Question

Consider a firm with production function f(K,L) = K +L. (a) Suppose that capital level is currently fixed at K = 10. Find the short term production cost function for producing outputs greater than or equal to 10 units when w = 1 and r=1. (b) Suppose wage goes up to w' = 2 while the price of capital remains same at r = 1. Find the new short-run cost function for producing output greater than or equal to 10 units.. (c) Find the long-run cost function at prices w' = 2 and r = 1.

Answer 1

Given production function, q = f(K,L) = K + L.

(a) Fixed capital level $\hat{K} = 10$,

We know Cost = wL + rK,

So we have short run cost where, $q\geq 10, w=1, r=1$

Putting the value of K in production function, q = 10 + L, and L = q - 10

C = L + 10 = q - 10 +10 = q which is greater than or equal to 10.

(b) When wages increases to 2, ${w}' = 2$,

C = 2L + 10 and we know L = q - 10 from part (a)

So, short run cost function, C = 2(q - 10) + 10 = 2q -10, where $q \geq 10$.

(c) Long run cost function,

C = wL + rK

We know ${w}' =2, r=1$

C = 2L + K, where q = K + L,

C = 2(q - K) + K = 2q - K.