whose columns (or rows) eachadd up to l. This property means Markov matrices are suitable for handling probabilitydishibution (or density) p.d. functions. In other words for stochastic models:

row = 1(orl00%)

row = I (or100%)

=l

[Review: P is a p.d. if P1 > 0 and I Pi = I ( like a Markov column)

iwk)

e.g. tossing of coin 2 outcomes H PH(H)=%> 0

T Pr(T) :% > 0

IPi =l 1

Defir: Stochastic process is a rule assigning probty that dynamic system will be in stateSn+r attime n*1, givenprobty of system being in state Sn, Sn-r, Sn-2, .... So inprevious periods.In particular, ifprobty that system will be in state Sn+r at time n+1, is dependent only onthe previous state Sn , then called Markov process (i.e. only immediate past state matte6).

Markov process Z n+ 1 = M., Zn\

Z6+grZ 6+ r)2

:

Z gt + r'sk

ml1m2l

:

mkl

r) column:l

M called hansition probability matrix or Markov

Econ interpretation: if econ is in state j (e.g. good econ, bad econ, depression, recession,boom cycle etc ) at time n then m;.; is the probability of econ will be in state i at time n+ I (i.e. probability ofintertemporal change in econ state).

Application:Let x n : 7o of companies in an economy growing at end of period n t

2 states of econ

! n : %o of companies NOT growing at end of period n d& let probability of growing companies at end ofperiod n to continue to grow at end ofperiodn+l:90% (caa arrange Markov M to letthis be represented by mrr =.9)

:> probability of growing companies at end of period n to be NOT growing at end ofperiod n + 1 : 10% (can arrange M to let this be represented by m u : .l):> column of M will add up to .9 + .1 = 1

Let probability of NOT growing companies end of period n to be growing at end ofperiodn + 1=40% (can arrange Markov M to let this be represented by m rz :.4)

Zrrrlz61z

I;lzr"rkl

matrix.

Q4)

:> probability of NOT growing companies at end of period n to continue NOTgrowing at end of period n + 1 : 600/o (can anange M to let mzz : .6):) column of M will add up to .4 + .6: I

Dynamic model describing above econ system Zn*r : A Zn

(2 dimensional) Xn+r :0.9xn +0.4yn/n+r :0.1 xn * 0.-6 yn

^rtt\note I column X column

-l-l

Znn'L = M Zn

:>(0.9re)(0.6-e) -0.04 :0.54-0.6e*0.9e+e2 -0.04 :0.5 - 1.5e*e2 :0:1 s, = [1.5+ 1 /2(1) = [1.510.5]12= lor%

F,,l : [: s{ FI:> one of Markov matrix M eigenvalues.: 1 and the other eigenvalue from solving

| %i-, ,lf

" l='

:) eigenvectors by solving M-e;I v:0

[on-' oo lf.,,l =iolt il t tlLo.t o.o-rJ luJ Lo.l [:

',:1; il []

-0Vl

VI

- 0.1 0.4

0.10.1 vr -vl

Vl + 0.4v2 = g

0.4v2 : g

u

v1 :4 v2

U

u

+ 0.4 vz

+ 0.7 v2

u

V2

normalized v12 + vrz : 1

l'1){ \

- i;] lil;J- [l fl,;;l(4:1 cc )

Alternatively

any vector 4:1 proportion will work

(1: -1 cc)

any vector 1: -1 proportion will work

By Theorem, solutionofZn+r = M Zn

: z^ :{- "l :cr ein l4r +c2 e2n L2tl

L'J- cr (t)' f+l , ", 1rra"l 'lL'J L 'I

As n --- "o this term (1/2) " ---, 0

llll['"J L'lWe wish to get this term into p.d. form (i.e. % terms) which can be achieved by lettingct: l/(4+1): 1/ 5

:> z"=["".| =rnl+]: loul=1aoxilllrfrirL'"J ['J = l,;l =L,yA

I coldmn I Jolumn:1 =l

Econ interpretation: over time (as n --r o )

80% of companies will continue to grow.20% of companies will stay stagnant (no growth).

Other possible economic scenarios:

In t}le above example, if we let gror'4h of companies in an economy be directlyproportional to growth ofthe economy then above conclusion can represent the long runeconomic grouth picture.

But note the conclusion figures depend entirely on our assumption of Markov probabilitymi.; . For instance, if we just change the above m 1r probability to 80% (probability ofgrowing companies at end ofperiod n to continue to grow at end of period n + I now:80% instead oforiginal 907o). This maybe due to business environment becomes non-accommodative (e.g. govemment imposing higher taxes or new sales tax, extremelyharsh environmental control, new labor legislations etc.) then conclusion becomes only66.67Vo of companies will be growing and 33.33% of companies remaining stagnant inthe long run and hence hindering economic growth. @lease check this as an exercise).

I.nA)

Chapter IV.A. iv). Discrete dynamic macro employment model to explain long rununemplol'rnent (R.8. Hall model for finding employment pattem).

Macro unemplolnnent rates are not useful for formulating macro policy because it doesnot give us information to which group our policy should be targeting.

We wish to know whether there are struchral or just short term reasons. E.g. low skillmanufacturing workers in HK maybe unemployed for a long time because low-skillmanufacturing jobs have moved to China (structural change in HK economy). DuringSARS and recent deflationary period in early 2000s, workers in all industries were losingjobs (short term reasons). 2005-2007 and 2008 saw some employment recovery and hintsof inJlation. Therefore typical macro employrnent models are tamed as follows:

Hall's Employrnent model for finding pattems of unemployment over time.

@. E Hall, "Turnover in the Labor Force". Brookings Papers on Economic Activity 3 ,1972)

Let p: probability worker will find ajob (on average in the LR)) l-p: probty w remain unemployed.

q : probty w remained employed next period) l-q : probty w will be unemployed next period

So if let x : workers employed this period) q x will rernain employed and ( i-q)x unemployed in next period.

Let y : workers unemployed this period) p y will be employed and (l-p)y will remain unemployed in next period.

I next period average number employed qx + pyI next period average number unemployed (l-q)x + (1-p)y

)dynamic system describing above average unemployment model for periods n & n+ I :

Ln-l * A Ln

(2 dimension&l) xr+l : q xo + py"yn+r : (1-q) x'' + (l-p)y,

[Hall estimated for male (white) workers in U.S. in 1966:

Xn+l : 0.998yn+r : 0.002

x " + 0.136 y "x " + 0.864 y "

We of course recogrize these p, q, ( I -q) are transitional probty that can be formulatedusing Markov.

Let x . = fraction of workforce employed at end of period n ,;s of econ

y n: fraction ofworkforce unemployed at end ofperiod n I

Assume probability ofemployed person at end ofperiod n to continue to be employed atend ofperiod n + | = 95% (e.g. can arrange Markov M to let this be represented by m 11

= .95)

:> probability of employed person to be unemployed next period,= (100-95)% :5%(e.g. can arrange Markov M to let this be represented by mzr = .05)

:> column of M will add up to .95 + .05 : Iprobability of unemployed person at end ofperiod n to be employed at end ofperiod n +l:45% (e.g. can arrange Markov M to let this be represented by mp : .45)

:> probty of unemployed person to remain unemployed at end of period n + 1= 55%(e.g. can arrange Markov M to let this be represented by mzz : .55:> column of M will add up to .45 + .55 = 1)

dl.namic system describing above econ model Z n * 1 = A Zn

= 0.95 x n + 0.45 yo= 0.05 x " + 0.55y"trI column X column

-l-l

:MZN

(2 dimensional)

i::r : [;: tgL;{:> one of Markov matrix M eigenvalues: 1 and the other eigenvalue frorn solving

0.95 - e 0.450.05 0.55 - e

:0

:>(0.95 - e) (0.55 - e) - 0.0225:0

0 = 0.5225 - l.5e + e' -0.0225 = 0.5 - l.5e + e2:> e i = tl.5 r \D75@3jf / 20) : t1.5 r {6:ttl t 2 = [r.s + 0.s1 t z : t or lz

X n+ 1

j n+ t

note

Zn+t

17tO)

:> eigenvectors by solvingI-rrrlo.ss-r o.4s llv, ltttl| 0.05 O.ss -l I I v, I\ -L-

- 0.05 -v1 + 0.45 vz = 0

0.05 -v1 - 0.45 v2 = 0

v

V1 =9V2

v

,:;:; [;J=i'lL'J

M-e 1I v:0'rf:io I lo.es-%

lol lo.otL.I Lu

0.45 v1 + 0.45

0.5 vr + 0.5

u

vl:-

U

v12 + yr2 : 1

v2=0

v2:0

uf",i | 't uzl

L"'l =L

"C"1:,1

il+c,,112)n \ tI'\[ {_ lt, _i

As n ---r oo this term (1/2) '

normalized

U".lie./ vaz

I:tl tl| {vaz IL.)

(9:1 cc )

Alternatively

any vector 9:1 proportion will work

(1: -1 c)

any vector l: -l proportion will work

---+ 0

By Theorem, solution ofZ n+r :

=2"=l*"-i ="tet 'y-rtlL'".l

= ", rrr i-qlL'.I

MZ"

+c2 ezlt y.J

We wish to get this term into p.d. form (i.e. % terms) which can be achieved by lettingcr=1/(9+1;:1719:> Zn =

Econ interpretation: over time (as n --+ o )

907o of employed workers will remain employedl0% ofunemployed workers stay unemployed (LR unemployment rate of 10%o)

Policy implication:

If there is a persistent LR unemployment rate of l0%, then this is not due to businesscycles and policy must be geared toward frrding out the cause of this structural hend andrelevant remedy.For example, we mentioned structural change of HK economy from manufacturing toservice (banking & finance, entrepot and logistics, tourism) sector in the last twentyyears. The cause is the loss ofHK comparative advantage (higher labor and other costs)to mainland in such manufacturing sector. HK Government policy seems to favor re-kaining ofthe displaced manufacturing workers or outright subsidization. Effectivenessof such policy is hampered by reluctance ofdisplaced workers to switch, especially tolow-paying jobs; &/or displaced worksrs cannot be re-trained for higher paying jobs; theusual mismanagement of bureaucrats includes wrong estimation ofpossiblejob vacanciesand placements, increasing licensing requirement barriers preventing workers ability toswitch jobs; waste due to bureaucratic rigidities (versus market allocation and discipline);abuse and sometimes fraud by subsidy recipients.

tJ :"'

|;]

Ll "''[:] vt r:r)

I column I column=l =1

Chapter IV. B. Bellman's dynamic programming: Discrete dynamic multi-stageoptimizati on models.

Basic notions and terminology (optimal decision sequences examples)

Dynamic models can be continuous time x(t) or discrete time x t

(t represented by real numbers) (t by integers).

In pure math econ, most models are continuous time. Main analytical tools aredifferential equations and optimal control.

However, most macro math models (e.g. macro growth, business cycles, employnenttheory) are of discrete time. Hence difference equations and Bellman's dynamicprogramming are the main tools.

fRmk: can actually assume such discrete period to be points of time and still usedifferential equations and get useful result.l

Bellman's dynamic programming DP:

Defn: Bellman's DP is basically a way of solving dynamic optimization problems.

