Question

Consider an economy described by the production function:

Y = F(K, L) = K^0.3L^0.7.

Assume that the depreciation rate is 5 percent per year. Make a table showing steady-state capital per worker, output per worker, and consumption per worker for saving rates of 0 percent, 10 percent, 20 percent, 30 percent, and so on. Round your answers to two decimal places. (You might find it easiest to use a computer spreadsheet then transfer your answers to this table.)

Steady State Values for Various Saving Rates
	s	k*	y*	c*
Depreciation Rate:	0.0
(0.05)	0.1
	0.2
	0.3
	0.4
	0.5
	0.6
	0.7
	0.8
	0.9
	1.0

What saving rate maximizes output per worker? What saving rate maximizes consumption per worker?

A saving rate of  percent maximizes output per worker. A saving rate of  percent maximizes consumption per worker.

Answer 1

Y = K^0.3L^0.7

Output per worker (y) = Y / L = (K/L)^0.3 = k^0.3 where k = K/L

When s: Savings rate, in steady state,

s / = k / y

s / = k / k^0.3

s / = k^0.7

Capital per worker (k*) = [s / ]^(1/0.7) = [s / ]^1.43

Output per worker (y*) = k^0.3 = [s / ]^(0.3/0.7) = [s / ]^0.43

Consmption per worker (c*) = yY - sy* = y* x (1 - s) = [s / ]^0.43 x (1 - s)

When = 0.05,

k* = (s / 0.05)^1.43

y* = (s / 0.05)^0.43

c* = (s / 0.05)^0.43 x (1 - s)

Therefore:

s	k*	y*	c*
0	0	0	0
0.1	2.69	1.35	1.21
0.2	7.26	1.82	1.45
0.3	12.96	2.16	1.51
0.4	19.56	2.45	1.47
0.5	26.92	2.69	1.35
0.6	34.93	2.91	1.16
0.7	43.55	3.11	0.93
0.8	52.71	3.29	0.66
0.9	62.38	3.47	0.35
1	72.52	3.63	0.00

A savings rate of 100% maximizes output per worker.

A savings rate of 30% maximizes consumption per worker.