Economic Growth and Development: Part I

Cecilia Garcıa Penalosa

September 2020

0

Neoclassical growth theory: The Solow Model

Consider a CRS production function

Y = F (K,L) → y = f (k)

For example

Y = Kα(AL)1−α

Lety = A1−αkα

·k = sy − δk

⎫⎬⎭ g =

·y

y

Then

g = α

·k

k= α

(sA1−αkα

k− δ

)= α(sA1−αkα−1 − δ

)

1

If 0 < α < 1 the economy converges to

k∗ = A(sδ

)1/(1−α)

Then

g∞ = 0

with

y∗ = A(sδ

)α/(1−α)

and we can express the rate of growth as

g = αδ

⎛⎝(y∗y

)1−αα − 1

⎞⎠

2

Output per worker Capital per worker Break-even investment per worker

k

)(ksf

)(kf

k*k

3

What can prevent the marginal product of capital from falling?

Population growth or exogenous technical change.

Let

Y (t) = K(t)α(A(t)L)1−α

where

A(t) = A0ext

Define

k(t) ≡ K(t)

A(t)L=

k(t)

A(t)

y(t) ≡ Y (t)

A(t)L=

y(t)

A(t)so that we can write

y(t) = k(t)α

4

Then

dy(t)/dt

y(t)= α

sy − (δ + x)k

k

= α(sk(t)α−1 − δ − x

)which converge to zero and defines

k∗ =(

s

δ + x

)1/(1−α)

Now

g =dy(t)/dt

y(t)

=dy(t)/dt

y(t)+

·A

A

5

g = α(sk(t)α−1 − δ − x

)+ x

= α(δ + x)

⎛⎝(y∗y

)1−αα − 1

⎞⎠ + x

and

g∞ = x

6

k

)(ksf

k*k

7

k

)(ksf

k*k

8

k

)(ksf

k*k *k

9

Recall

g = α(δ + x)

⎛⎝(y∗y

)1−αα − 1

⎞⎠ + x

Convergence:

- If the exogenous rate of technical change is the same for all

countries, then in the long run they will all grow at the same rate,

- Countries that are further from their steady state grow faster.

10

k

)(ksf

)(kf

kak bk *k

11

The Ramsey-Cass-Coopmans Model

Endogenizes the savings rate

Competitive equilibrium

Consumers

max

∞∫0

C(t)1−σ − 1

1− σe−ρtdt

s.t.·K(t) = w(t) + r(t)K(t)− C(t)

K(0) = K0

The hamiltonian

H(C,K, λ, t) =C1−σ − 1

1− σe−ρt + λ(w + rK − C)

12

The control variable is C and the state variable K

Variables C, K, w, and r are functions of time

First-order conditions

∂H∂C

= 0

∂H∂K

= −.λ

limt→∞λ(t)K(t) = 0

Then ·C

C=

r − ρ

σ

This is the ’Ramsey equation’ also called the ’Euler equation’

13

Firms

max π = F (K(t), L(t))− w(t)L(t)− r(t)K(t)

⇒ ∂F

∂L= w(t) and

∂F

∂K= r(t)

Then ·C

C=

∂F/∂K − ρ

σ

Let

Y (t) = K(t)α(AL(t))1−α

with 0 < α <1.

14

Then ·C

C=

α(AL)1−αKα−1 − ρ

σ,

which goes to zero as the economy accumulates capital

In steady state

·C

C= 0 and K∗ = AL

(α

ρ

)1/(1−α)

.

No long-run growth

15

Welfare analysis

max

∞∫0

C(t)1−σ − 1

1− σe−ρtdt

s.t.·K(t) = K(t)α(AL(t))1−α − C(t)

K(0) = K0

The hamiltonian

H(C,K, λ, t) =C1−σ − 1

1− σe−ρt + λ(Kα(AL)1−α − C)

16

First-order conditions

e−ρtC−σ − λ = 0

λα(AL)1−αkα−1 = −.λ

and

·C

C=

αAkα−1 − ρ

σ

Recall that this is exactly the expression we got for the competitive

equilibrium

Implication: since there is perfect competition and there are no

externalities, the competitive equilibrium is socially optimal

17

Towards Endogenous growth

Early attempts to endogenise technology

Why should the rate of technical change not depend on economic

decisions?

