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fertility and sustained growth robert e lucas, jr wuhan

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					Fertility and Sustained Growth Robert E. Lucas, Jr.

Wuhan University June, 2004

Problem • Ricardian theory rules out possibility of sustained growth: “dismal science” • Predicts that increases in knowledge eventually result in population growth only • Observe that this sometimes, but not always, occurs – Demographic transition • How reconcile with Ricardian theory of pre-industrial society?

World Population and Production
100000

Population (M), Production (B $1985)

Population
10000

Production

1000

100
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

GDP per capita, five regions
18000 15000 12000 9000 6000 3000 0 1750

1985 Dollars

1800

1850

1900

1950

2000

1990 Population in millions UK, USA, Canada, Australia, New Zealand 354 Japan 124 France, Germany, Netherlands, Scandinavia 184 Rest of Western Europe, Eastern Europe, Latin America 986 Asia (except Japan), Africa 3590

Demographic Transitions
2.5 2 1.5 1 0.5 0 -0.5 0 2 4 6 8 10 12 14 16 Per Capita GDP, Thousands of 1985 Dollars I II III IV V

Population Growth Rate

6.7

Population and Agricultural Real Wages in England and Wales, 1250-1990 from Javier Birchenall, "Escaping High Mortality." Chicago dissertation. 2004

6.2

5.7 Log(Wages)

1480 5.2

4.7

1650 4.2 1340

3.7 0.8

1.3

1.8

2.3

2.8

3.3

3.8

4.3

Log(Population)

• Add physical capital accumulation to theory? – Possibility explored by theorists from Ricardo to Solow

• Can’t explain sustained growth with physical capital only • Need growth in knowledge, human capital • How can human capital theory account for demographic transition?

Recommended readings:

Gary S. Becker, Kevin M. Murphy, and Robert Tamura. “A Reformulation of the Economic Theory of Fertility.” Journal of Political Economy, 1990. Gary D. Hansen and Edward C. Prescott. “From Malthus to Solow.” American Economic Review. 2002. Robert E. Lucas, Jr. Lectures on Economic Growth, chapter 5, sections 6,7.

A growth model with exogenous productivity growth

Every household endowed with one unit of time; divide between child care and production of goods. Technology is c ≤ h(1 − kn), where kn is time spent on children, 1 − kn is time spent on adult consumption, and h is the level of knowledge or human capital. Here assume that growth of human capital is exogenous. ht+1 = γht.

Every household has dynastic preferences W (c, n, u0) Assume that for all (c, n, u0), λ > 0 =⇒ W (λc, n, λu0) = λW (c, n, u0) State variable of economy is human capital level h. Each household has the Bellman equation: (*)

v(h) = max W (h(1 − kn), n, v(γh))
n

Conjecture a value function of the form v(h) = Bh. If so, Bh = max W (h(1 − kn), n, γBh)
n

Homogeneity property (*) implies Bh = h max W ((1 − kn), n, γB)
n

B = max W ((1 − kn), n, γB).
n

Solve for B and n : both independent of human capital level h.

Consider the example
´ η uβ 1/(1+β) W (c, n, u) = cn ´ η (λu)β 1/(1+β) = λW (c, n, u). W (λc, n, λu) = λcn ³ ³

Then

The optimal fertility level, the solution to
n

max(1 − kn)nη , is n= 1 η k1 + η

A growth model with endogenous productivity growth

Now suppose that the growth rate of human capital depends on time devoted specifically to this purpose: ht+1 = htϕ(rt), where c ≤ h(1 − (r + k)n). The Bellman equation for this case is: v(h) = max W [h(1 − (r + k)n), n, v(hϕ(r))].
n,r

Again conjecture a solution v(h) = Bh. If so, B satisfies B = max W [(1 − (r + k)n), n, Bϕ(r)]
n,r

´ η uβ 1/(1+β) and n and r are independent of h. For the case W (c, n, u) = cn

solve

³

max(1 − (r + k)n)nη [Bϕ(r)]β
n,r

Total time spent on children is η . 1+η How split between “quantity” n and “quality” r ? (r + k)n =

. Suppose ϕ(r) = Crε. Then FOCs imply 1η − ε n = , k1 + η

and r=k ε η−ε

Conclusion from these two examples?

• Why should successful societies have lower fertility levels than unsuccessful societies? Malthus and Ricardo predict the opposite

• Theory based on exogenous technological change can’t answer this question.

• Theory of endogenous growth has feature that increase in return to investment in human capital–in ε–can lead to increase in r, decrease in n.

Homework Exercise 2 : One-Child Policy (continued) Consider an economy with endogenous growth, in which r and n are constant at the values r0 ∈ (0, 1) and n0 ≥ 1, and consumption ct and human capital ht are growing at the rate γ defined by 1 + γ = ϕ(r). Suppose that the government now imposes the constraint n≤θ<1 on the birth rate. (a) What is the immediate (first generation) effect of the policy on consumption, human capital investment, growth, and family welfare W ?

(b) What are the dynamic consequences if this policy is maintained forever? (c) What are the dynamic consequences if this policy is maintained for T generations and then discontinued?

A theory of the demographic transition? Of the origins of the industrial revolution?

• At best, 2 theories: One for pre-industrial world; one for modern world. • Look to Hansen and Prescott for unified theory? No: No theory of fertility decline.

• Look to Becker, Murphy, and Tamura? No: No role for land, Malthusian population pressures

• But each of these models has interesting features. How combine?

• An outline of a unified theory • Two technologies: (Hansen-Prescott) • Traditional agriculture, labor-plus-land output = Axαθ1−α

• Modern, “Ak” technology output = Bh( − θ) Distribute workforce across these technologies

• Overall technology F is given by
h

i αθ 1−α + Bh( − θ) F (x, h, ) = max Ax θ

• Household’s Bellman equation is then v(x, h) = max W (c, n, v(x/n), h(ϕ(r)))
c,n,r

subject to c + kn ≤ F (x, h, 1 − rn).

• Call policy functions c(x, h), n(x, h) and r(x, h) Evolution of economy in the (x, h) plane given by xt xt+1 = n(xt, ht) and ht+1 = htϕ(r(xt, ht)) Look at possible (conjectured) phase diagrams

6.7

Population and Agricultural Real Wages in England and Wales, 1250-1990 from Javier Birchenall, "Escaping High Mortality." Chicago dissertation. 2004

6.2

5.7 Log(Wages)

1480 5.2

4.7

1650 4.2 1340

3.7 0.8

1.3

1.8

2.3

2.8

3.3

3.8

4.3

Log(Population)

What events or shocks might have initiated industrial revolution? Sustained human capital accumulation by large numbers of people?

• Increase in return ε due to scientific discoveries • Increase in ε due to increases in individual freedom • Increase in h for many people for non-economic reasons • Others?

European Shares of World Production and Population
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1750

1800

1850 Production

1900

1950

2000

Population


				
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