Problem Set 5: Economic Growth (I)
1. Country A and country B both have the production function
Y = F (K; L) = K 1=2 L1=2 :
a. Does this production function have constant returns to scale?
Answer. F (K; L) = (K)1=2 (L)1=2 = K 1=2 L1=2 = Y: Therefore Y = K 1=2 L1=2
displays CRS.
b. What is the production function per worker, y = f (k)?
on CRS above, set = 1=L and obtain y = Y =L =
Using the1=2results
1=2
Answer.
1=2
1=2
=L = K
= L
= k 1=2 :
K L
c. (Steady state properties) Assume that neither country has population growth or technological progress and that 5% of capital depreciates every year. Assume further that
country A saves 10% of output each year and country B saves 20% of output each year.
Use your answer from part (b) and the steady state condition that investment equals
depreciation to …nd the steady state level of capital per worker for each country. Then
nd the steady state levels of output per worker and consumption per worker.
Answer. We know the following facts about countries A and B :
= depreciation rate = 0.05
sA = saving rate of country A = 0:1
sB =
saving rate of country B = 0:2
y = k 1=2 is the per-worker production function
The growth of the capital stock per worker, k, is given by investment per worker sf (k)
minus depreciation per worker k. Therefore k = sf (k) k = sk 1=2 k: In steady
=
state, k = k such that s (k )1=2 = k : Therefore k = (s=)2 . In particular, kA
2
2
(0:1=0:05) = 4, and kB = (0:2=0:05) = 16: Steady state output per worker is given by
= 41=2 = 2, and y = 161=2 = 4: Finally, steady state
y = (k )1=2 : In particular, yA
B
consumption per worker is given by c = (1 s)y : In particular, cA = 2(1 0:1) = 1:8,
and cB = 4(1 0:2) = 3:2: Note that, while country B consumes a lower proportion of
disposable income (output per worker), they end up consuming more than country A in
steady state, due to higher output per worker in steady state in country B.
2. Three economists are debating whether an increase in the number of immigrants will increase,
decrease or have no e¤ect on the growth rate of the domestic economy. Use the Solow model
to justify each of their arguments.
Answer. If immigrants enter the country with an endowment of capital per head that is
lower than that of natives, then more intensive migration reduces the capital/labor ratio in
the economy (i.e. to the left of the steady-state), so growth becomes positive in the short
run, until capital per worker goes back to the pre-existing steady-state level. If immigrants
enter the country with an endowment of capital per head that is higher than that of natives,
more intensive migration raises the capital/labor ratio (i.e. to the right of the steady-state),
so growth goes negative in the short run, again until capital per worker goes back to the
pre-existing steady-state level. If immigrants enter the country with an endowment of capital
per head that is the same as that of natives, the capital/labor ratio does not change, nor the
short term growth rate of the economy. In the long run, growth is unchanged in the three
cases (i.e. it stays at zero).
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3. “Devoting a large share of national output to investment would help restore rapid productivity
growth and rising living standards”. Do you agree with this claim?
Answer. Devoting a large share of national output to investment means raising the saving
rate. Assuming that the economy is initially in steady state, this generates some positive
growth during the transition path to the new steady state, and a higher steady state level of
capital per head and income per head. So living standards will indeed be higher. But once
the new steady state is reached, growth in capital per head is indeed zero, so this cannot
sustain rapid growth in the long-run.
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