Endogenous Growth Models - economia.unipv.iteconomia.unipv.it/pagp/pagine_personali/lorenza.rossi/LECTURES... · Endogenous Growth and Learning Assuming that population L is constant

Endogenous Growth Models

Lorenza Rossi

Goethe University 2011-2012

Endogenous Growth Theory

Neoclassical Exogenous Growth Models

technological progress is the engine of growthtechnological improvements are automatic and unmodeled (exogenous)

Endogenous Growth Models

Try to explain the engine of growthIt is important to understand the economic forces underlyingtechnological progress

Endogenous Growth and Learning

IDEA: Capital accumulation embeds technological improvements(Arrow 1962 =)Romer 1982)Firms production function

Y (i) = AK (i)α L (i)1�α

where A is the Total Factor Productivity (TFP).

Technology A depends on Capital Stock. The higher the capital stockthe more the economy is able to use new technologies

A = BK 1�α

where K is the aggregate level of capital stock and B is the learningfactor (positive externality). Imposing symmetry across �rms andsubstituting in the production function, we get the aggregateproduction function

Y = BKL1�α


Assuming that population L is constant and equal to 1. Then, theaggregate production function becomes,

Y = BK

This production function is characterized by constant return to scale.The marginal productivity of capital is constant and equal to theaverage productivity of capital and is B.The low of motion of capital is

K = sY � dK

hence the growth rate of capital is

KK= s

YK� d = sB � d

given that YK = B =constant,KK =

YY . If sB > d =) the growth rate

is positive.


NOTICE!!!! IMPORTANT!! The rate of growth of A is

AA= (1� α)

KK= (1� α) (sB � d)

Contrary to the Solow model, the rate of growth of technologydepends on the rate of growth of capital. At the same timetechnology a¤ects capital. Growth is an endogenous process.

No transitional dynamics

An increase in savings means that the growth rate increasespermanently.


How to introduce a transitional dynamics

Suppose thatA = B0 + B1K 1�α

thenY = B0K α + B1K

and the rate of growth of capital

KK= sB0K α�1 + sB1 � d

the rate of growth of K is decreasing in K and converges to sB1 � d .


Endogenous growth plus transitional dynamics


Human capital and Endogenous Growth (Lucas 1988).

The production function

Y = K α (AL)1�α

whereA = H

human capital increases labor productivity, with L = 1

Y = K αH1�α


De�ne sK as the amount of GDP spend for capital accumulation. Forsimplicity and without loss of generality, we now assume that thecapital depreciation rate is d = 0. Hence,

K = sKY = sKKαH1�α

De�ne sH as the amount of GDP spent for human capitalaccumulation.

H = sHY = sHKαH1�α


De�ne γ = HK . substituting in the low of motion of capital and

dividing by KKK= sKγ1�α

SimilarlyHH= sHγ�α

Consider thatγ

γ=HH� KK

If HH >KK =)

γγ > 0 and γ increases. If γ increases KK increases,

while HH reduces, so that

γγ decreases. On the contrary if

HH <

KK =)

γγ < 0 and γ decreases. If γ decreases KK decreases,

while HH increases, so that

γγ increases. The process stops only when

HH =

KK and

γγ = 0.


If γγ = 0, then γ is equal to its steady state value, which is obtained

takingH/HK/K

=sHγ�α

sKγ1�α= 1

solving for γ

γ� =sHsK

Substituting this value in the low of motion of physical and human capital�KK

��= sα

K s1�αH =

�HH

��


The GDP growth rate is

YY= α

KK+ (1� α)

HH

hence in the steady state of γ�YY

��= sα

K s1�αH

Barro�s model of Endogenous Growth with GovernmentSpending and Taxation

Barro (1990) suggests a simple endogenous growth model withgovernment.

In the Barro model public spending goes for public investment(infrastructures, schools, sanitation etc.).

Public investments, which are �nanced through income taxes,complement private investments.

Since public investments raise the productivity of private investments,higher taxes can be associated with an increase or a decrease inoverall growth.


The model. Barro (1990) adds public spending to the Romer AKmodel.

