Question

Consider a firm that uses two inputs. The quantity used of input 1 is denoted by ?1 and the quantity used of input 2 is denoted by ?2. The firm produces and sells one good using the production function ?(?1, ?2) = 4?1 0.5 + 3?2 0.5 . The final good is sold at price ? = $10. The prices of inputs 1 and 2 are ?1 = $2 and ?2 = $3, respectively. The markets for the final good and both input goods are treated as competitive markets by the firm, that is, it takes prices as given.

e) Find the technical rate of substitution. Does the technology show diminishing technical rate of substitution? Explain. [2 marks]

Assume in the short run that ?2 is fixed at ?̅̅2̅ = 100.

f) Write down the firm’s profit function and the firm’s short run profit maximisation problem. Find the firm’s optimal use of input 1, the associated optimal quantity of the output good, and the firm’s profit level. [4 marks]

Now consider the long run, where the quantity of input 2 can be varied.

g) According to your answer in part a), does the firm have a profit maximising plan in the long run? If no, explain why. If yes, is the plan unique? [2 marks]

Answer 1