EXOGENOUS GROWTH MODELS -...

EXOGENOUS GROWTH MODELS

Lorenza Rossi

Goethe University 2011-2012

Course Outline

FIRST PART - GROWTH THEORIES

Exogenous GrowthThe Solow ModelThe Ramsey model and the Golden Rule

Introduction to Endogenous Growth modelsThe AK model - Romer (1990)Two sector model of Endogenous growth

SECOND PART - BUSINESS CYCLE

Introduction to NK modelThe BMW model as a static approximation of a forward-looking NKmodelThe BMW model in a closed economy: ination targeting versus TaylorrulesThe BMW model in an open economy: comparisons with the MundelFleming model

TODAY

Brief Review Growth Stylized Facts

Introduction to the Solow Model

DerivationsThe model performance and stylized facts

STYLIZED FACT 1

There is an enormous variation in the per capita income acrosseconomies. The poorest countries have per capita income that areless than 5 percent of per capita incomes in the riches countries.

The Rich and the Poor

World Population by GDP per worker

PER CAPITA GDP 1960

PER CAPITA GDP 2000

The world distribution of income in 1970

The world distribution of income in 2000

STYLIZED FACT 2

Rates of Economic Growth vary substantially across countries

STYLIZED FACT 3

Growth rates are not generally constant over time. For the world aswhole, growth rates were close to zero over most of the history buthave increased sharply in the twentieth century. For individualcountries, growth rates also change over time.

STYLIZED FACT 4

A country relative position in the world distribution of per capitaincomes is not immutable. Countries can move from being poor tobeing rich, or viceversa.

EXAMPLE: Venezuela vs Italy.

Fifteen Growth Miracles

Fifteen Growth Disasters

OTHER STYLIZED FACTS. FACT 5

In the US over the last century,

1 the real rate of return to capital, r , shows no trend upward ordownward;

2 the shares of income devoted to capital, rK/Y , and the share ofincome devoted to labor, wL/Y , show no trend;

3 the average growth rate of output per person has been positive andrelatively constant over time, i.e. the US exhibits steady, sustainedper capita income growth.

Real Per Capita GDP in the US

OTHER STYLIZED FACTS: FACT 6

Growth in output and growth in the volume of international tradeare closely related

Growth in trade and GDP 1960-1990

OTHER STYLIZED FACTS: FACT 7

Both skilled and unskilled workers tend to migrate from poor to richcountries or regions

ROBERT LUCAS: this movements of labor tell us something about realwages. The returns of both skilled and unskilled

Kaldor Stylized Facts and the Solow Model

The Solow Model

Solow seminal paper (1956). "A Contribution to the Theory ofEconomic Growth"

Check for the ability of the model to explain the stylized facts

Neoclassical model

Countries produce and consume one single good (units of GDP);There is no international trade (since there is only one good)Technology is exogenousPerfect competition in all markets

The Solow Model

The basic model is characterized by two equations1 a production function;2 a capital accummulation equation

The Solow Model

The Neoclassical Aggregate Production Function

Y (t) = F (A (t) ,K (t) , L (t))

where K (t) is physical capital, L (t) is labor and A (t) is a exogenoustechnology shift (TFP)

Technology is free; it is publicly available as a non-excludable,non-rival good.

The Solow Model: Key Assumption

Assumptions

F exhibits constant return to scale in K and L =) F is linearhomogeneous (homegeneous of degree 1)

The Solow Model: Key Assumption

Capital accumulation

K = I d K .

I = sY where s represents a constant savings rate.

d is capital depreciation rate

The Solow Model

The Inada Conditions

limK!0

FK () = ∞ and limK!∞

FK () = 0 for all L > 0 and all A

limL!0

FL () = ∞ and limL!∞

FL () = 0 for all K > 0 and all A

Important in ensuring the existence of interior equilibria.

The Solow Model

Firmsprots maximization

FOCs

From the assumption of homogeneity of degree 1

and thus rms prots are zero!

The Solow Model

The production function can be specied as follows:

Yt = K αt (AtLt )

1α with 0 < α < 1

it is Cobb-Douglas production function with constant return to scale,where A is a technology variable labour augmenting. AL are thee¢ cient units of labor. The rate of growth of technological progress isexogenous and dened as

dAdT

1A=ddtln (A) =

AA= g

The Solow Model

The Solow Model

The demand for capital implies

Rt = αK α1t (AtLt )

1α

and thus the capital share is

RKY= α

K α1 (AL)1α KY

= α constant

The labor demand implies

wt = (1 α)K αt (AtLt )

α At

and thus the labor share is

wLY= (1 α)

K α (AL)α ALY

= 1 α constant

The labor share and the capital share are constant in the longrun, in accordance with Kaldor stylized facts.