It is a multi-stage (: multi-period: sequential) optimal decision process.

Given a multi-stage problem, we find an optimal decision policy.) Based on this policy we generate a sequence of interrelated decisions that will

optimize our objective at every stage) and by following this decision policy, overall dynamic objective will also be

optimized.

A multi-stage optimization problem:

stagelperiod l " 2"d 3'd t th . .. . .. ...

policy decision transfer transfer transferstate-l A 51a1s-/ nr 51a1s-3 state-t /}

Find opt decisions.t. transfer will contributebest value to state variableat each stage, opt opt opt opt

value-l value-2 value-3 value-t

and best to overall value of state variable) overall objective optimized

DP Terminology: there are two variables in DP. One is decision (= choice = control)variable and the other state variable. Every path decision at period t ) affects value ofstate variable in all future periods t + 1,t+2, ......... hence affect overall value of statevariable.

How to do DP?Illustrate basic notions and terminology by an example of minimizing budgeted expense(abbreviated as exp) for economic development (growth):

You are appointed as minister for economic development in country Planningland. Youhave to draft a 4-year plan to develop the countqr from economic state A (manufacturing-based) to economic state O (service-based). There are altemative ways to achieve this asgiven by the following graph. The numbers represent respective exp levels of altemativeintermediate stages necessary to reach stage O.

stages/periods t IFeb/2009

2Feb/2010

34Feb/2011 Feb/2012

)e+

Objective: find optimal path to min overall exp to reach state O.A : manufacturing-based econ state3 : gov build IT network for all office buildings downtounC : gov build another theme parkD = subsidize private IT companiesE : increase funding of finance schools in universitiesF : special tax allowance for tourist and finance industriesG : build east, west, north, south Kowloon cultural centresH : expand airportI : build additional shipping terminalsO : service-based econ stateOne possible approach: grind out all possible paths, (in this case, all 18 paths) andcompare overall exp for each path. Tedious, even for our simple problem.

@,

One possible approach: grind out all possible paths, (in this case, all 18 paths) andcompare overall exp for each path. Tedious, even for our simple problem.

Another approach: pick min exp path every time:

A) B '.' 3 is smallestthen

B) F'.'5 is smallestthen

F) I '.'3 is smallestthen

I)A'.' only path (& exp:5)) totalexp:3+5+3+5:16

Butthispathisnotminexppath'.' e.g. A> C ) F ) I ) g523s) total exp: 15 < above 16

So need another approach: Bellman's DP.

DP Procedure:Bellman discovered the following technique, called Bellman's Principle of Optimality:

Various ways to state this Bellman's Principle of Optimality:A. Al optimal policy has following property: startiag from ANY initial and ANY optimal first decision,the remaining decisions constitute an optimal policy for the state resulting from the first decision.

B. Optirdze {optimized objective of current period + optimized objective of all subsequent periods -using recursion- based on current period decision) : get the solution for the overall problem.

C. Starting from any point on an optirnal trajectory, remainder trajectory is optimal for thecorresponding probl€m initiated from that point.

Above all mean that with Principle of Optimality, we can convert an n-period probleminto an equivalent 2-period problem, namely current period and all-future periods (orall- previous periods). All-future (all-previous) period treated as one period. Inaddition, we assume that we have already optimized the objective for all-future period(or for all-previous period). Then we only need to optimize the current period and tagon the already-optimized all-future period. We repeat this process for each period(either forward or backward in time) recursively and the problem is solved.

t-r + 't+t

lfconvertlrtl

optimize [ (t-1 & all previous) and t]

futurel.ti:\

' COnVerI

2-peri od probl em

optimize I & al I

Now we solve your min-exp problem using DP:

Define : x 1 : decision variable, in this case: immediate next stage at period t

)A=xo ) xr ) x2 )x3) xt:Q5

immediate next stage at penod 4 is @

S : current statemeans wg move from state S ) state x 1

4.4.S+ /{

,.4tFt

(He)-\/,2 \(L, )\/

total exp of already-

F. (s, zt)Letx r

* = 6""i.ion variable minimizing F,(S, x, ) : denoted by Fs*(S)

Value function F 1 can be written in the following form called policy decision function:

Fr(S,xt) = min {f(S,x) + Ft*r *(xt vtrich = for stages after S) }

ztsFe

DP terminology: an equation containing an unknown function is called a functionalequation. (Above equation contains the unknown frrnction F p r * (x t))

Terminology: above equation is recursive '.' both F t and F tr r are in the same -ion.

When working with time, we move backward stage by stage tbrough recursive equationuntil we get the first stage and we get the overall optimal solution for the whole problem.

Define: value function F t (S, x t ) for exp =policy from stage t onward.

Define: f( S, x 1) = exp of current period t

X1

min{ curent + alrcady-min t+l periods onward}

minimized overall

vt

(-D

We start from period 4 with destination state x a = Qt F 4 (S, x a) = min {exp cunent period 4 + already-min for stages 4+l onward}) =min{f(S,O) + F4*1*(no stage afterxa=A)}:F a* (x:)

E 0 '.' no stases 4*1 onwardcan tabulate:t=4

X4*

t = 3 (t 2 more stages to go) F3(S,x3):

ilF:*(S) X3 t

min{7,8}

t : 2 () 3 more stages to go) Fz(S,xz) =min {f( S, x 2)

X2+

EElG

t7

I

F

+ Fr

tl

z*

z)l

IIlI

\t/

*(x

(s)

StlB

F+*(S)

min {f (S,x3) + F+*(x:)}

l+4=5 4+5=9 min{5,9} =

=10 3+5:8 min{10,8} :8

8+5:13 5+8:13 6+7=13

3+5:8 2+8=10 4+7=11

1+8=9 5+7=12

t = | (t 4 more stages to go) [' I (S, X r ) rneans optimized for period I onward (i-e. overall)

)= min{f(s,x;) + pr*(*vF

Fr*(S) Xt'

3+13:16 5+8:13 4+9=13

) optimal path(s) for min effort for econ development

a

o

)

4)

n

)

)

I)

It

)

J

)

A

)

S

llA

)

5

t4)

xo*

A

orA

orA

x r*

C

D

x2r

E

x 3*

H

H

xq*

5

)a

iJ

ll3

tIJA

)DI

'F

3

'I

Remark A:Above DP can have infinite staees

) solution will be infurite sequence of interrelated decision variables { t , * f= o

Remark B:Often times there are no solution or no closed form solution (only approximation).

Remark C:Above DP require two pre-conditions:i) Principle of Optimality: At current stage, an optimal policy for the remaining stage is

independent ofprevious stage policy. This is analogous to EfFrcient MarketHypothesis in Finance that current stock price reflects all relevant information ofprevious periods so we don't need to know tlle previous period information.

@

ii) 3 recursive relationship: consecutive stages are related and such relationship is samefor all consecutive stages.

Remark D:Many time solution by common sense, not by elegant deep math manipulations.

Remark E.Most soluble problems need differentiability I limited to local results.

Dynamic programming and macro growth models.

As we mentioned, the latest development of grad level macro rely heavily on dynamicprogramming because it is employed to show rational eoonomic agents optimizing overtime periods (with or without risks) - proper description ofeconomic agents observedbehavior within GE (individuals optimizing to reach equitibrium) framework. Hence weget the micro (and concrete) foundation for macro. We also mentioned that macro GEmodels are different (not individual consumers or producers optimizing as in micro; butsocial planning maximizing some aggregate objective function subject to aggregateconstraints).

Example: simplified macro dl,namio progranming model

In an economy, 3 social planner elected by and always carrying out population wishesinstantaneously. Let c , : qurr.rl oonsumption in period t and is controlled bypopulation through the social planner. I os - control decision lariable ].

There is only one national product (a composite good) Y 1 in cunent period t which canbe consumed or saved-invested, two inputs labor (assume: population) n 1 and capitalk1. A production ftrnction F relating these 3 variables F( k1, nt ): Yt and F( k 1, n1) ishomogeneous of degree l, differentiable and concave.Assume labor constant over time periods so can write n 1 = 1 unit .

) | ln1 [F(k1, n)] = F(kt/nt, n1/n1):F(k1,1) ) fcnof k1only.

and next period capital : production this period - consumption this period

d€fined

) kt*r : F(kt,1) -ct = f(kr,ct) * recursive equation [2-5]Note k t is then the state variable [we are looking at dyramic evolution (or equation ofmotion) of states k,, k,*r ]. State variables are the recursive ones in the constraint.

Assume economy's utility function U is a function ofbothkl and c1 ) U(k 1, c, )and U differentiable and concave.

Social planner's problem for population is to maximize this period's, next period's, andall subsequent period's utility :

Ifwe are given constant discount rate [l for end of each period t, then we can discountdifferent future period utility back one period to PV and problem for any particularperiod becomes:

Max IU(kt,cs) + [3U(k6r, ct*r ) ] for all periods tand t+l

Suppose T + I is the last period, so we solve only for the periods T and T+l first and thensolve recursively backward to I "t period:

1"' step: Max B U(k t*t , c r+r ) wrt decision variable c r+r to get the optimal decisionfunction for c 1*1 in terms of a function h r+r of k r+r :

(by setting partial derivative = 0 )

DU(k r*r , c r*r ) :0 )get cr*r * =hr*r (kr*r)0"t-'t

V is analogous to F in previous example nothing after T+l pe.iod

2nd step: define a value function V r*r (k r*r ) : Max {U(k r*r , c r*r ) * fl V r*z (k r*z )}

= U(k r*r , h r*r (k r*r )) all same k r+t (T+1 period variables)

3'd step: set up objective firnction for the consecutive period problem:

U(k1,c1) + [3U(kr+r,cr*r ) : U(kr,cr) + [3V r*r (kr*r )

: U(kr,cr) + BVr*r(f(kr,cr)) from recursive equation [2-5]

4d step:

Max {U(kr,ct) + BVr*r((kr,cr)) } wrt c1

Setting partial = 0

aUGr,cr) D(kr,cr) ? Vr*r (krrr)+B I : 0 )getcr*:hr(kr)

? ct 0 c, 0 k r*r

i\J.u,

5m step: after getting optimal decision function c 1*:h1(k 1)repeat above procedure backward toward t = I and we obtaincr *:hr(kr) which

Max{U(k1,c1) + BVr*r(kr*r) } = Vr(kr) t2-61Cl

[2-6] is called Bellman Equation.

Furthermore if V t Gt ) converges to a steady state V ft1) [2-6] becomes

v(k,) = Max{u(kt,c) + t3v(kt+r) } [2-7]cl

[2-7] is a functional equation (i.e. same function V appearing on both sides oftheequation) and when we solve this functional equation for V, we get solution for thedynamic optimization. This method is Bellman dynamic programming.

