Answers to
Back-of-Chapter
Problems
Chapter 1
1. Managerial economics is the analysis of important management decisions using the tools
of economics. Most business decisions are motivated by the goal of maximizing the
firm’s profit. The tools of managerial economics provide a guide to profit-maximizing
decisions.
2. i) Multinational Production and Pricing. The global automobile company needs
information on demand (how many vehicles can be sold in each market at different
prices) and production costs.
ii) Market Entry. The office stores not only need information on local market demand,
they also need information on the ability and willingness of the other company to
compete. This means gathering information on the rival's cost structure, sources of
supply, access to capital, etc.
iii) Building a New Bridge. The authority should estimate usage of the bridge over its
useful life, the likely cost of building and maintaining the bridge, and other important
side-effects, pro and con – a including positive effects on business activity and the
impacts on air pollution and traffic congestion.
iv) A Regulatory Problem. Before deciding whether to promote the oil-to-coal
conversion, government regulators need information on how much oil would be
saved (and the dollar value of savings) and the cost of the chain of side-effects – not
only the direct cost of electricity provision but also pollution costs and environmental
damage.
v) Oil Exploration. Some of the information BP needs – such as current oil prices, rig
worker wages, and other operating costs – is readily available. Other information—
such as data gleaned from geological surveys, seismic tests, safety audits; wear and
tear on drilling components; short-term and long-term weather conditions; the
outlook concerning the global demand for oil – is probabilistic in nature.
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vi) An R&D Decision. The pharmaceutical company should quiz its scientists on the
chances of success (and the timetable for completion) for each R&D approach. The
company's marketing department would supply estimates of possible revenues from
the drug; its production department would estimate possible costs.
vii) David Letterman. Dave must carefully assess what he wants from a new contract
(in particular how much he values the earlier time slot). As the negotiations unfold,
Dave will glean valuable information as to the current competing offers of CBS and
NBC. Of course, Dave must also try to assess how far the two networks might be
willing to go in sweetening their offers.
3. The six steps might lead the soft-drink firm to consider the following questions. Step 1:
What is the context? Is this the firm’s first such soft drink? Will it be first to the
marketplace, or is it imitating a competitor? Step 2: What is the profit potential for such a
drink? Would the drink achieve other objectives? Is the fruit drink complementary to the
firm’s other products? Would it enhance the firm’s image? Step 3: Which of six versions
of the drink should the firm introduce? When (now or later) and where (regionally,
nationally, or internationally) should it introduce the drink? What is an appropriate
advertising and promotion policy? Step 4: What are the firm’s profit forecasts for the
drink in its first, second, and third years? What are the chances that the drink will be a
failure after 15 months? Should the firm test market the drink before launching it? Step 5:
Based on the answers to the questions in Steps 1 through 4, what is the firm’s most
profitable course of action? Step 6: In light of expected (or unexpected) developments in
the first year of the launch, how should the firm modify its course of action?
4. Decision vignettes
a. A couple who buy the first house they view have probably sampled too few houses.
Housing markets are notoriously imperfect. Houses come in various shapes, sizes,
conditions, neighborhoods, and prices. Personal preferences for houses also vary
enormously. The couple is likely to get a "better" house for themselves if they view
a dozen, two dozen, or more houses over the course of time before buying their
"most-preferred" house from the lot. Circumstances justifying the first-house
purchase include: (1) the house is so good that viewing others is a waste of time, (2)
the house is so good and the commitment must be made now or another buyer will
claim the house, (3) the couple must buy now (a job transfer has brought them to the
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area and schools open tomorrow), (4) they already have full information about the
types of other houses available (the wife's best friend is a real estate agent).
b. The company seems to be launching the product to avoid "wasting" the $6 million
already spent in development. This "sunk" cost is irrelevant and should be ignored.
What does matter for the reinvestment decision are the future revenues and costs of
continuing. (Reinvest if the net present value of future profits is positive.) Some
"close-to-home" examples of the sunk cost fallacy: i) A fellow pays $250 for a yearlong tennis membership but develops severe tennis elbow after two months. He
continues to play in great pain in order to get his money's worth. ii) Ms. K has a
subscription to a series of six plays for $150. She braves a snow storm so as not to
waste the $25 cost. On reflection, she admits that she wouldn't have gone had she
been given the ticket for free.
c. It's in the individual motorist's best interest to drive on. (Stopping is risky and
inconvenient). But it's in the collective interest of all the delayed motorists to have
someone stop and move the mattress. Here's an example of the potential conflict
between private and public interests (between private profit and social welfare). In
such circumstances, there is a potential role for government intervention.
d. Allowing the use of thalidomide had a disastrous outcome and more importantly was
a bad decision (besides its potential risk, the drug was of questionable benefit in
aiding sleep). The thalidomide disaster prompted a much tougher stance toward
prior drug testing in the U.S. and elsewhere.
e. The frantic couple should choose separate lines to take advantage of whichever line is
quicker. Whoever gets served first checks the baggage. The lesson here: DIVERSIFY.
f. To the extent that his actions and behavior were responsible for his marriage
breakup, the CEO’s mistake was to lose sight of the most important objective.
g. The cost per life saved is $400,000/20 = $20,000 for the ambulance service. It is
$1,200,000/40 = $30,000 for the highway program. Based on these average
measures, its seems strange that the ambulance budget is being cut and the highway
budget expanded. However, the real issue is the impact on lives saved from budget
changes at the margin. Perhaps, the ambulance budget has a lot of administrative
"fat" in it. It could be cut by 40% with very little impact on lives. By the same
token, a modest budget increase for highways might have a large impact on
additional lives saved. In short, the average cost per life may not tell the real story.
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h. FEMA’s prediction of the potential hurricane risk to New Orleans was timely and
prescient. However, the warning was not emphasized by the agency and certainly not
heeded by federal, state, or local policy makers. The decision error was a
combination of inattention, wishful thinking, and denial.
i. Compared to these extreme outcomes (abject surrender to terrorism or being a global
policeman) any option looks good. This is hardly an even-handed portrayal. The real
question is whether implementing increased security measures that sacrifice civil
liberties is better than other relevant alternatives.
j. According to the counts of pros and cons, the individual prefers: Home over Beach,
Beach over Mountains, but Mountains over Home. We have a cycle (i.e. intransitive
preferences). The individual is left going around in circles. The obvious way out of
this dilemma is to "score" each alternative by weighting the individual attributes.
The more important the attribute, then the greater is the weight. In addition, the
individual could use a broader scale (1 to 10) for each attribute as a way of
measuring relative strength of preferences between alternatives. (For a related
example, see Problem 4.4. In this context, the instructor may also wish to discuss
voting cycles and the Condorcet paradox).
Chapter 2
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 the same as marginal revenue, nor is
average cost identical to marginal cost. Indeed, if management followed the averagerevenue/average-cost rule, it would expand output to the point where AR = AC, in which
case it is making zero profit per unit and, therefore, zero total profit!
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 contrast, profit is maximized by expanding output
only to Q = 3.3 lots. Although the firm can increase its revenue by expanding output
from 3.3 to 4.5 lots, it sacrifices profit by doing so (since the extra revenue gained falls
short of the extra cost incurred.)
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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 lower alumni donations)? How should the university plan its
downsizing? Via cuts in faculty and administration? Reduced spending on buildings,
labs, and books? Less scholarship aid? How great would be the resulting cost savings?
Can the university become smaller (as it must) without compromising academic
excellence?
