Question 1. Short-Run Profit Maximization. Given the following short-run production function for the firm,

y = 2x₁^1/2x₂^1/3

where y is output, x₁ is the variable input 1, and x₂ is the fixed input 2, w₁ is the price of input 1, and w₂ is the price of input 2.

(a). What is the marginal product of the variable input 1?

(b). Does the variable input 1 exhibit diminishing marginal product (i.e. does the marginal product decrease as y increases)? Why?

(c). What is the short-run factor demand function for the variable input 1?

(d). What is the short-run supply function for output good y?

(e). In panel (b), you are given a graph of the short-run production function for a firm where y is output, x₁ is the variable input 1, and x₂ is the fixed input 2. You are also given an initial iso-profit line, Π₂, which represents the profit maximizing choice of input 1 and output when the price of output, p, is equal to p''. Using the relationship, p'< p''< p''', graphically derive the short-run supply function in panel (a). Please label all axes, curves and important points, and make sure you link the two graphs together in regards to y. Your supply function should be drawn through three points that indicate the profit maximizing output at p', p'', and p'''.

Question 2. Cost Minimization. Given the following long-run production function for the firm,

y = 4L^1/4K^1/4

where y is output, L is labor, K is capital, p is the price of output, w is the price of labor and r is the price of capital:

(a). Using cost minimization and the Lagrangian method, derive the firm's conditional factor demand equations for inputs 1 and 2. Show all your work for full credit and place a box around your answers.

(b). Using your conditional factor demand equations for inputs 1 and 2 from part (a), derive the long-run total cost function for the firm. Please simplify your expression and place a box around your final answer.

(c). Using your cost function from part (b) and profit maximization, derive the long-run supply function for the firm. Show all your work for full credit and place a box around your answers.

(d). Is your supply function from part (c) upward sloping, downward sloping, vertical or horizontal relative to the price of output, p. Why?

(e). Assume capital is fixed in the short-run at a level of K''. Using the long-run isoquant/isocost graph below, draw the short-run output expansion path for the firm. On the graph, clearly indicate the amount of labor, L, used to produce output y', y'', and y''', now that capital is fixed at K''. Draw in any new isocost curves that indicate whether the firm's short-run costs are equal to, less than, or greater than long-run costs at the points associated with L', L'', and L'''.

3. Cost Curves. Using the graphs below for a perfectly competitive firm, please answer the following:

(a). Given the short-run equilibrium price of output, pe, is this firm producing in the short-run? Why? If the firm is producing a positive level of output, indicate on the graph and label the short-run optimal output for the firm. If there are positive or negative economic profits, shade in the area of those profits, and clearly indicate whether they are considered profits or losses.

(b). Given the short-run equilibrium price of output, pe, is this firm producing in the short-run? Why? If the firm is producing a positive level of output, indicate on the graph and label the short-run optimal output for the firm. Is the firm making positive, negative, or zero economic profits?

(b). Given the potential market scenarios below, where up to 5 firms are considering entering the market, and the long-run cost structure of the representative firm, how many firms will choose to participate in the market in the long-run? Why? Clearly indicate and label which market clearing price would yield the long-run equilibrium price for the market. Is the long-run economic profit for the firms who choose to participate in the market positive, negative or zero? Why?