3. A profit-maximizing firm manufactures widgets (output is denoted by q) using machines (K) and full-time employees (E). In any given week, its output is given by the production function
q = f(E,K) = 1200K + 700E + 2EK - E2 -2K2
The marginal product of labor (MPE) is: 700 + 2K - 2E
What is the marginal product of capital (MPK)?
The way MPE changes with E satisfies a standard assumption or "law" about production functions. What is this "law" called? Explain what it means intuitively, using your own words.
The firm operates in perfectly competitive factor and product markets. The price of capital (r) is $1,000 per machine per week. Moreover, the firm sells its output at the going price (p) of $1 per widget. Consider the firm's short-run labor demand problem.
If the current capital stock is fixed at 250 units in the short run, how many full-time workers should the firm employ if the weekly salary of each full-time worker (w) is $400 per week? Compute the firm's level of output and profits per week in this short-run equilibrium.
Assuming the capital stock remains fixed at 250 units, what will the firm's short-run labor demand be if the weekly wage (w) decreases to $380 per full-time worker? What will the firm's level of output and profits be in this new short-run equilibrium?
What is the firm's short-run elasticity of labor demand as the wage falls from $400 to $380? Is this an elastic or inelastic response?
Consider now the long-run factor demand problem.
Suppose that the price of labor is again $400 per week. What is the firm's long-run demand for labor? What is the firm's long-run demand for capital? Compute the firm's level of output and profits per week in this long-run equilibrium.
What will the firm's long-run labor demand be when the weekly wage decreases to $380 per full-time worker? Compute the firm's demand for capital, level of output and profits in this new long-run equilibrium.
What is the firm's long-run elasticity of labor demand as the wage falls from $400 to $380? Is this an elastic or inelastic response? How does this long-run labor demand elasticity compare with the short-run elasticity you found in (e.)? Why the difference?