Wealth Dynamics in Competitive Markets

Wealth Dynamics in Competitive Markets

Larry Blume

Cornell University & The Santa Fe Institute & IHS, Vienna

Lectures at ScuolaSuperiore Sant’Anna

21-22 June, 2011

IntroductionWhy Biology

The Marginalist ControversyThe IssuesEvolutionA Contemporary Analysis

Evolution in FinanceIntroductionThe Kelly RuleSequences of BetsOptimal Proportional BetsLong-Run Performance of Optimal Portfolios

MarketsBiological Arguments in FinanceSimple Trading Rules

Exchange EconomiesBasic ResultsThe Survival of Noise Traders in Financial Markets

Biology and Economics

In order to pass the B.A. examination, it was, also, necessaryto get up Paley’s Evidences of Christianity, and his MoralPhilosophy . . . The logic of this book and as I may add of hisNatural Theology gave me as much delight as did Euclid.The careful study of these works, without attempting to learnany part by rote, was the only part of the Academical Coursewhich, as I then felt and as I still believe, was of the least useto me in the education of my mind. I did not at that timetrouble myself about Paley’s premises; and taking these ontrust I was charmed and convinced of the long line ofargumentation.

Charles Darwin

. . . when we come to inspect the watch, we perceive. . . that its several parts are framed and put together for a purpose, e.g.that they are so formed and adjusted as to produce motion, and that motion so regulated as to point out the hour of theday; that if the different parts had been differently shaped from what they are, or placed after any other manner or in anyother order than that in which they are placed, either no motion at all would have been carried on in the machine, or nonewhich would have answered the use that is now served by it. . . the inference we think is inevitable, that the watch musthave had a maker – that there must have existed, at some time and at some place or other, an artificer or artificers whoformed it for the purpose which we find it actually to answer, who comprehended its construction and designed its use.

William Paley, Natural Theology

ToC 2

Biology and Economics

In October 1838, fifteen months after I had begun mysystematic inquiry, I happened to read for amusementMalthus on Population, and being prepared toappreciate the struggle for existence whicheverywhere goes on, from long-continued observationof the habits of animals and plants, it at once struck methat under these circumstances favourable variationswould tend to be preserved, and unfavourable ones tobe destroyed. The result would be the formation of anew species.

Charles Darwin

. . . the most important book I read was Malthus’ Principle Of Population . . . It was the firstwork I had yet read treating any of the problems of philosophical biology, and its mainprinciples remained with me as a permanent possession, and twenty years later gave methe sought-after clue to the effective agent in the evolution of organic species.

Alfred Wallace

ToC 3

Why Biology?

The Mecca of the economist is economic biology ratherthan economic dynamics.

–Alfred Marshall (1898)

In economics every event causes permanentalterations in the conditions under which futureevents can occur . . . When any causaldisturbance has caused a great increase ofany commodity, and has thereby led to theintroduction of extensive economies, theseeconomies are not readily lost. Developmentsof mechanical appliances, of division of labour,and of means of transport, and of improvedorganisation of all kinds, when they have beenonce obtained are not readily abandoned.

ToC 4

Why Biology?

◮ Competition for resources, Robbins (1935).

◮ Invisible hand explanations, Nozick (1994), AER 84. A patternor institutional structure that apparently only could arise byconscious design instead can originate or be maintainedthrough the interactions of agents having no such overallpattern in mind.

◮ multiplicity of scales

ToC 5

Uses of Biological Thinking

◮ A source and testbed for economic hypotheses◮ Profit maximization◮ Behavioral economics

◮ Intertemporal market models with sufficient heterogeneityexhibit a population dynamic.

ToC 6

Features of Selection

◮ Path dependence

◮ Frozen “accidents”

◮ Discontinuity in phenotypic change◮ The systemic nature of the genome is important

◮ Evolution of evolvability◮ Self-organization

◮ Group selection

ToC 7

Why Not Cultural Evolution?

◮ Intentionality — gaming the system. This is not Penrose’s(1954) critique of biological analogy in economics.

◮ Cultural evolution involves the proliferation of packets ofinformation, codices in G. C. Williams’ terminology. Biologicalanalogies for evolution, he suggests, are more likely to befound in epidemiology than in population genetics, which hedefines as ‘that branch of epidemiology that deals withinfectious elements transmitted exclusively from parent tooffspring.’

◮ Unenforceable contracts in biological systems. Institutionalinfrastructure to trade in economic systems that we oftenignore.

◮ Biological emphasis on interaction processes which are oftenignored in market analysis.

ToC 8

On Exactitude in Science . . . In that Empire, the Art of Carto-graphy attained such Perfection that the map of a singleProvince occupied the entirety of a City, and the map of theEmpire, the entirety of a Province. In time, those Unconscio n-able Maps no longer satisfied, and the Cartographers Guildsstruck a Map of the Empire whose size was that of the Empire,and which coincided point for point with it. The following Ge n-erations, who were not so fond of the Study of Cartography astheir Forebears had been, saw that that vast Map was Useless,and not without some Pitilessness was it, that they delivere dit up to the Inclemencies of Sun and Winters. In the Deserts ofthe West, still today, there are Tattered Ruins of that Map,inhabited by Animals and Beggars; in all the Land there is noother Relic of the Disciplines of Geography.