Y = BK 1�αG α

whereG = τY

substituting into the production function

Y = K 1�α (τY )α

solving for YY = B

11�α

�τ

α1�α

�K = B (τ)K

where B (τ) = B11�α

�τ

α1�α

�


The low of motion of capital

K = s (1� τ)Y � dK

Then,

KK= s (1� τ) B (τ)� d = s (1� τ)B

11�α

�τ

α1�α

�� d

thus if s (1� τ) B (τ) > d =) KK > 0

Which is the e¤ect of taxation on growth? The economy faces aLa¤er CurveWhich is the optimal tax rate, i.e. the tax rate maximizinggrowth?We consider two models. 1) a model with exogenous savings; 2) Amodel with endogenous savings (Ramsey approach)

Barro�s model and the La¤er curve

Optimal taxation in a model with exogenous savings

It is su¢ cient to take the derivative of KK wrt τ and set equal to zero.

∂ KK∂τ

= 0 : �sB 11�α τ

α1�α +

α

1� αs (1� τ)B

11�α τ

�1+2α1�α = 0

solving forτ� = α

which is the optimal tax rate, i.e. the tax rate that maximizes growth.

Optimal taxation in a model with exogenous savings

The Barro model with endogenous savings

Optimal taxation in a model with endogenous savings

For simplicity, and without loss of generality, we assume thatpopulation is constant and equal to L = 1, and that capitaldepreciation rate is d = 0.

Given that L = 1 and constant, this means that per capita variablesare identical to variables in level, C = c ,Y = y ,K = k.

Then, the decentralized Ramsey problem is

maxfC ,K g

C 1�θ

1� θe�ρt

s.t. K = (1� τ)Y � CY = BK 1�αG α


The present value Hamiltonian associated is

H =C 1�θ

1� θe�ρt � µ

�(1� τ)BK 1�αG α � C

�FOCs wrt. consumption, capital and the costate variable are:

1.∂H∂C

= 0 : C�θe�ρt � µ = 0

2.∂H∂K

= �µ : �µ (1� τ) (1� α)BK�αG α � µ = 0

3.∂H∂µ

= K : (1� τ)BK 1�αG α � C = K

notice that G α = Bα1�α τ

α1�αK α.


Combining FOCs 1. and 2.

CC

=1θ

24(1� τ) (1� α)BK�αG α| {z }MPK

� ρ

35=

1θ

24(1� τ) (1� α)B11�α τ

α1�α| {z }

MPK

� ρ

35where MPK states for Marginal Product of Capital.


Notice that the MPK is

MPK = (1� τ)| {z }negative e¤ect of taxation

(1� α)BK�α G α|{z}positive e¤ect of public investment

Growth in consumption depends on: i) the gap between the MPK andthe rate of time preference ρ; ii) the intertemporal elasticity ofsubstitution θ.

Thus, Government a¤ects the MPK through two channels: i) increasein G raises the MPK to a point; ii) taxes always reduces the privatereturn of capital.

The main objective of a good Government is to balance these twoe¤ects.


The tax rate maximizing consumption is obtained by di¤erentiating CC

w.r.t. τ.

∂(C/C )∂τ = 1

θ (1� τ)�

α1�α

�(1� α)B

11�α τ

2α�11�α � 1

θ (1� α)B11�α τ

α1�α =

0

simplifying and solving for τ

τGR = α

the same value we found for ∂ KK∂τ


Is the Decentralized solution also the �rst best solution?It is important to compare the decentralized solution with the SocialPlanner one.

Which is the Social Planner solution?The Social Planner internalizes the e¤ect of G and thus the optimalproblem becomes

maxfC ,K ,G g

C 1�θ

1� θe�ρt

s.t. Resource Constraint

i .e. : Y = C + I + G

or : K = Y � C � G = BK 1�αG α � C � G


The present value Hamiltonian of the Social Planner is

H =C 1�θ

1� θe�ρt � µ

�BK 1�αG α � C � G

�The Social Planner FOCs wrt. consumption, capital and the costatevariable are:

1s.∂H∂C

= 0 : C�θe�ρt � µ = 0

2s.∂H∂G

= 0 : αBK 1�αG α�1 = 1 =) ∂Y∂G

= 1

3s.∂H∂K

= �µ : �µ (1� α)K�αG α � µ = 0

4s.∂H∂µ

= K : BK 1�αG α � C � G = K


Combining FOCs 1s. and 2s.

CC=1θ

h(1� α)B

11�α τ

α1�α � ρ

iNotice that (1� α)B

11�α τ

α1�α > (1� τ) (1� α)B

11�α τ

α1�α , hence the

MPK in the decentralized solution is (1� τ) ∂Y∂K , which is smaller

than what we get from the Social Planner solution, i.e. the socialmarginal product ∂Y

∂K , because of the tax rate. This gap betweensocial and private returns leads to a lower growth rate in thedecentralized solution.

