1

Chapter 11Growth and Technological Progress: The Solow-Swan

Model

© Pierre-Richard Agénor

The World Bank

2

Basic Structure and Assumptions The Dynamics of Capital and Output A Digression on Low-Income Traps Population, Savings, and Steady-State Output The Speed of Adjustment Model Predictions and Empirical Facts

3

Basic Structure and Assumptions

4

Solow-Swan model assumptions: closed economy, producing one good using both

labor and capital; technological progress is given and the saving

rate is exogenously determined; no government and fixed number of firms in the

economy, each with the same production technology;

output price is constant and factor prices (including wages) adjust to ensure full utilization of all available inputs.

5

Four variables considered: flow of output, Y; stock of capital, K; number of workers, L; knowledge or the effectiveness of labor, A.

Aggregate production function given by,

Y = F(K,AL).

A and L enter multiplicatively, where AL is effective labor, and technological progress enters as labor augmenting or Harrod neutral.

6

Assumed characteristics of the model: Marginal product of each factor is positive (Fh > 0,

where h = K, AL) and there are diminishing returns to each input (Fhh < 0).

Constant returns to scale (CRS) in capital and effective labor:

F(mK,mAL) = mF(K,AL). m 0 (1)

Inputs other than capital, labor, and knowledge are unimportant. Model neglects land and other natural resources.

7

Intensive-form production function: Output per unit of effective labor, y, and capital per unit of

effective labor, k, are related by setting m = 1/AL in (1),

F(K/AL, 1) = 1/AL [F(K, AL)] (2)

Let k = K/AL, y = Y/AL, and f(k) = F(k,1). Equation (2) is then written as

y = f(k), f(0) = 0 (3)

8

Intensive-form production function: In (3), f '(k) is the marginal product of capital, FK,

marginal product of capital is positive. marginal product of capital decreases as capital

(per unit of effective labor) rises, f "(k) < 0.

k0 k

Intensive-form satisfies Inada conditions:

lim f (k) = , lim f (k) = 0

9

Cobb-Douglas Function: A production function that satisfies Solow-Swan model characteristics:

Y = F(K, AL) = K(AL)1- , 0 < < 1, (4)

Intensive form Cobb-Douglas: Dividing both sides of (4) by AL yields,

y = f(k) = (K/AL) = k , (5)

and,

f '(k) = k - 1 > 0,

f "(k) = - (1 - )k - 2 < 0.

10

Labor and Knowledge Labor and knowledge determined exogenously with

constant growth rates, and :

L/L = , A/A = ; (7)

. .

Savings and Consumption Output divided between consumption, C, and

investment, I:

Y = C + I (9)

11

Savings, S, defined as Y - C, a constant fraction, s,

of output:

S Y - C= sY, 0 < s < 1 (10)

Savings equals investment such that:

S = I = sY (11)

12

with denoting the rate of capital stock depreciation Consumption per unit of labor, c, is determined by:

c= (1 - s)k , i = sk (13)

with, c = C/AL and i = I/AL See Figure 11.1 for graphical distribution of output allocated among

consumption, saving, and investment.

Savings and Consumption Capital stock, K, changes through time by:

K = sY - K (12), .

13

Figure 11.1The Production Function in the Solow-Swan Model

D

k

y

c = y - i

i

}1

0

B

A

y = k

i = sk

f'(k) = k

14

The Dynamics of Capital and Output

15

Differentiating the expression k = K/AL with respect to time yields,

Economic growth determined by the behavior of the capital stock.

k = K/AL - (K/AL)L/L - (K/AL)A/A

= K/AL - (L/L + A/A)k (14)

. . . .

. . .

Substituting (4), (8), and (12) into (14) results in

k = sk - ( + + )k, + + > 0, (15)

a non-linear, first-order differential equation.

.

16

Equation 15: the key to the Solow-Swan model; the rate of change of the capital stock per units of effective labor as the difference between two terms: sk: actual investment per unit of effective labor.

Output per units of effective labor is k, and the fraction of that output that is invested is s.

( + + )k: required investment or the amount of investment that must be undertaken in order to keep k at its existing level.

Figure 11.2.

17

Figure 11.2The Equilibrium Capital-Labor Ratio

(n++k

kk~

E

A

c~

sf(k) = (n++k~ ~

y

0

i = sk

y = k

18

Reasons why investment is needed to prevent k from falling (see (15)), capital stock is depreciating by k; effective labor is growing by (n + )k; therefore if sk is greater (lower) than (n + +

)k, k rises (falls).

19

k/s > 0, e.g. increase in savings rate increases k. (15) is globally stable: capital stock always adjusts

over time such that k converges to k. With k constant at k, capital stock grows at the rate,

~

~

~

~ Equilibrium point, k, found by setting k = 0, in (15), with the solution:

.

k =~ { s

+ + }1/(1 - )

gK = = A/A + L/L = + , ~

. .~

Kk = kK

.

Figure 11.3.

20

Figure 11.3Adjustment Process of the Capital-Labor Ratio

kk~

.k

0

.k > 0

.k < 0

21

At k, output grows at a rate

gY = A/A + L/L = + , . .

~

Since Y = ALy and y is constant at k , output grows at the rate of growth of effective labor.