Problem: in order to endogenise A, the decisions to increase it

must be rewarded. But,

• with CRS, Euler’s theorem implies that Y = FKK + FLL,

• if Y has CRS in K and L, it must have IRS in A,L, and K →cannot use competitive equilibrium.

18

Early attempts

1. Arrow 1962 → learning-by-doing in production

2. Shell 1973 → R&D done by the government

3. Uzawa 1965→ A represents human capital that can be produced

with labour services

19

The AK model

The Harrod-Domar model (1939, 1946)

Assumed a constant capital-output ratio

K

Y=

1

A→ Y = AK

Then

Y = AK.K = sY − δK

}.K = sAK − δK → g = sA− δ

Why should we assume constant returns to capital?

20

Romer (1986) introduces externalities in a neoclassical model

Yj = AKαj L

1−αj︸ ︷︷ ︸ → Y = AKαL1−α︸ ︷︷ ︸

firm p.f. aggregate p.f.

where the N firms are symmetric and

L =∑

j Lj and K =∑

j Kj

Assume

A = AKβ

ThenY = AKα+β.K = sY − δK

}g = sAKα+β−1 − δ

21

where for simplicity we suppose L = 1

Possible cases:

1. α + β < 1: Solow model

2. α + β = 1: CONSTANT SOCIAL RETURNS TO CAPITAL.

The aggregate production function is of the form

Y = AK

and

g = sA− δ

3. α+β > 1: increasing returns to capital, which imply exploding

growth

22

Romer considers the AK model in the Ramsey set up

max

∞∫0

C1−σ − 1

1− σe−ρtdt

s.t.·K = AKαL1−α − C

⇒·C

C=

αA(L/K)1−α − ρ

σRomer postulates

A = AKβ

and lets L = 1

Then

g =αAKα+β−1 − ρ

σ23

Possible cases:

1. α + β < 1: Solow model

2. α + β = 1: constant growth rate

g =αA− ρ

σ

3. α + β > 1: exploding growth

24

What about the rates of growth of capital and output ?

Define

gc =

·C

Cgk =

·K

Kgy =

·Y

Y

Consider the case α + β = 1

From the production function Y = AK, so

gy = gk

Rewrite the budget constraint as·K

K=

Y

K− C

K= A− C

K

In steady state

A− g∗k =C

K

25

where g∗k is the steady state growth rate of capital

Take logs

log(A− g∗k) = logC − logK

Note that g∗k is constant in steady state, so the l.h.s. of this expres-

sion is constant. Differentiate with respect to time to get

0 = g∗c − g∗kSo

g∗c = g∗k = g∗y26

Remarks on the AK model

• Constant growth rate

• Savings matter

• No convergence

• Shocks have permanent effects

• Scale effects

• Welfare implications

27

To compare the three cases, consider the simplest model with

exogenous saving rates and depreciation

Let Y = AKα and A = AKβ

Then

Y = AKα+β.K = sY − δK

}g = sAKα+β−1 − δ

28 29

30

Two Endogenous Growth Models

31

Infrastructure and Growth

Barro (1990)

Assumption: government expenditures affect the productivity of

privately owned factors

Production function:

Y (t) = AK(t)1−α (γ(t)L(t))α

y(t) = Ak(t)1−αγ(t)α

Proportional tax on income, τ

Balanced government budget

32

The competitive economy

max

∞∫0

c(t)1−σ − 1

1− σe−ρtdt

s.t.·k(t) = (1− τ )Ak(t)1−αγ(t)α − c(t)

⇒·c

c=

(1− τ )A(1− α)(γ/k)α − ρ

σ

Balanced budget condition

γ = τy ⇒ γ

k= (Aτ )1/(1−α)

and the competitive growth rate is

gc =(1− α)A

11−α(1− τ )τ

α1−α − ρ

σ33

The maximum rate of growth in a competitive econ-

omy

maxτ

g ⇒ maxτ

(1− τ )τα

1−α

⇒ τ∗ = α

34 35

Is the competitive equilibrium socially optimal?

