Question

Problem 1. Suppose that production function for the corporate sector is well presented by the Cobb-Douglas production function:

Y = K0.3L0.7 With this production function: MPL(K, L)=0.7K0.3L0.3 and MPK(K, L)=0.3K0.7L0.7

a) Suppose the capital stock of the firm is K¯ =10. Show that the demand for labor is given by:

b) Find the amount of labor demanded for W P = 0.1,W P = 0.3, W P = 0.5, W P = 0.7 and W P = 0.9. Plot the combinations of labor demand and real wages on a diagram in which labor is on the horizontal axis and W P is on the vertical axis.

c) Now suppose that the labor supply of the firm is L¯ = 10. Show that the demand for capital is given by:

d) Find the right amount of capital demanded for R P = 0.1, R P = 0.3, R P = 0.7 and R P = 0.9. Plot the combination of capital demand and real rental rates on a diagram in which capital is on the horizontal axis and R P is on the vertical axis

Answer 1

(a) For the given capital stock, we have the marginal product of labor as
MP = 0.765-0.3
or
MP = 0.7 * 1045-0.3
. The labor demand would be where
W = P + MP
, for P be the price of Y and W be the wage, and we have
POMP
or
Wp = MP
, where Wp is the real wage. We have
Wp = 0.7 * 100.5-0.3
or
WP L-03- 0.7 * 100.3
or
[0,3 - 0.7* 100 Wp
or
0.71/0.3 + 10 = .7 1/03
as the required labor demand.

Note however, the conditional input (labor) demand would be different. We have the production function as
27 pY = 1
or
Y = 100307
or
107 = 1003
or
y 1/0.7 L= 100.3/0.3
, ie
y 10/7 L = 1037
is the conditional labor demand, for the given capital level.

(b) The schedule and graph would be as below.

Wp	L*
0.1	6561.354
0.3	168.4959
0.5	30.6968
0.7	10
0.9	4.327

Wp 0-5 Labor Demand 2000 4000 6000

(c) For the given labor supply, we have the marginal product of labor as
MPK = 0.3K-01207
or
MPK = 0.3K-6.71007
. The labor demand would be where
R=P* MPK
, for P be the price of Y and R be the rental rates, and we have
P= MPK
or
Rp = NPK
, where Rp is the real rental rate. We have
Rp = 0.3K-6.7104.7
or
K-47= Rp 0.3* 1007
or
K0.7 -0.31007 RP
or
0.31/0.710 K* =
as the required capital demand.

Note however, the conditional input (labor) demand would be different. We have the production function as
ZpY=
or
Y = K31007
or
K03= 100.7
or
K= y1/03 100.7/0.3
, ie
y 10/3 K* = 107/3
is the conditional capital demand, for the given capital level.

(d) The schedule and graph would be as below.

Rp	K*
0.1	64.8212
0.3	13.4932
0.5	6.5041
0.7	4.0219
0.9	2.8088

Rp 0:5 Capital Demand K* 60