Suarez Miranda,Viajes de varones prudentes, Libro IV,Cap. XLV,Lerida, 1658

ToC 9

Profit Maximization Why Not?

◮ Do firms maximize profits?

Hall and Hitch (1939), “Price Theory and BusinessBehaviour”, Oxford Economic Papers 2, 12–45.

Lester (1946), “Shortcomings of Marginal Analysis forWage-Unemployment Problems”, American Economic Review36, 63–82.

ToC 10

Profit Maximization Why?

◮ Machlup, F. (1946), “Marginal Analysis and EmpiricalResearch”, American Economic Review 35, 519–54.

◮ Machlup, F. (1955), “The Problem of Verification inEconomics”, Southern Economic Journal 22, 1-21.

◮ Friedman, M. (1953), “The Methodology of PositiveEconomics”, in Essays in Positive Economics.

ToC 11

Methodology RobbinsThe propositions of economic theory, like all scientifictheory, are obviously deductions from a series ofpostulates. And the chief of these postulate are allassumptions involving in some way simple andindisputable facts of experience relating to the way inwhich the scarcity of goods which is the subject ofour science actually shows itself in the world of reality. . . . Themain postulate of the theory of production is the fact that there aremore than one factor of production. . . . These are not postulatesthe existence of whose counterpart in reality admits of extensivedispute once their nature is fully realized. We do not needcontrolled experiments to establish their validity: they are so muchthe stuff of our everyday experience that they have only to bestated to be recognized as obvious. . . . And it is from the existenceof the conditions they assume that the general applicability of thebroader propositions of economic science is derived.

An Essay on the Nature and Significanceof Economic Science, 1932 (1935)ToC 12

Methodology Friedman

To put this point less paradoxically, the relevant question to askabout the "assumptions" of a theory is not whether they aredescriptively "realistic," for they never are, but whether they aresufficiently good approximations for the purpose in hand. And thisquestion can be answered only by seeing whether the theoryworks, which means whether it yields sufficiently accuratepredictions. The two supposedly independent tests thus reduce toone test.

The Methodology of Positive Economics, p. 15

ToC 13

Biological Responses

Adaptation of he individual firm:

◮ Harrod (1939), “Price and Cost in Entrepreneurs’ Policy”,Oxford Economic Papers.

Selection across firms:

◮ Alchian (1950), “Uncertainty, Evolution, and EconomicTheory”, Journal of Political Economy 58, 211–21.

ToC 14


Whenever this determinant (of business behavior) happens to leadto behavior consistent with rational and informed maximization ofreturns, the business will prosper and acquire resources withwhich to expand; whenever it does not the business will tend tolose resources and can be kept in existence only by the addition ofresources from the outside. The process of natural selection thushelps to validate the hypothesis (of profit maximization) or, rather,given natural selection, acceptance of the hypothesis can bebased largely on the judgment that it summarizes appropriately theconditions for survival.

Friedman (1953), pg. 22

ToC 15


◮ Winter (1971), “Satisficing, Selection, andthe Innovating Remnant”, QuarterlyJournal of Economics 85, 237–61.

◮ Nelson and Winter (1982), AnEvolutionary Theory of EconomicChange.

Criticisms

◮ Koopmans (1957), Three Essays on theState of Economic Science

ToC 16

Model

Capitalists own firms and employ workers.

◮ J commodities.

◮ Each worker i has a time-stationary endowment ei ∈ RJ+/{0}.

Each capitalist h owns firm h and an initial output bundle ωh+0 .

◮ Individual k has utility function Uk (c) =∑∞

t=1 βk uk (ct ) onconsumption plans, where uk : RJ

+ → R and 0 < βk < 1. Theui are C2, strictly concave, strictly monotonic and satisfy anInada condition on the boundary.

◮ Firms are described by their technology sets Th ⊂ RJ . Forω ∈ Th , ω+ = ω∨ 0 ∈ RJ

+ are outputs, and ω− = ω∧ 0 ∈ RJ−

are inputs. For each firm, inputs and outputs are fixed.

Blume and Easley, “Optimality and natural selection in markets,” JET 107, 2002.

ToC 17

Decision Rules

Firms buy input at date t and sell output at date t + 1.

Definition. A decision rule for firm h is a function

dh : (p, q, y) ֒→ (ω−,ω+)

mapping input prices p, output prices q and input expenditures yinto production plans such that

1. dh(p, q, y) ∈ Th ,

2. p · dh−(p, q, y) = y,

3. d is u.h.c.

4. d is homogeneous; for all positive scalars α, β,

d(αp, βq,αy) = d(p, q, y).

ToC 18

Decision Rules

Example: Constrained profit maximization.

d(p, q, y) = arg maxω

qω+ + p ω−

s.t. ω ∈ Th

p ω− = y

This will be homogeneous if Th is a cone.