4. a. π = PQ - C = (120 - .5Q)Q - (420 + 60Q + Q2)
= -420 + 60Q - 1.5Q2.
Therefore, Mπ = dπ /dQ = 60 - 3Q = 0. Solving yields Q* = 20.
Alternatively, R = PQ = (120 - .5Q)Q = 120Q - .5Q2. Thus, we find MR = 120 – Q. In
turn, C = -420 + 60Q + Q2, implying: MC = 60 + 2Q. Equating marginal revenue and
marginal cost yields: 120 - Q = 60 + 2Q. Therefore, Q* = 20.
b. Here, R = 120Q; it follows that MR = 120. Equating MR and MC yields: 120 = 60 +
2Q. Therefore, Q* = 30.
5. a. The firm breaks even at the quantity Q such that π = 120Q - [420 + 60Q] = 0.
Solving for Q, we find 60Q = 420 or Q = 7 units.
b. In the general case, we set: π = PQ - [F + cQ] = 0. Solving for Q, we have: (P - c)Q = F
or Q = F/(P - c). This formula makes intuitive sense. The firm earns a margin (or
contribution) of (P - c) on each unit sold. Dividing this margin into the fixed cost
reveals the number of units needed to exactly cover the firm’s total fixed costs.
c. Here, MR = 120 and MC = dC/dQ = 60. Because MR and MC are both constant and
distinct, it is impossible to equate them. The modified rule is to expand output as far as
possible (up to capacity), because MR > MC.
6. a. If DVDs are given away (P = $0), total quantity soled is predicted to be: Q = 1600 (200)(0) = 1,600 units. At this output, firm A’s cost is:
1,200 + (2)(1,600) = $4,400, and firm B’s cost is: (4)(1,600) = $6,400. Firm A is the
cheaper option and should be chosen. (In fact, firm A is cheaper as long as Q > 600.)
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
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QA = 600 and PA = $5 (from the price equation). For firm B, we find that QB = 400
and PB = $6. With Firm A, the station’s profit is: 3,000 - [1,200 + (2)(600)] = $600.
With Firm B, its profit is: 2,400 - 1,600 = $800. Thus, an order of 400 DVDs from firm
B (priced at $6 each) is optimal.
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 book.) Setting MR = MC, we
find MR = 150 – 2Q = 50, implying Q* = 50 thousand books. In turn, P* = 150 –50 =
$100 per book.
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 maintain sales via a full $15 price hike. For
instance, in the case of a parallel upward shift, the new price equation is: P = 165 – Q.
Setting MR = MC, we find: MR = 165 – 2Q = 50, implying Q* = 57.5 thousand books,
and in turn, P* = 165 – 57.5 = $107.50 per book. Here, OS should increase its price by
only $7.50 (not $15).
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 MC, the intersection of MR
and MC occurs at a low optimal quantity. OS should reduce its targeted sales quantity
of the text and raise the price it charges per book. Presumably, the fixed cost savings
outweighs the variable cost increase.
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 MC: -8 = -20 + .5t, which implies: .5t = 12 or t* = 24 months. The
marketing manager’s 12-month target is too early. Delaying 12 more months sacrifices
revenue but more than compensates in reduced costs.
9. a. The MC per passenger is $20. Setting MR = MC, we find 120 - .2Q = 20, so Q = 500
passengers (carried by 5 planes). The fare is $70 and the airline’s weekly profit is:
35,000 - 10,000 = $25,000.
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 price to P = $80. Its total revenue
is (80)(400) + 4,000 = $36,000. Since this is greater than its previous revenue
($35,000) and its costs are the same, the airline should sign the freight agreement.
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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 intersection of MR and MC occurs at
a higher output, which, in turn, implies a cut in price. (Note that one should discount the
additional future profit from service and supplies to take into account the time value of
money.)
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:
P = a - b[(a - c)/2b] = a - (a - c)/2 = a/2 + c/2 = (a + c)/2.
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 is., if the marginal cost (c) increases. Price is
raised after a favorable demand shift (an increase in a) or after an increase in marginal
cost (c). Note that only $.50 of each dollar of cost increase is passed on to the consumer
in the form of a higher price.
*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 3 - Q/400 = 0,
so Q = 1,200 and P = $1.50. Total revenue is $1,800 and BQ’s share is 20% or $360.
The franchise owner’s revenue is $1,440, its costs are (.8)(1,200) = $960, so its profit
is $480.
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 = .80, we find Q = 800. In turn,
P = $2.00 and the parties’ total profit is (2.00 - .80)(800) = $960, which is
considerably larger than $840, the total profit in part (a).
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 MC. Thus, we have 3 - Q/400 = .80, so that Q
= 880. In turn P = $1.90, and total profit is: (1.90 - .80)(880) = $968.
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 to check (from the cash register receipts) but
costs are another matter. Individual franchisees have an incentive to exaggerate the
costs they report in order to lower the measured profits from which the parent’s split is
determined. The difficulty in monitoring cost and profit is the main strike against
profit sharing.
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Chapter 3
1. The fact that increased sales coincided with higher prices does not disprove the law of
downward-sloping demand. Clearly, other factors − an increase in population and/or
income, improved play of the home team, or increased promotion − could have caused
increased ticket sales, despite higher prices.
2. a. Q = 180 - (1.5)(80) = 60 pairs. R = ($80)(60) = $4,800.
b. At P = $100 and Q = 30 pairs, revenue falls to $3,000 per month.
c. EP =(dQ/dP)(P/Q). At P = $80, EP = (-1.5)(80/60) = -2; At P = $100,
EP = (-1.5)(100/30) = -5. Demand is much more elastic at the higher price.
3. a. Q = 400 - (1,200)(1.5) + (.8)(1,000) + (55)(40) + (800)(1) = 2,400.
b. EP = (dQ/dP)(P/Q) = (-1,200)(1.50)/2,400 = -.75.
EA = (dQ/dA)(A/Q) = (.8)(1,000)/2,400 = .333
c. Since demand is inelastic, McPablo’s should raise prices, increasing revenues and
reducing costs in the process.
4. a. The change in quantity sold is: ∆Q = (EP)(∆P) = (-1.5)(5) = -7.5 percent.
b. Because each firm’s output is one-fourth of the total, the individual firm’s elasticity is
calculated with a one-fourth smaller Q in the denominator, making the elasticity 4 times as
great or -6.
c. Using the formula ∆Q = (EP)(∆P) with EP = -1.5 and ∆Q = 9 percent, we solve to find: ∆P
= -6 percent. Price would be expected to fall by 6 percent.
5. The consultant should recommend an immediate price increase. As noted in the text, if
demand is inelastic, the firm can always increase profit by raising price, thereby raising
revenue and reducing cost.
6. a. This means that if the local population increases by 10 percent, ticket sales will
increase by (.7)(10) = 7 percent. The actual population increase of 2.5 percent implies
a sales increase of 1.75 percent.
b. The 10 percent increase in ticket price implies a (.6)(10) = 6 percent fall in ticket sales.
Because demand is inelastic, total ticket revenue increases.
c. Here, the increase in total revenue per admission (from $18 to $19) is only 5.55
percent. This is outweighed by the decline in admissions (6 percent) causing total
revenue to fall.
