2013-2014 – master 2 – macro i lecture 6 : r&d based ... · pdf file2013-2014...

2013-2014 – Master 2 – Macro I

Lecture 6 : R&D based models of endogenousgrowth

Franck Portier(based on Gilles Saint-Paul lecture notes)

[email protected]

Toulouse School of Economics

Version 1.130/09/2013

Changes from version 1.0 are in red

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[email protected]

Disclaimer

These are the slides I am using in class. They are not

self-contained, do not always constitute original material and do

contain some “cut and paste” pieces from various sources that I

am not always explicitly referring to (not on purpose but because it

takes time). Therefore, they are not intended to be used outside of

the course or to be distributed. Thank you for signaling me typos

or mistakes at [email protected].

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mailto:[email protected]

0. Introduction

I In neo-classical growth models technical progress is assumedto follow an exogenous trend. Therefore the model does notexplain the long-run growth rate of the economy, which (if TPis labor-augmenting) must be equal to the growth rate oftechnology.

I The purpose of R&D - based models of growth is toendogenize technical progress. This is called endogenousgrowth theory.

I First generation endogenous growth models assumed thattechnical progress was the by product of economic activity bysome mechanism such as learning by doing and productiveexternalities.

I In second generation models R&D is explicitly modeled as anactivity which produces technical change and is driven byeconomic incentives.

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1. On Knowledge

The state of technology is a form of knowledge. Knowledge has anumber of specificities :

I It is non-rival : if I use a piece of knowledge, it does notprevent anybody else from using it. This is unlike a standardgood like apples.

I Producers of apples own the apples ; they can make moneyfrom giving the apple to someone else in exchange for money.

I If the knowledge produced is visible, then because it is nonrival the producer cannot in principle prevent others fromusing it even though they have not paid for it.

I That is, a standard good is naturally excludable. This is notthe case for a nonrival good.

I In such a situation there are no private economic incentivesfor producing new knowledge and growth will stop.

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1. On Knowledge

I However one can make knowledge artificially excludable bygranting monopoly rights for its use, i.e. patents. (Othersolutions are possible (public contests))

I This increases the incentives for innovation and thus booststhe growth rate.

I But it comes at the cost of monopoly price distortions.

I Trade-o↵ between static and dynamic e�ciency.

I For those reasons in endogenous growth models theequilibrium and the optimum di↵er : we cannot derive theequilibrium as an optimum, unlike in the Ramsey model.

I In the model we will study, the number of goods Nt isendogenous ; new goods will be introduced by an R&D sector,and this will be the source of growth.

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2. A standard model of horizontal innovationThe consumer

I We consider a representative consumer whose utility isZ

+1

0

C

↵t � 1

↵e

�⇢tdt.

I In that formula, Ct is a Dixit-Stiglitz aggregate of acontinuum of individual goods of total mass Nt :

Ct =

✓Z Nt

0

c

�itdi

◆ 1

�

,

where cit is the individual’s consumption of good i and� 2 (0, 1].

I From the lecture on Dixit-Stiglitz, aggregate price level :

pt =

✓Z Nt

0

p

1��it di

◆ 1

1��

, (API)

where

� =1

1� �.

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2. A standard model of horizontal innovationThe consumer (continued)

I We will take the aggregate consumption index as thenumeraire. Therefore at any point in time

pt = 1.

I We know that the value of total consumption expenditures isptCt . Hence it is equal to Ct .

I One can borrow and lend at an instantaneous rate rt . That is,there exist instantaneous bonds that pay 1 + rtdt units of thenumeraire at t + dt in exchange for one unit of the numeraireat t.

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I Borrowers issue those bonds and lenders buy them.I The instantaneous interest rate rt adjusts so that there is

equilibrium on the market for loanable funds, i.e. the supply ofbonds is equal to the demand for bonds.

I The consumer has a flow of income denoted by Yt . He isendowed with L units of labor (equivalently L can beinterpreted as the size of the population), and he owns thefirms in the economy.

I Denoting by wt the wage at date t, we have that

Yt = wtL+ ⇧t ,

where ⇧t are aggregate profits.I Let Wt be the consumer’s net wealth at t : this is the total

value of the bonds held by the consumer. It evolves accordingto (remember that pt = 1).

Wt = rtWt + Yt � Ct (IBC)8 / 31


I The consumer problem can be decomposed in two stages :1. For a given Ct , what is my optimal allocation of Ct across

goods ? Answer from the Dixit-Stiglitz block.2. What is the optimal intertemporal allocation of Ct?

I To answer question 2, we integrate backward theinstantaneous budget constraint (IBC) and get

Wt = W

0

e

R t0

rudu +

Z t

0

(Yu � Cu)eR tu rv dv

du, (1)

where W

0

is the initial wealth of the consumer, inherited fromthe past.