The Solow Model

Production function in terms of output per worker:

y = kαA1α

where y = Y /L and k = K/LProduction funtion in terms of e¢ ecient unit of labor per worker is

y = kα

where y = Y / (AL) and k = K/ (AL) .

The Solow Model

To see the growth implications of the model, we take the log andthen di¤erentiate y = Akα, we nd

yy= (1 α)

AA+ α

kk

notice however that kk =K

L

LK . which is equal to

KL

LK=LKKL LKL2

thenkk=KK LL

where the labor force growth rate is exogenous and given by LL = n.

The Solow Model

We can express capital in terms of e¢ cient unit of labor per worker

k =KAL

thenkk =

KK

LL

AA

The Solow Model

Remember that K = I d K . This means that KK can be writtenas,

KK= s

YK d

if YK is constant, then KK is also constant.

While from kk =

KK

LL we can rewrite,

KK=kk+LL= s

Y /LK/L

d

thenk = sy (d + n) k = skαA1α (d + n) k

is the law of motion of capital per worker, whilek = sy (g + d + n) k = skα (g + d + n) k

is the law of motion of capital per unit of e¢ cient labor.

The Solow Model

An economy start with a given stock of capital per worker k0, a givenpopulation growth rate, n, a given technology growth rate, g , and agiven investment rate.

1 How does output per worker, y , (or output per e¢ cient units of laborper worker, y) evolve over time?

2 How does the economy growth?3 How does output per worker compare in the long-run between twoeconomies that have di¤erent investment rate?

4 How does output per worker compare in the long-run between twoeconomies that have di¤erent technology growth rate?

The Solow Model

With AA = 0, then capital per worker is

k = skα (d + n) k

in the steady state k = 0 and

k =

sd + n

11α

thus

y = (k)α =

s

d + n

α1α

In the Solow model countries with higher savings/investment rate willtend to be richer, ceteris paribus. Such countries have more capitalper worker and thus more output per worker.Countries with high population growth rate, in contrast will tend tobe poorer according to the Solow model.

The Solow Model

The Solow diagram determines the steady state value of output perworker

The dynamics converges to the steady state value of capital perworkers k.

The Solow Model

Balanced growth. Notice thatk = skα (d + n) k

kk

= 0 impliesKL= constant =) K

K= n

yy

= 0 impliesYL= constant =) Y

Y= n

cc

= 0 impliesCL= constant =) C

C= n

KY

= k1α =s

d + nin steady state

The Solow Model

The Solow diagram and household consumption

The Solow Model. The Golden rule

Consumption isct = F (kt ) sF (kt )

At the steady state consumption is

c = F (k) sF (k) = (k)α (d + n) k

consumption is maximum if

∂c∂k= 0 : F 0 (k) (d + n) = 0


∂c∂k = 0 implies

α (k)α1 (d + n) = 0and thus, solving for k

kGR =

α

d + n

11α

we call kGR the Golden rule capital stock, that is the value of k suchthat ∂c

∂k = 0 and consumption is maximum.


DYNAMIC EFFICIENCY. If kGR > k1 =) s1 < sGR and c1 < cGR .Increasing savings increases also the steady state per capitaconsumption. However, by increasing savings, consumption decreaseson impact.

DYNAMIC INEFFICIENCY. If kGR < k2 =) s2 > sGR andc2 < c

GR . The economy is oversaving and is said to be dynamicallyine¢ cient.

The Solow Model

An increase in the investment rate s

the steady state value of capital per worker increases.

The Solow Model

An increase in population growth

The Solow Model

Consider now AA 6= 0 = g . In the steady state

k = 0 and

k =

sd + n+ g

11α

thus

y = kα =

s

d + n+ g

α1α

while output per worker is

y (t) =

sd + n+ g

α1α

A (t)

where t is included just to remind that A is an exogenous growingvariable.

The Solow Model

Consider now AA 6= 0 = g . The Solow diagram becomes

The Solow Model

Balanced growth. Notice thatk = skα (g + d + n) k

kk

= 0 impliesKK= g + n and

kk= g

yy

= 0 impliesYY= g + n and

yy= g

cc

= 0 impliesCC= g + n and

cc= g

nally notice that given KK =

YY =)

KK

YY = 0 and thus

KY=

sg + n+ d

= constant

The Solow Model

THE BALANCED GROWTH PATH. In the Solow model thegrowth process follows a balanced growth path if the GDP perworker, consumption per worker, the real wage per worker andcapital per worker, all grow at one and the same constant rate,g , the labor force, i.e. population grows at a constant rate n,GDP, consumption and capital grow at a common rate, g + n,the capital-output ratio is constant and the rate of return oncapital is constant.

