Goethe University Lectures on Endogenous Growth

8/10/2019 Goethe University Lectures on Endogenous Growth

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Endogenous Growth Models

Lorenza Rossi

Goethe University 2011-2012
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Endogenous Growth Theory

Neoclassical Exogenous Growth Models

technological progress is the engine of growth

technological improvements are automatic and unmodeled (exogenous)

Endogenous Growth Models

Try to explain the engine of growth

It is important to understand the economic forces underlyingtechnological progress
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Endogenous Growth and Learning

IDEA: Capital accumulation embeds technological improvements

(Arrow 1962 =)Romer 1982)

Firms production function

Y(i)=AK(i) L (i)1

where Ais the Total Factor Productivity (TFP).TechnologyAdepends on Capital Stock. The higher the capital stockthe more the economy is able to use new technologies

A=BK1

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
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Assuming that population L is constant and equal to 1. Then, the

aggregate 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

K

K =s

Y

K d=sB d

given that YK

=B=constant,KK

=YY

.IfsB> d=)the growth rate

is positive.
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NOTICE!!!! IMPORTANT!! The rate of growth ofA is

A

A=(1 )

K

K =(1 ) (sB d)

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

No transitional dynamics

An increase in savings means that the growth rate increasespermanently.
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How to introduce a transitional dynamics

Suppose thatA=B0+ B1K

1

thenY =B0K

+ B1K

and the rate of growth of capital

KK =sB0K1 + sB1 d

the rate of growth ofKis decreasing in Kand converges to sB1 d.
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Endogenous growth plus transitional dynamics
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Human capital and Endogenous Growth (Lucas 1988).

The production function

Y =K

(AL)1

whereA=H

human capital increases labor productivity, with L= 1

Y =KH1
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Dene sKas 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 =sKKH1

Dene sHas the amount of GDP spent for human capitalaccumulation.

H=sHY = sHKH1
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Dene = HK

. substituting in the low of motion of capital and

dividing by KK

K =sK

1

SimilarlyH

H =sH

Consider that

=

H

H

K

K

If

H

H >

K

K =)

>

0 and increases. If increases

K

K increases,while

HH

reduces, so that decreases. On the contrary ifHH d=)KK > 0

Which is the eect of taxation on growth? The economy faces aLaer Curve

Which is the optimal tax rate, i.e. the tax rate maximizinggrowth?

We consider two models. 1) a model with exogenous savings; 2) A

model with endogenous savings (Ramsey approach)
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Barros model and the Laer curve

Optimal taxation in a model with exogenous savings

It is sucient to take the derivative ofKK

wrt and set equal to zero.

KK

=0 :sB 11

1 +

1

s(1 ) B 11

1+21 =0

solving for =

which is the optimal tax rate, i.e. the tax rate that maximizes growth.
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Optimal taxation in a model with exogenous savings
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The Barro model with endogenous savings

Optimal taxation in a model with endogenous savingsFor 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,Kg

C1

1 et

s.t. K = (1 ) Y C

Y = BK1G
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The present value Hamiltonian associated is

H= C1

1 et

(1 ) BK1G C

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

1.H

C = 0 :Cet =0

2.H

K

= : (1 ) (1 ) BKG =0

3.H

= K :(1 ) BK1G C= K

notice that G =B 1

1 K.
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Combining FOCs 1. and 2.

C

C =

1

24(1 ) (1 ) BKG| {z }

MPK35

= 1

2

4(1 ) (1 ) B

11

1

| {z }MPK

3

5where MPK states for Marginal Product of Capital.
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Notice that the MPK is

MPK= (1 )

| {z }negative eect of taxation

(1 ) BK G|{z}positive eect of public investment

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

Thus, Government aects the MPK through two channels: i) increase

in Graises the MPK to a point; ii) taxes always reduces the privatereturn of capital.

The main objective of a good Government is to balance these twoeects.
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The tax rate maximizing consumption is obtained by dierentiating CC

w.r.t. .

( C/C) =

1

(1 ) 1(1 ) B 11

211

1

(1 ) B

11

1

=0

simplifying and solving for

GR =

the same value we found for

KK
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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 eect ofGand thus the optimalproblem becomes

maxfC,K,Gg

C1

1 et

s.t. Resource Constraint

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

or : K=Y C G=BK1G C G
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The present value Hamiltonian of the Social Planner is

H= C1

1 et

BK1G C G

The Social Planner FOCs wrt. consumption, capital and the costate

variable are:

1s.H

C = 0 : Cet =0

2s.H

G

= 0 :BK1G1 =1=) Y

G

=1

3s.H

K = : (1 ) KG =0

4s.H

= K :BK1G C G= K
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Combining FOCs 1s. and 2s.

C

C =

1

h(1 ) B

11

1

iNotice that (1 ) B

11

1 > (1 ) (1 ) B

11

1 , hence the

MPK in the decentralized solution is (1 ) YK , which is smallerthan what we get from the Social Planner solution, i.e. the socialmarginal product Y

K , because of the tax rate. This gap between

social and private returns leads to a lower growth rate in thedecentralized solution.
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Endogenous Growth and R&D Sector

The Romer model try to explain why and how advanced countries of

the world exhibit sustained growth.Technological progress is driven by R&D sector in advancedworld.

Romer endogenizes technological progress by introducing an R&D

sector, i.e. search of new ideas by researcher interested in protingfrom their invention.