We see the parallel similarities of the previous example and above problem:

F1(S, x1) =Min {effs*1 +F1*1 *( x 1= L,n "tae€

t+l onward )} - policy decision fcn p-4]X1

V(k,) : Max{U(k1,c) + BV(kp1) }-policydecisionfcn12-71C1

Economic application example:

Planningland govemment now appoints you to be education minister. The govemmentdeems the best policy for economic growth is by investing in Planningland University butwants you to minimize the University expenditure. You did an econometric study andfound:l. Whenever govemment budgets education expenditure E, in period t , the actual

expenditure will be E i .

2. When E I is spent, "education capital" K1 consisting of camp^us building &infrastructure, library etc. will be I expanded/upgaded by &'

3. intertemporal relationship between Kt+r and K1 (dynamics of the system) is given by:

K*r-4Kt:E,

;: ri

4-^.0-A,L

You know at present K, represented by K6 = 5 (billions of $). You are glad the costfunction is quadratic which means I a soln. Your appointment is only for 2 years (T = 2)so you only wish to solve the following problem:rOlMin : (p,'+ K2) * Kr2E t=0

s.t. Kt*r-4Kt:& [IIB-2]

Ko:5 [IB-3]

To solve:

We start from T-1 period ) t : t

Min p12 + K1 2; + K22F-

s.t. Kz - 4 Kr : Er [IIB-5]Kr : Kr to be determined [118-6]

lrrB-rl

lrrB-4I

bo*."1

T;L

K""

1118-5-61 into [IIB-4] tMin (Ett+ Kr2) + @1 +4S)2

I't order condition for min: (take derivative wrt E1 and set to = 0)

zBt+ 2(Er+ 4K1)(l):0t Eq : - 2 K1_ [IIB-7] control variable for t:l equals to -2times state vble

) ValuetunctionVr(E^r)=Min(El 2+ &1^ + 1B1 +4&)2=(_2Kt\. + K,,* (_2Kr+4K,\'= 4Ki1 q-+ 4Kr 2

: e Kl IrrB-81

For T-2 period ) t:0

Min (Eo' + Ko') + Vr @r ) IIIB-91

s.r. & -Ko:

I Min (E62) Min (Eo2

Eo UrB-t0lurB_31

I 9Kt2+ gGko +Eo)2

aItu =5

+ 2s)+ 25)

I't order condition:) 2Eo + 18 (4Ko +EoXl) =0

> 2F4 + 18 (4(5) +EoXl):0) Eo=-18 [IrB-11]

I following :

& : 4 Ko + E6 = 29 a 1-t*, -, from [IIB-10]

Et: - 2Kr :-4 from[IIB-7]

Kz= 4Kr + \= 4(2)+(-4)= 4 from [IB-s]

Vr@r)= 9k: :36 from [IIB-8]

Vo@o):Min(E02+Ko2)+Vr (Er ) =(-18)2+52 + 36: 385 fiom[IIB-3-8-9-11]

In other words, you will budget to cut initial expenditure Eo : -18 and next period cutexpenditure Er = - 4 and the total minimized expenditure will be Vo(Eo ): 385.

Remark:We note that we did not discount the FV in the various periods. The implicit assumptionis interest rate i in Planningland is zero, then there is not need to discount the FV to PV.(discounting factor B': I / (l+tt : 1)

PLEASE READ STOKEY & LUCAS CIIAPTERS 1 AND 2 FORFAMILIARIZATION OF DYNAMIC PROGRAMMING CONCEPTS.

Bellman's Principle of Optimality in metric spaces

Givenl. metric space (X, d1) called state space:2. initial state x0 € X (x" if€ X is nth state)3. transitioncorrespondencef:X3 X; f continuous and compact-valued

[this self conespondence tells us that given state x " ,

what will be all the possiblenext state x 61]

4. discount factor for each period n : 0<p<1 (assumed same for all periods)5. one-period return (e.g. utility fimction ) function u: Gr( D ) R ( R including +oo);

u continuous & bdd

set ofall sequences in X1Feasible plan (: strategy) is represented by a seq(x") € Xsequences

s.t. xl ef (x6) and xn+r €f (x") n:1,2,3,...:N++veintegers

fNote: x" in the seq is the state ofthe system for period n+1]

Typical DP can then be expressed as:

Max u(xo, x r) * L- p, u(x,, x i+r )

seq(xJ

IDPU

s.t. xr€f (x0) and x"11 €f (x") n:1,2,3,... =N*

Basic assumption for Bellman DP:(to avoid endless oscillation)

lim XN po u1xn, x,,11) € IR*) - n=o

for any feasible plan seq(x") € Xsequences tA-DP-l]

[This assumpion means we can find real number representing the PV of intertemporalreturns stream offeasible policy decision.l

Define correspondence F p (x ) : X J Xt"au"n""":{seq(x")€ Xsequences : xr€f (x) and xo+r€f (x. ) n€N+}

[= set of all feasible plans]

q >zu)

Define new objection function H14 "1

(seq (x"), x) : {F (x ) x {x}with x e X} ) Raad look at altemative DP problem:

Max H6.t(seq (x" ), x) : u(x, " ')

* _l-l pn u(x", xo11)

s.t. seq(x")€F1(x) tDp2j

[this [DP2] is same as [DPl] except initial x6 in [Dpl] is replaced by any x e X in[Dp2] l[By assumption [A-DP-l], HE,1 (seq (x, ) , x) is well-defined real valued function. Sinceu is bdd, H1q ,1 also bdd.l

Define value function V(x) :X ) R: sup { Hp,l(seq(x,),x) : seq(xn) eF1(x)} [Dp3]

I f solution to [DP2] itr above sup : maxl[V(x) gives us the sup H s.t. feasibility]

Thm: (Bellman)Given any [DPl] problem with any initial state xo € X and see x(n*) e F r (xo )]

V(x6) : H6 "1

(seq (x* ) , xo)

) V(xo): u(xo*,xr*)+ FV(xr*)and V(xo*): u (xn*, xntr*)* 0V(x,*r*) V netl"

And conversely ifu is continuous and bdd.

Pf: Deft of V(x6) means for any seq x(n*) e F r (xo )

u(x6*,x1*)+1,-p"u(xn*,xn11*) Z u(*0, x1)+x- Fo u(xo, xo*r)o=l n=l

Vx(")eFr(xo)

Now due to recursive sfucture of the DP problem, for feasible (xz, x:, ...) eFp(x1*)) (x 1*, x2, x3. ...) e F 1(x6)[if (x2, x3. ...)is feasible plan induced by initial state x1* implies(x1*,x2, x3. ...)isfeasible plan induced by initial state xa with next state xr*l

I u(*o*, x1*) +t - pou(xn*, xo11*)n=1

Z u(*o,xr*)+ F u(x1*,x2) * :; p'u(xo,x"+r)

) H[-,.1(seq(x2*, x3*, x4'*,... ), xr*) ]H6ut(seq (x2, x3, &,... ),xr*)V feasible seq(x2, x3 , & , ... ) € F I (x1* )

But V(x1*) : H1Il ut (seq (x2* , X:* , &* , ... ) , xr*)

) V(xo) : Hrrq(seq (*"* ), xo)

: u(xo,xr*)+ PIu(xr*, *r*) * X- B"-r u(x"*, xo*1*)]n=2

: u(x6,x1*)+ F lHrr,ul (seq (xz*, x:*, &* , ... ), xr*) ]

: u(xo, x r*) + p[V(xr* )]

And by iteration:V(xn*): u (xo*, xo*r*) * 0V(x^*r*) V nell*

Conversely, ifu is continuous and bdd:

We have V(xo) : u(xo , x 1*) + B fv(xt* )l

: u(xo, x1*)+ pu(x1*, x2*)+ p'V(*r* )

: u(x6,xr*)+ Bu(x1*, x2*)+ p2 u(x2*, x3*)+p3V(x3*)

: u(xs,x1*)+ 1!r F ju(x,*,

x1*r*) I + F'*tv1"r*,*y V reN*

Since u is continuous and bdd by assumptionI Visbddt

= UB>0s.t. lVl < UB

* the term p r*lV(xr*r*) ) 0 as J) oo ('.' V bdd and p is a +ve fraction)

t V(xo): u(xo, x r*) * P,- p, u(x,*, x.;*r*) (J) co ;

= H1q 't

(seq (x'* ), xo)

[The converse part of the Thm tells us in case u is continuous and bdd and if f solution(a set of maximizer sequences) to [DPl], tien we can go from the value firnction to theset of maximizer seq. In other words, we can use the value ftnction to solve Dpproblems, under the stated assumptions.]

Principle of Optimality:

Optimal policy correspondence P: X 3 X for DP problems:P(x) :argmax { u(x,x1)+ 0V(xr): x1 ef (x)}

[reminder: arg max stands for argument of the maximume.g. given f(x), arg max: set of x* that maximizes (x).

so above P(x) gives the set of solutions maximizing the value function V(x)l

Bellman [1957j: "an optimal policy has the property that, whatever the initial state aaddecision are, the remaining decisions must constitute an optimal policy with regard to thestate resulting from the first decision."[Means: policy function is opt for the infinite summation ) policy, whatever initialstate and decision are, remaining decisions as per Bellman equation must be optimalpolicy with regards to resultant state from initial state/decision.l

Terminology: BtX)= { f(x): X) R :sup{ l(x)l withxeX} < -} }: {all bdd real firnctions defined on X}Note: B(X) space is metrized by the sup-metric d- i.e. B(X) :a metric space (X, d-)

Thm: (Bellman, Prinoiple of Optimality)Given any DP problem and frrnction Q(x) e A(!:

Q(x): max { u(x,x1)+ FQ(xr): xl ef (x)} V xeX [DP4]

t Q(x) will:max { Hpol (seq (x"), x) : seq(xn) eF1(x)} V xeX

Pf:For [DP4] Pick any arbitrary x € X and for any arbitrary seq(x,) e F p (x), we have

Q(x) >u(x, 11)+ pe(xr) lZ-\

but Q(x 1) = max {u (x1, xu) + B Q(x2): x2 e f (x1 )}so Q(x1) >u(xr,xz)+ lQGz) V-21

) incorporating lz-2)inIo lZ-Il wegel

Q(x) >u(x,x1)+ BQ(x1) -u(x,x1)+ F{ u(x,,x2)+ pe(x2) } tz-l-Al

but Q (x2) : max {u (x2, x3) + 9Q(xs): x3 € f (x2 )}so Q(x z) )u (xz, x:) + F Q (x:) [Z-3]

) incorporatin g lZ-21 into [Z-I-A] we get.... and by recursive iteration:

/-Q(x) Z (u(x,xr)+ xr gj u(*i,x1*r) ) * F'tte(*r*,) I:1,2,....

j=1

and since.Q is bdd ) asasJ) "o real seq (p t Q("t) converges to 0 ('.'p fraction)so lim p 'Q(x j) : 0 l : 1,2, ... .

then asJ ) oo

/Q(x) > (u(x.xr)+ X- pj u(x;,x;*r))

: HEul (seq (xo ) , x) + 0

By [DPa] we can pick a seq (xn* ) e F 1 (x ) with

Q(x):u(x,xr*)+ I Q(x 1*) and Q(xn*) = u (x"*, x,,+r*) + F Q(x"*r*) n:1,2, ...