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7. a. i. In pricing Triplecast, NBC faced a pure selling problem, the marginal cost of each
additional subscriber being insignificant.
ii. Unfortunately, management dramatically misjudged its demand curve as well as the
point of maximum revenue along it. Once it recognized the depressed state of demand,
management instituted a dramatic price cut (trying to reach the demand point at which
EP = 1). This was its best course of action to capture what revenue was available. Over
time, the partners reduced their package price from $125 to $99 to $79 and the daily
price from $29.95 to $19.95 to $11.95. However, these actions at best were able only to
stem large losses.
b. i. The main benefit of AOL’s new pricing plan was attracting new customers. Indeed,
the company raised its customer base over 18 months from 8 million to some 11
million subscribers. It also increased revenues from retailers, advertisers, and
publishers, who would pay for access to AOL’s customers. The main risk of the new
plan was that some current customers would pay less each month for the same online
use and others would greatly increase their use at the lower effective price.
ii. This is exactly what happened. Current customers more than doubled their daily
time online. Constrained by a fixed capacity, AOL’s system overloaded. Customers
received busy signals and experienced interminable waits for access. (One
commentator likened the new pricing policy to offering a perpetual all-you-can-eat
buffet to food lovers, who once seated would eat through breakfast, lunch, and dinner,
fearing they would not get back in if they gave up their table.) Customers were
disaffected, and AOL was forced by regulators to give widespread refunds while it
scrambled to increase its network capacity at a cost of $350 million.
8. a. During this period, Apple computers, although technologically superior, were priced
out of the range of many consumers. As a result of vigorous competition in the IBM PC
clone market, prices of PCs were significantly lower. Soon these standardized PCs
offered by numerous suppliers came to dominate the market. As a result of network
externalities (making it beneficial to have the same computer platform as everyone
else), the Mac rapidly lost market share.
b. Using the markup rule, we can see that with a price elasticity of -4 the profit-maximizing
markup is 25% (expressed as a percentage of price). And note that this only reflects shortterm profit maximization. An even smaller markup may be optimal when one considers
long-run demand. Thus the 50% markup goal was unrealistic and far from profit
maximizing.
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c. It does make sense to concentrate in niche markets where demand may be less elastic
and Apple already has significant market share. Likewise, software (and even
hardware) that make Apple compatible with PCs will help break the network
externalities enjoyed by PCs and encourage consumers to buy Macs. However, since
network externalities continue to exist even with these innovations, it may not be
prudent to abandon the low end of the market. Indeed, the introduction of the iMac
indicates that Apple has not given up on the consumer market.
9. a. With demand given by P = 30,000 - .1Q and MC = $20,000, we apply the MR = MC
rule to maximize profit. Therefore, MR = 30,000 - .2Q = 20,000 implies Q = 50,000
vehicles and P = $25,000. GM’s annual profit is: (25,000 – 20,000)(50,000) –
180,000,000 = $70,000,000.
b. According to the markup rule (with MC = $20,800 and EP = -9),
P = [-9/(-9 + 1)][20,800] = $23,400. Because of very elastic demand, GM should
discount its price in the foreign market (not raise it by $800).
c. This is a pure selling problem (the trucks have already been produced) so the goal is to
maximize revenue. Setting MR = 0 implies 30,000 – 2Q = 0, or Q = 15,000 vehicles
and P = $15,000. GM should discount the price (rather than hold it at $20,000) but not
so low as to sell the whole 18,000 inventory. It should sell only 15,000 (and perhaps
donate the other 3,000 to charity).
10. The key point here is that the optimal prices in summer and winter depend upon the
relative elasticities. Higher winter prices are warranted as long as winter demand is more
inelastic. There is no contradiction between more inelastic winter demand and a lower
occupancy rate. For instance, it is likely that the overall market is smaller in the winter
(fewer people take extended vacations and this accounts for the lower occupancy rate)
but the winter market is also less price sensitive (skiing is for the relatively wealthy).
11. How should the manager set prices when taking different levels of costs into account? The
answer is to apply the markup rule: P = [EP/(1 + EP)]MC. For instance, if changes in
economic conditions cause the firm’s marginal costs to rise, the correct action is to
increase price (even though there may have been no change in price elasticity). For the
same reason, an electric utility is justified in charging higher electric rates in the summer
when supplying sufficient electricity to meet peak demand is very costly.
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12. If a firm can identify market segments with different elasticities, it can profit by charging
different prices (even though marginal costs are the same.) It should set the price in each
segment according to the optimal markup rule. The existence of substitute products
should make demand for the firm’s product more elastic. Accordingly, the firm should
reduce its markup.
13. a. The garage owner should set prices to get the maximum revenue from the garage. The
owner should offer higher hourly rates for short-term parking and all-day rates (at a
lower average cost per hour). This pre-vents short-term parkers from taking advantage
of the all-day discount.
b. Start by setting MR = 0 for each segment. (This maximizes revenue in each separate
segment.) The resulting optimal quantities are QS = 300 and QC = 200. Notice that the
garage is not completely filled. The optimal prices are PS = $1.50 per hour and PC = $1
per hour.
c. Because there are only 400 places in the garage, the strategy in part b is not feasible.
The best the operator can do is to fill up the garage and maximize revenue by ensuring
that the marginal revenue is the same for the segments. Equating MRS = MRC and
rearranging implies QS = QC + 100. Together with the fact that QS + QC = 400, one
finds QS = 250 and QC = 150. The requisite prices are PS = $1.75 per hour and PC =
$1.25 per hour.
14.a. Frequent flier and frequent stay programs are primarily designed to induce strong
allegiances and repeat business. It is also a subtle way of offering discounts
(selectively) to enrollees (presumably, the most price sensitive customers).
b. Discount coupons deliver lower prices to the most price sensitive consumers, while the
average consumer pays the regular, full price.
c. A guarantee to match a lower price is a way of winning sales from customers with the
most elastic demand (who take the trouble to seek out a lower price), while maintaining
higher prices for typical customers.
Chapter 4
1. Survey methods are relatively inexpensive but are subject to potential problems: sample
bias, response bias, and response accuracy. Test marketing avoids these problems by
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providing data on actual consumer purchases under partially controlled market
conditions. Test marketing is much more costly than survey methods and suffers from
two main problems. First, some important factors may be difficult to identify and control.
Second, test market results are not a perfect guide to actual market experience down the
road.
2. a. Coca Cola’s management is likely to conclude that consumers will prefer New Coke to
Coke Classic. However, as part (b) shows, they may be very wrong.
b. Yes, these rankings are consistent with the information in part (a). Consumers prefer
Pepsi to Coke Classic by 58 to 42 (types A and C) and New Coke to Pepsi by 58 to 42
(B and C). However, a blind taste test between Classic and New Coke would have
Classic preferred 84 to 16 (A and B)!
c. It would be a big mistake to replace Classic by New Coke. The obvious strategy is to
retain Classic but also offer and promote New Coke. New Coke will attract type C
consumers away from Pepsi. As the text indicated, blind taste tests do not tell the
whole story about consumer buying behavior. Brand-name allegiance and loyalty are
also extremely important.