Endogenous Growth and R&D Sector

The Romer model try to explain why and how advanced countries ofthe world exhibit sustained growth.

Technological progress is driven by R&D sector in advancedworld.Romer endogenizes technological progress by introducing an R&Dsector, i.e. search of new ideas by researcher interested in pro�tingfrom their invention.

The aggregate production function in the Romer model is

Y = K α (ALY )1�α

Capital accumulation is

K = sKY � dK

population growth is LL = n.


The key equation of the Romer model is the one describing the R&Dsector.

According to Romer A is the number of ideas, or the stock ofknowledge accumulated up until time t.

The number of new ideas A is equal to the number of people devotingtheir time in discovering new ideas LA, multiplied by the rate at whichthey discover new ideas, i.e. δ. Thus,

A = δLA

Labor is used either to produce good, LY , or to produce new ideasLA. So the economy faces the following resource constraint:

L = LY + LA


The rate at which new ideas are discovered, δ, might be constant, oran increasing function of A

δ = δAφ

where δ and φ are constants.

Notice that with φ > 0 the productivity of research increases with thestock of ideas that have already been discovered. On the contrarywith φ < 0, discovering new ideas becomes harder over time. Withφ = 0 the discovery rate is independent from the stock of knowledge.


It is possible that new ideas are more likely when there are morepersons engaged in research. Thus, the e¤ect of LA is notproportional. Hence, it can be assumed that it is Lλ

A that enter in theproduction function of new ideas, with 0 < λ < 1. The generalproduction function of new ideas is

A = δLλAA

φ

Assuming that 0 < φ < 1. Dividing by A

AA= δ

LλA

A1�φ

which is the rate of growth along the BGP?


Along the BGP AA = gA = constant. Thus, the numerator and the

denominator should growth at the same rate, which means

λLALA� (1� φ)

AA= 0

along the BGP LALA= n and thus

AA= gA =

nλ

1� φ

In this model, as in the Neoclassical model, even if growth is anendogenous process, policy maker cannot do nothing to increase thelong-run growth rate. Indeed bot λ and φ are parameters independenton policies, such as subsidies to R&D


Introducing Microfoundation. Romer (1990 JPE)Romer (1990) explains how to construct an economy ofpro�ts-maximizing agents that endogenize technological progress.

The economy consists of three sectors:1 A �nal good-producing sector2 An intermediate good-producing sector: producing capital goods3 A research sector

The research sector sells the exclusive right to produce a speci�ccapital good to an intermediate-good �rm. The intermediate-good�rm, is monopolist, manufactures the capital good and sells it to the�nal good sector which produces output.


The �nal-good sector is composed by a large number of perfectlycompetitive �rms that combine labor and capital to produce the �nalgood, Y . There is more than one type of capital in the productionfunction, thus it is speci�ed as follows

Y = L1�αY

N

∑j=1xαj

where the capital goods xj , come from the intermediategood-producing sector.

Inventions, or new ideas correspond to the creation of new capitalthat can be used by the �nal-good sector to produce the �nal output.


The �nal-good sectorIf A is the number of capital goods. Then N = A and the productioncan be rewritten as

Y = L1�αY

A

∑j=1xαj

if the number of goods is continuos

Y = L1�αY

Z A

0xαj dj

For simplicity we will use the second de�nition. Notice that, whetherwe use a discrete number of goods or a continuos number, resultsremain unchanged.


Final good price P is normalized to 1.Firms in the �nal-good sector, choose labor and capital to maximizepro�ts,

maxfLY ,xJ g

L1�αY

Z A

0xαj dj � wLY �

Z A

0pjxjdj

where pj is the rental price for capital-goods and w the wage paid forlabor.

The FOCs imply:

w = (1� α)YLY

pj = αL1�αY xα�1

j for each j

As usual prices of inputs equate their marginal product.


The intermediate good sector consists of monopolists who producethe capital goods to sell to the �nal sector.Firms gain their monopoly power by purchasing the design for aspeci�c capital good from the R&D sector. Because of patentprotection only one �rm manufactures each capital good.Each �rm uses a very simple production function. One unit of rawcapital (purchased in the R&D sector) translates into one unit ofmanufactured capital.The pro�t maximization problem of the representativeintermediate-good �rm is

maxxjpj (xj ) xj � rxj

where pj (xj ) is the demand function of the capital good,corresponding to pj = αL1�α

Y xα�1j and r is the interest rate, or the

rental rate of capital.