Social planner

max

∫ ∞

0

c(t)1−σ − 1

1− σe−ρtdt

s.t.·k(t) = A(1− τ )(Aτ )α/(1−α)k(t)− c(t)

⇒ gs =A

11−α(1− τ )τ

α1−α − ρ

σRecall

gc =(1− α)A

11−α(1− τ )τ

α1−α − ρ

σ

Planner takes externality into account: more capital increases tax

revenue and hence raises the productivity of capital

36

Human capital

Lucas (1988)

Key assumptions

Output depends on human capital (education) of the labour force

Y = AKβ(Le)1−β

where , Le = uhL

Individuals have 1 unit of time at each instant and u ∈ [0, 1] is time

spent at work

Accumulation of human capital: the main input is “time”·h = zh(1− u)

is the production function for human capital

37

Write per capita output as

y = Akβ (uh)1−β

Then

maxc,u

∫ ∞

0

c1−σ − 1

1− σe−ρtdt s.t.

·k = Akβ(uh)1−β − c

·h = z(1− u)h

Two control variables, c and u, and two state variables, k and h.

The Hamiltonian

H(c, u, k, h, λ, μ, t) =c1−σ − 1

1− σe−ρt + λ(Akβ(uh)1−β − c)

+μ(z(1− u)h)

and the agents chooses consumption and the allocation of time be-

tween education and production.38

First order conditions∂H∂c

= 0 ,∂H∂u

= 0 ,

∂H∂k

= −·λ ,

∂H∂h

= − ·μ

Then ·c

c=

Aβkβ−1(uh)1−β − ρ

σCan show

g∗c = g∗k = g∗y = g∗h = g∗ and g∗c =z − ρ

σwhich implies that the steady state level of (1− u) is

1− u∗ = z − ρ

zσThat is

g∗ = z(1− u∗)39

Solow versus the AK model

40

Solow versus the AK model

Two crucial differences between the two that could be tested em-

pirically:

1. determinants of long-run growth rates

2. returns to capital

Cross-country evidence

• Barro and Sala-i-Martin (1995)

g60−85 = f (edu, I/GDP, Gov/GDP)

But:

• “some” evidence of “conditional convergence”

• reverse causation, eg. education

What are the returns to capital?

41

The augmented Solow model

Mankiw, Romer and Weil (1992)

Yt = AtKαt H

βt L

1−α−βt

Investments in the two types of capital

skYt

shYt

Then

gt = λ (log y∗ − log yt)

42

where

λ = (1− α− β)(x + n + δ)

y∗ =

⎛⎝ sαksβh

(x + n + δ)α+β

⎞⎠1/(1−α−β)

and

x technical change

n population growth

δ depreciation

43

44

Poverty Traps

45

Poverty Traps I: The Big Push

Murphy, Shleifer, and Vishny (JPE 1989)

A Simple Model with a Unique Equilibrium

Utility

U = exp

[∫ N

0ln x(q)dq

](1)

Labour, L, is paid a wage 1

Aggregate income

Y (n) = πn + L (2)

n : number of sectors that have industrialised

46

Two types of firms for each good

- Cottage production:

c(x) = 1 · x ⇒ p = 1

- Mass production: IRS

c(x) = F +x

α, α > 1

Limit pricing p = 1

Expenditure on each good is y = Y/N

Profits are

π(n) = y(n)−(F +

y(n)

α

)=

α− 1

αy(n)− F ≡ ay(n)− F

47

Write profit as a function of aggregate income

π(n) = aY (n)

N− F (3)

Equations (2) and (3) together imply

Y (n) =L− nF

1− an/N(4a)

π(n) =aL−NF

N − an(4b)

Multiplier

dY (n)

dn=

aL/N − F

(1− an/N)2=

π(n)

1− an/N

48

Unique Nash Equilibrium

No industrialisation

π(0) = ay(0)− F = aL

N− F

Then

π(0) < 0 ⇔ aL

N< F (5a)

All industrialise

π(N) = ay(N)− F = aL−NF

N − aN− F = α

(aL

N− F

)Then

π(N) > 0 ⇔ aL

N> F > 0 (5b)

aL

N> F ⇒ all firms industrialise at n = 0

aL

N< F ⇒ no firm industrialises at n = N

49

A Model with a Factory Wage Premium

Same production structure as above

Crucial assumption: compensate workers to move to industry

Utility in cottage production

Uc = exp

[∫ N

0ln x(q)dq

](1a)

Utility in mass production

Um = exp

[∫ N

0ln x(q)dq

]-v (1b)

50

Competitive factory wage:

w = 1 + v (6)

Monopolist’s profits

π =

(1− 1 + v

α

)y − F (1 + v) (7)

Assume

α− 1 > v (8)