3. a. Northwest does have a better overall on-time record than Delta. Its frequency of late
flights is: 412/2,058 = .20 or 20 percent. By comparison, Delta’s frequency is:
626/2,898 = .22 or 22 percent.
b. Delta’s management will tout city-by-city on-time comparisons. In New York, its
frequency of late flights is: 484/1,987 = .24 or 24 percent, compared to Northwest’s:
120/399 = .30 or 30 percent. In Chicago, Delta’s frequency of late flights is 16
percent, compared to Northwest’s 20 percent. Finally, in Memphis, Delta’s frequency
of late flights is 12 percent, compared to Northwest’s 13 percent. Delta has superior
on-time performance in all three cities.
c. The disaggregate comparisons provide the more accurate measure of on-time
performance. Here, Delta wins hands down. The overall record is misleading. Delta’s
overall on-time percentage looks worse because it flies many more flights than
Northwest in and out of New York where poor weather and airport congestion cause
frequent delays. To get an accurate picture, one must control for the differences in
airport delays experienced by different cities.
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4. a. False. A high R2 indicates that the equation closely tracks the past data, but this is
only one part of performance. A complete evaluation would address these questions: i)
Does the equation make economic sense? ii) Are the signs and magnitudes of the
coefficients reasonable? iii) How well does it forecast, short-term and long-term?
b. Partly True. More data is better as long as the time-series relationship is stable.
However, such behavior often changes over time. If two time periods (say, one decade
versus another) show very different behavior, one should estimate separate time-series
for each.
c. False. Throwing in everything but the kitchen sink is bad on theoretical grounds and
empirical grounds. Including irrelevant or insignificant variables will lower the
adjusted R2 (a better measure of performance) and will typically worsen the
equation’s forecasting accuracy.
d. Partly True. But there are exceptions. i) Even forecasts that accurately track the past
can produce implausible long-term predictions. ii) No matter how good the past fit,
an equation will generate poor forecasts if it relies on explanatory variables that are
themselves difficult to predict.
5. a. According to the t-statistics, all explanatory variables are significant except income.
b. This coefficient measures the price elasticity of demand, EP = -.29. A 20 percent price
hike implies a 5.8 percent sales drop.
c. With EY = -.09, sales hardly vary with income.
6. a. Both t-values (based on 60 months of data) are much greater than 2, implying that both
coefficients are significantly different from zero.
b. The equation says that the expected return on Pepsi’s stock roughly follows the
expected return on the S&P 500. (The coefficient .92 is the stock’s “beta.”)
Nonetheless, there remains a large random element in any individual stock’s return.
Day-to-day stock prices follow random walks. Explaining even 28 percent (R2 = .28)
of the variation in the stock’s monthly return is impressive.
c. Setting RS&P = -1 implies RPEP = .06 - .92 = -.86 percent expected return over the next
month.
7. a. Although the time coefficient is negative (b = -.4), its t-value is well below 2,
indicating that the coefficient is not statistically different from zero. The water table
has been stable over the decade.
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b. Think of yearly rainfall as one thinks of tosses of a coin. Even though each coin toss is
random and independent of the other tosses, it is still possible to have an unusually
large number of heads or tails in 10 trials by pure luck. Thus, the second expert is
foolish to claim that dry years and wet years necessarily will cancel each other out.
8. a. The t-statistics for each of the explanatory variables are:
Price
Comp. Price
Income
-5.11
4.97
11.70
Population
1.29
Time
3.85
Using a cutoff of 1.68 (41 degrees of Freedom), we see that all the explanatory
variables are statistically significant except population. The regression model explains
93% of the total variation.
b. Price elasticity is (dQ/dP)(P/Q) = (-3,590.6)(7.50/20,000) = -1.35.
Cross-price elasticity is
(dQ/dPc)(Pc/Q) = (4,226.5)(6.50/20,000) = 1.37
c. According to the regression, pie sales should increase by approximately (4)(356) =
1,424 pies next year. (Remember one year equals 4 quarters.)
d. You might be fairly confident in predicting sales for the next quarter given that 93% of
the variation is explained by the regression but only if accurate information about the
explanatory variables can be obtained. Of course, you control your own price.
However, competitors’ prices and other variables are not in your control. Two years
from now, predictions as to the values of the explanatory variables become even more
difficult. Furthermore, the demand relationship itself is subject to change as tastes
change over time. This makes prediction two years from now much more uncertain.
9. a. To estimate price elasticity, we compare 2011 and 2014, two years in which the level
of income was the same: EP = [(1.90 – 2.00)/2.00]/ [(22 – 20)/20] = -5%/10% = -.5.
b. Comparing 2011 and 2013 when prices were constant, we find:
EY = [(1.94 – 2.00)/2.00]/ [(97 – 100)/100] = -3%/-3% = 1.0.
c. dQ/Q = EP(dP/P) + EY(dY/Y)
= (-.5)[(24 – 22)/22] + (1.0)[(105 – 100)/100]
= -4.5 + 5 = .5 percent.
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The forecast calls for a very slight increase in sales, whereas actual sales were
unchanged.
2
d. The OLS regression produces the equation: Q = 1 - .05P + .02Y with an R of 1.00!
Surprisingly, this equation provides a perfect fit of the five years of observations.
Obviously, this degree of accuracy is more than a bit unrealistic.
10. a. Focusing on years 3 and 4 exaggerates the growth trend (because sales were depressed
in year 3.)
b. The second manager is correct in principle. Using the average change over the period
offers a better (more stable) prediction of annual growth.
11. a. The estimated trend equation is S = 95 + 5.5t, using OLS regression.
b. Although the equation’s R2 is .69, the t-value on the time trend is only 2.12. With 2
degrees of freedom, the critical value for significance is 4.30. With only four
observations, there is not enough data to say whether there is a true upward trend.
c. Suppose you take the estimated coefficient at face value (even though it lacks
statistical significance.) Then, the forecast for year 5 is 95 + (5.5)(5) = 122.5 (a slight
increase from sales of 120 in year 4). From the regression output, the standard error of
this forecast is 5.81. This error is so large that sales could well increase or decrease in
year 5.
12. a. As the economy improves, we would expect firms to stop laying off workers, then
increase overtime hours, then begin hiring temporary workers, and finally initiate new
permanent hiring.
b. Yacht sales probably will not rebound until well after the upswing is in progress. The
manager should not plan for greater yacht sales until employment, wage earnings, and
consumer income have permanently increased.
13. a. Yes, the equation makes economic sense. Growth in tire sales is fueled by growth in
miles driven and growth in new car sales.
b. The equation performs well in explaining the past data (R2 = .83). The coefficients of
the two explanatory variables are highly significant, and the Durbin-Watson statistic
indicates no serial correlation.
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c. The t-statistics for the respective coefficients are: (1.41 - 1)/.19 = 2.15 and
(1.12 - 1)/.41 = .29. The first coefficient is significantly different than one; the second
is not. If the second coefficient is taken to be one, this means that tire sales are
proportional to new auto sales.
d. The forecast is: .45 + (1.41)(-2) + (1.12)(-13) = -16.93. An actual drop of 18% would
not be surprising; it’s well within the margin of forecast error.
e. Your confidence would depend on how well these test markets represent the national
market.
Chapter 5
1. Maximizing average output per hour is typically non-optimal. First, we should emphasize
that maximizing total output and maximizing average output are two different things. For
instance, in Table 5.2, the firm’s maximum output is 403 units using 120 workers. In
contrast, the firm would maximize its average product by using 10 workers producing
only 93 units. Second, optimal use of an input requires comparing extra output (and
revenue) against the input’s extra cost. As we have seen, optimum input use typically
means producing below the level of maximum output.
2. The production function, Q = 10L - .5L2 + 24K - K2, has marginal products: MPL = 10 - L
and MPK = 24 - 2K. Both marginal products decline; therefore, there are diminishing
returns. Starting from any L and K, doubling the use of both inputs generates less than
double the level of output. Thus, the production function exhibits decreasing returns to
scale.