ToC 19

Constrained Equilibrium

Definition. A constrained equilibrium is a sequence(

p∗t , (ci∗t )

Ii=1, (c

h∗t ,ωh∗

t )Hh=1

)∞

t=1with p∗t ∈ RJ

+/{0} such that

1. For all workers i, (c i∗t )∞t=1 solves

maxc

U(c)

s.t. for all t , p∗t · (ct − ei) ≤ 0

ct ∈ RJ+.

ToC 20



p∗t , (ci∗t )

Ii=1, (c

h∗t ,ωh∗

t )Hh=1

)∞


+/{0} such that

2. For all capitalists h, (ch∗t ,ωh∗

t )∞t=1 solves

maxc,ω

U(c)

s.t. for all t , p∗t (ct −ωt ) ≤ 0,

(ωh∗−t ,ωh∗+

t+1 ∈ d(p∗t , p∗t+1, p

∗t ·ω

h∗−t ),

c ∈ RJ+.

and (ωh∗+0 )H

h=1 is given.

ToC 20



p∗t , (ci∗t )

Ii=1, (c

h∗t ,ωh∗

t )Hh=1

)∞


+/{0} such that

3. At every date t ,∑

i

c i∗t +

∑

h

ch∗t −

∑

h

ωh∗t −

∑

i

ei = 0.

ToC 20

Constrained Equilibria

◮ Prices can be rescaled period by period.

◮ Constrained equilibrium is recursive. Starting a constrainedequilibrium at date t gives a constrained equilibrium.

ToC 21

Competitive Equilibrium

Definition. A competitive equilibrium is a sequence(

q∗t , (ci∗t )

Ii=1, (c

h∗t ,ωh∗

t )Hh=1

)∞

t=1with q∗t ∈ RJ

+/{0} such that

1. For all workers i, (c i∗t )∞t=1 solves

maxc

U(c)

∞∑

t=1

p∗t · (ct − ei) ≤ 0

ct ∈ RJ+.

ToC 22



q∗t , (ci∗t )

Ii=1, (c

h∗t ,ωh∗

t )Hh=1

)∞


+/{0} such that

2. For all capitalists h, (ch∗t ,ωh∗

t )∞t=1 solves

maxc,ω

U(c)

∞∑

t=1

p∗t (ct −ωt) ≤ 0,

(ωh∗−t ,ωh∗+

t+1 ) ∈ Th,

ct ∈ RJ+.

and (ωh∗+0 )H

h=1 is given.

ToC 22



q∗t , (ci∗t )

Ii=1, (c

h∗t ,ωh∗

t )Hh=1

)∞


+/{0} such that

3. At every date t ,∑

i

c i∗t +

∑

h

ch∗t −

∑

h

ωh∗t −

∑

i

ei = 0.

In a competitive equilibrium, every firm maximizes profits, andequilibrium profits are 0.

ToC 22

Selection

Let ρh(ph−, ph+

, y) denote firm h’s revenue function.

Assume that on an equilibrium path, ρy exists, and that for all t ,lim infy→0 ρy(p∗t , p

∗t+1, y) , 0.

Theorem. In any constrained equilibrium, and for any twocapitalists h and k with discount factors βh and βk ,

t−1∏

τ=1

βkρkyτ

βhρhyτ→ 0 implies lim

t

ptωk−

∑

i ptωi−t

and ckt → 0.

ToC 23

Proof of Theorem

The capitalist’s solution solves

max∑

t

βt vh(pt , zt )

s.t. for all t , zt + yt = mt

mt+1 = ρh(pt , pt+1, yt )

where mt is revenue at the beginning of t , zt is consumptionexpenditure, yt is input expenditure and vh is an indirect utilityfunction.

ToC 24

Proof of TheoremF.O.C.

vhz (pt , z

ht ) = βhρ

hyt v

hz (pt+1, z

ht+1)

so

vhz (pt+1, zh

t+1)

vkz (pt+1, zk

t+1)=βkρ

kyt

βhρhyt

vhz (pt , zh

t )

vkz (pt , zk

t )

=t−1∏

τ=1

βkρkyτ

βhρhyτ

vhz (p1, zh

1 )

vkz (p1, zk

1 )

and so for each good l,

Dluh(cht+1)

Dluk (ckt+1)

∝

t−1∏

τ=1

βkρkyτ

βhρhyτ

If the RHS goes to 0, so does the LHS, and so ckt → 0.

ToC 25

Selection

Let rht = ρ(pt , pt+1, ptω

h−t )/ptω

h−t denote the date t average

return on investment.

Assume. For every firm h and prices (p, q), the revenue function isconcave in y.

Corollary. Suppose capitalist h is a constrained profit-maximizer,and suppose that for capitalist k , ρh

y is non-increasing in y, and

t−1∏

τ=1

βk rkτ

βhrhτ

→ 0.

Then the theorem’s conclusions still hold.