I Exercise 1 : Derive this by noting that (IBC) is a lineardi↵erential equation and that the above is the unique solutionsuch that Wt=0

= W

0

. [Hint : Derive this by making use of

the following quantity : Wt = Wte�

R t0

rudu and byre-expressing the IBC in terms of d

dt Wt .]

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I We impose the solvency condition

limt!+1

Wte�

R t0

rudu � 0.

I Substituting into (1), this delivers the intertemporal budgetconstraint of the consumer

W

0

+

Z+1

0

(Yt � Ct)e�

R t0

rv dvdt � 0. (ITBC)

I The present discounted value of net borrowing Ct �Yt cannotexceed initial wealth.

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I We are now in a position to write the Lagrangian

L =

Z+1

0

C

↵t � 1

↵e

�⇢tdt+�

✓W

0

+

Z+1

0

(Yt � Ct)e�

R t0

rv dvdt

◆.

I The FOC isC

↵�1

t e

�⇢t = �e�R t0

rv dv .

I Taking log derivatives with respect to time we eliminate � andget the Euler equation

Ct

Ct= ⌘(rt � ⇢), (EE)

where

⌘ =1

1� ↵.

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I Consider a BGP where C grows at rate g . Then along thatBGP r must be constant and (EE) implies a positiverelationship between r and g :

g = ⌘(r � ⇢). (MGR)

I The greater the growth rate, the greater must the interestrate be to induce consumers to plan for a greater consumptionin the future relative to now.

I The response of r to g is smaller, the greater ⌘ : The more Iam willing to substitute intertemporally, the smaller theincrease in r that will induce me to postpone consumption bya given relative amount.

I This is just the modified Golden rule but g is now treated asendogenous.

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2. A standard model of horizontal innovationThe production sector

I Individual goods are produced using one unit of labor, i.e.

yi = li .

I Consequently, the cost function of firm i at t is

ct(yi ) = wtyi .

I From the dixit stigliz block we then get that

pi t = µwt = p

0t , with µ =�

� � 1.

I Using (API) we have that

p = 1 = p

0tN

1

1��t = µwtN

1

1��t .

I That is

wt =N

1

��1

t

µ. (RW)

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2. A standard model of horizontal innovationThe production sector (continued)

Interpretation :

Iwt is the consumption wage.

I When there are more products, the consumption wage goesup. This is because one unit of labor produces a greatervariety of goods, and therefore more units of welfare.

I The e↵ect is stronger, the greater the taste for variety, i.e. thesmaller �.

I If goods are perfect substitutes (� ! 1), new goods do notbring any extra welfare and the consumption wage does notgo up with N

I If µ is larger, goods are more expensive relative to the wageand the equilibrium wage falls.

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2. A standard model of horizontal innovationThe innovation sector

I New goods are invented using labor.

I At any point in time a fraction of the workforce is employed inthe R&D sector.

I To invent 1 new good per unit of time I need 1/(�Nt) units oflabor.

I Here Nt stands for an externality : for N to grow in asustained way, we need the cost of producing new goods tofall over time. Otherwise we will just be able to produce aconstant flow of new goods, which will fall relative to N,implying that N/N will gradually fall to zero over time.

I The flow of new goods is

Nt = �NtLRt ,

where LRt is total employment in the R&D sector.

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2. A standard model of horizontal innovationThe innovation sector (continued)

I How much R&D do firms do ? For this we have to computethe costs and benefits of R&D.

I The costs are given by the wage that must be spent forinventing one extra good. This is equal to

wt

�Nt.

I The benefit is given by the present discounted value of theprofits from innovation. We assume an innovator gets a patententitling it to the whole stream of monopoly profits associatedwith the new good. Let Vt be the value of this patent. Then

rtVt = ⇡t + Vt , (PV)

where ⇡t is a firm’s profit at date t, given by

⇡t = (µ� 1)wty t . (PR)

I Here xt is the output of any individual firm (by symmetrythese are the same across firms).

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2. A standard model of horizontal innovationThe innovation sector (continued)

II In equilibrium, entry into the R&D sector occurs until the costof an innovation equals its benefit.

I Therefore we must have

Vt =wt

�Nt. (2)

I Exercise 2 : One may now make use of the condition fromDixit-Stiglitz which says that y t = Ct(µwt)��. Show thatusing this condition delivers (RW) again.

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2. A standard model of horizontal innovationEquilibrium

I In equilibrium, we must have that

L = y tNt + LRt .

I Consequently,

y t =L� LRt

Nt=

L� 1

�˙NtNt

Nt. (3)

I Substituting this and (RW) into (PR) we get

⇡t = (µ� 1)N

2��1

t

µ

L� 1

�

Nt

Nt

!. (4)

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2. A standard model of horizontal innovationEquilibrium (continued)

I Consider now a BGP. In a BGP, the numerator of the RHS of(3) is constant. Therefore,

gy = �gN .