3. a. The marginal products for labor and capital are given by: MPL = 10 - L and MPK = 24
- 2K. For L equal to K (in the range 0 to 10), capital's marginal product is greater than
labor's. At the same input prices, the firm will use more capital than labor.
b. Setting the price of each input equal to its marginal revenue product implies: 100 - 10L
= 40, or L = 6 units, and 240 - 20K = 80, or K = 8 units.
4. a. Labor’s MPL is dQ/dL = 1 - L/400. Setting MRPL = wage implies 40 - .1L = 20, or
L* = 200 labor hours. In turn, Q = 150 dresses and π = $2,000.
b. With P = $50, MRPL becomes 50 - L/8. The new solution is L* = 240 labor hours and
Q = 168 dresses. With the price increase, optimal output increases. If input and output
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prices change in equal proportions, there is no effect on any of the firm’s optimal
decisions.
c. The 25% increase in productivity implies: MPL = (1.25)(1 – L/400). Setting MRPL (at P
= $50) equal to the unchanged wage implies: 62.5 - .15625L = 20. Thus, L* = 272
labor hours and Q = 179.52 dresses.
5. The law of diminishing returns states that an input’s marginal product declines as one
increases its use past some point (holding other inputs constant). Decreasing returns to
scale states that increasing all inputs in proportion generates a less-than-proportional
increase in output. A production function can exhibit diminishing returns without
decreasing returns to scale, or vice-versa.
6. a. In the short run, the restaurant should hire more wait staff (and possibly chefs) and
squeeze in some extra tables so as to produce the maximum number of (delicious)
meals per night. Clearly, given the restaurant’s small quarters (fixed capacity),
applying extra labor inputs will quickly run into diminishing returns.
b. In the longer run, the restaurant would be wise to increase its entire scale of operation.
Perhaps, it could expand its space in the new building. If not, it might even consider
opening a second restaurant in a carefully scouted location. In either case, it is much
more efficient to produce gourmet meals using the “right” mix of culinary labor and
distinctive restaurant floor space.
7. a. The isoquant for the 200-pound steer has the usual convex curvature.
b. The cost of the 68-60 mix is: ($.10)(68) + ($.07)(60) = $11.00 per day. The cheapest
diet is a 56-70 mix; its cost is $10.50 per day.
c. For a 200-pound steer, the cheapest mix is 56-70. Given constant returns to scale,
feeding a 250-pound steer would require (250/200) = 125 percent of this amount. A
70-87.5 mix (at a cost of $13.125) is needed.
8. In all likelihood, Chrysler's move to 24-hour production was prompted by the high capital
cost of building new factories combined with a slowdown in wage growth. With labor
cheap relative to capital, a switch to a greater ratio of labor to capital makes economic
sense. Increased 24-hour production was in response to the soaring demand for minivans
and Jeeps.
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9. a. Production of steel by electric furnace has the lowest average cost per ton ($325).
Therefore, its share of production would be expected to increase over time.
b. A tripling of energy prices would leave continuous casting ($400) as the least-cost
production method.
c. A fall in the price of steel scrap would favor production by electric furnace (the only
process that uses scrap).
10. Here is a graphical explanation. The firm's initial (optimal) input mix occurs where the
lowest isocost line is tangent to its isoquant. If the price of labor increases, this changes
the slope of the isocost line (so that labor “trades” for fewer units of capital). The new
tangency with the same isoquant must occur at a mix using less labor and more capital.
11. a. The grade improvements offered by extra hours of studying finance are 8, 5, 5, 2, and
2 points. For economics, the improvements are 6, 4, 2, 2, and 1 points.
b. The “first” hour should be devoted to finance (an 8-point increase), the next hour to
economics (6 points), the next 2 hours to finance (5 points each hour), and the “last”
hour to economics (4 points). The student’s predicted grades are 88 and 85.
c. This allocation is optimal. Devoting her first 5 hours to finance and economics offers
the greatest point opportunities. Then, devoting 2 additional hours to accounting will
produce more extra points (3 points each hour) than devoting an additional hour to
finance (2 points) or economics (2 points).
12. The optimal input condition is: MPK/MPL = PK/PL. The inputs’ respective marginal
products are: MPK = ßLKß-1 and MPL = αLα-1Kß. Thus, the ratio of the marginal products
is:
MPK/MPL = (ß/α)(L α/L α-1)(K ß-1/K ß) = (ß/α)(L/K).
Setting this equal to PK/PL and rearranging yields: L/K = (α /ß)( PK/PL). Other things
equal, an input's use increases the greater its output elasticity or the lower its price.
13. a. For N1 = 16 and N2 = 24, the average catch at the first lake is Q1/N1 = [(10)(16) .1(16)2]/16 = 8.4 fish, and the average catch at the second lake is Q2/N2 = [(16)(24) -
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.4(24)2]/24 = 6.4 fish, respectively. Lured by the greater average catch, some number
of fishers will leave the second lake for the first.
b. Movement between lakes will cease when all individuals obtain the same average
catch. Equating the average catches at the lakes implies 10 - .1N1 = 16 - .4N2. In
addition, N1 + N2 = 40. Solving these two equations simultaneously implies N1 = 20
and N2 = 20. The total catch at the two lakes is 320 fish.
c. The commissioner seeks to maximize Q1 + Q2 subject to N1 + N2 = 40. The optimum
solution to this constrained maximization problem implies that the marginal catch of
the last fisher should be equal across the lakes. Here, MQ1 = dQ1/dN1 = 10 - .2N1 and
MQ2 = dQ2/dN2 = 16 - .8N2. Setting MQ1 = MQ2 and using N1 + N2 = 40, we find that
N1 = 26 and N2 = 14. The marginal catch at each lake is 4.8 fish; the maximum total
catch is: [(10)(26) - (.1)(26)2] + [(16)(14) - (.4)(14)2] = 338 fish.
Chapter 6
1. The fact that the product development was lengthier and more expensive than initially
anticipated is no reason to charge a higher price. These development costs have been
sunk and are irrelevant for the pricing decision. Price should be based on the product’s
relevant costs (the marginal cost of producing the item) in conjunction with product
demand (as summarized by the product’s price elasticity).
2. This statement confuses average quantities and marginal quantities. Though average total
cost is always greater than average variable cost, marginal cost certainly can exceed
average cost. For instance, when short-run production is pushed past the point of
diminishing returns, marginal costs tend to turn steeply upward and exceed average cost.
3. a. The profit associated with an electronic control device (ECD) is
πE = 1,500 - [500 + (2)(300)] = $400. If the firm sells the two microchips separately
(instead of putting them into an ECD), its total profit is πM = (550 - 300)(2) = $500.
Thus, the firm should devote all of its capacity to the production of microchips for
direct sale. Producing ECDs is not profitable.
b. If there is unused microchip capacity, the firm earns $400 in additional profit for each
ECD sold. Producing ECDs now becomes profitable.
c. If $200 (of the $500 average cost) is fixed, each ECD’s contribution becomes πE =
1,500 - [300 + (2)(300)] = $600. The firm should produce ECDs in the short run; this
is more profitable than selling chips directly.
