endogenous growth models -...
TRANSCRIPT
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
Endogenous Growth and Learning
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.
Endogenous Growth and Learning
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.
Endogenous Growth and Learning
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 and Learning
Endogenous growth plus transitional dynamics
Endogenous Growth and Learning
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
Endogenous Growth and Learning
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
Endogenous Growth and Learning
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.
Endogenous Growth and Learning
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
��
Endogenous Growth and Learning
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.
Barro�s model of Endogenous Growth with GovernmentSpending and Taxation
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
�
Barro�s model of Endogenous Growth with GovernmentSpending and Taxation
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 Barro model with endogenous savings
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.
The Barro model with endogenous savings
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.
The Barro model with endogenous savings
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 Barro model with endogenous savings
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
The Barro model with endogenous savings
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 Barro model with endogenous savings
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
The Barro model with endogenous savings
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.
Endogenous Growth and R&D Sector
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
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D Sector
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?
Endogenous Growth and R&D Sector
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
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D 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
Endogenous Growth and R&D Sector
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
Endogenous Growth and R&D Sector
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
Endogenous Growth and R&D Sector
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
Endogenous Growth and R&D Sector
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
.
Endogenous Growth and R&D Sector
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.
Endogenous Growth and R&D Sector
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.