3 Homework 3 - Homepage Usask

3. Unemployment and growth. Consider how unemployment would affect
the Solow growth model. Suppose that output is produced according
to the production function Y = K α [(1−u ∗ )L] 1−α ,whereu ∗ is the natural
rate of unemployment. There is no technological progress. Assume again
that the labour force equals population.
(a) Express output per worker, y, as a function of capital per worker, k,
and the natural rate of unemployment. Describe the steady state of
this economy.
Output per worker equals
y = Y
L = Kα (1 − u∗ ) 1−αL1−α =(1−u
L
∗ ) 1−α
µ α
K
=(1−u
L
∗ ) 1−α k α .
A decrease in the natural rate of unemployment increases output
per worker y at all levels of capital per worker k since there is more
output to divide about: the production function curves shifts up.
(Remember that workers are everybody in the labour force, not only
the employed ones.)
Since output per worker increases at all levels of k, investmentincreases
at all levels of k, other things being equal (the investment
curves shifts up); therefore, the capital per worker in the steady state,
k ∗ , depends on the natural rate of unemployment.
(Analytically, the steady state condition is the same:
substituting
sy = d 0 k;
s(1 − u ∗ ) 1−α k α = d 0 k
and solving for k
k ∗ ³
s
=
d0 You were not required to do this part.)
´ 1
1−α
(1 − u ∗ ).
(b) Suppose that some change in government policy reduces the natural
rate of unemployment. Describe how this change affects output both
immediately and over time. Is the steady-state effect on output larger
or smaller than the immediate effect? Explain (Your answer should
include a graph).
The immediate effect is the increase in output per worker y at the
same level of capital per capita k ∗ 1 (the old steady state capital per
capita), from y∗ 1 to y0 2 =(1−u∗ 2) 1−αk∗ 1. However, this immediate
effect also increases savings from s · y∗ 1 to s · y0 2 >d0 · k∗ 1 , greater than
depreciation; therefore, there exists net investment which will start
the process of accumulating more capital and moving to a higher
steady state; i.e., the eventual effect (steady state) includes the extra
increase in output due to the consequent increase in capital.
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