GMC is producing cars using machines (K) and workers (L). The technology is represented by the production function: q = (L^ 1/2) (K^3/4). The value of GMC's physical capital is K = $16 billion (in calculations, ignore billions). Because we are measuring capital in dollars, r = 1. We wish to analyze the firm's behavior in the short run (and so K cannot be changed). Suppose the price of a car is p and the wage rate is w (parameters).

(a) Write down the profit as a function of L (and the exogenous parameters w and p).

(b) On a graph with L on the horizontal axis and $ on the vertical, plot two components of the profit function: total revenues (pF(L, K)) and total labor costs (wL). When drawing these, you can assume p = 1 and w = 2. On the graph, indicate the profit level as the difference between these two curves (for any given L).

(c) In order to find the maximum of a function, we use the necessary first-order condition (NFOC). Explain intuitively why the first-order condition allows us to find the optimum? What does the first-order condition in the case of profits tell you about the slopes of the revenue and labor cost curves at the optimum?

(d) Compute the NFOC for the profit equation you found in part (a). Use this to prove that the optimal L occurs where MPL = w p . Explain the economic intuition behind this condition.

(e) Find the optimal quantity of labor, L , to maximize profit as a function of the real wage, w p . Call this the firm's labor demand. Find the value of L for the following parameter values: p 1 1 1 w 8 4 2 Plot the demand for labor on the graph with real wages on the vertical axis and L on the horizontal axis. Using the parameter values in the table, mark the coordinates of those three points on your graph.

(g) What is the maximum profit for each of the three real wage values in the table above?