The Solow Model

An increase in the savings rate

The Solow Model

An increase in the savings rate and conditional convergence.

The Solow Model

Output per worker along the balanced growth path isdetermined by technology, investment rate and the populationgrowth rateChanges in the investment rate and the population growth ratea¤ect the long-run level of output per worker, but do not a¤ectits long-run growth rate.Policy changes do not have long-run growth e¤ects.Policy changes can have level e¤ects, that is a permanentpolicy change can permanently raise (or lower) the level of percapita output.Conditional convergence.

The Ramsey Growth Model

The Solow model:

The rst general equilibrium model with production side.

It is empirically testable.

Lacks microfoundations (saving is not determined exogenously)

Exogenous technological progress explains all.

The Ramsey Model

Introduces endogenous savings/consumption decision

Optimal consumption

The decentralized equilibrium can be compared with the Paretoe¢ cient equilibrium


The decentralized model

Two agents

Households. maximize their life-time utility subject to an intertemporalbudget constraintFirms: maximize prots subject to their factor accumulation constraint

General Equilibrium. Demand=Supply.

The Social Planner model

The social planner maximizes the householdsutility subject toaggregate resource constraint of the economy.

If the decentralized solution coincide with the Social Planner solution,then the outcome is Pareto optimal.


The Household Problem

One innitely-lived household maximizing the intertemporal utilityZ ∞

0u (ct ) Lteρtdt =

Z ∞

0u (ct ) Lte(nρ)tdt

where ρ is the discount factor. A higher ρ implies a smallerdesiderability of future consumption in terms of utility compared toutility obtained by current consumption.

where u0 (c) > 0 and u00 (c) < 0 and limc!0 u0 (c) = ∞ andlimc!∞ u0 (c) = 0 (Inada Conditions)

Population growth rate is constant (equal to n) and at time t = 0there is only one individual in the economy (i.e. L0 = 1), so that thetotal population at any time t is given by Lt = ent and thus LL = n.



The household budget constraint is

B = wtLt + rtBt Ct

where B are assets, w is the wage rate, r is the interest rate. In percapita terms

b = wt + (rt n) bt ct



The trasversality condition

limt!∞

0BBBBB@btetZ0

(rsn)ds

1CCCCCA 0

the present value of current and future assets must be asymptoticallynonnegative

households cannot borrow innitely until the end of their economiclife cycle

the householdsdebt cannot increase at a rate asymptotically higherthan the interest rate



Writing the Hamiltonian

H =Z ∞

0u (ct ) e(nρ)tdt + µt (wt + (rt n) bt ct )

FOCs with respect to ct (control variable) and bt (state variable).

∂H∂c

= 0 : u0 (ct ) e(nρ)t = µt

∂H∂b

= µ : (rt n) µt = µ

∂H∂µ

= b : wt + (rt n) µt ct = b (is the constraint)



From the rst FOC

u0 (ct ) e(nρ)t = µt =) µt = u00(ct ) e(nρ)t c+(n ρ) u0 (ct ) e(nρ)t

combining with the second FOC, i.e. with (rt n) µt = µ, we get

(rt n) µt = u00(ct ) e(nρ)t c + (n ρ) u0 (ct ) e(nρ)t

or

(rt n) u0 (ct ) e(nρ)t = u00(ct ) e(nρ)t c + (n ρ) u0 (ct ) e(nρ)t

simplifying and rearranging we nd the Euler equation

rt = ρ u00(c) cu0 (c)

!cc



Suppose that the utility is u (c) = c1θ

1θ . =) u00(c )c

u 0(c ) = θ and theEuler equation becomes:

cc=1θ(rt ρ)

The higher r , the more willing households are to save and shiftconsumption in the future.

The higher the rate of return to consumption is, the more willinghouseholds are to sacrice future consumption for more currentconsumption and thereby less current saving.


Firms Problem

Prots Maximization

π = F (K , L) (r + δ)K wL

with F (K , L) Cobb-Douglas with constant return to scale

Focs of the static problem

f 0 (k) = (r + d)

f (k) kf 0 (k) = w


Equilibrium

The law of motion of per capita capital

k = f (k) (n+ d) k c

The law of motion of per capita consumption

cc=1θ

f 0 (k) d ρ

it is a 2x2 system of di¤erential equation which denes theequilibrium.