) foranyJ eN+

Q(x): u(x,x1*)+ p u(xr*, xz*) + ..... + pru(x;*,x:*r*)* 0'*t Q(*r.,*)

: (u (x, x r*) + !r Fr u(x j*, x1*'*) ) + p r+r q1;,-'*;

-lim piQ(xr) J:1,2,....

As J ) oo since Q bdd we get

Q(x): \u (x, x r*) + X- F, u(*;*,xi*r*) )

: max { HEq (seq (x"), x) : seq(x,) eFp(x)} V xeX

[Above thm says if 3 solution Q*(x) for [DP4], then et is the value function for our Dpprobleml

Chapter IV. C. Discrete Time Recursive Optimal Models

i) Deterministic Optimal Gro'*'th Model: Finite horizon case

Indirect dgramic study using recursive methods:

Model: function mapping state variables onto other variables, with intertemporaltransition fimction on states variables to generate time series.

Mathematics for continuous time models are more elegant and better developed.However, economists often times use discrete models for various reasons, one being theempirical nature of economics. Most human (including economic) activities are discrete.For instance markets, factories, offices are open for certain hours everyday; hence areperiodic and discrete. Economic data, mostly times series, are collected at discrete timeintervals, e.g. GDP tabulated monthly, annually (hence growth rate of GDP also discrete).Commodities are discrete (however, we can look at the service attribute of a commodityto change it into a continuous case. For example,2.l566 hours of car service, instead ofone car, two cars etc.).

In 1970s economists like Lucas (Nobe1 prize 1995: rational expectation) think ifgovernment policy changes (announced at discrete time point, applicabie to discreteperiods, e.g. increase in income tax for the fisca.l year 2008-2009, raising of benchmarkinterest rate between policy meetings), then rational economic agents will reactaccordingly by changing their decision rules for each period.I econometricians cannot assume fixed behavioral equations for empirical work. Moreimportantly, policy implication is that people's reaction based on expectation ongovemment policy car render policymakers' effort useless.

Counter argument:1. Such changes in behavioral equation parameters are either too smali or too slow

to seriously affect the usefulness of optimal control calculations.2. Sims [1980]: govemment policy rule can be framed in the form ofa reaction

function (e.g. monetary policy as a function of observed state like differentialbetween TIPS and straight treasurys, inflation rates, credit expansion . .. .).

With the above caveat, we will proceed with discrete growth models and re-visitBellman's dynamic programming for applying to such models.

Let us start with some preliminaries of a deterministic (no risk) optimal growth model:

Given: Time t

ID'

Capital kt J> inputs 1

Labor n, J t linked by production function y t = F (k tI

Product y, J ouput J,t' r F: continuouslY differentiable;

, tt)

h r: uolllllluuusly ult.teterrlraurr !

I strictly f; quasi-concave;

I homogeneous ofdegree I

I F(0,n)=0I nk,.Fn,>0; kt,ntI li. Fr, =o| *' *'I

I lim Ft, =-I k' -'"{/

allocated byconsumption-savings decision

to

F(&,nt)

0<6<1[N-0])

constant depreciation rate

ka

/\,currentconsumption grossinvestments

C1 lg-.--:'--" /

assume s.t. constmints: - cr * irS Yt

and kr.r =kr - -6k, * i,lfl

machy depreciatedmchY new machy

)c1+kt*r -(1 - 6)k, S Y,

Assume labor inelastic (fixed) each period i n , =

Given society's util fcn U(co, cr, c2' ."" ,

and U additively seParable

I (normalized) V t

= tllU(ct)t=0

: [!0 u(co ) + [3'U(c1lL

0<[l<l

)+t!'?u(cz) +P.V. [3'U(ct ) +

u(c.) u(

{"(tl

h",rG) ul&,)

P'u(t')-

U also bbd, cont diffble, strictly t , concave, lim U ' (c r; : 5

c t*- 0

To apply dynamic progamming:

Wenote [IV-0] kt*r:kt - 6kt + it - A recursive equation!!

And society's cunent utility function U(c 1 ) is given to depend only on c 1 , is separable

and is additive over discrete time period. Discount rate Bt for each period t are given so

that we can discount different future period utility to PV

) dynamic programming can be used to fmd optimal consumptive-investment allocativepolicy to Max society's utility over planning time horizon (finite or infinite) as follows:

Social planner wishes: find sequence consumption and capital{c r , k t. n, }oo r to:

@

Max IBtu(ct) 0<f3<1 IIV-Ut=0

s.t. feasibility constraint ct * kt*r -(l- 6)kt < F(k1,n1)

at opt, assume output not wasted ) c1+ kt*r -(1 - 6)k, : F(kt,n,)at opt, assume all labor employed ) nt= I ) F(kt,nt): F(kt'1)

) can defure fcn - f(kt )=F(kt,I )r (l- 6) k,tunction of k, filnrv ."rr"r. k I = beginniDg-of-period capiial

)c,: f(kt) - kt*r

Lemma: flkt) is cont. diffble, strictly 1, concave, (0)=0, f '(k,) >0,

lim f'(kr):1- 6 lim f'(kt): oo

kr-'- k r--' 0

Pf: Since F continuously diffble, strictly 1, concave

(1 - 6)kt cont diffble, stictly 1, linear (i.e. both concave and convex)

) f:F+(l - 6)kt also continuously diffble, strictly 1, concave

f (0) : F(0,1 ) +(1- 6)0 : 0

f'(kt)= Ft,(kt'1)+ (1 -6)>0>0 >0

er,

lim f '(k,)

kt-'co

lim f '(k,)

k, '0

lim Frt + lim (1 - 6) = (1 - 6 )k,-- - kt-.

"o

: lim F rt + lim (1 - 6):o+ (1 - 6 )=-k,.-'0 k,...'0

Note: similarity between above F & f and Solow Growth Model F & f.

) can rewrite [IV-l] as

c, = f(k1) - k,*1 fiom feasibility equality constaint

@lltIV-21 =MaxI[l'

t=0

s.t. feasibility constraint 0 < k,*r

ulf(kt)-kr*rl o<[!<1

< f(k,) -{'.' 0 < ct :f(kt)-kt*rI tk.r S (k,)and

L o< k.''

glven ko and ko > 0.

&")=o*.T

In particular, if [IV-2] frnite time horizon, stops at time T, we get

T

tIV-31 = Max t [3'ulf(k,)-k,*r] 0<B<1r=0

s.t. feasibility constraint 0 < kt*r S f(kt) given ko and ko > 0.

Note: at T, k r+t : 0 (no more capital after T, all consumed)

&-t

J(et)

1*rr'#ft

' "11\\-"

Since cont. diflble ) l"t order condition:

I

--{tulf(ko) - k rFB' ulf(kr) - kzl + ... + BL'u[(k t-r) -kj+ Btut(k,)- k*1]+ ......1:6ak

only these 2 terms have k1

t [3F'u' [(k ur) -k] Gt) + B' u'[f(k,)- k,*1] f '(k t) = 0

.) [rv-4] f3u'[f(kt)-kr*r]f'(k)= u' [(kt-r)-kd t:1,2,.....,7

) from bdary condition [IV-5] k t*t:6 given k 6 > 0

to determine specific solution

TD

t) : -14, -{t-

tI-'

-') o( & t

o( -t ) = l-

6l3ou'eult* -6.kr*ttt= *-l-{yr 9+- t* .

/ft-t

a_ttca4 & ltt =a,t=&

:- l- 4- +<+ 7= t -ot$ * s61 =(-.tF) c -v+ t- fi-*A)rr*6:o

-)To( p ( /t-.{, -r- o<.lt +

,- . +(F/+ <A+

/+a(p +gpf- vg(r+ YP"+G

- o<6(,+* r)rt <6 +("a6)'+for

t +<p,t$cf)

Ig-c-r8-,rJ

+ - --- +(*3+ - - -- t("crs)Ft +U

>A - "<s( IT-t+l

- > (r-<F) ar t =l4C -:l

e9

<(\ //r-

,-"(F

=l 2 - -- - Lg-t'

ff=1) V e.'+r"t* q,*7.-*

fu {r"-c-}"EL

[o -t']f r {"zF)r*t+rl

, :k Fr'- t :' -."F * Grp)'-'*"

u'( ft&tt - L*", ) *

oa a4-K. Cr-LqFf-LJ

r-QrPlT*trl

t-{Epf't*t -o{A +t<

'-Qcl3i*:*'

t-

#sF)(t- *

: 421 T+A p,,- 4&a:.<. U.arL& SO

T+o ,-EF

=4.*7 L; lw-1-t a*Lt- L /*r/t*.r

Dc A,*, (^,o.q, uq- L AM.e a***dat

Chapter [V C. ii) infinite time horizon case usins

d Lagrangian method

lReminder note: constrained optirnization is called mathematica] programming (lco,nputerprogr:amming which is a branch of numericar anarysis). Lagrangiau is used tJhansform aconstJained optimization problem into an unconstrained optimization one.Interpretation ofthe La,grangian multiFlier l, : at the optimal level, 2"* shows the approximatechaage in the objective function when there is marginal change in the constraint.l

Given cobb'Douglas production function f with unit labo! hclurring capital good depreciation:

We want to choose consumption c t (control variable) to:T

k t = capital good quantitry at beginning ofperiod tko> 0 andis given

0<B<l

k t state variable (','controlled by c Jct= 0

cJ

f(kr) = kf

aldkr*r =kf - ct

Max I B, Int=0

s.t. kt+r =kto

Lagralgian:T

O= : Bt ln cr - Bi+l

0< q < 1

Ct

- Cr

\ kt*r - kto +

ll

L t*r (kt*r' kto +

l"t order condition

lrv-91 ao Bt

Act

where .L t*r is the Lagraagian multiplierSince I +r is margina-l contribution of A k t*r to multi-period objectivefunction at period t +1

) I t+t must be discounted back to period t I B t*rtrt*r

I[lt*r).tnr =0 ) .-"- Blt*r=0

t=0, 1, 2, ..., T

llv-lol a@

= - [3t].t - Bt+l lt+1 ( ' o k to-1; = B t ( - tr t + [3 ] t*r o k 1o-1; = edkt

I can solve for c t * , .tr t*

We solve recursively backwards, starting from time T.

