Question

3. The land of Grim can be described by the following production func- tion Y = KÈL Moreover, there is no population growth nor is there technological progress. (a) Find the steady-state capital stock per worker, output per worker, and consumption per worker as a function of the saving rate and the depreciation rate. (b) Research shows that the depreciation rate in the Land of Grim is 10% per year. Make a table showing steady-state capital per worker, output per worker, and consumption per worker for sav- ing rates of 0%, 10%, 20%, 30% ... up to 100%. What saving rate maximizes output per worker? What saving rate maximizes consumption per worker? (c) Derive the formula for the marginal product of capital. Calcu- late the marginal product of capital for each value of the saving rate from part (b). What do these calculations tell you about the relationship between MPk and steady-sate consumption per worker?

Answer 1

Y/L = (K/L)^1/2

y = k^1/2 [y = output per worker and k = capital per worker]

(a)

If s be savings rate and d be depreciation rate, then in steady-state,

s / d = k / y

s / d = k / (k^1/2)

s / d = k^1/2

k = (s/d)²

y = s/d

c = (1 - s) x y = (1 - s) x (s/d) = (s - s²) / d

(b)

d	s	k	y	c
0.1	0	0	0	0
0.1	0.1	1	1	0.9
0.1	0.2	4	2	1.6
0.1	0.3	9	3	2.1
0.1	0.4	16	4	2.4
0.1	0.5	25	5	2.5
0.1	0.6	36	6	2.4
0.1	0.7	49	7	2.1
0.1	0.8	64	8	1.6
0.1	0.9	81	9	0.9
0.1	1	100	10	1.1E-15

Output per worker is maximized when savings rate = 100% (= 1)

Consumption per worker is maximized when savings rate = 50% (= 0.5)

(c)

MPK = $\partial$ Y/$\partial$K = (1/2) x (L/K)^1/2 = (1/2) / (K/L)^1/2 = (1/2) / (k^1/2)

d	s	k	k^(1/2)	MPK
0.1	0	0	0
0.1	0.1	1	1	0.50
0.1	0.2	4	2	0.25
0.1	0.3	9	3	0.17
0.1	0.4	16	4	0.13
0.1	0.5	25	5	0.10
0.1	0.6	36	6	0.08
0.1	0.7	49	7	0.07
0.1	0.8	64	8	0.06
0.1	0.9	81	9	0.06
0.1	1	100	10	0.05

While MPK is steadily decreasing, stead-state consumption per worker first increases, reaches a maximum and then decreases.