ToC 26

Selection

Corollary. If h and k are profit maximizers, if

lim supt

t−1∏

τ=1

rkτ

rhτ

< ∞ and ifβk

βh< 1,

then limt ckt = 0 and lim yk

t /∑

h yht → 0.

ToC 27

Selection

Assume. Common discount factor, all inputs are endowed and allconsumer goods are produced.

Theorem. Every competitive equilibrium consumption plan isstationary from date 2 on.

Theorem. A stationary competitive equilibrium is a constrainedequilibrium for some assignment of initial outputs (ωh

0)Hh=1.

ToC 28

Selection

Theorem. The allocation resulting from any stationary and interiorconstrained equilibrium is a competitive allocation.

Theorem. If a constrained equilibrium is locally stable, then thelimit allocation is competitive.

ToC 29

Dynamics

Two produced consumption goods, x and y, one factor z. Linearproduction with unit outputs (1, 0.1) and (0.001, 1).

u(x, y) =∞∑

t=1

βt log (xρt + yρt )1/ρ

ToC 30

Dynamics

Four firms. with unit output vectors (1, 0.1), (0.05, 1), (0.9, 0.15),(0.3, 0.7, ).

ToC 31

Evolution in Finance

“The trading floor is a jungle,” he went on, “and the guy you end upworking for is your jungle leader. Whether you succeed here or notdepends on knowing how to survive in the jungle.”

Michael Lewis, Liar’s Poker

ToC 32

Kelly’s Betting System

You are offered a gamble with probability 2/3 of winning and 1/3of losing. You may bet up to your current wealth. The amount youbet is either doubled or lost. You will bet for 50 days.

x0 given.

0 ≤ bk ≤ Xk−1

xk =

Xk−1 + bk with prob. 2/3,

Xk−1 − bk with prob. 1/3.

ToC 33

Kelly’s Betting SystemMaximize end date wealth:

E {X50|X49} = X49 +23

b49 −13

b49

= X49 +13

b49

so take b49 = X49.

E {X50|X49} =43

X49

E {X50|X48} =43

E {X49|X48} =169

X48

.

.

.

E {X50|X0} =

(

43

)50

X0

ToC 34


E {X50|X0} =

(

43

)50

X0 = 1.77 × 106X0

BUT

X50 =

250X0 with probability 0.00000000157,

0 with probability 0.99999999843.

ToC 35


A Proportional Betting System: Invest fraction π in the bet.

Xn = (1 + π)Zn(1 − π)n−ZnX0 Zn is # wins

= exp{Zn log(1 + π) + (n − Zn) log(1 − π)}X0

= exp n

{

Zn

nlog(1 + π) +

(

1 −Zn

n

)

log(1− π)

}

X0

≃ exp n

{

23

log(1 + π) +13

log(1− π)

}

X0

Maximize the rate: π = 1/3.

growth rate is23

log43+

13

log23≈ 0.056633

X50 ≈ 16.97X0

ToC 36


Let π , 1/3 be another proportional betting rule. The SLLN impliesthat for any exponential confidence bands around the mean growthpath, Xt will almost surely be inside the bands for all large t .

Xt

t

π∗

π

ToC 37

Breiman on Kelly

“Assume that we are hardened and unscrupulous types with aninfinitely wealthy friend. . . . ”

◮ X takes values in S = {1, . . . ,S}. Pr {X = s} = ps .

◮ There are J assets. Asset j pays off A(s, j) in state s.

◮ A bet is a vector {α1, . . . ,αJ}, where if wealth share bj is beton asset j, and if the event {X = s} is realized, we receive∑

j αjA(s, j). W.l.o.g. we bet everything.

◮ Wn is our fortune after n plays.

Definition. A game is favorable if there is a strategy such thatWn → ∞ a.s.

ToC 38

Optimal Gambling for Favorable Games

αnk The fraction of Wn−1 invested in asset k .

Bn The set of date n feasible portfolios.

Vn The return on portfolio αn,

Vn =∑

j

αnj A(s, j) if Xn = s;

Wn+1 = Vn+1Wn.

Gn Growth rate, Gn = log Vn. Depends on αn.

ToC 39

A Single Stage

The expected growth rate is

G(α, p) ≡∑

s

ps log∑

j

αjA(s, j).

The maximal expected growth rate is

G(p) = maxα∈B

G(α, p)

ToC 40

Properties of G(α, p)

Lemma. W(α, p) is concave in α and linear in p. G(p) is convexin p.

Lemma. If B is convex, then set of log-optimal portfolios is convex.

Lemma. Any two portfolios α′ and α′′ that maximize the expectedgrowth rate G have the same rate of return in each state i; for all i,

∑

s

α′j A(s, j) =∑

s

α′′j A(s, j) for all s.

ToC 41

Optimal Portfolios

Suppose that each Bn is the simplex.

Theorem. A portfolio α∗ is log-optimal iff

E

A(s, j)∑

j α∗j A(s, j)

=≤

1if α∗j > 0,

if α∗j = 0.