Wenotekr"r is assumed to be not providing utiJity ) kr+r = 0

Forperiod I cr* =kr " tomaxutility(U' > 0) (&om tie constraint equation when t = T)

and [rV-g] 1

= B.tr r*r

Irv,10]t- ).r +(1 / cr)( o kr'-')= -trr+(1 /kr") ( a kr'1)=-Ir + (o/kr) =0

t Ir*= (a/kr)

And for period T - 1

trv-91 I= f3Lr = f3(o /kr)= aB[1/(kr,," -cr.rrl

c T.1

) (kr-r' / cr.t) ' I = aB

t cr-r * = kr,t" / ( 1+ qB)

from IIV.1ol

-|1.1 +B),r.r*ro kr'ro-1 =0I hr = f3 ).ro k r.ro-1

=(tlcrJ cr kr-ro-1

= (r+crB)/ k r-1oo k r-r"-1 J

) )rr.r* = q( 1 + cx[!) /kr.r

And for period T - 2

tw-el 1

= fJ.Lr.r = f3(q(1+a13 ) /kr.r)c T.2

= qf3(1+qB) /[ kr,r" - crz]) cr.z * = kr_2" /[1 + dB(1+qfl)]= kr-r" / lr + ct B+ tcrB)2 1

[v.1ol

) l\ r.z * = [11+qf!1r+atl )) q] / krz = [cr (1+crB + 1aB)z) J / kr2

::

;;

And for period T - t

cTr* = kr,.' /tt + cxB+ 1oB;z + .... + (a&)t I try-ll]Ir.'*= [a ( l+q B + (a [!)z a . . . + {qt!)r] / kr.t tiv-i2l

ForT)co 1+o B + 1c t3)z + . . . + (cx B,)r + . . . . . = f /(1, crt3)lrv- il1 t crt* = (t- ot!) kr_t" tIV-lBl[IV-12] t lr,t* = o / t (1+crB) kr.tl tfv.l4l

) optirnal control consumption IIV.lgl = &action (1- oB) ofproduction k r,1"

And optimal kt*r * = kt" - ct= k,o - t(t-dt3)kt. l

= kf (1 _r(1-cB)l

= crBkf * whichis [V-g.r]

[Note in the^above approach, we erc letting T) o during the derivation of the solution. Thisis different from taking the l_imit of solutio"n as T) oI

This approach is also convenient for us to intooduce technologicar level into the model byadding a technology factor At in period t, assuming A, to be a glven parameter.

) production function f (k t )= At kf) above [fV-13] becomes c r., * = (1.q8)Ar kr_to [V.lA-A]

above [V-14j becomes ]. r.t * = a /[11+at3) krt] IfV.t4-Al

tech A ) to allow technological progress (A or shock) we change At parameter into avariable.

stochastic model ) to introduce ri,"ks, we change At into a random variabre (with aprobability distribution/density function attached) ind we will be -r*i_d;;l;;";;;;;;value of the above objective function

(e

Cfrot/J ousi t,ts Lyctas 6JTo begin to answer this question, examine how the decision or control

variables including consumption c,,, leisure W, demand jrrl for input 7 in theproduction of good I, and labor input L, in producing good r' behave, given theparameters 0, of the utility function, a, in the production functions, andthe derived paramelers y, from the former parameters through equation (3.1g).Equation (3.22) srates thar c, equals (0,,{,)y,,, and equation (3.24) srates thatx,,, = (py,a/'l)Y

1,, p being the discount factor. Thus, consumption o[ sood i is

proportional ro lhe out.put Y,r of good i, and so is the demand lor gooJ i in theproduction of good k. Equations (3.23) and (3.25) state that leiiute W, andlabor input l, used in producing good f are both given fractions of total hoursH. If random shock 2,, is introduced in the production of Y,, output of eachgood i is subject to this shock. Consumption cu of each good i and the use ofgood i as input r^ir in rhe production of good /< will be affected proportionally,because both are given fractions of Y/r. Because H is fixed, and leiiure W. andlabor input L,, are both given fractions of H, they will not be affected by therandom shock z,i. Thus, the Cobb-Douglas utility and production functionsimpose very restrictive behavior on cn, xr*Wu and Lu if the only shock is amultiplicative shock 2,, in the production of output y,r.

The above dynamic characteristics of the economic variables generated bythis multisector model are examined by Long and plosser (19g3) who empha-size the ;mplied comovements of economic variables:

If the output of commodity i is unexpectedly high at time, ldue to a stochasticshock on the production functionl, then inputs of commodity i in all of itsproductive employments will also be unexpectedly high at time ,, Assuming thatthe commodity has at least several alternative employments, this not only propa-gates the output shock forward in time, it also spread the future effects of theshock across sectors of the economy. At the most simplistic level of analysis, thisis the primary explanation of persistence and comovement in the consumDtion.input, and output time series in our example. (p. 49)

Long and Plosser (1983) also examine the dynamic properties of pnce, wagerates, and interest rates. Prices and wages are studied by using ihe partLlderivatives of the value function with respect to the state variables output y!and leisure W,. Using the Lagrange method to solve the dynamic optimizationproblem for the centralplanner (or equivalently for the market economy), onecan obtain prices and wages by the Lagrange multipliers themselves. The rateof interest is found as the ratio of prices of goods in I and , + 1, as discussed in

Keynesians question the ability of these two basic mechanisms to do so satis_factorily (see the discussions oi prose..lresljanJ r'r."ti* iis"8;), ""

,".,n",iepresenting an RBC viewpoint, and the f.ti", u ""r" i"yriisl"an vrewpoint;.

As pointed out in section 5.1, Keynes did not, und rh" _;l;;;'ieynesians donot, believe that an equilibrium theory wittr full empklyment-of ,e.ou."es3l.ways

prevailing can explain observed economic fluciuaiions. l"-or example,Markiw (1989, p. 85) questions whether mosr ,"";;;;-;;;';, associatedwith some exogenous deteriorarion in the economy,s f;;;";;; capabilities.r, ltll:uch the model of section 3.2 may be

"bt" r.; *,;i;i. ;h;

-simuttaneous

)menhof macroeconomic variables ro some exlent, it might be inad_in explaining the leadJag relationships umong e.onomi"

"iii"U[r. O,other hand, such leadJag rllations.appear to be well captured, throughtral analysis, by a simple linear multiplier_accelerato. fiquiJity_pr"f"r"n".consisting of .an aggregate consumption r"*ti.", i*"'r'gir"gri" i._nt functions based on the ilves

. tment,accelerator, and a liquidity-

'erence relarion. See Chow and Leviran (19691, Cf,o* ifS;,lh;t", sl,chow (1993b). The key question is whether the dynamiJ"ir"r""t'-.lti* "rT:"::::il"T"_11iables through business cyctes can be adequatety

.Dy mooers ol optlmizing agen6 in equitibrium. Research is still:jl,,"g:il:: "":wer

this questLn. No .onr"r,ru, or opi"ion n".i""nred by the economics profession. The remainde,

"f thi, ;h;;;;*_".

the resulrs of some recent studies, \4,ith th" p".p;;;iili;ni so_eon this question.

section 4.4, 4.5, and 4.6. As the optimal decision functions for the control::

Esllmotlng Economlc Etfecls ot polllicot Evenls In Chlno

'and Chow (1996).Chinese economy is modeled as if there were central planners whoize, at any initial year zero,

!E,p,rnc,r-0

optimization models are useful not only for studyinq .economic

:'-,:*:.!fi"-._1T:T"l b:t arso fo' studyinsec.l,.fir" .rrriei,aagveloniye

counrries, To illustrate, I provide " .;;;il;;;ii,o .,rroy

:1,":j:.:nTJ:',l 9lT:*l to measure the economic-effeciJii poriticarinctuding the.creat Leap Forward *d rh"a;ft,";i ilf"i;r#:serves as an introduction to sections 5.4 and 5.5. It araws,fJm

ln z,*, = ln z, + p+ e,., (s.4)

Per capita consumption or investment is determined by the social plannerssolving the dynamic optimization problem. One might object that this model istoo simple, but it is useful in providing a crude answer to the question: If theGreat Leap had not occurred, what would per capita output in 1992 be as

compared with actual output? (By the way, the answer is, Twice as large.)To arrive at this answer, estimate the optimization model by using annual

Chinese data from 1952 to 1993. The model consists of the random walkequation for ln z and an equation for lnk,,, derived from solving the dynamicoptimization problem. Because the variables are nonstationary, first detrendby dividing by 2,, yielding k,.1= k,r1|2,,7,= z/zt and d,= c12,. The model interms of the detrended variables consists of

rsso re55 1950 re65 1970

with lnk,*, as the control variable, and lnZr and ln&, as two state variables. Thecoefficients (g, G,, Gr) are derived from optimization, given the structuralparameters o, 0, and T = sp. The residual e, is added to the optimal controlequation for ln k,.r to account for the fact that the simple model cannot explainthe actual data on lnk4r cornpletely. Given the structural parameters, andtime-series data on per capita output q, and per capita capital stock k, we canconstruct z, by solving q,= zf kl-) and derive the coefficients (9, G,, G) andhence the residuals q and e, in the above statistical model. Assuming (g, e,) tobe jointly normal, we can evaluate the likelihood function and maximize itwith respect to the parameters o, p, T and p.

The results are, with standard errors in parenlheses,

(a, 0, l)=[o.zrs (o.oroa), o.wr (o.ooor), o.ozrs (o.oozs)]

The labor exponent 0.7495 in the Cobb-Douglas production function isreasonable. The large p = 0.9999 shows that the Chinese planners place greatweight on future consumption as compared with current consumptiqn, thusdevoting a large fraction of output to investment, which is in the neighborhoodof 0.35 for most years from 1952 to 1993. The annual labor augmented techno-logical progress of2 percent is reasonable if I include the years 1979-1993 afterthe economic reform. In Chow (1993c), a Cobb-Douglas production functionwith constant total factor productivity was shown to fit China data from 1952to 1980, excluding certain abnormal yeari, but total factor productivity in-creased substantially after 1980. The current production function incorporatesa random total factor productivity and fits ihe data well enough for the entiresample from 1952 to 1993 to be included in estimating the parameters.

Figure 5.1 shows the estimated residual0, in the log productivity equation.The impacts of the Great Leap and the Cultural Revolution on this residual isobvious. To estimate the economic effects of the Great Leap, smooth out theresiduals €, and e, in the Great [-eap period and simulate the model using the

' smoothed residuals. The result is a tactor of 2.0 in per capita ootput in L992I ,using the residuals free of the effects of the Great I-eap, and a factor of 1.2:,using the residuals free of the effects of the Cultural Revolution. The latter

estimate appears reasonable, because this estimate is concerned only withmeasured physical output and not with emotional sufferings.

E,| Esllmollng ond lestlng o Bose-llno Reol Buslness Cvcle Model

Returning to the study of economic fluctuations in the United States, I presentin this section a base-line real business cycle model discussed in King, ilosser,and Rebelo (1988a,b). The model is slightly more complicated than the onepresented in section 5.3, having another control variable, the number of work-ing hours per capita ur,, in addition to investment t,r,,. The representativeeconomic agent is assumed to maximize

FIcuRE,5.1 Observed and simulated residual e

to the constraint on the evolution of capital stock Jr,,

",.,_, = (t -6)r,, +r,,

lnz, =p+e,

ln i<,., =g+G,ln! + G"lnE, + e,

(s.5)

(5.6)

eie,p,[tnr, + eln(l - a..)ldt _.