51

No industrialisation equilibrium

π =

(1− 1 + v

α

)L

N− F (1 + v) < 0 (9a)

Industrialisation equilibrium

All firms industrialise,

y = π +1 + v

NL

and positive profits are earned

π = α

(L

N− F

)− L

N(1 + v) > 0 (9b)

52

Multiple equilibria

Re-express above as

L

N

α− (1 + v)

α< F (1 + v)

L

N

α− (1 + v)

α> F

If

L

N

α− (1 + v)

α(1 + v)< F <

L

N

α− (1 + v)

α

then, multiple equilibria exist53

Poverty Traps II: Threshold effects

Azariadis-Drazen (1990)

OLG model + human capital accumulation Continuum of two-

period OLG families Mass 1, and no population growth

Human capital is inherited from the previous generation

h1,t = h2,t−1

Individuals of a same generation are identical

Human capital is accumulated during 1st period of life

h2,t = (1 + γ · vθt )h1,t where 0 < θ < 1

54

Linear preferences,

Ut = c1,t + δc2,t, where δ =1

1 + r

Output per worker is y = h per unit of time. Then

maxv

Ut = (1− vt)h1,t + δh2,t

s.t h2,t =(1 + γ · vθt

)h1,t

which gives the unique solution:

v∗ = [δθγ]1

1−θ

g∗ =h2,t

h2,t−1− 1 = γv∗θ

55

Similar to Lucas, as θ and γ matter

But: different welfare effects due to intertemporal externalities

Assumption :

Education technology has positive threshold externalities,

γ(vt−1) =

⎧⎨⎩ γ if vt−1 ≤ v0

γ if vt−1 > v0,

where 0 < v0 < 1 and γ < γ.

56

Low-growth equilibrium

v = argmaxv

{(1− v)h1,t + δ(1 + γvθ)h1,t

}that is

g = γ · vθ = γ(δθγ)θ

1−θ

High-growth equilibrium

v = argmaxv

{(1− v)h1,t + δ(1 + γ · vθ)h1,t

}that is

g = γ · vθ = γ(δθγ)θ

1−θ

57

These two equilibria coexist iff

v < v0 < v

Then:

(1) multiple steady-state growth paths,

(2) the outcome is determined by history,

(3) policy implications: education subsidies can avoid low-development

traps

58

Poverty Traps III: Income Distribution and Macroeco-

nomics

Galor and Zeira (1993)

OLG economy with constant population, L

Single good that can be produced with two CRS technologies, one

uses unskilled and the other skilled labor

Agents can work as unskilled both periods, or invest in human cap-

ital when young and work as skilled workers when old

One unit of labor is supplied in each period

Investment in human capital is indivisible, h > 0

59

Utility function

Ut = (1− β) log ct + β log bt

Borrowing rate i > risk-free rate r

Reason : Cost of monitoring investment in an intangible asset

Notation xjt = bjt−1

Production: Two sectors

Unskilled

Yu = wuLut

Skilled

Ys = AKαt (Ls

t )1−α

Small open economy:

ws = (1− α)A (αA/r)α/(1−α)

60

There are three types of agents:

Not invest

Uu(x) = log [wu + (x + wu)(1 + r)] + γ

bu(x) = β [wu + (x + wu)(1 + r)]

where γ ≡ β log β + (1− β) log(1− β)

Invest and x ≥ h

Us(x) = log [ws + (x− h)(1 + r)] + γ

bs(x) = β [ws + (x− h)(1 + r)]

Invest and borrow x < h

Us(x) = log [ws + (x− h)(1 + i)] + γ

bs(x) = β [ws + (x− h)(1 + i)]

61

Assumption

ws − h(1 + r) ≥ wu(2 + r)

Those who have to borrow choose to study if

Us(x) ≥ Uu(x)

i.e. if

x ≥ f ≡ wu(2 + r) + h(1 + i)− ws

i− r

that is, if inheritance is “large enough”

62

What is the level of output?

Recall that xt = bt−1 . Then the distribution of bequests left by

individuals born in period t− 1 , Dt−1, determines the number of

skilled and unskilled workers in dynasty t,

Lst =

∫ ∞

fdDt(xt) and Lu

t =

∫ f

0dDt(xt) ,

and hence the level of long-run output is

yt = 2wuLut + wsL

st

63

Dynamics:

bt+1 =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

β [wu + (bt + wu)(1 + r)] if bt < f

β [ws + (bt − h)(1 + i)] if f ≤ bt < h

β [ws + (bt − h)(1 + r)] if h ≤ bt

Assume (1 + r)β < 1

64

0 f z h

65

0 z

66

0 z

67

0 z

68

0 z

69

Long-run number of skilled workers

Ls∞ =

∫ ∞

zdDt(bt) .

