8.11. 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., 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?