ToC 42

Proof

Let α∗ be optimal, and consider a deviation in direction of feasibleportfolio α. Take αλ = (1 − λ)α∗ + λα. Then

ddλ

G(αλ)

∣

∣

∣

∣

∣

λ↓0≤ 0

for all feasible portfolios α. Computing

ddλ

G(αλ)

∣

∣

∣

∣

∣

λ↓0= limλ↓0

1λ

E

log(1− λ)

∑

j α∗j A(s, j) + λ

∑

j αjA(s, j)∑

j α∗j A(s, j)

= limλ↓0

E

1λ

log

1 + λ

∑

j αjA(s, j)∑

j α∗j A(s, j)

− 1

= E

∑

j αjA(s, j)∑

j α∗j A(s, j)

− 1

ToC 43

Proof (cont.)

Thus for all α ∈ B,

E

∑

j αjA(s, j)∑

j α∗j A(s, j)

≤ 1.

This expression is linear in α, so it suffices to check it for theextreme points of Bn. Thus

E

A(s, j)∑

j α∗j A(s, j)

≤ 1.

If α∗j > 0, the line from asset j extends below 0, so the inequality istwo-sided, that is, it holds with equality. If α∗j = 0, it cannot beextended, and we have the weak inequality.

ToC 44

More on Optimal Portfolios

Let Vα =∑

j αjA(s, j) denote the random return from an arbitraryportfolio.

Theorem.

E

{

VαVα′

}

≤ 1 for all α ∈ B iff E

{

logVαVα′

}

≤ 1 for all α ∈ B.

ToC 45

Proof.

The second condition is log-optimality, so Only If follows from thefirst-order condition. If follows from Jensen’s inequality:

E

{

logVαVα′

}

≤ log E

{

VαVα′

}

≤ log 1 = 0.

ToC 46

Asymptotic Optimality

Lemma. Let Wn∗ be the wealth of an investor after n rounds of

using a log-optimal portfolio, and let Wn denote the wealth fromany other portolio rule. Then

E{

log Wn∗

}

= nW(p) ≥ E{

log Wn}.

The proof is obvious.

ToC 47


Theorem.

lim supn→∞

1n

logWn

Wn∗

≤ 0 with pr. 1.

Proof. From the Kuhn-Tucker conditions,

E

{

Wn

Wn∗

}

≤ 1.

From Markov’s inequality

Pr{

Wn > tnWn∗

}

= Pr

{

Wn

Wn∗

> tn

}

<1tn,

so,

ToC 48


Pr

{

1n

logWn

Wn∗

>1n

log tn

}

<1tn.

Take tn = n2, and summing,

∞∑

n=1

Pr

{

1n

logWn

Wn∗

>2 log n

n

}

<

∞∑

n=1

1

n2=π2

6.

Borel-Cantelli says

Pr

{

1n

logWn

Wn∗

>2 log n

ninfinitely often

}

= 0,

ToC 49


Thus for almost every sequence (x1, . . .) there is an N such that forn ≥ N,

1n

logWn

Wn∗

≤2 log n

n,

from which the theorem follows.

ToC 50

The Cost of Being Wrong

Suppose you believe the true distribution is q, when the truth is p.How badly off will you be?

Theorem. If αq is log-optimal for probability distribution q, then

∆W = W(αp, p) −W(αq

, p) ≤∑

s

ps logps

qs

ToC 51

Proof

∆W =∑

s

ps log

∑

j

αpj A(s, j)

−∑

s

ps log

∑

j

αqj A(s, j)

=∑

s

ps log

∑

j αpj A(s, j)

∑

j αqj A(s, j)

=∑

s

ps log

∑

j αpj A(s, j)

∑

j αqj A(s, j)

qs

ps

ps

qs

=∑

s

ps log

∑

j αpj A(s, j)

∑

j αqj A(s, j)

qs

ps

−∑

s

ps log

(

ps

qs

)

ToC 52

Proof

∑

s

ps log

∑

j αpj A(s, j)

∑

j αqj A(s, j)

qs

ps

≤ log∑

s

ps

∑

j αpj A(s, j)

∑

j αqj A(s, j)

qs

ps

≤ log∑

s

qs

∑

j αpj A(s, j)

∑

j αqj A(s, j)

≤ log 1

= 0

This result gives an upper bound on the value of additionalinformation for the optimal portfolio.

ToC 53

ToC 54

Evolutionary Arguments in Financial Markets

“I know you miss the Wainrights, Bobby, but they were weak and stupidpeople—and that’s why we have wolves and other large predators.”

ToC 55

Given the uncertainty of the real world, the many actual and virtualtraders will have many, perhaps equally many, forecasts. . . If anygroup of traders was consistently better than average inforecasting stock prices, they would accumulate wealth and givetheir forecasts greater and greater weight. In this process, theywould bring the present price closer to the true value.

Robert Cootner

. . . dependence in the noise generating process would tend toproduce ‘bubbles’ in the price series. . . If there are manysophisticated traders in the market, however, they will be able torecognize situations where the price of a common stock isbeginning to run up above its intrinsic value. If there are enough ofthese sophisticated traders, they may tend to prevent these‘bubbles’ from ever occurring.

