Chapter 4
ECONOMIC GROWTH
Main Objective
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To explain how an economy grows over time with the Solow
growth model.
Three important determinants of economic growth:
saving
population growth
technological progress
Solow Growth Model
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We start by looking at how saving affects the growth of an
economy, labor and technology are given for the moment.
Two components: the production function and the consumption
function.
A constant returns to scale production function:
Y = F(K, L)
zY = F(zK, zL), z = any positive constant
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If z = 1/L, then
Y/L = F(K/L, 1), or
y = f(k), where
y = Y/L is output per worker
k = K/L is the capital per worker ratio
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Easier to work with, with only one variable and also because y is
a better approximation to living standard.
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MPK = f(k+1) - f(k), (see fig. 4-1)
Start with (assume G = T = 0)
y=c+i
c = C/L is consumption per worker
i = I/L is investment per worker
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The Consumption function:
c = (1 - s)y,
s = S/Y is the average saving rate.
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y = (1 - s)y + i, or
sy = i , o r
sf(k) = i,
saving (per worker) equals investment (per worker)
Fig. 4-2 shows the relationship between output, consumption,
and investment
The Steady State Level of k
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Investment increases the stock of capital, but depreciation
decreases the stock of capital.
The change in capital stock is the net result of investment and
depreciation:
Dk = i - ak,
D (delta) = a change in,
a (alpha) = is a constant depreciation rate (fig. 4-3).
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Now
Dk = sf(k) - ak.
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The steady state level of k is that level of k where Dk = 0, or
equivalently,
sf(k) = ak.
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Fig. 4-4 shows the steady state level of k (k*), and deviations
from k*.
Case study, Japanese and German Growth.
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How does the depreciation rate affects k*
How the saving rate affects k* (fig. 4 - 5).
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Conclusion:
Economy with a larger saving rate (thus a larger investment rate)
will have a larger capital stock (higher k*) and a higher level of
output (see fig. 4-6).
The Golden Rule Level of k*
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Suppose s can be altered by the government, is there an optimal
level of k*?
What do we mean by optimal?
Define optimal as being the maximum c that can be obtained.
Now
c = y - i, or
c = f(k) -ak.
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Since we are interested only in the steady state where i = ak
Thus, maximum c* means (f(k*) - ak*) is the biggest.
See fig. 4-7 for a graphical determination of k*gold.
The mathematical condition at k*gold is
MPK = a.
Comparing Steady States
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See fig. 4-8.
If k* > k*gold, s needs to decrease (fig. 4-8.1).
if k* < k*gold, s needs to increase (fig. 4-8.2).
Transition To Golden Rule Steady State
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If k* > k*gold
–
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See fig. 4-9 and 4-8.1.
If k* < k*gold
–
See fig, 4-10 and 4-8.2.
Conclusion
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When the economy starts above the Golden Rule, moving to the
Golden Rule is the right policy since it increases the
consumption of the present and all future generations.
When the economy starts below the Golden Rule, the policy is
not so obvious. Reaching the Golden Rule requires increasing
saving and thus reducing consumption for the current generation,
but will increase consumption for all future generations.
The Story So Far
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k* is a constant, y* is also a constant. Thus, so far we have not
been able to explain long-run growth.
There is only short-run growth when the economy adjusts to the
steady state.
Population Growth
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Suppose the population (labor force) grows at a rate of n.
Then
Dk = i - (a + n)k, or
Dk = sf(k) - (a + n)k.
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Two factors now reduce k:
–
–
depreciation (ak), and
capital stock now has to be shared among a larger labor force (nk).
The Effects of Population Growth
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The condition for steady state (see fig. 4-11):
Dk = 0,
i = (a + n)k.
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Economies with higher population growth rate will have a lower
k*, and therefore lower y (fig. 4-12, Case Study 4-3).
The Golden Rule with Population Growth
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The condition for the Golden Rule is now
MPK = a + n.
Conclusions With Population Growth
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The steady state k is still a constant, but capital must grow at the
same rate as labor.
The steady state y is also a constant, but output must grow at the
same rate as labor.
Thus, we can now explain economic growth but not growth in
standards of living.
The End
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