Long-run distribution of income : two point distribution, where the

mass of agents at each level is uniquely determined by the initial

distribution of wealth

Long-run average level of output

y∞ = 2wu + (ws − 2wu)Ls

L,

which depends on the initial distribution of bequests

70 71

72 73

74 75

76 77

78

Recap: So far we have

Growth

•Neoclassical growth model: diminishing returns to cumulable

capital

• Solow:– long-run growth due to exogenous technological change

– convergence

– optimality of competitive equilibrium

•Mankiw, Romer and Weil: human capital

– conditional convergence

79

• Endogenous growth model: AK

– Constant returns to cumulable factors

– Sources

1. Learning-by-doing (externality)

2. Infrastructure (externality)

3. Human capital

– Lon-run growth is endogenous & depends on pref., technology,

policy

– C.E. may not be socially optimal

80

Poverty traps

• Traps possible both for ’income levels’ and ’growth rates’

•Different types of traps1. Big Push: coordination problems, due for example, to pecu-

niary externalities

2. Threshold effects in education: history matters

3. Distribution: initial inequality can determine the level of long-

run output

•Key implication

– at least two levels of output/growth are compatible with the

same fundamentals

– different types of intervention are required81

Many of these mechanisms are relevant for

•Middle income countries (e.g. infrastructure)

•Very poor economies (poverty traps)

• But what about growth in rich economies?

What we are missing

•Growth in rich economies is mainly driven by technological progress

• Is it exogenous?• Is it due to an externality?

•Or is it intentional?

82

80

Note : Gross Domestic Expenditure on Research and Development (GERD) istotal intramural expenditure on R&D performed on the national territoryduring a given period

Source : OECD Main Science and Technology Indicators

83

Country Business sector Higher Education sectorRussia 29.49 9.68Canada 41.12 41.73

United States 62.37 12.85China 76.63 7.41Korea 76.64 8.22Japan 79.05 11.56

Share of GERD financed by the business and H.E. sector, selected OECD countries, 2018

84

Technological Change: Increasing Product Variety

85

Technical Change:

Expanding Product Variety

Increase in variety of inputs (Young 1920) → growth based on in-

creasing returns due to specialization

Growth due to growth in K but agents are now rewarded

Monopolistic competition under product differentiation (Dixit and

Stiglitz)

R&D activity

Internal IRS: imperfect competition

Intermediate goods

Romer (1990)

86

Three sectors:

final good

intermediate goods

R&D

Two types of labour: L and H

Final output sector

Y = Hα1 L

β∫ A

0x1−α−βi di (1)

Competitive

Symmetry Y = (AH1)α(AL)βK1−α−β

where K = xA

Intermediate goods sector87

Sunk cost of producing a new variety, PA

Fixed cost → monopolistic competition

Same production function as the final good

Machinery does not depreciate

• flow cost rxi

• flow revenue p(xi)xi

Research sector ·A = zH2A (2)

Sell new design for PA to one intermediate goods producer

88

Solving the model

Competitive factor pricing in the final goods sector

pi(x) = (1− α− β)Hα1 L

βx−α−βi

w1 = αHα−11 Lβx1−α−βA

wL = βHα1 L

β−1x1−α−βA

Profit function for intermediate goods producers

π(x) = p(x)x− rx

= (1− α− β)Hα1 L

βx1−α−β − rx

Choose x

x =

((1− α− β)2

rHα1 L

β

)1/(α+β)(3)

89

Hence

π = x(p− r) =xr(α + β)

(1− α− β)

The research firm extracts the entire monopoly rents

PA =π

r=

α + β

(1− α− β)x (4)

Recall that research output is·A = zH2A

So skilled labour is paid its marginal value product

w2 = zAPA

90

Labour market equilibrium

w1 = w2 ⇒ PA =α

z

x1−α−βLβ

H1−α1

(5)