The Steady State

k = 0 : f (k) c = (n+ d) k

The law of motion of per capita consumption

cc= 0 : f 0 (k) = d + ρ


The Dynamics

Notice that ∂k∂c = 1 < 0, while

∂c∂k =

1θ f

00(k) c < 0


The Dynamics

Remember that c is a jumping variable while k is a predeterminedvariable.


Variation in the rate of time preferences ρ : a decreases in ρdenotes that households have become less impatient and that theyconsider the value of future consumption higher. Thus, they arewilling to sacrice present consumption.

For a given level of k, a decrease in ρ lowers the consumption growthrate c , and hence, shifts the c = 0 schedule to the right.


To maintain consumption constant the capital stock must increase sothat, in the new long-run equilibrium, consumption, capital andoutput per capita attain a higher level (k 0 > k, c 0 > c).Consumption jumps along the saddle path, whereas k at the samelevel k. The reduction in consumption boosts saving and therebyleads to a increase in capital stock, until it reaches k 0 . =) incomeper capita level rises, with positive consumption growth rate untilreaching the higher levels of consumption and capital (c 0, k 0), andincome.


The Modiced Golden Rule

In the Solow model the Golden rule implies

f 0 (k) = n+ d

In the Ramsey model, the modied gold rule, which is obtained fromconsumer problem implies

f 0kRamsey

= ρ+ d

Given that the transersality condition implies that ρ > n.

+

In the Ramsey model, the long-run equilibrium level of capital perworker is lower that in the Solow model. In the Ramsey modelhouseholds save less since there is discounting of future utility.


Is the decentralized equilibrium of the economy also the rstbest-outcome (the Pareto e¢ cient equilibrium)?

The social planner aims at maximizing householdsintertemporalutility subject to the aggregate resource constraint, which shows howthe GDP is allocated to di¤erent uses.

In a closed economy with no government, the aggregate output Y isused either for consumption C or for investment in physical capital, I ,thus : Y = C + I

Using that I = K + dK

Y = F (K , L) = C + K + dK

In terms of variables per worker

f (k) = c + k + (d + n) k


The Social Planner Problem is

maxfc ,kg

Z ∞

0

c1θ

1 θe(nρ)tdt

s.t. f (k) = c + k + (d + n) k

and the Hamiltonian associated is

H =c1θ

1 θe(nρ)t µ [c + (d + n) k f (k)]

FOCs wrt. consumption and capital

∂H∂c

= 0 : cθe(nρ)t µ = 0

∂H∂k

= µ : µ (d + n) µ = 0


combining the FOC

cc=1θ

f 0 (k) (ρ+ n)

+ the low o motion of capital accumulation

k = f (k) c (d + n) k

in the steady state c = k = 0 and

c = f (k) (d + n) kf 0 (k) = (ρ+ n)

the Social Planner solution coincides with the decentralized solution.=) The competitive equilibrium in the Ramsey model is Paretooptimal.

The Solow Model with Human Capital

Mankiw, Romer and Weil (MRW 1992), "A Contribution to the Empiricsof Economic Growth".

They evaluate the performance of the Solow modelThey claim that the t of the model can be improved by extendingthe model to include human capital HWe consider a simplied version of MRW model.The Cobb-Douglas production function becomes

Y = K α (AH)1α

A is the labor augmenting technology, which grows exogenously at arate g and H is human capital, described by

H = eψuL,

where ψ is a positive constant and u is the fraction of individualstime spent learning skills. L denotes the total amount of labor used inproduction in the economy.


Notice that if u = 0 then H = L,that is all labor is unskilled.

By taking the rst the log and then the derivative of H with respectto u

log (H) = ψu + log (L)

thend log (H)du

= ψ =) dHdu

= ψH

example u = 1 (one additional year of schooling), with ψ = 0.1 thenH increases by 10%.

This forumlation is intended to match the empirical literature onlabor economics that nds that an additional year of schoolingincreases the productivity of the worker and than its wage by 10%.


Physical capital accumulates according to

K = skY dK

where sK is the investment rate in physical capital and d is thedepreciation rate.The production function in terms of output per worker is

y = kα (Ah)1α

where h = eψu is a constant.We can redine the production function in terms of y/(Ah) = y

y = kα

and the capital accumulation becoms

k = sk y (n+ g + d) k


THE STEADY STATEk = 0 implies

ky=

sk(n+ g + d)

substituting in the production function

y =

sk(n+ g + d)

α1α

in terms of output per worker

y (t) =

skn+ g + d

α1α

hA (t)

where t is included just to remind that A is an exogenous growingvariable.