The aggregate production function in the Romer model is

Y =K (ALY)1

Capital accumulation is

K=sKY dK

population growth is

L

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

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

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

A= LA

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

L= LY+ LA
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The rate at which new ideas are discovered, , might be constant, oran increasing function ofA

=A

where and are constants.

Notice that with > 0 the productivity of research increases with thestock of ideas that have already been discovered. On the contrary

with < 0, discovering new ideas becomes harder over time. With=0 the discovery rate is independent from the stock of knowledge.

G & S
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It is possible that new ideas are more likely when there are morepersons engaged in research. Thus, the eect ofLA is notproportional. Hence, it can be assumed that it is LA that enter in theproduction function of new ideas, with 0 < < 1. The generalproduction function of new ideas is

A=LAA

Assuming that 0 < < 1. Dividing by A

A

A=

LAA1

which is the rate of growth along the BGP?

E d G h d R&D S
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Along the BGP

A

A =gA =constant. Thus, the numerator and thedenominator should growth at the same rate, which means

LA

LA(1 )

A

A=0

along the BGP LALA

=n and thus

A

A= gA =

n

1

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 and are parameters independenton policies, such as subsidies to R&D

E d G h d R&D S
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Introducing Microfoundation. Romer (1990 JPE)

Romer (1990) explains how to construct an economy ofprots-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 specic

capital good to an intermediate-good rm. The intermediate-goodrm, is monopolist, manufactures the capital good and sells it to thenal good sector which produces output.

E d G th d R&D S t
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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 specied as follows

Y =L1Y

N

j=1

xj

where the capital goods xj, come from the intermediate

good-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.

E d G th d R&D S t
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The nal-good sectorIfA is the number of capital goods. Then N=A and the productioncan be rewritten as

Y =L1Y

A

j=1

xj

if the number of goods is continuos

Y =L1Y

Z A0

xjdj

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

E d G th d R&D S t
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Final good price Pis normalized to 1.

Firms in the nal-good sector, choose labor and capital to maximizeprots,

maxfLY ,xJg

L1Y

Z A0

xjdj wLYZ A

0pjxjdj

where pjis the rental price for capital-goods and wthe wage paid forlabor.

The FOCs imply:

w = (1 ) YLY

pj = L1Y x

1j for each j

As usual prices of inputs equate their marginal product.

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The intermediate good sector consists of monopolists who produce

the capital goods to sell to the nal sector.Firms gain their monopoly power by purchasing the design for aspecic 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 prot maximization problem of the representativeintermediate-good rm is

maxxj

pj(xj) xj rxj

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

1Y x

1j and r is the interest rate, or the

rental rate of capital.

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The FOC of the intermediate-good rm is.

p0j(xj) xj+ pj(xj) r = 0

2L1Y x1

j| {z }pj

r = 0

Imposing symmetry and solving for p

p= 1

1+ p0(x)xpr=

1

r.

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

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Equilibrium and Aggregation

The total demand for capital from the intermediate good sector mustequal the total capital stock in the economy. Thus,Z A0

xjdj=K

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

x= K

A

The nal good production function can be rewritten as

Y =L1Y

Z A0

xdj=L1Y Ax

substituting for x= KA

Y =K

(ALY)

1

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In the Research Sector new design are discovered according to

A=LAA

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 prots earned by an

intermediate-good rm.Let the discounted value of prots earned by an intermediate-goodrm be PA, where prots are:

= (1 )Y

A

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The research sector

How 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 prots and then sell the patent. Inequilibrium the return of these two alternatives must be the same.

Thus,rPA =+ PA

Which gives

r=

PA

+PA

PAAlong the BGP r is constant and thus and PA must grow at the

same rate, which is the population growth rate n (when =1 and=0). Thus, along the BGP

PA =

r n

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Share of population working in the R&D and good producingsector

Once again we can use the arbitrage concept. It must be the casethat at the margin, individual are indierent between working in thenal-good sector or the R&D sector.

We know that in the nal-good sector

wY =(1 ) Y

LY

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

wR = PA

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Because there is free entry in the two labor markets it must be that

wY =wR, then

(1 ) Y

LY= PA =

r n=

(1 ) YAr n

then

1LY=

(1 )r n YA (1 )Y = r n

A

Rearranging and considering that A= LA =)AA

= LA

A =gA along

the BGP, then1

LY =

r n

gA

LALALY

= gArn =

sR1sR

and sR = LA

L is

sR = 1

1+ rn

gA

.

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OPTIMAL R&D. Is the share of population involved in R&D

sector optimal?

The answer is no. Why? The economy is characterized by threedistortions

1 The market does not endogenize the fact that new research may aect

the productivity of future research. > 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 > 0 the market provides too littleresearch and the fraction of population hired by R&S is too low. Thiseect is called spillover eect or "standing on the shoulders eect".

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

3 Consumer surplus eect. The monopoly prots are less than theconsumer surplus. This eect tends to generate too little innovations.

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OPTIMAL R&D

Classical economic theory: imperfect competition and monopoly arebad for welfare and eciency because they generate adeathweight-loss in the economy. This happens because prices are

higher than marginal costs. However, the literature on the economicof ideas suggests that it is the possibility to make prots, 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 and

long-run gains.

Concluding. In deciding antitrust policies, the regulator has toweight the deathweight losses against the incentive to innovate.
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Goethe University Lectures on Endogenous Growth