Solow Model
Problem 1:
Consider a closed economy with population growth rate of 1% per year. The per-worker production function is
y = 6k.5,
where y is output per worker and k is capital per worker. The depreciation rate of capital is 14% per year. Households consume 90% of their income and save the remaining 10% of it. There is no government.
(a) What is the steady state in this economy?
Answer:
(a) In steady state, sf(k) ’ (n + d)k
0.1 ( 6k.5 ’ (0.01 + 0.14)k
0.6k.5 ’ 0.15k
0.6 / 0.15 ’ k / k.5
4 ’ k.5 k ’ 42 ’ 16 ’ capital per worker y ’ 6k.5 ’ 6 ( 4 ’ 24 ’ output per worker c ’ .9 y ’ .9 ( 24 ’ 21.6 ’ consumption per worker
(n + d)k ’ .15 ( 16 ’ 2.4 ’ investment per worker
(b) Suppose the government wants to double the steady state value of output per person by using policies to change the saving rate. What policies the government can use? What value of the saving rate the government needs to reach? Calculate it and show in the diagram the effect of such change.
Answer:
(b) To get y ’ 2 ( 24 ’ 48, since y ’ 6k.5, then
48 ’ 6k.5, so k.5 ’ 8, so k ’ 64.
The capital-labor ratio would need to increase from 16 to 64. To get k ’ 64, since sf(k) ’ (n + d)k, s ( 48 ’ .15 ( 64, so s ’ .2. Saving per worker would need to double.
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Problem 2.
According to Solow model, what happens with consumption per worker in the long run, if
a) a portion of nation’s capital stock is destroyed because of war b) energy prices permanently increased c) population growth rate permanently increased due to increased immigration d) population growth increased only temporarily
Answer.
(a) No effect on the steady state, because there has been no change in s, f, n, or d. Instead, k is reduced temporarily, but equilibrium forces eventually drive k to the same steady-state value as before.
(b) The rise in energy prices reduces the productivity of capital per worker. This causes sf(k) to shift down from sf1(k) to
Consider a closed economy with population growth rate of 1% per year. The per-worker production function is
y = 6k.5,
where y is output per worker and k is capital per worker. The depreciation rate of capital is 14% per year. Households consume 90% of their income and save the remaining 10% of it. There is no government.
(a) What is the steady state in this economy?
Answer:
(a) In steady state, sf(k) ’ (n + d)k
0.1 ( 6k.5 ’ (0.01 + 0.14)k
0.6k.5 ’ 0.15k
0.6 / 0.15 ’ k / k.5
4 ’ k.5 k ’ 42 ’ 16 ’ capital per worker y ’ 6k.5 ’ 6 ( 4 ’ 24 ’ output per worker c ’ .9 y ’ .9 ( 24 ’ 21.6 ’ consumption per worker
(n + d)k ’ .15 ( 16 ’ 2.4 ’ investment per worker
(b) Suppose the government wants to double the steady state value of output per person by using policies to change the saving rate. What policies the government can use? What value of the saving rate the government needs to reach? Calculate it and show in the diagram the effect of such change.
Answer:
(b) To get y ’ 2 ( 24 ’ 48, since y ’ 6k.5, then
48 ’ 6k.5, so k.5 ’ 8, so k ’ 64.
The capital-labor ratio would need to increase from 16 to 64. To get k ’ 64, since sf(k) ’ (n + d)k, s ( 48 ’ .15 ( 64, so s ’ .2. Saving per worker would need to double.
[pic]
Problem 2.
According to Solow model, what happens with consumption per worker in the long run, if
a) a portion of nation’s capital stock is destroyed because of war b) energy prices permanently increased c) population growth rate permanently increased due to increased immigration d) population growth increased only temporarily
Answer.
(a) No effect on the steady state, because there has been no change in s, f, n, or d. Instead, k is reduced temporarily, but equilibrium forces eventually drive k to the same steady-state value as before.
(b) The rise in energy prices reduces the productivity of capital per worker. This causes sf(k) to shift down from sf1(k) to