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4. a. In the short run, the merged banks hope for sizeable cost savings by eliminating
redundant operations, for instance, closing branch banks. In the longer term, the banks,
by combining the provision of several related services, hope to benefit from economies
of scope on both the supply side and the demand side. On the supply side, similar
services may take advantage of the same inputs. For example, it may be easier to train a
financial services officer to do a number of related transactions than to train separate
officers for each type of transaction. Additionally, the information to process one type
of transaction, such as a customer's credit history, may also be relevant for other
transactions. On the demand side, one-stop shopping allows consumers to conserve on
transaction costs. Rather than going to several different firms for each type of
transaction, the customer can do everything in one stop.
b. It is possible for national banks to operate more efficiently than regional banks or state
banks, but this depends on both production and transaction costs. Since banking is an
information intensive industry, banks may be able to take advantage of economies of
scope and scale in information collection and transmission. For example, national
banks could provide a customer with access to his or her personal accounts nationwide
and could use the same information for a variety of related transactions. On the other
hand, the mortgage market depends on local information about the real estate market.
National banks may not possess any informational advantages in these markets. In
addition, national banks may have higher administrative costs associated with
monitoring a large organization. Determining whether national banks have cost
advantages requires a careful study of costs.
c. Some of the mergers are based on economies of scope, particularly where the
institutions have different but related products, such as banking and insurance. Other
mergers are based on geographical expansion. Given that different products may be
offered in different regions, these expansions could be said to represent economies of
scope and economies of scale. Simple economies of scale may be driving some of the
mergers.
5. a. Setting MR = MC implies 10,000 - 400Q = $4,000. Thus, Q* = 15 games.
b. The contribution is R - VC = ($150,000 - 45,000) - ($4,000)(15) = $45,000. The
opportunity cost of the entrepreneur’s labor is $20,000, and the required annual return
on the $100,000 investment is 20 percent or $20,000. Thus, her economic profit is
$45,000 - 20,000 - 20,000 = $5,000.
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6. a. The running shoe producer's demand is P = 48 - Q/200. and its costs are C = 60,000 +
.0025Q2. We confirm the table entries simply by computing prices, revenues, and
costs for appropriate levels of output Q.
b. The firm maximizes profit by setting MR = MC. Therefore, MR = 48 - Q/100 and MC
= .005Q. Setting MR = MC implies: Q* = 3,200. In turn, P* = $32.
7. a. To maximize profit set MR = MC. Therefore, 10 -.5w = 5, or w = 10 weeks. Profit is
75 – 50 = $25 thousand.
b. The “total” marginal cost (including the opportunity cost of lost profit) of showing the
hit an extra week is 5 + 1.5 = $6.5 thousand. Setting MR = MC = 6.5 implies: w = 7
weeks.
c. On the cost side, there are economies of scale and scope. (With shared fixed costs, 10
screens under one roof are much cheaper than 10 separate theaters.) Demand
economies due to increased variety probably also exist. Filmgoers will visit your
screens knowing that there’s likely to be a movie to their liking.
d. Obviously, DVD sales compete with (and potentially cannibalize) theater revenues.
The delay makes sense as long as the extra theater profits from extending the run
exceed the video profits given up.
8. a. Setting MR = MC implies 96 - .8Q = 16 + .2Q, or Q* = 80. In turn, P = $64, and π =
5,120 - 2,080 = $3,040.
b. She is correct that QMIN = 40 units. At this output, AC = 960/40 = 24 and this exactly
matches MC = 16 + (.2)(40) = 24. Her second claim is incorrect. Optimal output is Q*
= 80 where MR = MC.
c. Yes, it is cheaper to produce 80 units in two plants (each producing at QMIN = 40).
Total cost is (AC)(Q) = ($24)(80) = $1,920. This is cheaper than the single-plant cost
($2,080) of part (a).
9. a. We are given that MC = $20,000, and from the price equation, we derive MR =
30,000 - .2Q. Setting MR = MC implies Q = 50,000, confirming that GM’s current
output level is profit maximizing.
b. Fixed costs should not be mixed with variable costs in determining output and price
decisions. Removing the allocated fixed cost means taking out 160,000,000/40,000 =
$4,000 per unit. Thus, the true marginal cost per unit is $22,000 - $4,000 = $18,000.
Note that the actual MC in the West Coast factory is lower than the MC in the
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Michigan plants. Thus, GM should expand its West Coast output (to 60,000 units to
be exact).
10. a. Given that C = 175,000 + 300Q + .1Q2, Firm K’s marginal cost, is: dC/dQ = 300 +
.2Q. Setting MR = MC implies: 800 - .3Q = 200 + .2Q. Therefore, Q* = 1,000 and P*
= $650. The profit associated with coats is: π = R – C = 650,000 – 575,000 = $75,000.
Firm K’s total profit is $125,000.
b. If Firm K continues to produce 1,000 regular coats (as in part a) it will be able to
fulfill only 100 coats of the 200 corporate coats ordered, implying an opportunity cost
of $200 per coat (in foregone profit). Thus, the full MC of producing the last regular
coat is 200 + [200 + .2Q]. Setting MR = MC implies Q* = 600. Thus, if the firm had
sufficient corporate coat orders, it would be willing to cut output from 1,000 to 600
coats. But, with only 200 corporate coats orders (and surplus capacity for producing
100 coats) regular coats need only be cut from 1,000 to 900.
Rather than cut winter production of regular coats, the firm should consider
whether it is more profitable to make additional coats during the summer (when there
is plenty of unused capacity) and store them in inventory to sell in the winter. This
will free up winter capacity and will be optimal if inventory costs are sufficiently low.
c. When demand falls permanently to P = 600 - .2Q, the firm’s new optimal output and
price (after setting MR = MC) are Q* = 500 and P* = 500. Firm K’s profit from coats
drops to: π = R - C = 250,000 - 350,000 = -$100,000. Total profit is -$50,000. During
the winter (while the firm is committed to the factory lease), the firm should adopt
this price and production plan in order to minimize its loss. When its lease expires in
June, the firm should shut down.
11. a. We have: MCE = 1,000 + 10Q and MC = 3000 + 10Q. Setting MR = MC implies:
10,000 - 60Q = 3,000 + 10Q. Thus, Q = 100 cycles and P = $7,000.
b. Purchasing engines implies a marginal cost of 2,000 + 1,400 = $3,400 (compared to
the MC in part a of $4,000). Again setting MR = MC implies Q = 110 and P = $6,700.
However, the firm should continue to produce some engines itself (up to the point
where MCE = 1,400). Setting 1,000 + 10QE = 1,400 implies QE= 40 engines. The firm
should produce 40 engines and buy the remaining 110 – 40 = 70 engines.
*12. a. MPL = 120,000 watches/60,000 labor hours = 2 watches per hour. The marginal labor
cost is: ($8/hour)/(2 watches/hour) = $4/watch. Total MC is: $6 + $4 = $10/watch. To
maximize profit, the firm sets MR = MC. Therefore, 28 - Q/10,000 = 10, or Q =
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180,000 watches. However, the firm’s limited capacity makes this output impossible.
The best the firm can do is to produce up to its capacity, Q* = 120,000 watches. To
sell this quantity, the firm sets P* = 28 - 120,000/20,000 = $22. The firm's
contribution is: (22-10)(120,000) = $1,440,000.
b. Marginal labor cost on the night shift is $12/2 = $6. The relevant MC if the night shift
is used is: MC = $6 + $6 = $12. Setting MR = 12, one finds Q* = 160,000 watches
and P* = $20. Contribution is:
R - VC = $3,200,000 - [1,200,000 + 480,000] = $1,520,000.
c. Demand drops permanently to P = 20 - Q/20,000. A good bet is that the firm will now
use only the day shift, implying MR = MC = 10. It follows that 20 - Q/10,000 = 10 or
Q* = 100,000 watches. In turn, P* = 20 - 100,000/20,000 = $15. Contribution is now
$500,000. In the short run (when its $600,000 in costs are fixed), the firm minimizes
its losses by producing 100,000 units. If these losses continue, the firm should shut
down in the long run, i.e., when its lease is up.
