Econ 375

a)         Yes.
b)         y = k^1/2
c)         From our derivations in class, we found the steady-state level of capital per worker to be: k* = (s/d)²

Thus Country A: k* = (s/d)² = (0.10/0.05)² = 4
Country B: k* = (s/d)² = (0.20/0.05)² = 16

To find the steady-state levels of consumption per workers, recall that c = (1-s)y:

Country A: c* = (1-s)y* = (1-0.1)(2) = 1.8
Country B: c* = (1-s)y* = (1-0.2)(4) = 3.2

d) Assuming k = 2 for both countries in Year 1 the following time path can be calculated. Note that it will take five years before consumption in Country B is higher than consumption in Country A.

Year	k	y	c	i	dk	D k = i - dk
1	2.000	1.414	1.273	0.141	0.100	0.041
2	2.041	1.429	1.286	0.143	0.102	0.041
3	2.082	1.443	1.299	0.144	0.104	0.040
4	2.122	1.457	1.311	0.146	0.106	0.040
5	2.162	1.470	1.323	0.147	0.108	0.039

Year	k	y	c	i	dk	D k = i - dk
1	2.000	1.414	1.131	0.283	0.100	0.183
2	2.183	1.477	1.182	0.295	0.109	0.186
3	2.369	1.539	1.231	0.308	0.118	0.189
4	2.559	1.600	1.280	0.320	0.128	0.192
5	2.751	1.658	1.327	0.332	0.138	0.194

5. First, consider steady states. A slower population growth rate will shift the capital replacement line [=(d +n)k] downward. The new steady state has a higher level of capital per worker, hence a higher level of output per worker.

What about steady-state growth rates? In steady state, total output grows at a rate n+g, whereas output per person grows at rate g. Hence, slower population growth will lower total output growth, but per-person output growth will be the same.

Now consider the transition. We know that the steady-state level of output per person is higher with low population growth. Hence, during the transition to the new steady state, output per person must grow at a faster rate than g for a while. In the decades after the fall in population growth, growth in total output will fall while growth in output per person will rise.

6. In the United States, the rate of growth in output per person fell from 2.2 percent per year for the period 1948-1972, to 1.7 percent per year for the period 1972-1991. Other countries experienced even more severe declines in growth. It appears that this slowdown in output growth is attributable to a slowdown in productivity growth--the rate at which the production function is improving over time. Various explanations for this slowdown have been proposed, but the slowdown remains a mystery. The period since 1991 has seen rising productivity growth.

8. a) For any production function, the MPK is the additional output produced per worker if each worker has an extra unit of capital. That is: MPK = f(k+1) - f(k).

For the production function y = Ak, we find: MPK = A(k+1) - Ak = A + Ak -Ak = A.

b) To show that a higher saving rate leads to a permanently higher growth rate of output per worker, we first look at how the saving rate affects the growth rate of capital per worker; we then look at how this growth rate of capital affects the growth rate of output per worker.

To convert this into a growth rate for capital per worker, divide by k: D k/k = sA - d - n

Next, we want to see how this growth in capital translates into growth in output. We can express the production function y = Ak in terms of growth rates as: D y/y = D A/A + D k/k

Since A is a constant, D A/A = 0. Hence, the growth rate of output per worker equals the growth rate of capital per worker: D y/y = D k/k = sA - d - n

This expression shows that if the saving rate s increases, then the growth rate of output will be permanently higher.

10. Another growth model question.
a) You can calculate the MPK in two ways:
If you use calculus, MPK = dY/dK = .3AK^.3-1L^.7 = .3(AK^.3L^.7)K^-1

Since we are told that the ratio of capital to output is 3, this makes Y/K = 1/3. Thus MPK = .3 (1/3) = 0.10

The second hint told you to make use of the definition of capital's share of output.
Thus capital's share = (MPK)(K/Y)

Capital's share of output is represented by the exponent of K in the production function, in this case, 0.3. Thus

e) To find the saving rate necessary to achieve the Golden-Rule, use the equation for steady-state K (see part (b) above) and your answer to part (d): s = (d + n + g)K/Y = (.04 + .03)(4.3) = 0.30

11. Economic policy can influence the saving rate by either increasing public saving or providing incentives to stimulate private saving. Public saving is the difference between government revenues and spending. If spending exceeds revenue, the government runs a budget deficit, which is negative saving. Policies that decrease the deficit (such as reductions in government spending or increases in taxes) increase public saving, whereas policies that increase the deficit decrease saving. A variety of government policies affect private saving,. The decision by a household to save may depend on the rate of return; the greater the return to saving, the more attractive saving becomes. Tax incentives such as tax-exempt retirement accounts for individuals and investment tax credits for corporations increase the rate of return and encourage private saving.