Eugene Fama

ToC 56

Arrow Securities

◮ There are S securities, one for each state. Each security paysof $1 in its state, and $0 otherwise. The price of security s isps .

◮ Investor i invests fraction αis ≥ 0 of his wealth w i in asset s.

His budget constraint is∑

s αis = 1.

◮ Assets are in 0 net supply, so the total amount returned by theasset that pays off in state s is the total amount invested in allassets, w =

∑

i w i . Market clearing implies

∑

i

αisw i

ps= w ps =

∑

i

αis

w i

w

ToC 57

Dynamics

Suppose income is reinvested repeatedly.

w it =

t∏

τ=0

S∏

s=1

(

αisτ

psτ

)1s (ωτ)

w i0

Suppose that states are iid, that Pr {ωt = s} = qs , and thatinvestors use constant share rules.

w it =

t∏

τ=0

S∏

s=1

(

αis

psτ

)1s(ωτ)

w i0

log w it =

t∑

τ=0

∑

s

1s(ωτ)(

logαi − log psτ

)

+ log w i0

ToC 58

Comparing Investors

Suppose that i and j both use constant share rules.

logw i

t

w jt

=t

∑

τ=0

∑

s

1s(ωτ)(

logαi − logαj)

+ logw i

0

w j0

From the law of large numbers,

1t + 1

logw i

t

w jt

→∑

s

qs

(

logαi − logαj)

=∑

s

qs

(

log qs − logαj)

−∑

s

qs

(

log qs − logαi)

= I(q;αj) − I(q;αi)

ToC 59

Selection

The SLLN give a simple market selection result:

Theorem. If for trader i, I(q,αi) > minj I(q,αj), thenw i

t∑

j w jt

→ 0.

◮ So being relative-entropy minimal is necessary for survival.

◮ Is it sufficient?

ToC 60

Generalizations I

Suppose that S = R ×S where R has public states and S hasstate components that cannot be traded on. Suppose that thereare |R | assets, and asset r pays off a(s, r) in state (s, r). Then

w it =

t∏

τ=0

S∏

s=1

R∏

r=1

αirτ

(

a(s, r)

prτ

)1sr (ωτ)

w i0

log w it =

t∑

τ=0

∑

sr

1sr(ωτ) logαirτ + 1sr(ωτ) log

a(s, r)

prτ+ log w i

0

◮ The log-optimal portfolio is independent of prices.

◮ Markets are incomplete.

◮ Selection will depend upon the marginal distribution of r .

ToC 61

Generalizations II

For an arbitrary asset structure, the log-optimal rule will be pricedependent, and thus time-dependent. Nonetheless,

Theorem. If lim inft

1t

t∑

τ=1

W(αiτ, q) −W(αj

τ, q) > 0 thenw j

t

w it

→ 0.

ToC 62

Generalizations III

◮ If traders are constrained in their portfolios, log-optimality stilldominates.

◮ Adjustment for savings rates.

If trader i saves at rate δi , define the survival rate

κi = log δi − I(q,αi)

A maximal survival rate is necessary for long-run survival.

ToC 63

Long Run Asset Pricing

Suppose some trader uses the log-optimal portfolio.

◮ First order conditions imply

E

{

ak (s)/pk∑

j αjaj(s)/pj

}

= 1 if αk > 0.

◮ The long run equilibrium condition is that for all s

∑

j

αjaj(s)

pj= 1.

Thuspk = E

{

ak (s)}

.

ToC 64

Survival: Sufficient Conditions

Suppose traders 1 and i both have maximal survival index.

logw i

t

w1t

=t

∑

τ=0

∑

s

1s(ωτ)(log δi + logαis) − . . .

=∑

s

(

ns(ωt) − tps

)

logαis + t{log δi + ps logαi

s)} − . . .

=∑

s

(

ns(ωt) − tps

)

(logαis − logα1

s)

ToC 65

Survival: Sufficient Conditions

logw2

t

w1t

.

.

.

logw I

t

w1t

=

logα21 − logα1

1 · · · logα2S − logα1

S...

.

.

.

logαI1 − logα1

1 · · · logαIS − logα1

S

·

n1(ωt) − tp1...

nS(ωt) − tpS

=

logα2

1

α2S

− logα1

1

α1S

· · · logα2

S−1

α2S

− logα1

S−1

α1S

.

.

.

.

.

.

logαI

1

αIS

− logα1

1

α1S

· · · logαI

S−1

αIS

− logα1

S−1

α1S

·

n1(ωt) − tp1...

nS−1(ωt) − tpS−1

ToC 66

logw2

t

w1t

.

.

.

logw I

t

w1t

= A ·

n1(ωt ) − tp1...

nS−1(ωt ) − tpS−1

◮ If there is an x such that A · x ≪ 0, then there is an open coneC such that A · y ≪ 0 for all y ∈ C. In this case, for all ǫ > 0,w1

t > 1 − ǫ infinitely often.