Chapter 7
1. a. According to the “law” of supply and demand, the existence of a large body of
Picasso’s artwork will tend to lower the value of any individual piece of work.
b. If demand for Picasso’s work is inelastic, increasing the number of pieces sold (by
driving down prices) will reduce total revenue. The artist’s heirs should try to limit
supply by spreading sales of his artwork over long time periods.
2. a. At harvest time, supply is fixed (regardless of the price) so the supply curve is nearly
vertical.
b. At the beginning of the growing season, supply is quite flexible implying an upwardsloping supply curve.
c. In the long run, the supply curve is nearly horizontal.
3. a. Setting QD = QS implies 184 - 20P = 124 + 4P or 24P = 60. Therefore, P = $2.50 and
Q = 134 pounds per capita.
b. This increase represents only .7 percent of total supply and will have little price effect.
The new quantity supplied is (1.007)(134) = 135. Rearranging the demand curve, we
have P = 9.20 - .05Q. Therefore, we find that P = 9.20 - (.05)(135) = $2.45. Montana
farmers’ revenue should increase by about 8 percent (based on a 10 percent quantity
increase and a 2 percent price drop).
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c. If the total harvest is 10 percent above normal, QS = (1.10)(134) = 147.4 pounds per
capita and P = 9.20 - (0.5)(147.4) = $1.83. Farm revenue drops from (2.50)(134) =
$335 to (1.83)(147.4) = $269.74, a 19.5 percent drop. Demand is inelastic. A modest
quantity increase caused a large price drop and this is detrimental to farmers’ incomes.
Since varying harvest conditions can cause significant price and revenue changes,
today’s farm profits quickly can become tomorrow’s losses.
4. a. The tax per bottle on beer producers increases the cost per bottle and shifts the
industry supply curve upward. By itself, this effect implies a lower total quantity sold
at a higher equilibrium price. The fall in consumer income implies a decline in beer
demand, shifting the industry demand curve to the left. By itself, this effect implies a
lower total quantity sold at a lower equilibrium price. The twin factors move total
volume in the same direction towards lower output levels. The two effects move price
in different directions, so the net price effect – up or down – is uncertain, depending
on which effect is larger.
b. The emerging economic recovery means an increased demand for trucking transport
services (a rightward shift in the industry demand curve), therefore, increasing
trucking volume and transport prices. The reduction in capacity means that the
industry supply curve shifts to the left. The increases in diesel fuel costs and
regulatory costs mean that the supply curve shifts upward (and leftward). These
separate supply effects are all in the same direction, resulting in lower trucking
volume and higher transport prices.
The four factors all move trucking rates (prices) in the same direction, upward. As
far as trucking volume is concerned, it’s likely (though not certain) that the three
significant adverse supply shifts outweigh the favorable demand shift, causing
trucking volume to fall.
5. a. The Green Company’s marginal cost is MC = dC/dQ = 4 + 2Q, and the price is P = $40.
Setting MC = P implies 4 + 2Q = 40, or Q = 18 units. More generally, setting MC = P
generates the supply curve 4 + 2Q = P, or Q = (P - 4)/2.
b. With the increase in fixed cost, the firm should continue to produce 18 units. Its profit
is π = R – C = (40)(18) - [144 + (4)(18) + (18)2] = 720 - 540 = 180. Of course, the firm
will supply no output if price falls below the level of minimum average cost. We set
MC = AC and find that average cost is a minimum at Q = 12. In turn, ACMIN = 28.
Thus, the firm’s supply is zero if price falls below 28.
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c. In part a (when fixed costs are 100), ACMIN = 24 at a quantity of 10 units for each firm.
Thus, the original long-run equilibrium price is 24. With elevated fixed costs, one
would expect the long-run price to rise to 28 (the new minimum level of AC). At this
higher price, total demand is reduced. However, each firm’s output would rise from 10
units to 12 units. With reduced total demand and greater output per firm, the number
of firms must decline.
6. a. Depicting the market effect of declining milk demand means reversing the result shown
in Figure 7.4. Now the new demand curve (call this D´´) shifts to the left of the original
demand curve D. The intersection of D´´ and the upward-sloping, short-run supply curve
occurs at a lower price and output (current dairy farmers reduce their cow herds).
b. At the reduced milk price, dairy farmers are making economic losses, so dairy farmers
exit the industry. Over time, enough dairy farmers exit to shift the supply curve of
current producers to the left where the price returns to 1 cent per ounce (where suppliers
make zero economic profits) in long-term equilibrium. Note that the long-run supply
curve is horizontal (as in Figure 7.4) There is a reduction in total milk supply that just
matches the reduction in demand.
7. a. The typical firm’s total cost is C = 300 + Q2/3, which implies MC = (2/3)Q and AC =
300/Q + Q/3. Setting AC = MC implies: 300/Q + Q/3 = (2/3)Q, or 300/Q = Q/3. After
rearranging, we have 900 = Q2. Therefore, Qmin = 30. In turn, ACmin = 300/30 + 30/3 =
$20.
b. Each firm’s supply curve is found by setting P = MC = (2/3)QF. Therefore,
QF = 1.5P. With 10 firms, total supply is QS = 10QF = 15P. Setting QD = QS implies 900 - 15P
= 15P. Thus, we find P = $30. By substitution, we have: Q = 450 and QF = 45. At QF = 45,
each firm’s total cost is: C = 300 + (45)2/3 = 975. Thus, the typical firm’s profit is: R – C =
1,350 – 975 = $375.
c. In long-run equilibrium, free entry ensures that P = ACmin = $20. In turn, Q = 900 (15)(20) = 600. The long-run number of firms is Q/Qmin = 600/30 = 20.
8. a. Setting P = MC implies 16 = 4 + 4QF, or QF = 3. Thus, π = (16)(3) - 80 = -$32.
b. QD = 200 - (5)(16) = 120 units. The number of firms is: QD/QF = 120/3 = 40.
c. Firms will exit the industry because all are making losses. In the long run, PC = ACMIN
= 120/5 = $24. (With QMIN = 5, we have calculated, ACMIN = C/QMIN.) At PC = $24,
then QD = 80, and the # of firms is: 80/5 = 16.
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d. Price increased enough to allow each remaining firm a zero economic profit. Each
firm’s output increased from 3 units to 5 units (the point of minimum AC), while the
number of firms declined from 40 to 16.
9. a. Here, MC = AC = $5. Thus, PC = $5. From the price equation, 5 = 35 - 5Q, implying
QC = 6 million.
b. The industry displays constant returns to scale (constant LAC). The real microchip
industry probably displays increasing returns to scale (declining LAC). For
competition to be viable, returns to scale must be exhausted at volumes well below
total market demand.
c. Total profit is zero. Consumer surplus is (.5)(35 - 5)(6) = $90 million.