(s.7)

(s.8)

a Cobb-Douglas production function for per capita output, per capitarDtion is

c, = sr"(z,ur,)" -uu (s.e)

in.which the first state variable s,, = 117, f6llows a random walk with drift y.

the,conslrainl on

cstcd in multisagc dccisionB.lltnrn died on Mcch 19, l!in his own wods sinc. be l€ftinforiouivc autobiograpbt

in 19,,16 at the agc of 5, despite vaiqr! war-relatedactiyiti€s during World War Il-inchrding being assigaedby tbe Arny to the Manhattan project in Irs Alamos.He had already €xhibiEd ouBtanding ability both in pul€matbematics and in solving applied problems arising ftomthe physical world" Assured of a successfirl convcntionalacademic carecr, Beulca& dudng the period under con-sideratiorl cast his lot instead 'r'ith tbc kind of appliedmathematics later to h€ hown as opcf,atioDs rar."t"i. tothose days applied practitiooers w€re regarded as distinctlysecond-class citizcns of the matbematical Aatcmity. Alwaysone to enJoy conEoverst when invited to spea& at vari-ous univenity mat[ematics depdtmeft seminars, Bellmandelighted in jusdfying his choice of applied over purc math-eoatics as being motivaEd by the real world's greater chal-lenges aDd mrtbematical dcmands.

Following ar€ excerp6, tak€n cbjonologicaly ftomRichard Bcllman's autobiogaphy. The page numbers arcgiveu after cach. Th€ excerpt s€ction titles arc mine. Theseexc€rpts are far morc serious tban most of the book whichis full of entcrtaining anccdotes and outra€eous bchsviorsby 80 exceptionally hunot bebg.Stuafi Dr€tlrls

BELLI'AN'S INTRODUCTION TO MULTISTAGEOECISION PROCESS PROBLEMS

"I was vay eag€f, to go to RAND in tbe sumrer of1949 . . .I b€csse fricndly wir! Ed Paxson and asked him

Sbj.c, ctarri'rotb{: Dr!.ric For'tEEils: bindrr. prf..sbn t: coocrn o[.Afra d! tukw: A,vxrvlr!^jry ItsJ! (SrEr^!I

Op.rdd n....rt O 2qn INFORMS\,b1. 50. No. l. JllorFF€huq ?,rt\fr. a8-5t 48

(trd tDe nane,l95os wqc not

gmd ycars for maitematical rcsegrcb- We had a very inEr-csting gentlemal i! Washitrgton named wilson. He wasSecretary of Defens€, and he actually bad a pathologicalfea and hatrEd of thc wm4 rcsefich. I'm not using theterm Iightly; I'm u$ng it precisely. His facc would sufttse,he would tum rcd, ard be would get violent if pecple used

tbc t€rm, rcsearcb. in his prcsencc. You can imagine how hefe& tbea about the term" matlenatical Tbe RAND Cor-pcration was cnployed by the Air Force, and the Air Forcehad Mlson as ia boss, essentirlly. Henc€, I felt I hsd to dosomething to shield Tilson and tbe Air Force from the factthat I was rcally doing mathematics inside the RAND Ccrr-pcration Wlat title, wbat name, could I choose? In the fustplace I was intercsted in plaaning, in decision making, inthinking. But planning, is Dot a good word for vrious rea-sons. I decided therEfqe to use the wo4 programmiag.'I wanted to get adoss th€ idea that this was dynamic, thiswas multistage, tbis was time-varying-I tbought, let's kiltwo birds witb one storc. IJt's take a word that has an

absolutely precise neaaing, namely dynamic, in tbe c.las-

sical physical sense. It also has a very interesting propertyas an adjcctive, and that is it's impossible to use the word,dynaEic, in a pejorative sense. Try tbinking of some com-bination that will possibly give it a pcjorative me3dn8.It's impossible. Thus, I thought dynadic pmgrarudng wasa gd nam€. It was soEethirg not svcn a Congressmancould objcct to. So I used it as al umbrella for my activities" (p. 159).

EARLY AI{ALYNCAL RESULTS-Ihe sumner of l95l was old-hooc-week- Sam Kadin andHaI Shqiro wer€ at IiAND.

m3{!364xi12l5m104 t&t.@152&J{d} clcdto.ic [;sN

..ot ::"::

f>,rs)

publisher bas generously approvcd cxtjnsive excerpting.During tbc summer of 1949 Bellmen a Eour€d asso-

ciate profcssor of mathcmatics at Staniod Univcrii; wi6a derclopilg intcrcst in analytic numbcr thcory, was con-sulting for the second suD-m€f, at the RAND Ccoorationin Santa Monica- Hc had rcceived his ph"D. from *nccton

i.:,fl-',*nd to'solvc

mber tbcory,

whicb

firDc-doing tlis.aay oth€rs

whcn I found

se9n Prwas clcar

Ivinq Clictrb€rCllngeD'ous mattrceaticirns-THE MOOERN MAT}IEMAIICAL IMTELLECTUAL

'I !4d !o mrLe 1 6alor decision. Should I rctum to StaDfordor. stay at RAND? I bad thought about this question inttrDc€ton, but it was Dot an easy decisiotr !o makc siocetierc were sbong arguments on each side.

, "At Stanfond, I bad I tenured positioq good for anotberuury-€lgbt yeas. Tbc retircment age at StaDftrd was sev-cnty. I also had I good teaching position, with not too mucbteachtng, aDd a fine bouse, which I have described abovc.

:",. T*_ *T nor te importa consid€r.tiotrs. Ar Stan-

IoT , * I chanc. to do analytic trumber thetry, which I

nad wa.nted b do since I was sixteen.

_ "Howev€r, I had to face be fact that I could not do wbatI-laDted b do. Possibly the state of matheoratics did aotallow this. C€rtaiDly, ny state of howledge was not upto iL

"I. $ _sfTt cnough timc in Los Angeles to know that Iwould_enjoy living there. I also knew th; Los Aagetes hadmany fiae bouses. although it was not undl 196g that I hadone that was b€t€r than lhc one up in Stanford.

^ "I was htrigued by dynanic progasming. It was clear

:o T" * th:rc *.r a good deal of good analysis tbere.F.u1tb€rTorc, I could see many applicatioDs. It was a clcarcDolce. I could either be a tradidoual iDte[ectual, or a mod_cm. inElle€tua.I using 6e rcsults of .y ,"r"*l foiO"problems of

-conemporary society. Thii **

" a"rrgo*iparn. Elther I could do too much rcs€arch and roo litrlcappligtion, or too littlc rcsealch and too

^u"t "ppU"u-tion. I had conEdencc that I could do &is aalcate aiivity,pic a la modc" (p. 173).

THE PRINCIPLE OF OPTIIIAUTY AND rrsASSOCIATED FUNCTIONAL EOUAIOIS'I, dcqided to itrvestigate tbrec areas: dyoamic program_mrng, control tbcory, aad time-lag processes.

':me toot we uscd ias tbe calculus of yadations. Wlatwe foud was that very simple problems rcquired grearingeDuity. A sEall change in 6e problem caused a greatchange in thc solution.

'Clearly, something w8s wrong. Tber€ was an obviouslack of balancc. Reluctrntly, I was forced to the conclusiontbat the calculus of variations Eas not an efectivc tool forobtaining a solution" (e,p. 174-175).

FORMULANON OF TTIE MARKOV DECTSIONPROCESS PROBLEM

"I spent a geat deal of tine and effort on the firnctionalequations of dynanic Fogramning. I was able to solvesomc equations and to dctermitre tbe propenies of theftnction and &e policy for others. I dev€Ioped sone newtbeories, Markovian dccision processcs, and was able torcint€rpr€f an old thecry like the calculus of variations, ofwhich I will speat morc about below,' (p. 17g).

OYNAMIC PROGRAMIIING AilD OPTIMALCONTROL THEORY

"A number of mathematical models of dynamic program-ning typ€ were analyzed using the calculus of variations.The treatmeDt was not rortiae since we suffered cithcr Aomtbc prcsenc€ of constraiDts or from an excess of lincrity.Ar itrterBsting fact that cmerged ftom this detailed scrutinywas that tie way on€ utilized rcsourccs deprnded criticallyupon the levcl of these rcsources, and 6e tinc remainingin tbc process. Nanrrally this was surlnising only to soE€-one u €rscd in ecoaooics such as mvself. But this wasmy condition wi6 rbe rcsult that the observation of rbisphenomenon came as quite a sbock" Again the indglingthoughe A solution is not mercly a sct of functions of time,

as a p obdm closcty rctari'As I result of a &tailcd i

over-involvd sincc all along I had Do desir€ to wod scri_ously io the calqllus of variations. A corsc ir tbe sub.ject in college had grven me simultan€ously a rather lowopinion of its intrinsic int€rest and a healthyrcspect fd itsinticacic.s. It ryFared to be [Ued with comDlicated €xis-teDce and uniqueness tbeorems with self-imposcd rcstric-tions, none pointbg ia any particular dirertion. This isP€rtnent to a commeat made by Felir Klein, the greatt:erman mathcmaticia4 concerning a c€rtain rypc of math_ematician Wlen this iadividual discorms that he cal jumpaqoss a streaE, he rcnuls to the oth€t sidc, ties a chair

lo hit l"S, and sees if he can st'rll jump across the stream.

lory T"y enjoy this spcq others, like myself, may feellnat rt is &ce firn to see if you can jump across biggersE€ams, or at l€ast diftrcnt ones.

-, "Despite Ey personal ;selings, tbe cballenge remaiaed-

trow drd one obtain the nurcrical solutions of ootimiza-tion problems? Were there reliable methods? As oointcdgut abov.e, I_ did not wish to gapple witb this thorny ques_o:1 -d J had ccrtainly rot contenplarcd tle appicationor.dynaEic progratnming to contsol processcs of determin-tsfic typcs. OriginaUy, I bad developed tbe thcory 8s a roolfor stochatic_ decision processes. However, the thought was_llruy torccd upon me that the &sired solution in a con-Eol Foccss was a. po.licy: .Do thus-and-thus if you 6nd'yourself in this portion of stare space wirh this s-mount of!m-_cJcft.''Cowenely, once it was realizcd tbat th€ cooc€prof potcy was fi.6rrqmen6t in conrol tbeary, the natbi-paticization of thc basic engh€ering concept of fecdbackconBol,' tben the empbasis upon a state variable forDrula-

e": ry"." natural. We s€€-iEE-ErEr,-irErBritrgEer-acuotr bctweetr dyna.mic goga'nming and contol theorylDls gntorces tbc point tbar whcn working in tbc field of

A SYST$'ANC METHODOLOGICALAPPROACH TO MAIHEMAfICS"As pointed out abovc, as of 1954 or so I had stumblcd intosome importaat types of pmblems and had b€en pushe4willy-nilly, into answcring somc significant kinds of gues-tions. I could handle deterministic conaol ptocesses tosome extent and stochastic decision orocess in ecolomicsand operations rcscarch as well Where next? At 6is point,

.I began !o thiD& in a logical fashio4 using a systeEaticmethodological approach. The point about th€ suitable ph.i-losophy Plcpaing ouc for the fortunate accident sbould bck€pt in nind.

