A firm produces a product with labor and capital as inputs. The production function is described by Q = LK. The marginal products associated with this production function are MPL = K and MPK = L. Let w = 1 and r = 1 be the prices of labor and capital, respectively.
a) Find the equation for the firm's long-run total cost curve as a function of quantity Q.
b) Solve the firm's short-run cost-minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive the equation for the firm's short-run total cost curve as a function of quantity Q and graph it together with the long-run total cost curve.
c) How do the graphs of the long-run and short-run total cost curves change when w = 1 and r = 4?
d) How do the graphs of the long-run and short-run total cost curves change when w = 4 and r = 1?