◮ If for all x there is a row of A such that a · x > 0, then there isalways some trader i , 1 who “beats” trader 1. If S > 3, thenfor all ǫ > 0 the event w1

t > ǫ is transient.

ToC 67

There is an x such that A · x ≪ 0 iff there is no non-trivial,non-negative linear combination λ of the rows of A such thatλ ·A = 0.

That is, there is no non-negative vector of weights λ summing to 1such that

∑

i≥2

λi

logαi

s

αiS

S−1

s=1

=

logα1

s

α1S

S−1

s=1

.

That is, lim supt w1t = 1 iff

(

log α1s

α1S

)S−1

s=1is not in the convex hull of

the remaining vectors of log-odds ratios.

ToC 68

Survival Summary

If S > 3,◮ Interior traders vanish.◮ For extremal traders,

lim sup t w it

/∑

j w jt = 1.

Survival requires much patience andextreme beliefs.

ToC 69

A Simple Exchange Economy I

◮ One non-storable good.

◮ Uncertain endowments

◮ Long-lived consumers maximize the expectation of thepresent discounted value of utility streams.

◮ Consumers are heterogeneous — they may differ in beliefs,discount factors and payoff functions.

◮ Markets are complete.

ToC 70

A Simple Exchange Economy II

States: S , a finite set.

A path is denoted σ = (σ1,σ2, . . .), σt ∈ S .There is a “true” probability p on (Σ,F ). For anyprobability distribution q,

qt (σ) ≡ q(

{σ1, . . . ,σt } ×S × S . . .)

.

Consumers: Beliefs pi on (Σ,F ).Discount factors 0 < βi < 1.Consumption plans c =

(

c1, . . .)

, ct t-meas.Endowments ei, strictly positive consumption plans.Payoff functions ui : R+ → R.Preferences

Ui(c) = Epi

{ ∞∑

t=1

βt−1i ui

(

ct(σ))

}

ToC 71

Assumptions

A.1. (Payoff Functions) The payoff functions ui are C1, strictlyconcave and monotonic, and satisfy an Inadacondition at 0.

A.2. (Endowments)

0 < f = inft ,σ

∑

i

eit (σ) ≤ sup

t ,σ

∑

i

eit(σ) = F < ∞.

A.3. (Truth) For all i, t and σ, if pt(σ) > 0 then

pit (σ) > 0.

We will assume A.1-3 throughout.

ToC 72

Optimal Allocations

If c∗ = (c1∗, . . . , c I∗) is an interior Pareto optimal allocation of

resources, then there is a vector of welfare weights(λ1

, . . . , λI) >> 0 such that c∗ solves the problem

max(c1

,...,c I)

∑

i

λiUi(c)

such that for all t and σ,∑

i

c it(σ) −

∑

i

eit(σ) = 0,

and ∀i c it(σ) ≥ 0.

ToC 73

First Order Conditions

For all σ and t :

There are numbers λi > 0 and ηt(σ) > 0 s.t., for all i:

λiβt−1i u′i

(

c it (σ)

)

pit(σ) − ηt (σ) = 0.

So for any i and j and for all σ and t :

u′i(

c it (σ)

)

u′j(

c jt (σ)

) =λj

λi

βt−1j

βt−1i

pjt(σ)

pit(σ)

.

ToC 74

Analysis of the IID Casepi

t (σ) generated by iid draws with distribution pi = (pi1, . . . , p

iS).

pit(σ) =

S∏

s=1

(pis)

nst (σ)

Taking logs and dividing by t gives

1t

logui′(c i

t (σ))

uj′(c jt (σ))

=1t

logλj

λi+

logβj

βi+

1t

∑

s

nst (σ)(log pj

s − log pis)

From the SLLN, the lhs converges a.s. to(

log βj − Ip(pj)

)

−

(

log βi − Ip(pi)

)

,

where Ip(q) = Ep logp(s)

q(s)is the relative entropy of p with respect

to q.ToC 75

Long-Run Fates in the IID Case

Definition. Consumer i survives iff lim sup c it > 0. She vanishes iff

limt c it = 0.

Trader i’s survival index is κi = log βi − Ip(pi).

1t

logui′(c i

t (σ))

uj′(c jt (σ))

→ κj − κi

Theorem. If κi , maxj κj, then trader i vanishes almost surely.

ToC 76

Bayesian Learning

◮ States are iid with distribution p on S .

◮ Consumer i considers a model set Θi and has prior beliefs µi

on Θi .

◮ Each θ ∈ Θi describes a distribution pθ on Σ.

◮ The collection of pθ’s and µi induce a joint distribution onΘi ×Σ.

◮ pit is the marginal distribution on the t-fold product S × · · · × S .

ToC 77

Bayesian Learning

Theorem. Suppose traders i and j are Bayesians with model setsΘi that are manifolds. Suppose for each k = i, j that:

1. βi = βj ;

2. The parameter θ ∈ Θk , is identified;

3. that each Θk is a manifold of dimension dk ;

4. and that trader k ’s prior belief has strictly positive density onΘk with respect to Lebesgue measure.