'Th€f,e are sevenl ways in which a math€matician can

Pmce€d to extend his rcsearch eforts, particularly one whois deeply intercsted in problems arising from the physicslworld He can, on onc hand, examine thc equations he hasbeen vorking witb and modify thea in a vriety of ways.Or hc can ask questions tbat have not been asked beforcconceming the Datul! of the solution of the original cqua-tions. This is basically a very dificult way to carry outrcscarch- It is very casy to cbaagc tbe form of at cqua-tion in a largc number of ways. Th€ glEar orajorify of then€s, equatioDs ee not Eesningful, and, i-n conscgueoce,lcad to both difficult and unimporant problems. Similarly,&cre are many questions that are diffcult to answcr, buthardly worth asking. The weU-taincd mathematician docsnot Ecasule tbe valuc of a problem solely by ia int-actabil-ity- The challenge is thec, but cnen very small boys do notacc?t all dares.

uisc df!gF,'altqongo! q

' of tbc cols tle assunptlon Uat rilthc staE v{iableg caD bc

'"In thc ibsl wcl4 n;fomly r4id Oftcn paand comlltltss cannotproblcns of society, as I'

in temptation to r€turn to nu-ob€f, the-howwer, does Dot seem to be rcwad-

dontinual e.ffon. The prcbleos are too

of tbe

it was

ical algorith'n".this Sorccreds

sid€rd thc last r€sort of an incomp€tent mathcmaticia!.The opposirc, of course, is tue. Once working in the area,it is vqy quickly rcalized that far Eore ability and sophis-tication is rcqut€d to obtain a numffical solution than bestablish the usual *istcace and uniquaess theorems. Itis far more difficult to obtaitr an efective algorirhm thanone that stops with a denonstation of validity- A final goalof any scient'ric theory must be the derivation of nunbers.Theories stand or fall, ultimately, upon numbers. Thus Ibecame int€restcd in computers, not as electronic toys butrath€r because of whar tbey signified mathcmatica.lly aldscientifically. This interest led in oany uDexp€cted direc-tions, as I will indicate subsequendy. This is a significantpafi of the story of scietrtific methodology. It is usually, ifnot always, impossible to predict where a theoretical iaves-tigation will end onc€ started But wbat one can be certainof is that the investigation of a meaningfirl scientific areawill lead to meaningfrl mathematics. Inevitably, as soon as

one pursues the basic thcm ofobtaining num€rical arswCsto numerical questions, one will be led to all kinds ofintercstiag and signifcant problems in purc mathematics"

Crp. 182-185).

boondoggte ald high encrgr-ligl cost ptues rn tE tict thal wc dont Lnow hdx, topl€x systems of soci*y involving people, wc don't under-sta.nd cause and ctrcct, which is to say the cons€qucnccsof decisions, and we don't erren klow how to make ourobjcctives reasonably p{€cise. Nooc of the rcquirements ofclassical scienc€ are met. Gradlally, a ncw -"thodologyfor dealing with these 'fuzzy' problems is being developci,but tbe path is not e.sy.

'Upon 6rst gaeing upoa thc complexities of tbe real

difficult and &e victori€s too few. Taking up the challengeof complexity, J felt ih'r the apprcpriate +t"g to do wasto start with deterministic control prccesses and to mod-iS them stage by stage to obtain tbeories which could beus€d to deal with basic uncrrtainties in a morc soohisti_catcd fashion.

-*fo this end, we can begin by inroducing well-beiavedwrc€rtainty of the type exteasively treated by the classi_caLllEo+f@iilitJFTbi{ leads to thc modern theoryof stochastic contol processpwhere uncefiainry is rcpre_sedltdbv-randnln v+eblefwith known probabitiry disri-butions, and where the objective is to maximize €xDectedvalues. This gives rise to an clegant 6eory with a good dealof alts'active analysis. It is a new part of pure matbematics-

wortr

Chapter IV C. ii)

b) infinite time horizon case using Bellman Functional Equation

In previous sections, we covered general principle of Bellman's Dp. we illustrated itwith simple examples. ln discrete period growh analysis, we oan again use Dp. Nowotll coverage is on more advanced level since we already have concept maturity in Dp.

For above optimal savings (= optimal consumption q) model, social planner is looking atthe following problem: (c s control variable, k t state variable ...

controlled by c )We want to choose infinite sequence {c, , k,*r },=o- to:

I BtU(ct){ ct, kt*r},-6' ,:o

s.t. kt+r =g(ct,kt)=(k) -ct ) r'u-trr

t

Max 0<B<1

1i'.'-'- {o76 *u(co)J

glven ko>0 k1e K1c R*t2

e"

I

t3 $-y &s

pu cct)

Q u (c")

f .r(c6;1

Max Bb(c 6) + BlU1c,; + B2U(c2) + ... + ntU1c,; + .. ..

s.t.kt*r=(k)-ct

If can find value fcn V(k J = max (all future utility) fmm time t = I onwardss.t. lerr=f(kJ-ctthen [IV-15] =

Itv-f6l ( uo u(co)- BV(kr) o"?"oov(t

o)T "'\crI s.t. co+ kr :f(ko)

I stven ko > o\

That is, we convert [IV-15] Max t Bt U( c t )to [IV-16] MaxU + BV without any I

:(44,

But how do we solve for value function V:

First we notice above [v-16] reasoning applies to all periods (not just initial period 0).i.e. social planner actually max the following for EVERy period t (recursive):

Max U(c)+ BV(k*r)s.t. kt+r =f(kt) -ct

[v-17]ko> 0 [ry-18]

o t 23

'-1.., j

t t+t t*z

&( tt*t

ij-l,rax alL fu- pz'"*aL+ uz-I^-z-

We have two equations [IV-17] and [V-18] but 3 variables c1,k1 , and k1a1

) need one more equation.

Social planner needs some policy function h(k ) = c 1 [V- I 9]to max in each period t by controlling the consumption c 1 , depending on k 1. We noteagain the recursive nature of the policy function which is same for EACH period t (calledtime-invariant policy).

[IV-I7-18-19] means

Max U(h(k) )+ BV(f(k) -h(kt))given ka > 0

this functional equation all in terms of k) can solve for k1 and then get kpq and q from [V-18-19]

) Dynamic Programming DP: frnd policy function h (state k1) : control qthen iterate following 2 equations from initial k0 ('.'recursive) :

h(k) =c1 [V-19]

kt*r: g(kt, c1) (which = f ft) - c l in our example) [IV-18]

the iteration will generate oo sequence { " " }=o

- which solves original problem [IV-l5].

The value functional equation linking value and policy functions:

Vft) :maxU(c) + B V(f (k) -c) [V-20] is called Bellman Equation

The fi:nction tl'at solves Max on RHS of [IV-20] is the policy function h(k) whichsatisfi es functional equation:

u(h(k)) + B v(f (k) - h(k) [rv-21]

There are 3 methods to solve the Bellman Equation, depending on U and g, namely:

1) value function iteration method (since value function V is recursive, we can useiteration as we did in previous - the recursive compound interest equation).

2) Guess-and-veriff (Guess-and-verify a solution V for the Bellmaa Equation) and3) Howard's improvement algorithm (or policy improvement algorithm). We iterate

the policy function \ for each successive period until h1 converges to the optimalpolicy function.

Here we will just cover method 1) value function iteration. (We have seen a crudeversion ofthis method in our min effort example in previous chapter):

From Vo , construct

V;*r (k) : max {U (c) + B q G)} for j: 0, r,2, . .

s.t. k-: g (c, k), given k

Iterate V.;tr until sequence of value functions converges and we solved the Dp problem.Retuming to our example (2 alternative approaches), we know by the Lagrangianapproach, the optimal path for capital in the oo horizon planning case is

k+r: o0kt" [IV-09-1]

We want to check if this is also correct by Bellmaa functional equation approach.

rr\

| ^ 04t-+oz at-, = ol Bk, u *]2.fr-eJ,

st. Ca+ k-,. = s!,0!4r,t4@-

--) v&) - nax L t (--(+@D-

^ L -

t 'alrfl K+.,; <A {ii

r/-,-(kt.(,-<

,z Ltt+94t +1,

(t t-*(,-<

" /.- (L"

t-("p + <"/^.(Lo)

2 *il A(+) r (

+[E:-*] ,= [g:"-a] >

lsf P2AJ.v z-aa*<'; h+,1u+ pV

t. .4- t -"a,

V'= k., i;:

r' ((ct l -

(/-) = ts' tL&')

L, t"It sK>

€k'tt pKr

Kr

-- ot t',"L' l*(t+$p>: + Fk,+ fr. 4t)+ frr</-k - fr,t^{+*")

= (rrgr,)v t^-k, -(elr)L,Q+Akz) + gk,+ (r.

rv:-r:- A] v-Lt)u-' tn*-= l(>, l-r-lc-, t ?)K+ a'.- u:a-f € el*&-k, rc-,-

+ tlpEt-)4 =

k, = - [r+Ak,-) IAJ(l+pt!)tAt, +

tr{o 7-= \;--------s

+ F k, +Fkz t1r-<F

? tt6k'= w *-J o

-r-*F /= :- ./ = h!-!D +pr,+l*-[u*l - r^(,-<tt-

+ or, t<l?kf= /-(r'<p) +

\7{...,.........-:-- u.

i - : ; - -F =*P-)2 s, = -+-l l^, Q'<F) + frl

From above, I optimal savings policy function g(k) (: f(k)maximizes value function

-q=k1*1)which

v (k) : Max U( (k) - g(k) ) + 0 V (g(k))

and we solved it and got optimal policy function:

FKz qg(k) = ------- h" where Kz

1+ BK2 l-crF

In fact when we substitute value of Kz into g function we get

0 [ o/ (l - cxp)]

cG,) =1+ BIo/(1-qF)]

F I cxl (1 - cxp)]

(1-oF + Fcx) /(l-oF)

0B k" which : ktnr

In particular, if we assume g(k):s f 0 < s< I (analogous to Solow Growth Model,constant c€ s of output is saved

and invested)

Since fis continuously differentiable and strictly concave) s f= gft) also cont. differentiable & strictly concave, get following phase diagram:(difference equation phase diagram)

Aj&t)= 1Gt)

Kt-

{&tr , !(111

?6'

, : .gi:'l&t' tx t ft.r,'

) stationary solutions (= steady states, rest points, fixed point, equilibdum): where gG,) : k (wffie g crosses the 45o line)

) steady states at kt = 0 and at k * but k1 > 0 by model assumption

I only k * steady state

Stability: pick arbitrary kt point (e.g. kr , ka ) and trace out lines vertically to g, thehorizontally to 45o line, vertically to g ..... to see if convergent or divergent.

From above phase diagram, ifka > or < k *, all k convergent to k * as t ) o(note: divergent away from k1 :0)

k,*r fraction fraction.\ {{

In particular g (k) : cx B f (lq) 0<o B<1

) optimal savings policy function g ft) has steady state and infinite sequence

{ k r}'=o - converges to that steady state .

Chapter IV. D. Stochastic Optimal Growth Models

We extend above deterministic (=certainty) model to include risks) stochastic optimal growth models.