Using (3), (4) and (5) we get employment in the production sector

H1 = θr

z

where

θ =α

(α + β)(1− α− β)Then

H2 = H −H1 = H − θr

z91

Steady state growth

Recall

Y = AHα1 L

βx1−α−β

⇒·Y

Y=

·A

A= zH2

The rate of output growth:

g = zH2 = zH − θr

The Ramsey equation : ·c

c=

r − ρ

σ

In steady state g =·c/c

g =zH − θρ

1 + θσ, r =

zσH + ρ

1 + θσ

92 93

Welfare analysis

Two sources of market failure

(i) monopoly markup

(ii) research spillovers

The problem faced by the social planner is

Max U =

∫ ∞

0

C1−σ − 1

1− σe−ρtdt

s.t.·K = K1−α−βAα+βHα

1 Lβ − C

·A = zAH2

H ≥ H1 +H2

94

The Hamiltonian of this problem is

H =C1−σ − 1

1− σe−ρt + μzAH2 + λ

(K1−α−βAα+β(H −H2)

αLβ − C)

The solution

g∗ = zH/ϕ− ρ

1/ϕ− 1 + σwhere

ϕ ≡ α

α + βRecall

g =zH/θ − ρ

1/θ + σso that

ϕ = θ(1− α− β) < θ

That is

g∗ > g95

.

Techonological Change : Quality Ladders

96

Technical Change: Increasing Product Quality

Aghion and Howitt (1992)

Assumptions

Consumers

L ∞ ly-lived agents

Utility

u(y) =

∫ ∞

0yτe

−rτdτ

where τ denotes time

Can work in either• research, n

• manufacturing, m

}L = n +m

97

Final good sector

Competitive

Output depends upon the input of a single intermediate good, x,

and on the quality index, A

y = A · xα 0 < α < 1 (1)

Intermediate good sector

Monopolistic: uses only labour x = m

Labour market equilibrium condition

L = x + n

98

Research sector

Innovations increase the quality index

At = γAt−1 where γ > 1

At = A0γt

Let t be the quality level or ”vintage”

Competitive (free entry) → patent race

Random R&D process: Poisson with parameter λ > 0

Poisson arrival rate λ · nInnovations are drastic

99

The intermediate goods monopolist

The profit flow πt of the intermediate producer

πt = maxx

[pt(x) · x− wt · x]Competitive final good sector implies

pt(x) = At · αxα−1

That is

xt = argmaxx

{Atαxα − wtx}

Then

xt =

(α2

wt/At

)1/(1−α)

πt =1− α

αwt · xt

100

Let ωt = wt/At

So

xt = x (ωt) and πt = At · π (ωt)

and∂x (ωt)

∂ωt< 0 and

∂π (ωt)

∂ωt< 0

101

The research sector

cost = wtzdτ

expected benefit = λzdτVt+1

The free entry gives the arbitrage condition :

wt = λ · Vt+1Asset equation:

rVt+1 = πt+1 − λnt+1 · Vt+1So

Vt+1 =πt+1

r + λnt+1(2)

102

The model is now fully characterized by both:

- the arbitrage equation

ωt = λ · γπ(ωt+1)r + λnt+1

(A)

- the labour market clearing equation

L = nt + x(ωt) (L)

103

The Steady State level of Research

Steady state: (A) and (L), with ωt ≡ ω and nt ≡ n

ω = λ · γπ(ω)r + λn

(A)

n + x(ω) = L (L)

(A) and (L) can be combined as

1 = λγ1−α

α (L− n)

r + λn(3)

This yields

n =λγ1−α

α L− r

λγ1−αα + r

104 105

The Steady State Rate of Growth

The steady-state flow of consumption goods

yt = Atxα = At(L− n)α

⇒ yt+1 = γyt

The path of the log of final output ln y(τ ) is a random step function,

of size ln γ > 0

106 107

Taking the time interval between τ and τ + 1, we have

ln y(τ + 1) = ln y(τ ) + (ln γ) · ε(τ )

Given that ε(τ ) is distributed Poisson with parameter λn, we have

E(ln y(τ + 1)− ln y(τ )) = λn ln γ

The average growth rate in steady-state is

g = λn ln γ (G)

108

Welfare Analysis

A social planner maximizes the discounted flow of income

U =

∫ ∞

0e−rτ · y(τ ) · dτ

=

∫ ∞

0e−rτ

⎛⎝ ∞∑t=0

Π(t, τ )Atxα

⎞⎠ dτ

The Poisson process with parameter λn implies

Π(t, τ ) =(λnτ )t

t!· e−λnτ

the constraint is

L = x + n

Using At = A0γt and L = x + n welfare becomes

maxn

U(n) =A0(L− n)α

r − λn(γ − 1)