The FOC of the intermediate-good �rm is.

p0j (xj ) xj + pj (xj )� r = 0

α2L1�αY xα�1

j| {z }αpj

� r = 0

Imposing symmetry and solving for p

p =1

1+ p 0(x )xp

r =1αr .

which is the optimal price set in the intermediate-good sector.


Equilibrium and AggregationThe total demand for capital from the intermediate good sector mustequal the total capital stock in the economy. Thus,Z A

0xjdj = K

Since the capital goods are each used in the same amount, x , theprevious equation can be used to determine x

x =KA

The �nal good production function can be rewritten as

Y = L1�αY

Z A

0xαdj = L1�α

Y Axα

substituting for x = KA

Y = K α (ALY )1�α


In the Research Sector new design are discovered according to

A = δLλAA

φ

When a design is discovered, the inventor receives a patent from theGovernment for the exclusive right to produce the new capital good.The patent last forever.The inventor sells the patent to an intermediate good �rm and usesthe proceeds to consume and save.What is the price of a new patent?Anyone can bid for a patent. The potential bidder will be willing topay the discounted value of the pro�ts earned by anintermediate-good �rm.Let the discounted value of pro�ts earned by an intermediate-good�rm be PA, where pro�ts are:

π = α (1� α)YA


The research sectorHow does PA change over time? Firms can put money (an amountequivalent to the value of a patent, PA), in a bank, earning theinterest rate r . Alternatively, they can purchase patent for one period,manufacture capital, earn pro�ts and then sell the patent. Inequilibrium the return of these two alternatives must be the same.Thus,

rPA = π + PAWhich gives

r =π

PA+PAPA

Along the BGP r is constant and thus π and PA must grow at thesame rate, which is the population growth rate n (when λ = 1 andφ = 0). Thus, along the BGP

PA =π

r � n


Share of population working in the R&D and good producingsectorOnce again we can use the arbitrage concept. It must be the casethat at the margin, individual are indi¤erent between working in the�nal-good sector or the R&D sector.

We know that in the �nal-good sector

wY = (1� α)YLY

in the R&D sector, real wages are equal to the marginal product oflabor δ, multiplied by the value of new ideas created, i.e. PA, thus

wR = δPA


Because there is free entry in the two labor markets it must be thatwY = wR , then

(1� α)YLY

= δPA =δπ

r � n =δα (1� α) YA

r � nthen

1LY

=δα (1� α)

r � nYA(1� α)

Y=

α

r � nδ

A

Rearranging and considering that A = δLA =) AA =

δLAA = gA along

the BGP, then1LY

=α

r � ngALA

LALY= αgA

r�n =sR1�sR and sR =

LAL is

sR =1

1+ r�nαgA

.


OPTIMAL R&D. Is the share of population involved in R&Dsector optimal?The answer is no. Why? The economy is characterized by threedistortions

1 The market does not endogenize the fact that new research may a¤ectthe productivity of future research. φ > 0, implies that productivity ofresearch increases with the stock of ideas. Researcher are notcompensated for their contribution toward improving the productivityof future researcher. Thus, with φ > 0 the market provides too littleresearch and the fraction of population hired by R&S is too low. Thise¤ect is called spillover e¤ect or "standing on the shoulders e¤ect".

2 With λ < 1 research productivity is lower because of duplications.Thus, too many people are hired by the research sector. This e¤ect iscalled "stepping on toes e¤ect".

3 Consumer surplus e¤ect. The monopoly pro�ts are less than theconsumer surplus. This e¤ect tends to generate too little innovations.


OPTIMAL R&DClassical economic theory: imperfect competition and monopoly arebad for welfare and e¢ ciency because they generate adeathweight-loss in the economy. This happens because prices arehigher than marginal costs. However, the literature on the economicof ideas suggests that it is the possibility to make pro�ts, and thus toset a markup over marginal costs, that incentives �rms, or the R&Dsector, to produce more ideas.

This means, that there is a trade-o¤ between short-run losses andlong-run gains.

Concluding. In deciding antitrust policies, the regulator has toweight the deathweight losses against the incentive to innovate.

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Endogenous Growth Models - economia.unipv.iteconomia.unipv.it/pagp/pagine_personali/lorenza.rossi/LECTURES... · Endogenous Growth and Learning Assuming that population L is constant