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Chapter4
Working with the Solow Growth Model
Macroeconomics
Chapter 4
1
Key Equations
Solow Growth Model
 ∆k/k=






s· (y/k) − sδ − n
k is capital per worker
y is real gross domestic product (real
GDP) per worker
y/k is the average product of capital
s is the saving rate
δ is the depreciation rate
n is the population growth rate.
Macroeconomics
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Solow Growth Model
Steady State
s·(y*/k*)=sδ+n



We assumed that everything on the righthand side was constant except for y/k.
In the transition to the steady state, the
rise in k led to a fall in y/k and, hence, to a
fall in ∆k/k.
In the steady state, k was constant and,
therefore, y/k was constant. Hence, ∆k/k
was constant and equal to zero.
Macroeconomics
Chapter 4
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Solow Growth Model
Change in savings rate (s)
Macroeconomics
Chapter 4
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Solow Growth Model
Change in savings rate (s)


In the short run, an increase in the saving
rate raises the growth rate of capital per
worker.
This growth rate remains higher during the
transition to the steady state.
Macroeconomics
Chapter 4
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Solow Growth Model
Change in savings rate (s)


In the long run, the growth rate of capital
per worker is the same—zero—for any
saving rate.
In this long-run or steady-state situation, a
higher saving rate leads to higher steady
state capital per worker, k∗, not to a change
in the growth rate (which remains at zero).

A·f(k*)/k*=δ+n/s
Macroeconomics
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Solow Growth Model
the effect of s on consumptions




In the short run, consumption decreases
and k arises.
c*=y*-δk*-s(y*-δk*) = y*-δk*-nk*
max c*=y*-(δ+n) k*
k*
FOC: dc*/dk*=dy*/dk*-(δ+n)=0
MPK= δ+n
Macroeconomics
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Solow Growth Model
the effect of s on consumptions


⊿c*= ⊿y*-(δ+n)⊿k*=(MPK-δ-n)⊿k*
In the long run, whether the consumption
in the steady state increases depends on
MPK.

Dynamic inefficient

“Golden Rule”
Macroeconomics
Chapter 4
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Solow Growth Model
Change in technology level (A)
Macroeconomics
Chapter 4
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Solow Growth Model
Change in technology level (A)


In the short run, an increase in the
technology level, A, raises the growth rates
of capital and real GDP per worker.
These growth rates remain higher during
the transition to the steady state.
Macroeconomics
Chapter 4
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Solow Growth Model
Change in technology level (A)


In the long run, the growth rates of capital
and real GDP per worker are the same—
zero—for any technology level.
In this long-run or steady state situation, a
higher technology level leads to higher
steady-state capital and real GDP per
worker, k∗ and y∗, not to changes in the
growth rates (which remain at zero).
A·f(k*)/k*=δ+n/s
Macroeconomics
Chapter 4
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Solow Growth Model
Change in the labor input
Macroeconomics
Chapter 4
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Solow Growth Model
Change in the labor input


In the short run, an increase in labor input,
L(0), raises the growth rates of capital and
real GDP per worker.
These growth rates remain higher during
the transition to the steady state.
Macroeconomics
Chapter 4
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Solow Growth Model
Change in the labor input



In the long run, the growth rates of capital
and real GDP per worker are the same—
zero—for any level of labor input, L(0).
The steady-state capital and real GDP per
worker, k∗ and y∗, are the same for any L.
In the long run an economy with twice as
much labor input has twice as much capital
and real GDP.
Macroeconomics
Chapter 4
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Solow Growth Model
Change in population growth rate
Macroeconomics
Chapter 4
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Solow Growth Model
Change in population growth rate
Macroeconomics
Chapter 4
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Solow Growth Model
Change in population growth rate


In the short run, a higher n lowers ∆k/k
and ∆ y/y.
These growth rates remain lower during the
transition to the steady state.
Macroeconomics
Chapter 4
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Solow Growth Model
Change in population growth rate



In the steady state, ∆k/k and ∆y/y are zero
for any n.
A higher n leads to lower steady-state
capital and real GDP per worker, k∗ and y∗,
not to changes in the growth rates, ∆k/k
and ∆y/y (which remain at zero).
A change in n does affect the steady-state
growth rates of the levels of capital and real
GDP, ∆ K/K and ∆Y/Y.
Macroeconomics
Chapter 4
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Solow Growth Model
Sum up
k* =
k* [ s, A, n, δ, L(0) ]
(+) (+) (−) (−) (0)
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence

One of the most important questions about
economic growth is:
whether poor countries tend to converge
or catch up to rich countries.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence


Economy 1 starts with lower capital per
worker than economy 2—k(0)1 is less
than k(0)2.
Economy 1 grows faster initially because
the vertical distance between the s·(y/k)
curve and the sδ+n line is greater at
k(0)1 than at k(0)2.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence


That is, the distance marked by the red
arrows is greater than that marked by the
blue arrows.
Therefore, capital per worker in economy
1, k1, converges over time toward that in
economy 2, k2.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence




Economy 1 starts at capital per worker
k(0)1 and economy 2 starts at k(0)2, where
k(0)1 is less than k(0)2.
The two economies have the same steadystate capital per worker, k*, shown by the
dashed blue line.
In each economy, k rises over time toward
k*. However, k grows faster in economy 1
because k(0)1 is less than k(0)2.
Therefore, k1 converges over time toward
k2.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence



y= A· f(k) and ∆y/y= α·(∆k/k)
∆k/k was higher initially in economy 1 than
in economy 2.
Therefore, ∆y/y is also higher initially in
economy 1. Hence, economy 1’s real GDP
per worker, y, converges over time toward
economy 2’s real GDP per worker.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence


The Solow model says that a poor
economy—with low capital and real GDP
per worker—grows faster than a rich one.
The reason is the diminishing average
product of capital, y/k.
The Solow model predicts that poorer
economies tend to converge over time
toward richer ones in terms of the levels of
capital and real GDP per worker.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence

Economy 1 starts with lower capital per worker
than economy 2


Assume that economy 1 also has a lower saving
rate;




k(0)1 < k(0)2.
s1 < s2.
The two economies have the same technology
levels, A, and population growth rates, n.
Therefore, k*1 is less than k*2 .
It is uncertain which economy grows faster
initially. The vertical distance marked with the
blue arrows may be larger or smaller than the one
marked with the red arrows.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
33
Solow Growth Model
Convergence




Economy 1 starts with lower capital per worker than
economy 2
 k(0)1 < k(0)2.
The two economies now have the same saving rates,
s, and technology levels, A, but economy 1 has a
higher population growth rate, n;
 n1 > n2 .
Therefore, k*1 is less than k*2 .
It is again uncertain which economy grows faster
initially. The vertical distance marked with the blue
arrows may be larger or smaller than the one
marked with the red arrows.
Macroeconomics
Chapter 4
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Solow Growth Model
Convergence
Macroeconomics
Chapter 4
35
Solow Growth Model
Convergence



Economy 1 has a lower starting capital per worker—
k(0)1 < k(0)2—and also has a lower steady-state
capital per worker— k*1 (the dashed brown line) is
less than k*2 (the dashed blue line).
Each capital per worker converges over time toward
its own steady-state value: k1 (the red curve)
toward k*1 ,
And k2 (the green curve) toward k*2 . However,
since k*1 is less than k*2 , k1 does not converge
toward k2.
Macroeconomics
Chapter 4
36
Solow Growth Model
Convergence

Key Results

k* = k*[ s,
A,
n, δ, L(0) ]
(+) (+) (−) (−) (0)

∆k/k = ϕ[ k(0) , k*]
(−) (+)
Macroeconomics
Chapter 4
37
Solow Growth Model
Convergence

Conditional convergence:


a lower k(0) predicts a higher ∆k/k,
conditional on k∗.
Absolute convergence

the prediction that a lower k(0) raises
∆k/k without any conditioning is called.
Macroeconomics
Chapter 4
38
Solow Growth Model
the speed of Convergence
Macroeconomics
Chapter 4
39
Solow Growth Model
the speed of Convergence
y  Ak 
 y / k  Ak (1 )
 0
 1
•Calibration:
 k  1 / 3, n    3%
The half-life is roughly 18 years.
Macroeconomics
Chapter 4
40
Solow Growth Model
Endogenous population growth

Malthus (1798)
the increase of y (or k) leads to a higher
growth rate of population, which reduces
the level of income per capita.

Modern growth theory:
the higher income per capita reduces the
population growth rate.
Macroeconomics
Chapter 4
41