Answers to Selected Problem Set 3 Questions

1. Just because hiring another worker causes diminishing returns does not mean you shouldn't hire that worker. What is important in such a decision is the marginal cost of the worker compared to the marginal revenue produced by the worker. As long as the MR > MC, the business executive should continue to hire workers.

2. The cost minimizing rule for hiring inputs is to equate the
Marginal Product per dollar spent on each input. Since

MP_{K}/r > MP_{L}/w, the firm could lower its
cost (for a given output) by using more capital and less labor.

3. AP must be rising.

4. We did this one in class.

5.

6. Note: The question states that MP_{L} = 6K. This
should be MP_{W} = 6K.

a) Optimal waste water to capital ratio is found by setting Pw/Pk =
MP_{W}/MP_{K} and solving for W/K. Thus, 7.5/30 = 6K/6W
==> W/K = 4/1.

b) K = 5000; W = 20000; and Q = 400m.

c) K = 7071; W = 14142; effluent fees = $106,065. Total Cost =
$318,195.

7. Diminishing marginal returns is the primary reason for the shape of the MC curve.

8. Economic costs = 7,000,000 + 700,000 + 40,000,000(.08) + 10,000,000(.08) = 11.7 million

9. Since MC is a constant $1000, so is AVC. ATC = 1000 + 10,000/Q. To minimize ATC, you would want to be very big.

10. Embalmed bodies:

Bodies | TC | FC | VC | ATC | AVC | AFC | MC |

0 | 40 | 40 | 0 | -- | -- | -- | -- |

1 | 45 | 40 | 5 | 45 | 5 | 40 | 5 |

2 | 60 | 40 | 20 | 30 | 10 | 20 | 15 |

3 | 79 | 40 | 39 | 26.33 | 13 | 13.33 | 19 |

4 | 105 | 40 | 65 | 26.25 | 16.25 | 10 | 26 |

5 | 150 | 40 | 110 | 30 | 22 | 8 | 45 |

6 | 200 | 40 | 160 | 33.33 | 26.67 | 6.67 | 50 |

11. True or false:

a) False. Compare this question to #1 above.

b) False. Minimizing cost is equivalent to maximizing output for
a given cost. The optimal hiring rule is the same in each case
(see #2 above).

c) False. ATC will rise. AVC and MC will not be affected.

d) False. Economic models typically assume that firms maximize
profits. In the long run, competition forces firms to locate at
the minimum of their long run average cost curve (the zero
profits condition).

12. If the firm can produce one chair with either four hours of labor or four hours of capital, or any linear combination, then the isoquant is a straight line with a slope of -1 and intercept at K=4 and L=4. The isocost line, TC = 22L + 110K has a slope of -22/110 = -0.2 when plotted with capital on the vertical axis and has intercepts at K = TC/110 and L = TC/22. The cost minimizing point is a corner solution, where L = 4 and K = 0. At that point, total cost is $88.

13. This is similar to #2 above. The bang per buck rule
(tangency rule) is not being satisfied. Since the MP_{c}/P_{c} = .33 > MP_{L}/P_{L} =
.0125, the firm can produce the same output with a lower cost if it uses more
cement and less land (i.e., builds up rather than out).

14. Engine manufacturing.

a) Since K = 10, the production function become Q = 40L or we can write
this as L = Q/40.

Define costs as C = rK + wL + 2000Q = 12000K + 3000L + 2000Q (after substituting
in the input prices)

Substitute in for K and L to get: C = 12000(10) + 3000(Q/40) + 2000Q
= 120,000 + 2075Q

AC = 120000/Q + 2075

MC = 2075

b) L = 80/40 = 2

AC = 3575

c) At the current input levels, the K/L = 10/2 = 5/1. The optimal
input levels require w/r = MPL/MPK = 3000/12000 = 1/4. Thus, less K and
more L should be used.

15. Currently, the short-run average
costs are: ATC (10,15) = [15(10)^{2}+12(15)]/10 = 28.00. If
she uses 4 fewer hair dryers in the short run, her short-run ATC becomes:
ATC(10,11) = [15(10)^{2}+12(11)]/10 = 26.84. If she uses 4
fewer dryers and produces 10 units, her short-run ATC decreases. Her
long-run ATC are: LRAC = C_{LR}(q)/q = 26.8q/q = 26.8. We
see that her LRAC is constant. This implies that her cost
curve exhibits constant returns to scale.

16. Think about the difference between economic and accounting profit.

17. Competitive Firm:

a) MC = 2Q

b) Set P = MC: 60 = 2Q ==> Q = 30

c) Profits = (60)(30) - [100 + (30)^{2}] = 800

d) The lowest price the firm would produce at (in the short run)
is the shut down price. The shut down price corresponds to the
minimum point on the AVC curve. Set MC = AVC to solve for this
price: 2Q = Q ==> Q = 0.

18. Assuming the industry starts from a position of zero economic profits, the increase in market demand will cause the price to rise in the short run. The higher price will encourage existing firms to expand their output and enable them to earn a positive profit. The positive profits will encourage the entry of new firms, driving the price back down to minimum average total cost.

19. Another competitive firm story.

a) In long run equilibrium, the firm will locate where MC = AC:
Thus 2Q = Q + 100/Q ==> Q = 10.

b) Since P = MC, we know that P = 20 (plug Q = 10 into MC
equation). If P = 20, then market output Q = 7000 and there are
700 firms, each producing 10 units.

c) P = 40; Profit = 200

d) In the long run, price will be driven back down to 20; the
number of firms will be 8000, with each earning 0 profits again.

20. This is for you to ponder.

21. You should be able to do this one with a blindfold on.

22. True or False:

a) False. In the textbook model, there is perfect information
under competition, therefore there are no trade secrets that will
enable a firm to enjoy a significant cost advantage for any
extended period of time.

b) False. Firms maximize their profits.

c) False. Firms will temporarily suspend product if total revenue
falls below total variable costs.

d) False. Cereal companies would probably just spend the
"saved" money on some other form of advertising to
position their products.

23. Set Q_{d} = Q_{s} to solve for the market price of $30.
Set MC = $30 to solve for the individual firm's optimal output of 5 units.
The market output at P = 30 is Q = 700. Thus, there must be 140 firms in
the industry (= 700/5). This is an increasing cost industry.

24. For the output before the tax, set MC = P and solve for q: .06q =
.7 ==> q = 11.67. Profit = 4.08.

For the output after the tax, set MC + t = P and solve for q: .06q + .01 = .7
==> q = 11.5. Profit = 3.97

25. Competitive firm.

a) Set Supply = Demand and solve for Q = 25; P = 37.5

b) Set MC = P and solve for q. This will give q = 3.5

c) Q = 37.5; P = 43.75; q = 4.125

d) No. The higher price in part (c) would generate profits for the
existing firms. In the long run, new firms will enter the market and drive price
down to the zero profits level.

26. We did the first part in class.

27. Bud Wizer.

a) Q = 2500; π = 1125 (this is an
economic profit since the cost function is an economic cost function)

b) Q = 2000; π = 0 (Bud should continue
operating since this is a zero economic profit. The impact upon Bud's
competitors will be favorable or neutral. As he cuts output, 500 six packs worth
of business will either shift elsewhere or choose temperance.)

28. We did this in class.

29. See Figure 8.13 in the text for a similar example.

30. Not necessarily. What matters is the marginal revenue per flight versus the marginal costs per flight, not the fact that Continental fills only 60% of its seats on the 737. If the marginal revenue per flight is greater than the $2000 marginal costs, then it is profitable to fly the 60% full jets.