Transcript A = K

Chapter 11
Growth and Technological
Progress: The Solow-Swan
Model
© Pierre-Richard Agénor
The World Bank
1






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
2
Basic Structure
and Assumptions
3
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.
4


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.
5
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.
6
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)
7
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.

Intensive-form satisfies Inada conditions:
lim f  (k) = , lim f  (k) = 0
k0
k
8
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.
9
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)
10

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)
11
Savings and Consumption

Capital stock, K, changes through time by:
.
K = sY - K

(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.
12
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f
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)
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
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A
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k
13
The Dynamics of Capital
and Output
14
Economic growth determined by the behavior of
the capital stock.

Differentiating the expression k = K/AL with respect
to time yields,
.
.
.
.
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.
15


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.
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s
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~
k
k
17

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).
18

Equilibrium point, k, found by setting k = 0, in (15),
with the solution:
~
k=




.
~
1/(1 - )
s
++
{
}
Figure 11.3.
~
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,
.
~
gK = K
K
~
k=k
.
.
= A/A + L/L =  + ,
19
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k
k
20


Because K equals ALk, capital’s growth rate will
equal the growth rate of effective labor at k.
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)
~
~
21


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 
22
A Digression on Low-Income
Traps
23

Setting n = n(k) in (15),
.
k = sk - (n(k)+  + )


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.
24
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.
25

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.
26
Population, Savings, and SteadyState Output
27
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 H,
k increases until reaching >
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.


~
kL
28
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29
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k
30
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
31

c/s is positive (negative) when marginal product of
~
capital,

~ -1
k
,
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.
32
Speed of Adjustment
33


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
34

Solution of (24) given by
~
~
k - k  e t(k0-k)
and
y-ye
~

t(y
~
0-y)
(25)
(25) indicates that  * 100 percent of the initial
gap between output per worker and its steady state
level is closed every year.
35

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 - )/   - /
36
Model Predictions and Empirical
Facts
37
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.
38


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 capitaleffective 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.
39
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.
40
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).
41

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 SolowSwan.
42