From

y (t) =

skn+ g + d

α1α

hA (t)

we can state that some countries are rich because they:1 have higher investment in physical capital;2 spend a large fraction of time accumulating skills h = eψu ;3 have low population rate;4 have higher level of technology.


To see how well the model performs empirically in explaining whysome country are rich and other are poor, we can consider relativeincome with respect to the US economy

y =y

y US

or

y =skx

α1α

hA

where x = (n+ g + d) .


The Fit of the Neoclassical Growth model, 1997 (source: Jones book)


The lack of convergerce for the world.

Conditional convergence vs absolute convergence

The Green Solow Model and the Environmental KunetzCurve

Brock and Taylor (BT), 2010, Journal of Economic Growth

Environmental Kunetz Curve (EKC): is a humpshaped relationshipbetween environmental degradation and per capita income. At lowlevel of economic activity, the environment is worsening. As theeconomic activity increases environmental degradation peacks. Then,as a country becomes richer and richer, environmental degradationbegins to fall.

BT use a simple variant of the basic Solow model with DR in capitaland technologycal progress for abatement. They provide a theoreticalexplanation of the EKC and for abatement and emission intensity.They also derive an estimating equation for pollution convergence.


US evidence:1 Emissions per unit of output have been falling for a long period oftime


US evidence:2. Emission per unit of output were falling before absolute emissionsstarted to fall


US evidence:

3. Abatement cost per unit of output have been relatively constant


EU evidence:

European data show a strong evidence of an Environmental Kunetzcurve relationship

The evidence on cross-country data is mixed

The Green Solow Model

The Green Solow model augments the standard Solow model withpollution and abatement activityThe production function is

Y = F (K ,BL)

where B is the labor augmenting technology.the low of motion of capital is standard

K = sY dKNet Emission of Pollution E

E = ΩF ΩAF ,FA

= ΩF

1 A

1,FA

F

= ΩFa (θ)

where Ω is pollution from Y , A is the abatement with CRS productionfunction A

1, F

A

F

, and technology growth at exogenous rate gA. F

is total production activity, FA is total abatement activity. θ = F AF

and a (θ) = 1 A (1, θ) with a (0) = 1, a0 (θ) < 0, and a00 (θ) > 0.


Output available for consumption and investment is

Y = F FA = (1 θ) F

the model can be written in terms of e¢ cient units of labor (EUL) asfollows:

y = f (k) (1 θ)

k = sy (n+ d + gB ) ke = f (k)Ωa (θ)

where k = K/BL, y = Y /BL and e = E/BL.


Notice that a (θ) =constant. Given

e =EBL

= f (k)Ωa (θ)

we can rewrite

E = BLf (k)Ωa (θ) = BLkαΩa (θ)

and thus, taking the growth rates

EE=BB+LL+ α

kk+

ΩΩ


Easy to understand if we rewrite

EE= α

kk|z

transitionalcomponent

+ gB + n gA| z gE

Emission growthalong the BGP


The growth rate of emission along the balanced growth path, i.e.along k

k = 0 isgE = gB + n gA

we can dene substainable growth as the condition implying gE < 0

1) gB > 0

2) gA > gB + n

1) gB > 0 is needed in order to have a positive growth of per capitaincome; 2) the technological progress in emission abatement has tobe greater than the rate of growth of percapita GDP because ofgrowth of population


ConsideringEE= α

kk+ gB + n gA

andkk= skα1 (1 θ) (d + n+ gB )

Assuming substainable growth, i.e. gE < 0 and multiplying by α thesecond equation

αkk= αskα1 (1 θ) α (d + n+ gB )


PROPOSITION 1

If growth is substainable gE < 0 and k (0) < k (T ) , then emissionsgrow initially and then fall continuously. An EKC occurs.If growth is substainable gE < 0 and k (0) > k (T ) , then emissionsfall continuouslyIf growth is unsubstainable gE > 0, then emissions growth but at adecreasing rate.

PROPOSITION 2

Identical economies with di¤erent initial values produce di¤erent percapita income and emission proles over time. The peak level ofemissions and the associated level of per capita income are not unique.This can explain the mixed evidence on the EKC in cross-country data.It is thus important to control for initial conditions and unobservedeterogeneity.


The econometric results reported by Brock and Taylor (2010) in Table2 show that

1 the coe¢ cient on lneCis negative and statistically signicant across

countries. Thus conditional convergence in per capita emissionsemerges.

2 The coe¢ cient on ln(s) is positive and the coe¢ cient on ln(n+ g + d)is negative, as predicted by the green Solow model.

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