Question 2: Consider a firm with the following production function: y = f(x1, x2) =
2x1/4x2/2. The cost of x₁ and x2 are w₁ and w₂, respectively. This is a perfectly competitive
market, and the market price for a single unit of output is p. (25 points)
a. Find the marginal product of x₁ and x2. (2 points)
b. Show that the marginal product of x₁ is diminishing. (2 points)
c. What is the maximal output level possible from the input bundle (x1, x2) = (4,4)? (2
points)
d. Let's say the firm increases the input level of both x₁ and x2 by a rate of 4, so the new
input level is (16,16). What's the new output level? Does this technology exhibit
constant, increasing, or decreasing returns to scale? Show your work. (2 points)
e. Write the firm's profit function and the firm's maximization problem. (2 points)
f. Find the level of output and the input bundle that maximize the firm's profit. Show
your work (beginning with the first order conditions). (5 points)
g. In the short-run, the firm's input level of x2 is fixed at 4. What is the firm's short-run
production function? (1 point)
h. Recall, the cost of x₁ and x2 is w₁ and w₂, respectively. What is the firm's fixed cost?
Variable cost? Total cost? (3 points)
i. What is its short-run profit maximization problem? (2 points)
j. Write the equation of a short-run isoprofit line. (2 points)
