Problem 2 Prot Maximization- Short run GMC is producing cars using machines (K) and labor (L). The technology is capital intensive; the production function is given by f (K , L) = K3/4L1I'2. The value of GMC physical capital (machines, real estate etc.) is equal to K = $16 billon (in calculations ignore billions). We analyze rm's behavior in the short run (note: K Problem 3 2 cannot be changed in the short run). Suppose the price of a car is equal to p and the wage rate is w. a) Write down the prot as a function of L. b) On a graph with L on the horizontal axis and $ on the vertical axis, plot two components of prot function: total revenue, 3) f (K , L), and labor cost, wL, (when drawing, assume p = 1 and w = 2). On the graph, mark the prot level as the dierence between the two lines (for any given L). c) In order to nd a: that maximizes some function f (m), we take the rst derivative of the function with respect to a: and set it equal to zero (we call it a rst order condition or a at-hand principle). Please explain intuitively why this method allows us to nd the optimum. (1) Set the derivative of your prot function with respect to L equal to zero and derive the optimality condition (the equation that tells: M PL = 10/39). Explain the Table 1: parameters p111 w842 economic intuition behind the latter condition. e) Find analytically the optimal level of labor L* that maximizes the prot as a function of the real wage w/p. We call this function the rm's labor demand. Find the values of optimal L for the parameter values in the table. Plot your demand for labor in the graph with the real wage 10/}; on the vertical axis and L on the horizontal one. Using the table mark three points corresponding to the three values of 10/17. f) What is the maximal prot for each of the three values of w/p