Growth of capital per worker and output per worker (labor productivity), are given by,

~

gK/L = gK - L/L = ,

gY/L = gY - L/L = (20)

~.

~

~ ~ .

Because K equals ALk, capital’s growth rate will equal the growth rate of effective labor at k.

22

On the balanced growth path, the growth rate of output per worker is determined solely by growth of technological progress.

Rate of return on capital, r, and wage rate, w, are given by:

r = k - 1 - , and

w = (1 - )Ak

23

A Digression on Low-Income Traps

24

Endogenously determined labor supply can lead to a dynamically stable, low-steady-state level of y.

Figure 11.4 displays a situation where n(k) is negative at low levels of k, positive at intermediate levels of k, and again negative at higher values of k, leading to multiple equilibria and the possibility of a low level trap.

Setting n = n(k) in (15),

k = sk - (n(k)+ + ).

25

Figure 11.4Endogenous Labor Force in the Solow-Swan Model

Source: Adapted from Burmeister and Dobell (1970, p. 37).

k

E

E'

k~Uk~S

k

.k

0k~Uk

~S

0

[n(k)++ksk

26

Rosenstein-Rodan (1961) and Murphy, Shleifer, and Vishny (1989) show that a big push, e.g. an exogenous increase in savings, can bring the economy up to a higher, more stable equilibrium path.

27

Population, Savings, and Steady-State Output

28

With exogenous population growth: A decline in population growth from nH to nL causes

the following results in Solow-Swan See Figure 11.5: k increases until reaching kL > kH, output per

worker, Ak rises, but in the long run, Y/L grows at .

~ ~

If the saving rate, s, increases: sk shifts up and to the left, k rises from kL to kH ,

Ak rises, but in the long run, Y/L grows at . See Figure 11.6.

~ ~

~

29

Figure 11.5Reduction in the Population Growth Rate in the Solow-Swan Model

k

sf(k), (n++k

sf(k) = (n ++kH

sf(k) = (n ++kL

E

E'

H(n ++k (n ++kL

k~

Lk~

H

A

0k =0

sk

30

Figure 11.6Increase in the Saving Rate in the Solow-Swan Model

(n++k

k

sf(k), (n++k

s f(k) = (n++kL

s f(k) = (n++kH

A

E

k~

L k~

H

E'

0k =0

Hs k

Ls k

31

Savings and Consumption

Consumption per unit of effective labor, c, is given by

c = (1 - sL)k , i = sk (21)

On the balanced growth path, with sk = (n + + )k

c = k - (n + + )k

and,

c/s = [k - 1 - (n + + )] k/s>/<0

~ ~ ~

~ ~ ~

32

c/s is positive (negative) when marginal product of

capital, k - 1 , is greater (lower) than (n + + ).

~

~

Elasticity of y with respect to s, , given by:

= / (1 - ) (23)

Low implies that y/k is low, that the actual investment curve, sk, bends sharply and that FK is low.

33

Speed of Adjustment

34

Important to assess the speed at which adjustment to a new equilibrium proceeds.

Changes in k can be approximated by

k (k - k), (24) where

= - (1 - )(n + + ) < 0

~.

35

Solution of (24) given by

k - k e t(k0-k) and

y - y e t(y0-y) (25)

~

~ ~

~

(25) indicates that * 100 percent of the initial gap between output per worker and its steady state level is closed every year.

36

Adjustment ratio, , is the fraction of change from y0 to y completed after t years:

= (y - y0)/(y - y0)~

using (25)

= 1 - e t

Time taken to achieve a given fraction of the adjustment is given by

t* = ln(1 - )/ - /

37

Model Predictions and Empirical Facts

38

Predictions of the Solow-Swan Model: Capital-effective labor ratio, marginal product of capital,

and output per units of effective labor, are constant on the balanced growth path.

Steady-state growth rate of capital per worker K/L and output per worker Y/L are determined solely by the rate of technological progress. In particular, neither variable depends on the saving rate or on the specific form of the production function.

Output, capital stock, and effective labor all grow at the same rate, given by the sum of the growth rate of the labor force and the growth rate of technological progress.

39

Reduction in the population growth rate raises the steady-state levels of the capital-effective labor ratio and output in efficiency units and lowers the rate of growth of output, the capital stock, and effective labor.

Rise in the saving rate also increases the capital-effective labor ratio and output in efficiency units in the long run, but has no effect on the steady-state growth rates of output, the capital stock, and effective labor.

40

The Empirical Evidence:

Support Data from industrial countries suggests that growth

rates of labor, capital, and output are each roughly constant (Romer, 1989).

Growth of output and capital are about equal and are larger than the labor growth rate. In industrial countries output per worker and capital per worker are rising over time.

As shown in Figures 10.2 and 10.3, evidence on the relationship between the level of income and both population growth and savings are consistent with the model.

41

The Empirical Evidence:Baggage Strong empirical relationship between growth of per

capita income and both the savings rate and the share of investment in output. Solow-Swan predicts no association.

Observed differences in y are far too large to be accounted for by differences in physical capital per worker.

Attributing differences in output to differences in capital without differences in the the effectiveness of labor implies large variations in the rate of return of capital (Lucas, 1990).

42

The only source of variation in income per capita growth rates across countries is labor effectiveness, . But the model is incomplete because this driving force of long-run growth is exogenous to Solow-Swan.