Problem 2 [21 marks] Consider a firm that uses two inputs. The quantity used of input...

Problem 2 [21 marks] 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.

a) Show whether the production function has increasing, decreasing, or constant returns to scale. [2 marks]

b) Find the marginal product of each input. Show whether the production technology obeys the law of diminishing marginal products. [2 marks]

c) Draw the isoquant for an output level of 12. Clearly label the axes and the curve and show any two input bundles on the curve by indicating their coordinates. [2 marks]

d) Does the firm have convex production technology? Explain. [2 marks]

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 ?2 ̅̅̅ = 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]

h) Write down the firm’s profit function and the firm’s long run profit maximisation problem. Find the firm’s optimal use of input 1, input 2, the associated optimal quantity of the output good, and the firm’s profit level. [4 marks]

j) Explain why the profit level in the long run must be at least as high as the profit level in the short run. [1 mark]

Expert Solution

Rahul Sunny answered 2 years ago

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