Question

A production function is given by q = L1/3K1/3 and a total cost function is given by TC = q2/3. Under what conditions could these belong to the same firm?

Answer 1

The production function is . The cost of production would be , for w and r be price of labor and capital.

The marginal products would be as or or and or or . The optimal choice of inputs would be where or or or , which is the optimal combination of inputs. Putting it in the production function, we have or or , and since , we have or . The total cost function would be or or or .

For the total cost function to be , the condition must be as or or or . Hence, only for the quantity , the total cost would be for the same firm.

-----------------------------------------------------------------------

NOTE : The answer given above is according to the question. But a speculation on the validity of the question is as below.

Supposing a typo, the cost function be , the answer would be as below.

The condition must be as or or or is the required condition for both firms to be the same.

The cost function of is valid since the marginal cost would be , which is increasing as q increases. But, for the cost function be , the marginal cost would be , which continuously decreases as q decreases, and hence, the cost function can not be considered to be a valid one.

-----------------------------------------------------------------------