109

The socially optimal level of research is

1 =λ(γ − 1)

(1α

)(L− n∗)

r − λn∗(γ − 1)(6)

and the average growth rate is

g∗ = λn∗ ln γ

Recall

1 =λγ(1−αα

)(L− n)

r + λnand

1 =λ(γ − 1)

(1α

)(L− n∗)

r − λn∗(γ − 1)

110

Three differences

1. Intertemporal spillover effect: social discount rate < private

discount rate

2. Appropriability effect

3. Business-stealing effect

The business-stealing effect dominates when there is much monopoly

power (α close to zero) and innovations are not too large

Laissez-faire growth will be excessive !

111

Uneven Growth

Negative correlation between current and future research:

more research tomorrow nt+1 implies more creative destruction (r+

λnt+1 ↑) and less profits (πt+1 = At+1π(ωt+1) ↓) after the next

innovation (t + 1) occurs. This discourages current research ↓ nt

Equations (A) and (L) give

nt = ψ(nt+1), ψ′ < 0. (8)

112

)(: 1tt nAn

)(: tt nL

1n

1

L2n

2

113

114

Technological Change : Complementarities between

human capital and R and D

115

The Nelson-Phelps Approach to Education

Neo-classical approach (MRW) and Lucas: human capital is an

ordinary input in production

Nelson-Phelps (AER 1966): education increases individuals’ ca-

pacity to innovate and to adapt to new technologies → speeds up

technological diffusion

116

Predictions of the N-P approach

1. The rate of innovation should increase with the level of educa-

tion.

2. Marginal productivity of education attainment is an increasing

function of the rate of technical progress.

3. Complementarity between education and R&D activities:

(i) macro policies which affect innovation → affect the relative

labour demand and the skill distribution of employment and earn-

ings

(ii) subsidies to education ↑ profitability of R&D → speed up tech-

nological progress.

117

Complementarity between R&D and education invest-

ments

Redding (1996)

Workers

Continuum of two-period OLG workers

Linear utility Ut = c1,t + δc2,t

All individuals are born with h1,t = 1 ∀ t

Invest a fraction v of time when young in education,

h2,t = 1 + γ · vθ

where γ is constant and 0 < θ < 1.

118

Entrepreneurs

Continuum of 2-period OLG entrepreneurs with linear utility

Do research when young and produce when old:

yij,t+1 = Ait+1 · hj,t+1,

(i) Ait+1 : entrepreneur i’s productivity at t+ 1 (ii) hj,t+1 : human

capital of the j−worker

Innovation

By investing a non-monetary cost equal to

αμA,

entrepreneurs can increase productivity from A to λA with

probability μ

where λ > 1 and 0 ≤ μ ≤ 1 is effort119

Employment

Workers are self-employed when young:

(1− v)A

where A is the current leading-edge technology

When old they are randomly matched with firms, and get a frac-

tion β of output surplus.

120

Optimal Decisions

Entrepreneurs

Choose R&D effort to maxV (μ)

maxμ

{−μαA + δ(1− β)(μλ + 1− μ)(1 + γvθ)A}Then

μ∗ =

⎧⎨⎩ 1 if α < δ(λ− 1)(1 + γvθ)(1− β)

0 if α > δ(λ− 1)(1 + γvθ)(1− β)

effort depends on the worker’s education

121

Workers

maxv

{(1− v)A + δ · β · [μλ + 1− μ](1 + γvθ)A}

Then

v∗ = [βδθγ(μλ + 1− μ)]1/(1−θ)

increasing in the probability of innovation μ

Equilibrium

Symmetric equilibrium: same v∗ and same μ∗ for all agents

122

Multiple Steady States

Low-development trap: μ∗ = 0 and therefore v∗ = v

Can occur if

1 + γ(βδθγ)θ

1−θ <α

δ(1− β)(λ− 1)

The growth rate is g = g = 0.

High-growth equilibrium: μ∗ = 1 and therefore: v∗ = v

Can occur ifα

δ(1− β)(λ− 1)< 1 + γ(λβδθγ)

θ1−θ

The growth rate is g = g = lnλ

123