I Total output in physical terms is Ny and is constant overtime. This is normal since there is no physical productivitygrowth. But total welfare grows over time because of the tastefor variety.

I Furthermore, Ct = y tN

1

�t = y tN

��1

t .I Therefore,

gC = g = gy +�

� � 1gN =

1

� � 1gN .

I Hence the growth of welfare is larger in proportion to gN , thelower �,i.e. the more the new goods are complementary withthe existing ones and therefore ”useful”.

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I Hence gN = (� � 1)g .I Remark 1 : Note that along a BGP,

gw = gN��1

= gwL = gwL+⇧

= g . Thus wages and nominalincome also grow at g . And since p = 1 throughout, this isalso true of real income.

I This helps us to rewrite (4) :

⇡t = (µ� 1)N

2��1

t

µ

✓L� � � 1

�g

◆.

I Note that g⇡ = gV = 2��1

gN = (2� �)g .I If 1 < � < 2, profits grow over time : the quantity of a given

good being produced falls less fast than its price. The moregoods are substitute, the lower the growth rate of their price(and of wages), and the less profits grow over time. For � > 2profits fall over time.

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I In a BGP, (PV) implies that

Vt =⇡t

r � gV=

(µ� 1)

r � (2� �)g

N

2��1

t

µ

✓L� � � 1

�g

◆.

I In equilibrium this must be equal to

wt

�Nt=

1

�Nt

N

1

��1

t

µ=

N

2��1

t

�µ.

I Equating the two expressions delivers another equilibriumrelationship between r and g

(µ� 1)

r � (2� �)g

✓L� � � 1

�g

◆=

1

�,

I or equivalently (given that µ = ��1

)

g =(µ� 1)�L� r

� � 1. (RD)

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g =(µ� 1)�L� r

� � 1(RD)

I This now defines a negative relationship between g and r . Wehave made use of many equilibrium conditions to derive it.But it is chiefly driven by the incentives to do R&D.

Ig falls with r : a greater cost of capital discourages innovationand thus growth.

Ig grows with µ (falls with �). The mechanism is two-fold.Greater markups raise profitability, thus increasing incentivesto innovate. Greater complementarities make g greater givengN .

Ig grows with � : less costly R&D increases innovation.

Ig grows with L : more people around increase the equilibriumnumber of innovators and therefore the flow of new ideas.

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I Equilibrium in a BGP is determined by the intersectionbetween (RD) and (MGR)

I This delivers a growth rate g and and interest rate r .

I The comparative statics and their interpretation arestraightforward.

I Algebraically, we get

g =⌘

(� � 1)⌘ + 1[(µ� 1)�L� ⇢] ;

r =(µ� 1)�L+ (� � 1)⌘⇢

(� � 1)⌘ + 1.

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3. Some RemarksRemark 1 : The equilibrium is not e�cient

There is a number of reasons for that :

I Reason 1 : The monopoly distortion in price-setting tends topush prices above marginal cost. In equilibrium this meansthat real wages are too low relative to the social optimum.

I Reason 2 : Innovators do not entirely appropriate the socialsurplus from their innovation. They get profits but part of thesurplus goes to the consumers. This tends to make innovationtoo low compared with the optimum.

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3. Some RemarksRemark 1 : The equilibrium is not e�cient (continued)

Note :

I Reason 1 goes in the opposite direction of reason 2 : lowerwages reduce the cost of inventing new goods, so it boostsR&D.

I Another way to look at it is to say that the only alternative towork in the productive sector is the R&D sector.

I Therefore, any distortion that reduces incentives to work inthe production sector (such as the monopoly markup here)tends to be good for R&D.

I Things would be di↵erent if for example labor supply wereelastic. In this case the monopoly markup would reduce totallabor supply.

I Remark : In some models the two e↵ects exactly cancel outand the equilibrium is optimal.

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3. Some RemarksRemark 1 : The equilibrium is not e�cient (continued)

I Reason 3 : There is a productive externality in the productionof new goods. Innovators do not internalize the gains from thefact that their invention increases the productivity in inventingfurther goods.

I Remark : While such an externality is plausible, remember weintroduced it essentially for technical reasons, i.e. to getsustained growth.

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3. Some RemarksRemark 2 : scale e↵ect

I The model exhibits scale e↵ects : larger economies (i.e. with agreater L) grow faster, as the above formulae make clear.

I This is because the flow of innovation depends on the numberof people doing R&D, which is itself proportional topopulation size. Thus larger populations lead to moreinnovation and faster growth of GDP per capita.

I There is much debate on whether that is realistic.

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2013-2014 – master 2 – macro i lecture 6 : r&d based ... · pdf file2013-2014...

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