Intermediate Macroeconomics - Lecture 5 - Endogenous ...econ.sciences-po.fr/sites/default/files/file/barany/int_macro/lecture5.pdf · Intermediate Macroeconomics Lecture 5 - Endogenous

Intermediate MacroeconomicsLecture 5 - Endogenous growth models

Zsofia L. Barany

Sciences Po

2014 February

Recap: Why go beyond the Solow model?

I we looked at the Solow model with technological progress andfound that it matches the Kaldor facts well

I we looked at why economists moved beyond the Solow model:

I capital does not move from rich to poor countriesI growth accounting → technological improvements contribute

significantly to growthI development accounting → there are large differences in the

level of technology across countries

I this week we look at endogenous growth models

1. learning by doing2. human capital3. research and development

and look at international technology transfer

Learning by doing

Learning by doing

Based on Romer (1989).

I Main idea: skills or knowledge are accumulated during theproduction

I ⇒ the skills or knowledge accumulation is free and is aby-product of production

I the marginal product of capital diminishes at the firm level

I BUT when a firm invests, other firms learn from its experiencetoo, i.e. investment by a firm generates a positiveexternality for the economy

I ⇒ no diminishing marginal product of capital at theaggregate level, i.e. ’AK’ for aggregate capital

Model

I A representative firm i ’s production function

Yi = Kαi (BNi )

1−α

I α < 1 ⇒ diminishing marginal product of capitalI B is the stock of economy-wide knowledge, the firm takes this

as given

I The economy wide stock of knowledge is proportional to theeconomy-wide stock of capital

B = λK

I λ > 0 represents the idea of a positive externalityI the higher aggregate capital, K , and thus aggregate output,

Y , the higher is productivity, B

Since firm i is a representative firm, it represents aggregate outputand capital

K = Ki and N = Ni and Y = Yi

Use B = λK in the production function:

Y = Kα(BN)1−α = Kα(λKN)1−α = K (λN)1−α

⇒ this is an ’AK’ production function, with no diminishingmarginal returns to capital at the aggregate level

The capital accumulation equation

K ′ − K = sK (λN)1−α − dK

Let’s assume that the population is constant

k ′ − k = s(λN)1−αk − dk

the growth rate of capital per person

k ′ − k

k= s(λN)1−α − d = x = constant

⇒ if x > 0, then there is long run endogenous growththis is satisfied if the saving rate, s is sufficiently high

Endogenous growth in the learning-by-doing model

k

i

dk

s(λN)1−α

Assuming s(λN)1−α − d > 0

Implications of the learning-by-doing model

I there is endogenous growth because the stock of knowledgeis determined by the endogenous level of K throughlearning-by-doing

I the saving rate affects not only the level of income but alsothe growth rate, as x depends on s

I the growth rate is constant in both the short and the longrun, so there is no convergence

I there are ’scale effects’: the growth rate depends on the sizeof the populationlarger N implies stronger knowledge spillovers and thereforehigher growth rate, x

I Is this model prediction problematic?I one way to remove this scale effect is to replace B = λK by

B = λk, i.e. knowledge depends on capital per worker






I Is this model prediction problematic?

I one way to remove this scale effect is to replace B = λK byB = λk, i.e. knowledge depends on capital per worker






I Is this model prediction problematic?I one way to remove this scale effect is to replace B = λK by

B = λk, i.e. knowledge depends on capital per worker

Human capital

Human capital

I skills are required to put ideas or knowledge into practiceI for the OECD countries and in most parts of the world

I average years of schooling are increasingI fraction of college graduates is increasing

I as opposed to the process of learning-by-doing, there arecosts and returns to education

Human capital

Based on Lucas (1988).

I introduce a production function for human capital

I the production of new human capital is proportional toexisting human capital⇒ ’AK’ for the production of human capitalno diminishing marginal product in the production of humancapital

I how good someone is in accumulating human capital (A)might depend on his years of education

I prediction: positive growth in the long run

ModelThe consumer consumes all his wage income:

C = wuHs

I w - real wage per efficiency unit of labor

I u - fraction of time devoted to working (exogenous)

I Hs - stock of human capital supplied

I ⇒ uHs total units of efficiency labor supplied

Future human capital

Hs ′ = b(1− u)Hs

I depends on current human capital, Hs

I on the time devoted to training and education, 1− u

I b - efficiency of human capital accumulation

Model

The representative firm’s production function:

Y = zuHd

I z TFP

I uHd amount of efficiency units of labor in production

the profits are:

π = Y − wuHd = zuHd − wuHd = (z − w)uHd

The competitive equilibrium:

I the labor market has to clear: Hd = Hs = H

I the goods market has to clear: C = Y = zuH

I human capital accumulation: H ′ = b(1− u)H

I the equilibrium growth rate of human capital is:

H ′ − H

H= b(1− u)− 1

⇒ if b(1− u) > 1 ⇒ human capital increases forever⇒ there is endogenous growth

Endogenous growth in the human capital model

H

H ′

45◦

b(1− u)H

Assuming b(1− u) > 1

The role of u

remember, u is the fraction of time devoted to working, while1− u is the fraction of time devoted to studying, and

C = zuH

what is the effect of a decrease in u on Y and C ?

immediate effectfewer hours worked (at constant H) ⇒ a decline in Y ⇒ a declinein the level of C

long run effectafter the immediate change u is again constant, thus

C ′ − C

C=

H ′ − H

H= b(1− u)− 1

⇒ lower u ⇒ higher 1− u, more time devoted to studying ⇒higher growth rate of H and C

The role of u

Is a lower u necessarily better?

Is there convergence?imagine country A and country B have the same characteristicsuA = uB , zA = zB , bA = bB

but country A has a higher initial level of human capitalHA(0) > HB(0) → will they converge?

No convergence – countries grow at the same rate







Convergence in the Solow modelin the Solow model what happens to two countries, A and B,which have the same characteristicssA = sB , nA = nB , dA = dB , same technology zF (K ,N)but country A has higher initial capital per person than country B:kA(0) > kB(0)?

Conditionalconvergence –poor countrygrows faster





Research and development

Research and developmentI new ideas and knowledge are developed in the market through

devoting resources to research and developmentI for the OECD countries

I R&D spending as a fraction of GDP is increasing over timeI number of researchers as a fraction of the total employment is

increasing over time

Slide #25

Research and DevelopmentResearch and Development

Slide #26

Research and DevelopmentResearch and Development

But what is the key difference between ideas and physical capital?

Like capital, ideas are an economic good:

there is a cost of producing them

they can be used in production

there is a price for them (the value of a patent)

Capital vs. IdeasCapital vs. IdeasCapital vs. Ideas

What is the key difference between ideas and physicalcapital?

Like capital, ideas are economic goods

I there is a cost to producing them

I they can be used in production

I there is a price for them (the value for a patent)

There are some differences along the following attributes

I rivalrous vs non-rivalrous

I degree of excludability

Can you come up with examples for each?

Examples

rivalrous non-rivalrous

excludable

non-excludable

One country R&D model

Based on Romer (1990).

I introduce a production function for ideas

I production of new ideas is proportional to the existing stockof ideas, i.e. ’AK’ for the production of ideas

I the coefficient depends on, for example, the number ofresearchers

I there is no diminishing marginal product in the production ofideas

Model

The production function is

Y = ALY

where LY is the number of workers engaged in producing outputand A is the level of knowledge (or technology)

The output per worker is then

y =Y

L=

ALY

L= A

LY

LY + LA= A

(1− LA

LA + LY

)= A(1− γA)

where γA is the fraction of workers engaged in R&D

Model

We assume that the production of new knowledge leads to thefollowing growth rate of knowledge

A ≡ A′ − A

A=γaµ

L

I proportional to the number of workers engaged in R&D, γAL

I µ captures the cost of new inventions

If γA is constant ⇒ y is proportional to A since y = A(1− γA)

⇒ the growth rate of y is the same as the growth rate of A

y = A =A′ − A

A=γaµ

L

The role of γA

γA is the fraction of labor which works in R&D

what is the effect of an increase in γA?

Immediate effectfewer people working in production ⇒ a decline in Y ⇒ a declinein C

long run effectthe growth rate of A increases ⇒ the growth rate of Y increases

The role of γA

Slide #33

R&D R&D –– OneOne--Country ModelCountry ModelShifting Labor into R&DShifting Labor into R&DShifting Labor into R&D

An increase in γAleads to a permanentincrease in the growth rate

Ln(A)

**Remember the advantage of drawing figure in log from handout 1

Slide #34


Ln(y)

Slide #33


An increase in γAleads to a permanentincrease in the growth rate

Ln(A)

**Remember the advantage of drawing figure in log from handout 1

Slide #34


Ln(y)

Summary of the one-country model

I there are scale effectsthe growth rate depends on the size of the population, La larger L implies more workers engaged in R&D (for given γA)

I this implies that countries with larger populations have highergrowth rates, higher levels of technology and are richer→ are these good predictions?

I one interpretation of the model is that a country’s level oftechnology depends on R&D done around the worldthis is a reasonable assumption if there is internationaltechnology transfer→ two-country model

Two country R&D model

There are two countries, labelled 1 and 2.

