Question

51. Suppose that a firm has the production function

Q = 2K1/2 L1/2

Assume that the capital stock is fixed at 9 units (i.e. K =9). If the price of output (P) is $6 per unit, and the wage rate (w) is $2 per unit, determine the optimal or profit-maximizing rate of labor to be hired. What labor-input rate is optimal if the wage rate increased to $3 per unit?

Answer 1

$Q=2*9^{1/2}*L^{1/2}$

$L=(\frac{Q}{6})^2$

$C=9+L*2$ ; C-cost

$C=9+\frac{Q^2}{18}$ ; $E=PQ=6Q$; E-earnings

$I=E-C$; I-Income

the quantity that gives max income  :

$\frac{dI}{dQ}=(6Q-9-\frac{Q^2}{18})'=6-\frac{2*Q*18}{18^2}=6-\frac{2Q}{18}$

$0=6-\frac{2Q}{18}$ ; $Q=54$; $L=81$

W=3:

$C=9+3*L$; $C=9+\frac{Q^2}{12}$

$0=6-\frac{2Q}{12}$; $Q=36$

$L=36$