Consider again the two ways in which we can view the producer’s profit maximization problem.
A: Suppose a homoethetic production technology involves two inputs, labor and capital, and that its producer choice set is fully convex.
(a) Illustrate the production frontier in an isoquant graph with labor on the horizontal axis and capital on the vertical.
(b) Does this production process have increasing or decreasing returns to scale? How would you be able to see this on an isoquant graph like the one you have drawn?
(c) For a given wage w and rental rate r, show in your graph where the cost minimizing input bundles lie. What is true at each such input bundle?
(d) On a separate graph, illustrate the vertical slice (of the production frontier) that contains all these cost minimizing input bundles.
(e) Assuming output can be sold at pA, use a slice of the isoprofit plane to show the profit maximizing production plan. What, in addition to what is true at all the cost-minimizing input bundles, is true at this profit maximizing plan?
(f) If output price changes, would you still profit maximize on this vertical slice of the production frontier? What does the supply curve (which plots output on the horizontal and price on the vertical) look like?
(g) Now illustrate the (total) cost curve (with output on the horizontal and dollars on the vertical axis). How is this derived from the vertical slice of the production frontier that you have drawn before?
(h) Derive the marginal and average cost curves and indicate where in your picture the supply curve lies.
(i) Does the supply curve you drew in part (f) look similar to the one you drew in part (h)?
B: Suppose that the production technology is fully characterized by the Cobb-Douglas production function x = f (ℓ,k) = Aℓαkβ with α+β < 1 and A,α, and β all greater than zero.
(a) Set up the profit maximization problem (assuming input prices w and r and output price
p). Then solve for the input demand and output supply functions. (This is identical to
parts B(b) and (c) of exercise 12.2 —so if you have solved it there, you can simply skip to part (b) here.)
(b) Now set up the cost minimization problem and solve for the first order conditions.
(c) Solve for the conditional labor and capital demands.
(d) Derive the cost function and simplify the function as much as you can. (Hint: You can check your answer with the cost function given for the same production process in exercise 12.4.) Then derive from this the marginal and average cost functions.
(e) Use your answers to derive the supply function. Compare your answer to what you derived in (a).
(f) Finally, derive the (unconditional) labor and capital demands. Compare your answers to
those in (a).