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Answers to Back-of-the-Chapter Problems
1. This statement confuses the use of average values and marginal values.
The proper statement is that output should be expanded so long as
marginal revenue exceeds marginal cost. Clearly, average revenue is not
2. The revenue function is R = 170Q - 20Q2. Maximizing revenue means
setting marginal revenue equal to zero. Marginal revenue is: MR =
dR/dQ = 170 - 40Q. Setting 170 - 40Q = 0 implies Q = 4.25 lots. By
3. In planning for a smaller enrollment, the college would look to answer
many of the following questions: How large is the expected decline in
enrollment? (Can marketing measures be taken to counteract the drop?)
How does this decline translate into lower tuition revenue (and perhaps
4. a. = PQ – C = (120 - .5Q)Q - (420 + 60Q + Q2) = -420 + 60Q -
1.5Q2. Therefore, M = d /dQ = 60 - 3Q = 0. Solving yields Q* = 20.
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b. In the general case, we set: = PQ - [F + cQ] = 0. Solving for Q, we
have: (P - c)Q = F or Q = F/(P - c). This formula makes intuitive sense.
c. Here, MR = 120 and MC = dC/dQ = 60. Because MR and MC are both
6. a. If DVDs are given away (P = $0), demand is predicted to be: Q = 1600
- (200)(0) = 1,600 units. At this output, firm A’s cost is: 1,200 + (2)
b. To maximize profit, we simply set MR = MC for each supplier and
compare the maximum profit attainable from each. We know that MR
= 8 - Q/100 and the marginal costs are MCA = 2 and MCB = 4. Thus,
for firm A, we find: 8 - QA/100 = 2, and so QA = 600 and PA = $5
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7. a. The marginal cost per book is MC = 40 + 10 = $50. (The marketing
costs are fixed, so the $10 figure mentioned is an average fixed cost per
b. When the rival publisher raises its price dramatically, the firm’s demand
curve shifts upward and to the right. The new intersection of MR and
MC now occurs at a greater output. Thus, it is incorrect to try to
c. By using an outside printer, OS is saving on fixed costs but is incurring
a higher marginal cost (i.e., printing cost) per book. With a higher
8. The fall in revenue from waiting each additional month is: MR = dR/dt
= -8. The reduction in cost of a month’s delay is: MC = dC/dt = -20 + .
5t. The optimal introduction date is found by equating MR and
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9. a. The MC per passenger is $20. Setting MR = MC, we find 120 - .2Q =
b. If it carries the freight, the airline can fly only 4 passenger flights, or
400 passengers. At this lower volume of traffic, it can raise its ticket
10. The latter view is correct. The additional post-sale revenues increase
MR, effectively shifting the MR curve up and to the right. The new
11. Setting MR = MC, one has: a – 2bQ = c, so that Q = (a - c)/2b. We
substitute this expression into the price equation to obtain:
The firm’s optimal quantity increases after a favorable shift in demand
either an increase in the intercept (a) or a fall in the slope (b). But
quantity decreases if it becomes more costly to produce extra units, that
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*12. The Burger Queen (BQ) facts are P = 3 - Q/800 and MC = $.80.
a. Set MR = 0 to find BQ’s revenue-maximizing Q and P. Thus, we have
b. The franchise owner maximizes its profit by setting MR = MC. Note
that the relevant MR is (.8)(3 - Q/400) = 2.4 - Q/500. After setting MR
c. Regardless of the exact split, both parties have an interest in
maximizing total profit, and this is done by setting (full) MR equal to
d. The chief disadvantage of profit sharing is that it is difficult,
time-consuming, and expensive for the parent company to monitor the
reported profits of the numerous franchises. Revenue is relatively easy
Discussion Question
Suppose the firm considers expanding its direct sales force from 20 to,
say 23 sales people. Clearly, the firm should be able to estimate the
marginal cost of the typical additional sales person (wages plus fringe
benefits plus support costs including company vehicle). The additional
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Spreadsheet Problems
S1. a and b. Setting MR = MC implies: 800 – 4Q = 200 + Q. Therefore,
c. To confirm these values on a spreadsheet, we maximize cell F7 by
S2. a. Given = 20[A/(A+8)] –A, it follows that M = 20[8/(A+8)2] – 1.
b. Confirm this value on your spreadsheet by maximizing cell F7 by
S3. a. To confirm these values on our spreadsheet, we maximize cell F7 by
changing cell B7. The optimal sales volume is: Q* = 2.4 million
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b. We extend the spreadsheet by including contribution from sales of
profit to compute total profit in cell H7. Maximizing total profit, we
find the new optimal solution to be: Q* = 3.829 million units and P*
Appendix Problems
1. When tax rates become very high, individuals will make great efforts
to shield their income from taxes. Furthermore, higher taxes will
discourage the taxed activities altogether. (In the extreme case of a
2. a. B(t) = 80 - 100t. Therefore, R = 80t - 100t2. Setting MR = dR/dt = 0,
b. B(t) = 80 - 240t2. Therefore, R = 80t - 240t3. Setting MR = dR/dt = 0,
c. B(t) = 80 - 80t.5. Therefore, R = 80t - 80t1.5. Setting MR = dR/dt = 0,
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b. The Lagrangian is L = + z(8 - x - y). Therefore, the optimality
c. The Lagrangian is L = + z(7.5 - x - .5y). Therefore, the optimality
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