The production function for each country j = 1, 2 is

Yj = Aj(1− γAj)Lj

Countries acquire new technologies either by invention or byimitation.The option of imitation is open only to the less developed country,the ’technology follower’.

Assume L1 = L2 = L and γA1 > γA2 .⇒ country 1 will be the technology leader and country 2 thefollower in the steady state

The cost of imitation

Assume that the cost of imitation is

µc = c

(A1

A2

)

I c(·) is downward sloping

I c(·) tends to zero as A1/A2 tends to infinity

I c(·) tends to the cost of invention, µi as A1/A2 tends to one

⇒ the cost of imitating a given technology is less than the cost ofreinventing the technology

µc < µi

The cost of imitation

Slide #37

R&D R&D –– TwoTwo--Country ModelCountry Model

Assume the cost of imitating (μc) a given technology is less than the cost of reinventing (μi) the technology

The cost of imitation,

Assume c(.) is downward sloping c(.) tends to zero as A1/A2 tends to infinity c(.) tends to the cost of invention, μi, as

A1/A2 tends to one

2

1

A

Acc

Slide #38

R&D R&D –– TwoTwo--Country ModelCountry ModelCost of ImitationCost of ImitationCost of Imitation

The steady stateWhat can the growth rates of technology (and hence output) be inthe two economies?

the growth rate has the same form as before: Aj =γAjµj

L

country 1 is the leader ⇒ µ1 = µicountry 2 is the follower ⇒ µ2 = µc if A2 < A1

Slide #39


Li

A

1,

Steady StateSteady StateSteady State

In the steady state, the growth rates of A1 and A2 are equal . There is a steady state level of A1/A2 .

Lc

A

2,

Growth rate of technology

Slide #40


i

A

Ac

i

A

Ac

c

A

i

A

c

LL

1,

2,

2

1

2

1

1,

2,2,1,

A

A

:.c using solved becan A

A of level statesteady The

:same theare ratesgrowth thestate,steady In the

Steady StateSteady StateSteady State

The steady state

In the steady state, the growth rates are the same

γA1

µiL =

γA2

µcL ⇒ µc =

γA2

γA1

µi

The steady state level of relative technologies, A1/A2 can be foundusing c(·)

c

(A1

A2

)= µc =

γA2

γA1

µi

The role of γA2

what happens if the follower increases R&D effort, i.e. γA2

increases?

steady state effectmore resources in R&D ⇒ growth curve shifts up ⇒ lower level ofsteady state A1/A2

Slide #41

R&D R&D –– TwoTwo--Country ModelCountry ModelAn increase in R&D in the followerAn increase in R&D in the followerAn increase in R&D in the follower

An increase in γA2 shifts up the growth rate of A2. The new steady state level of A1/A2 is lower.

Slide #42


An increase in γA2 leads to a temporaryincrease in the growth rate for the follower

How does this compare to the one-country model?

The role of γA2

what happens if the follower increases R&D effort, i.e. γA2

increases?

steady state effectmore resources in R&D ⇒ growth curve shifts up ⇒ lower level ofsteady state A1/A2

Slide #41



Slide #42



How does this compare to the one-country model?

The role of γA2

immediate effectfewer people in production ⇒ drop in output, Ythentemporary increase in the growth rate of the follower

Slide #41



Slide #42



How does this compare to the one-country model? Slide #43


Ln(y)

Slide #44

Appropriate TechnologyAppropriate Technology

The technologies developed in the leader might not be “appropriate” to the follower, e.g. the leader (the rich) tends to have more physical and human capital

Different types of technological change:Neutral Capital-biased

The role of γA2

immediate effectfewer people in production ⇒ drop in output, Ythentemporary increase in the growth rate of the follower

Slide #41



Slide #42



How does this compare to the one-country model? Slide #43


Ln(y)

Slide #44

Appropriate TechnologyAppropriate Technology

The technologies developed in the leader might not be “appropriate” to the follower, e.g. the leader (the rich) tends to have more physical and human capital

Different types of technological change:Neutral Capital-biased

Recap of economic growth models

1. Malthusian ModelI population growth increasing in per capita consumptionI stagnation in the long-run

2. Solow ModelI capital accumulationI no (endogenous) growth in the long-runI conditional convergenceI model with labor-augmenting technological progress matches

the Kaldor facts well

3. Endogenous Growth ModelsI role of technology and its originI no convergence

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Intermediate Macroeconomics - Lecture 5 - Endogenous ...econ.sciences-po.fr/sites/default/files/file/barany/int_macro/lecture5.pdf · Intermediate Macroeconomics Lecture 5 - Endogenous