Question

Consider a firm with production function Q(L,K) = min( 6L , 3K ).
The wage rate is w = 6, and the rental rate of capital is r = 3.
(a) Calculate the optimal capital to labour ratio
(b) Derive the long-run cost function of the firm.

need help in all! Thank you!

Answer 1

 Q(L,K) = min( 6L , 3K ).

Wage rate w = 6 rental rate r=3

a) optimal combination of labor and capital is given by 6L = 3K

K= 2L

$\frac{K}{L} = 2$

Optimal capital labor ratio = $(\frac{K}{L} = 2)$

b)Substitute the ratio K= 2L in Q

Q = min{6L,3(2L)} = min{6L,6L}

Q= 6L

$L= \frac{Q}{6}$

$K= \frac{2Q}{6}$ = $\frac{Q}{3}$

Cost(C) = WL + rK

C= $\frac{Q}{6}(W)$ + $\frac{Q}{3}(r)$

C=$\frac{Q}{6}(6)$ + $\frac{Q}{3}(3)$

C= 2Q

therefore long run cost function C(Q)

C(Q) = 2Q