Brief review ofProbability Theory, Theory of statistics and dynamics of distributions.

Ref Wu, L, Economics of Finance Academy Press 1990

Terminology:

Random experiment / transaction: Given experimenVtransaction under identicalconditions, if outcomes will not be the snme every time (: no certainty outcome), thenexperimenVtransaction is called random.

(= experimenVtransaction whose outcomes cannot be predicted with certainty before theexperiment/transaction. )

Examples of random experiments

COINTOSSING(by same person, same coin)

H, T, T, H, T, H, T, 8......

ABC company STOCK PRICE RETURNS(same management)

z00r U 03 04 0s 06 07

3.6% 4% -5o/o 6%o -7% -2% 5%

Since outcome are not certain, we wish to find a way to assign probability (chance) P ofeach outcome occurring.

e.g. P(Hoccurs) :.4999 OR49.99% P(ABCRETURN2006: 5%): 12.760/0e.g. P(coin landing on edge):0%

P is called a probability function. To define P properly we need a random variable.

Definition: Random ( = stochastic) variable t: value offr known probabilistically (notcertainty outcome) = outcome governed by a probability function.

Random variables can be discrete fr; or continuous t.

Population: complete set of measwements to be analyzed.

,..,f*:tu j

BASIC POSTULATE: ! probability function P which govems outcomes of events in a

random experiment.

=l P which assigrrs probability to each fl = q (a quantity or number) in population.

Probability function: is a function ofrandom variable. Two types:

probability function fordiscrete random variable fl i

probability distribution function Pwith properties :

P(t; : Q: q e PoPulation) > 0

r P(fii):lVq

e.g. coin flipping experiment

P(ii' : !I) : P(nz : T) :'/z

'.\-l

can check PGI) = P(T) > 0

andX P(itt)=%+nH,T

probability frrnction forcontinuous ranilom variable t

probability density function Pwith properties:

P(fi) > 0

I r 1u; dii :1-ao

e.s. normal (Gaussian) distributionl f G:] .-r' 'r A| - z- \-Af-l )n eKP(0)=

-- Q.

eJ2Ir

P1fi\=> 0'\-,/@

J P(u)ou-@

*

Vfi

We just denote all probability (distribution/density) function by p'd' function and pick the

ttt" ipptop.iut" one according to which type of variables we are working with'

,i.r1

);

1t

(* t4\.* ,

Adr,,\ J.ote,-

a6t

Y) = ELzlt raTl utt-.LC-a-.

t&.,...c.. = q(i)t-sPzEc

)/,1 \

i - g:!:\l

rH +6tT-31

! 6-r r -o;-,

el.-*sl $o {-{ I

l-:"J 7t>=W

t.rlif.

{l or7

q, *4.ao

N..#..-{

v*a,tbt X

AIA-.-.& a E+ql;- (r-r,-,.- . )

+ b !_(ll:rl: e)-= e

. /llv.t

7.--t*,-1t-- lar- L-

l!e

i-c. a" T+@ ,

k tl L. ,.e'Ea^1 el-',"c- {s l,Dt * '

=l

lA4)

r-..--.- 4 I,€*; €r l-

(t4 t-te ,

t/J4.'L..-t

l.r-.)n 6t*eLT a*lJ/il'rJLtsll4 ar+tL A.

\y

L\- l"'L (Le4

( t,c, t '^.."u-d,^. o".n T,*-,

= Q^-51"

'L 1-

t{ut = Yi t"p L) (.=

v<J€

_t t>1

&,ta-Ll

+r,.. r\ at'|ltt'T2

Above is a brief review of probability and statistics theory. Based on that we proceed toa short inhoductory note on stochastic models:

Recall Ramsey's Model on optimal savings (continuous, infinite horizon case):

Max J PV(society's utility U (c t ))

s.t. technology in the form of feasibility constraint'

We find the caPital stock k 1 level tomax consumption c t ) U (c t) maximized.

tech is viewed as

A production set

tr represented by

Current cap stock k 1 = given current state k IB( k , ) : k,*r is optimal investment policy

J o=s)optimal savings PolicY

) recursive method based on curent cap stock k tfind g and get next period opt cap stock k r*r

But if the model is stochastic (uncertain outcomes after action), k t*r depends on both k 1

AND as-yet-to realize technology shock

) we need Anow-Debreu state-contingent concept

i.e. k. is expressed as

{k,itt', k,*tt, ..., k,*"n}

and these states depends on history oftech shocks'

Hence to extend a we introduce risks throughstochastic model,

\ \ \

toadeterministic model

./A. technology affecting outPut

function fB. some other factors affecting

other variables in the mode..

1l-'

We look at approach A. by assuming y 1: z 1 f(k )

where sequence \ z1 | arc independently & identically distributed (i'i'd) random

variables, representing technological shocks.

Note: if z1 negative ) some damaging shocks' like machinery breakdown, reducing

output.

Furthermore, assume households use expected values ofutility to rank stochastic

consumption sequences (Von Neumann- Morgenstem Expected Utility Thm)

Finite time horizon case:

I social planner

T

MaxE[ lt3'u(cs)] o<B<1 [Iv-24]t=0

s.t.feasibilitykt*r* ct S zt(kt) ct, kt*t} 0 Vt'V{ zt\AY-251

Assume

Begiruring ofperiod beginning of next period

t t+1

shock z t realized

) z t(k ), cs known k r+t

(k t, z t ) state of econ at time t

a deterministic model

SocialPlannerMaxbv tct , tt+t 5choosing +--0

stochastic model

- - T continsent on realization

lct,L**,1 ofzrlz2,......,z1for- - +?o each period

(i.e. a sequence of contingent plans

for each period)

r'. lrl

P-- - ?n

v,t, u(ed= Fou

+?:t. h € [t,G, = !(Co u[C1Car) uQ(a;) ...u(c+z q 3,,1. (, u3r?r>\+ (, u(c,@,D --. +?-u(Ci@"\

,))+P

=2 dt ,'<- t-.r4r* P14*1 ft*, A* l

,:t,)

?, u(C, @i))

.T

Ri, ci ---x elpli.*4

c = {4 e *+ c,(tt)e c-t+-teK'i tr(ar) e &'i

0- '- {+, eL+ , k,qt ) e e2, kn(Zr) e

) L*t..-+a *A ,*.agr szfc(ko,1i ={c. ( ctgts an,,Lf,)e 9(kt ,6t), t--o. t,..-,t\

Lop+ +r C-^tyzf U^'z.f ScS . *o I'o+--

L ? i { n7c,1a ;," )) + Fl- - -)

td n^/.t-., Lt.4rfr;. D+f ItU4- &rt-V-' -

V+ 7E |,il riJ ,.* '- ' q

,"J (cl gvta

dt^L &-- t* G^l*e" 6t^tLcl-, ttlt-S

L on- E l,a-,,ar*, trool^--il--;

h d-,Vl,-..-,

L, ct + FEV(LI*')

: : , r)o y,+^E; -r1..,.-- a^* 3 "^-fu*&, E hL; aaav<-

6.J/* ^ .F-f.^i 1a"5.;-. 711- A u*;4c4. .\ bt

2<, k\ = +4!bQ + ktLA+# Lz/iF-,.--'l

+ vQt*,, 1.,,)= tq+ lcz d^' (&ll,,cr) +E,l^?t*,

k, r K:. t*(l"i*r-cc)+ P. EI[-LI.*,

t tt"^<,*

t*C* - l^(+fz*) +

a L^.L+ + bTt * A( rAk') Er*

= "t!^-L++L-\t - b(+fb'-) r L, + ('t,

+frL)"/. L" + (tf f'L- )t . - L-( i 5c-1 + Se''

il_r,* Fr: l^ti;

'ir r 5./

L

(r * Er=L

E->1-t-te-raa-r1

t.1 t^t"f+

bG-;tt *O4,,4rt!.-c[ = A* 4 -L-"DE 6e __

L: j-? l

4{-;L 2j-.. =-5-t ltl/ al.ut LoweLt -i

u' A+ t&+> - t(L+,y\ = F ELY 1c.,t.>, 6' >f

i1 u 1-ct\= b (t a-.A (4)= d

- Ll t +., +Iy]ItL Lra

@uo ua-t ts v-'4-, v (kr,kt A*'e+'" LE-sr7

- =P!E)]i-;t. -E'-;1;{'rtti;;i) --

(.1"i- t

b

o"{tv + l"Zn tr .i3.y' I ' - L - ? co

o:l - . I

t-l '

-t(,,v3)+t3 ?l)" 1.

a ')o-:p

+'L{J =

n=

n=

-

tI

'"- +

).-

P-

"1-lV*

4t-t-o^ ,f4-ib

<+-r-, Ef +

= F 4,6' .+- n+ f e'l- sa<-

(lB?. j

"\.17"* dt"'tt>-'tn

--.L

o?tn>o

+aF

3l-rl

1.r.-^-7-r-L L'J baL frclqrc; /.ft.--+ et'*t'g,'-'

a,,+. a,l'l.c<L tw- aLp Hft-epov 0aac'4 t

/Jc l: ca-tL i U'€a+ {- . A frLtT'-L ?L' 14 u atJ

9ntt4+

iz.^f t&\

u;u u ,^ fu-, S a+

t/./'.t

=/

= EoLt^t+7

E,[t.tg\P)+ Lfu-yl ]LL

tgr} L;-a 4gd]-+-ry'- L r "t W! b a:-r t&

frv-+c'\

)8t.1

t+a+,.}t-..+J-',

-@'Y'l

,--f 14

l-.<E/ +J)

d.'!

Jtz* O<oa<, *"1 .+e e

= v*n^[ t-tr

*-L + i;. r.,&.*, +1+ |

t+oh^-r-- o 1{ a ILor

.ltO

cY I

-+S -.rb

O t+ o , 0* tnt'^ dzv tlaq. Au+U .

; nr.uJ, ,J(

Lo >o a2 ulaf F-.^ kt fut^ vlr'|-a- u7t4- c";"'t'*-t

tUtl-*"le-a;t#fL

= t.t4 ( k, r 4\ = f*t (uFtoa! La-

oL*,*)=a(*pE-*-*:-

H ( a. E\ = (,,ut (L*",3 al Q=t')-= 6 $ a,6>o

.G

h,rrzr<-s 6 a- l'^'4 LcA: /r'-.1'* (

/u4 f A ruJ-L t^,.,a''"* d6fi)J2.<Jd C^rb^ h,-ci.*.;tl .+ V q^--, t^.ta*inz a-sr-+f-. a 4 rt*t-.*;rtt

--=v-a.-2 4t4 *; *t ' r 4 ^vt+r-- cl

.,aa.t,

d-;9: 4+<. a-t+

Yp

4 (lr> L CIL>

= Va al t4<.r<-dr+--}L

/.i-ZLr a,- l;A- 4951/^Lz;

- "t'rL

,.:1.{j

tL-e .^t 6' 1P z-ta-'"

)-rl +6EIv(qr'tev&)

E*^l-E;^