A: Suppose you have a homothetic production technology and you face input prices (w,r).
(a) On a graph with labor „“ on the horizontal and capital k on the vertical axis, illustrate a ray along which all cost minimizing production plans might lie for a given set of input prices. Does your answer depend on whether the production technology has increasing or decreasing returns to scale (or some combination of these)?
(b) Illustrate in your graph an isoquant corresponding to some output level x. What has to be true at the intersection of the ray and the isoquant?
(c) Show what happens to the ray of cost minimizing input bundles if w increases to w€². Then illustrate how you would derive the conditional labor demand curve for producing .
(d) From this point forward, suppose that the production technology has decreasing returns to scale. Illustrate how you would derive the firm€™s long run cost curve for the original input
prices.
(e) What happens to the cost curve when w increases to w€²?
(f) Suppose that you are initially producing at the intersection of your original isoquant (corresponding to ) and the original ray. If w remained unchanged, where would your (short run) expenditure curve fall on your graph with the long run cost curve?
(g) Translate your cost/expenditure curve graph to a graph with the average (long run) cost and average (short run) expenditure curves.
(h) How does the average (long run) cost curve change when w increases to w€²? If you also graphed a cost curve that removed the substitution effect, where would it generally lie relative to the original and final cost curve? What would its precise location depend on?
(i) Now suppose that instead of wage increasing, the rental rate on capital r fell to r €². What
happens to the conditional labor demand curve that you graphed in part (c)?
(j) Repeat (h) for the change in the rental rate.
B: Suppose again (as in exercise 13.1) that the production process is defined by the Cobb-Douglas production function x = f („“,k) = A„“αkβ.
(a) For input prices (w,r), calculate the long run conditional input demand functions.
(b) Do you need to assume 0(c) Derive the long run total, marginal and average cost functions.
(d) Suppose output price is p. Use your answer to derive the firm€™s (long run) profit maximizing output supply function. Do you need to assume 0 (e) From your answer, derive the firm€™s profit maximizing long run labor and capital demand functions. (You can again check your answers with those you derived through direct profit maximization in exercise 13.1.)

(g) Derive the short run (total) cost function as well as the short run marginal and average cost functions.
(h) Derive the short run supply curve.
(i) True or False: As long as the production function has decreasing returns the scale, the (short run) average expenditure curve will be U-shaped even though the short run average cost curve is not.
(j) What is the shape of the long run average cost curve? Can the Cobb-Douglas production
function yield U-shaped long run average cost curves?