Question

Consider the Solow Growth model with constant population and technology, as follows.

Yt = F(Kt , Lt) = K 1/3 t L 2/3 t . (1)

Yt = Ct + St (2)

St = σYt (3)

Kt+1 = (1 − δ)Kt + It (4)

St = It (5)

Where Yt is output in period t, Kt is capital stock in period t, Lt is labor in period t, Ct is consumption in period t, St is savings in period t, and It is investment in period t.

a) Find output per worker in period t, denoted as yt , as a function of the capital per worker in period t, that is, yt = f(kt).

b) Write the equation (1) ~ (5) in per capita terms.

c) Write the fundamental equation of capital accumulation, that is, capital accumulation equation as a function of kt and kt+1. (Hint. Use five equations from Q2).

d) Country A and country B both have the production function. Assume that neither country experiences population growth or technological progress and that 20 percent of capital depreciates each year. Assume further that country A saves 10 percent of output each year and country B saves 30 percent of output each year. Find the steady state level of capital per worker for each country. Then find the steady-state level of income per worker, savings per worker, investment per worker, and consumption per worker.

e) Suppose that both countries start off with a capital stock per worker of 1 (k A 1 = k B 1 = 1. Here, k X t denotes the capital per worker in period t in country X). What are the levels of income per worker and consumption per worker in period 1?

f) Show how the capital per worker will evolve over time in both countries for 7 years (t = 1, · · · , 7). For each year, calculate income per worker and 2 consumption per worker. Present your answers in the following table. How many years will it be before the consumption in country B is higher than the consumption in country A? Why country B has a lower consumption than country A for the initial period? Explain.

Answer 1

Country A	Country B
t	k	y	c	t	k	y	c
1	1	1	0.9	1	1	1	0.7
2	0.9	0.965489	0.86894	2	1.1	1.03228	0.722596
3	0.816549	0.934675	0.841208	3	1.189684	1.059605	0.741723
4	0.746707	0.907228	0.816506	4	1.269629	1.082827	0.757979
5	0.688088	0.882839	0.794555	5	1.340551	1.102625	0.771837
6	0.638754	0.861214	0.775093	6	1.403228	1.119548	0.783684
7	0.597125	0.842083	0.757875	7	1.458447	1.134045	0.793831

After the 5th year the consumption in country B will be higher than that of country A.

The consumption of country A was initially higher because it has higher rate of consumption (lower savings rate) than country B. But as country B has higher savings rate, it will generate higher output through higher investment in time. This will increase consumption over that of country A eventually.