Question

Suppose a firm has a production function given by q = 3L + K.

The firm can purchase labor, L at a price w = 24, and capital, K at a price of r = 5.

What is the firm’s total cost function?

Answer #1

Minimize : Cost(C) = wL + rK = 24L + 5K

Subject to 3L + K = q -----------------(1)

We can see from the production function that it considers L and K as perfect substitutes and values L times over K.

Cost Minimizing Criteria in this case is :

(i) If w > 3r, then he will hire only Capital(K)

(i) If w < 3r, then he will hire only Labor(L)

(i) If w = 3r, then he will hire any combination of L and K such that (1) satisfies.

Here w = 25 > 3r( = 3*5 = 15). Thus he will hire only Capital and hence L = 0

Putting this in (1) we get :

3*0 + K = q => K = q

Hence, Cost(C) = 24*0 + 5q = 5q

Hence, firm's cost function is given by :

**C = 5q**

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