The point K = 25, L = 400 also lies on the q = 2000 isoquant. If L = 401, what must K be for this input combination to lie on the q = 2000 isoquant? What is the approximate value of the MRTS at K = 25, L = 400?

Problem Set #3

1. Suppose that the hourly output of chili at a barbecue (q, measured in pounds) is characterized by q = 20√KL where K is the number of large pots used each hour and L is the number of worker hours employed.
a. Graph the q = 2000 pounds per hour isoquant.
b. The point K = 100, L = 100 is one point on the q = 2000 isoquant. What value of K corresponds to L = 101 on that isoquant? What is the approximate value for the MRTS at K = 100, L = 100?
c. The point K = 25, L = 400 also lies on the q = 2000 isoquant. If L = 401, what must K be for this input combination to lie on the q = 2000 isoquant? What is the approximate value of the MRTS at K = 25, L = 400?
d. For this production function, the MRTS is MRTS = K/L.

Compare the results from applying this formula to those you calculated in part b and part c. To convince yourself further, perform a similar calculation for the point K = 200, L = 50.

e. If technical progress shifted the production function to q = 40√KL all of the input combinations identified earlier can now produce q = 4000 pounds per hour.
Would the various values calculated for the MRTS be changed as a result of this technical progress, assuming now that the MRTS is measured along the q = 4000 isoquant?

2. Consider the production function f (L, K) = L + K. Suppose K is fixed at 2. For this function, the marginal product of labor, MP(L), is 1. (a) Find the algebraic expression for the average product of labor AP(L). (b) Graph the total product of labor function TP(L), the average product of labor AP(L), and the marginal product of labor MP(L). Does your answer to part b violate the rule describing the relationship between average and marginal values? Explain.

3. For each of the following production functions, determine if the technology exhibits increasing, decreasing, or constant returns to scale.
a. f (L, K) = 2L + K
b. f (L, K) = √L + √K
c. f (L, K) = L2 + K
d. f (L, K) = √LK + L + K

4. A firm uses capital and labor to produce output according to the production function q = 4√KL, for which MPL = 2√K/L and MPK = 2√L/K.
a. If the wage w = $1/labor-hr. and the rental rate of capital r = $4/machine-hr., what is the least expensive way to produce 16 units of output?
b. What is the minimum cost of producing 16 units?
c. Show that for any level of output q, the minimum cost of producing q is $q.
d. Explain how a 10% wage tax would affect the way in which the firm choose to produce any given amount of output. (That is, the new wage is (1 + .10) times the original wage.)

5. Suppose that the Acme Gumball Company has a fixed proportions production function that requires it to use two gumball presses and one worker to produce 1000 gumballs per hour.
a. Explain why the cost per hour of producing 1000 gumballs is 2v + w (where v is the hourly rent for gumball presses and w is the hourly wage).
b. Assume Acme can produce any number of gumballs they want using this technology. Explain why the cost function in this case would be TC = q(2v + w), where q is output of gumballs per hour, measured in thousands of gumballs.
c. What is the average and marginal cost of gumball production (again, measure output in thousands of gumballs)?
d. Graph the average and marginal cost curves for gumballs assuming v = 3, w = 5.
e. Now graph these curves for v = 6, w = 5. Explain why these curves have shifted.

6. For each of the following total cost curves, determine if the technology exhibits increasing, decreasing, or constant returns to scale.
a. TC(q) = q2
b. TC(q) = √q
c. TC(q) = 2q
d. TC(q) = 2q + 4

7. A competitive firm has a short-run total cost curve STC(q) = 0.1q2 + 10q + 40.
a. Identify SVC and SFC.
b. Find and plot the SAC and SAVC curves.
c. For this function, the SMC curve is given by SMC(q) = 0.2q + 10. Include this curve in your diagram for part b.
d. Write the equation for the firm’s short-run supply curve and indicate it in the above graph.
e. Find the firm’s break-even price pe (at which profits are zero) and its shut-down price ps

8. Abby is the sole owner of a nail salon. Her costs for any number of manicures (q) are given by
TC = 10 + q2,
AC = 10
q + q,
MC = 2q.
The nail salon is open only 2 days a week – Wednesdays and Saturdays. On both days, Abby acts as a price taker, but price is much higher on the weekend. Specifically, P = 10 on Wednesdays and
P = 20 on Saturdays.
a. Calculate how many manicures Abby will perform on each day.
b. Calculate Abby’s profits on each day.
c. The National Association of Nail Salons has proposed a uniform pricing policy for all its members. They must always charge P = 15 to avoid the claim that customers are being “ripped
off” on the weekends. Should Abby join the Association and follow its pricing rules?

9. The following figure depicts the short-run per unit cost curves for a competitive firm. If the market price is p◦, indicate the firm’s revenue, STC, SVC, and profits.

10. For the firm in problem 9, indicate the firm’s short-run supply curve. What will be the effect of an increase in fixed costs on the firm’s supply curve and on its profits?

11. Suppose there are two technologies for producing steel. Under technology A, a firm’s short-run total cost curve is STCA(q) = 1
2 q2 + 100q + 10 (for which SMCA(q) = q + 100), and using technology B it is STCB (q) = 2q2 + 6 (SMCB (q) = 4q). Assuming there are 100 firms using technology A and 400 using B, determine the short-run market supply curve.