Suppose di > dj and Θj ⊂ Θi . Then for all θ ∈ Θj , c it → 0 in

probability. If βi , βj , the consumer with the lower discount factorvanishes.

ToC 78

References

◮ Blume and Easley, “Evolution and market behavior,” JET 58,1992.

◮ , “If you’re so smart, why aren’t you rich? Beliefselection in complete and incomplete markets,” Econometrica74, 2006.

◮ , “The market organism: Long-run survival inmarkets with heterogeneous traders,” JEDC 5, 2009.

◮ , “Market Selection and Asset Pricing,” in Handbookof Financial Markets: Dynamics and Evolution, ed. by T. Hens,& K. Schenk-Hoppe, 2009.

◮ , “Heterogeneity, Selection, and Wealth Dynamics,”Annual Reviews of Economics 2, 2010.

Other work: See the Handbook of Financial Markets for relatedsurveys.

ToC 79

The Survival of Noise Traders?

DeLong, Shleifer, Summers and Waldmann, “The survival of noisetraders in financial markets,” Journal of Business, 1991.

◮ In our exchange model, traders with better information driveout those with worse information, ceteris paribus. DSSL claimotherwise.

◮ This paper is cited as a justification for much current research.

◮ It illustrates the misuse of “continuum-of-agent models”.

◮ Similar modelling issues arise in other continuum models,e.g. Bewley Models.

ToC 80

The Survival of Noise Traders?

◮ There are two continuua of traders: Group Ip and Iq.

◮ Initial endowments are 1 for each group, uniformly distributed.

◮ Traders bet on the flip of identical i.i.d. coins, one for eachtrader. The true H probability is p.

◮ For every dollar i bets correctly, he earns $2.

◮ Traders have identical CRRA utility functions with parameterγ < 1 and a common discount factor 0 < β < 1.

◮ How does the wealth of each group evolve?

ToC 81

Let αi denote the share of wealth invested in H. For the trader withbelief r ,

α =r1/1−γ

r1/1−γ + (1 − r)1/1−γ

and

δ = 2γ/1−γβ1/1−γ(

r1/1−γ + (1− r)1/1−γ)

ToC 82

The stochastic process of i’s wealth is

w it = (2δi)

tt

∏

τ=1

α1H(ωi

τ)i (1− αi)

1−1H(ωiτ)

For traders i and j,

log rt ≡ logw i

t

w jt

= t logδi

δj+

t∑

τ=1

1H(ωiτ) logαi +1T (ω

iτ) log(1−αi)−

t∑

τ=1

1H(ωjτ) logαj + 1T (ω

jτ) log(1 − αj)

ToC 83

Suppose i ∈ Ip and j ∈ Iq.

Ep

{

log δi + 1H(sit )αi + 1T (s

it )}

=

11− γi

(γi log 2 + log βi − Ip(qi) + C(p))

where C(p) = p log p + (1 − p) log(1 − p). By the SLLN, almostsurely:

1t

log(rt)→1

1 − γi

(

log βi − Ip(qi) + γi log 2 + C(p))

−

11 − γj

(

log βj − Ip(qj) + γj log 2 + C(p))

If the r.h.s. > 0, then p-almost surely,w i

t

w jt

→ ∞.

ToC 84

Let wt be the wealth of the rational group at time t .Let xt be the wealth of the irrational group at time t .Let rt = wt /xt . Take w0 = x0 = r0 = 1.

wt+1 = pδrat (2pwt) + (1− p)δrat

(

2(1 − p)wt

)

= 2δrat wt

(

p2 + (1 − p)2)

xt+1 = pδirr(2qxt ) + (1 − p)δirr(

2(1 − q)xt

)

= 2δirrxt

(

pq + (1 − p)(1 − q))

so

rt =wt

xt=

(

δrat

δirr

p2 + (1 − p)2

pq + (1− p)(1 − q)

)t

=

(

p1/1−γ + (1 − p)1/1−γ

q1/1−γ + (1 − q)1/1−γ

p2 + (1− p)2

pq + (1 − p)(1 − q)

)t

ToC 85

Results

Suppose p > 1/2.

Fact: If γ > 0, then for all q sufficiently near to 1, rt → 0 p-a.s.

Fact: If γ < 0, then for all p sufficiently near to 1/2 and qsufficiently near to 1, rt → 0 p-a.s. In this case, δrat > δirr .

How can it be that rt → ∞ and rt → 0?

ToC 86

In the DSSW analysis, wealth dynamics are derived by “applying”the SLLN to each group at each date.

◮ Analysis in the DSSW style computes

limt→∞

limN→∞

wt

xt.

◮ A correct analysis computes

limN→∞

limt→∞

wt

xt.

◮ These two limits are different:

limt→∞

limN→∞

wt

xt= 0

limN→∞

limt→∞

wt

xt= ∞

ToC 87

ToC 88

Wealth Dynamics in Competitive Markets

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Transcript of Wealth Dynamics in Competitive Markets