10. a. Setting demand equal to supply, we have 200 - .2Q = 100 + .3Q. Solving this, we find
Q = 200. Substituting this value for Q, we find P = $160.
b. If a tax of $20 per unit is levied on suppliers, the industry supply curve undergoes a
parallel upward shift. Increasing the curve’s price intercept by 20 implies the supply
equation, P = 120 + .3Q. Setting the demand curve equal to the new supply curve
implies 200 - .2Q = 120 + .3Q or Q = 160. Consumers pay the price: P = 200 – (.2)(160)
= $168. Eight dollars of the $20 tax increase (or 40%) has been passed on to consumers.
The price that producers receive (net of the tax) is $148.
c. If a tax of $20 per unit is levied on consumers, the demand curve undergoes a parallel
downward shift and becomes P = 180 - .2Q. Setting the new demand curve equal to the
supply curve, we have: 180 - .2Q = 100 + .3Q or Q = 160. The price before tax is
P = 180 - (.2)(160) = $148. The price inclusive of the tax is 148 + 20 = $168. The price
and quantity results in parts (b) and (c) are identical. Whether a given tax is levied on
suppliers or consumers makes no difference in the ultimate competitive equilibrium.
11. a. Setting QD = QS implies 28 – 4P = -12 + 6P, so P = $4 and Q = 12 million bushels of
rice. To find consumer surplus and producer surplus, we graph the demand and supply
curves, noting there vertical intercepts ($7 and $2 respectively). Thus, consumer
surplus is .5(7 – 4)(12) = $18 million, and producer profit is: ½(4 – 2)(12) = $12
million.
b. With free trade, the world price establishes the new prevailing price, P = $3. From the
demand and supply equations, we find QD = 16 and QS = 6, so imports = 16 – 6 = 10
million bushels. In turn, consumer surplus is ½(7 – 3)(16) = $32 million, and producer
profit is: ½(3 – 2)(6) = $3 million. Free trade has increased total welfare.
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c. Offering rice growers a $1 per bushel subsidy means a (subsidized) price of $4 and
restores domestic supply to QS = 12 million bushels (as in part a). Producer profit is
once again $12 million, but the cost to taxpayers of the subsidy is ($1)(12) = $12
million. Thus, total welfare is: 32 + 12 – 12 = $32 million (less than under the free
trade “level playing field” of part b).
Chapter 8
1. a. The merger should mean the end of the prevailing cutthroat competition. The merged
firm should set out to achieve the available monopoly profit.
b. Formerly, cutting rates made sense in order to claim additional clients from one’s rival.
After the merger, the newspapers will raise rates (again seeking the monopoly level).
2. If the company is currently charging the optimal monopoly price, any cut will reduce its
profit – the larger the price cut, the larger the profit reduction. Here is an illustration of
how to compute the profit impact of a 20 percent price cut. Suppose the company’s
current price and output are P = $.20/pill and Q = 1 million pills and that MC = $.10/pill.
Then, the company’s current contribution is: (.20 - .10)(1) = $.1 million.
Suppose the company lowers price to P = $.16 and the elasticity of demand turns out
to be EP = -1.5. Then dQ/Q = (-1.5)(-20%) = 30%, implying Q = 1.3 million. The firm’s
new contribution is: (.16 - .10)(1.3) = $.078 million, implying a 22% fall in profit.
3. Packing the product space with a proliferation of differentiated items is a classic example
of strategic entry deterrence. The slower selling brands are not profitable in themselves.
However, they raise the firms’ overall profits by leaving no product niche for a new rival
to profitably enter the market.
4. a. A profit-maximizing cartel sets MR = MC. Thus, 500 - (2/3)Q = 200, or QM = 450
thousand trips. In turn, PM = 500 - 450/3 = $350 per trip.
b. Under perfect competition, PC = LAC = $200. Thus, QC = 900 thousand trips.
5. a. We know that P = 11 - Q and C = 16 + Q. Setting MR = MC, we have 11 - 2Q = 1.
Thus, the monopolist sets QM = 5 million and PM = $6.
b. The regulator sets P = AC. Thus, 11 – Q = 16/Q + 1. After multiplying both sides by Q,
this becomes a quadratic equation with two roots: Q = 2 and Q = 8. Naturally, the
regulator selects the larger output level, so we have QR = 8 million and PR = $3.
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c. Under marginal-cost pricing, P* = MC = $1 and Q = 11 – P = 10 million. At this
quantity, AC is 26/10 = $2.60. The shortfall of price below average cost is 2.60 – 1 =
$1.60 per unit.
6. a. The monopolist sets MR = MC, implying 1,500 - .2Q = 300 + .1Q, or QM = 4,000 tons.
In turn, PM = 1,500 - (.1)(4,000) = $1,100.
b. Total profit is: 4,400,000 - [1,400,000 + 1,200,000 + 800,000] = $1,000,000.
7. a. OPEC maximizes its profit by setting MR = MC. We have 165 - 5Q = 15.
Therefore, Q* = 30 million barrels per day. In turn, P* = 165 – (2.5)(30) = $90
per barrel.
b. If OPEC sets P = $65, it sells: Q = 66 – (.4)(65) = 40 million barrels per day.
Profit (per day) is: π = (65 - 15)(40) = $2.0 billion. Instead, if it sets P1 = $90 for
the first five years, its initial profit is: π1 = (90 - 15)(30) = $2.25 billion per day.
In the second 5-year period, its optimal quantity and price are: Q2 = 24 million
barrels per day and P2 = $75. (Check this by using the long-run demand curve
and setting MR = MC.) Thus, its profit is: π2 = (75 - 15)(24) = $1.44 billion per
day. OPEC’s average profit over the decade (ignoring discounting) is $1.845
billion per day – lower than the $2 billion per day achieved if it holds its price to
$65.
8. a. We have P = 35 - 5Q and MC = AC = 5. Setting MR = MC, we find QM = 3 million
chips and PM = $20.
b. The monopolist’s profit is: πM = (20 - 5)(3) = $45 million. Consumer surplus is: (.5)(35 20)(3) = $22.5 million.
9. a. At P = $15, 2.5 million trips are demanded. In the text, we saw that each fully utilized
taxi had an average cost per trip of $10 and, therefore, earned an excess profit of (15
- 10)(140) = $700 per week. The commission should set the weekly license fee at L =
$700 to tax away all this excess profit. Assuming that 17,857 taxis operate (just
enough to meet the 2.5 million trips demanded), the commission collects a total of
$12.5 million in license fees.
b. The rearranged demand curve is: P = 20 - 2Q. We saw that the extra cost of adding a
fully occupied taxi is $1,400 per week, or $10 per trip. The relevant MC per trip is
$10. Setting MR = MC, we have 20 - 4Q = 10. Thus, QM = 2.5 million trips and PM =
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$15. The current, regulated fare happens to be profit maximizing ($12.5 million in
profit per week) for the industry.
c. If the market could be transformed into a perfectly competitive one, the result would
be PC = ACmin = $10, QC = 10 - (.5)(10 = 5 million trips, and number of taxis would
double to 35,714.
d. Taxi trips are not perfect substitutes. If a taxi charges a fare slightly higher than the
industry norm, it will not lose all its sales. (Customers in need of a taxi will take the
one in hand, rather than wait for a slightly cheaper fare.) Since there is room for
product differentiation and price differences, the taxi market probably is best
described as monopolistic competition. In this setting, all cabs make zero profit (due
to free entry). If price settles at P = $12.80, then AC = $12.80 for each cab. One can
check that this AC occurs at 100 trips per week; each taxi is 71 percent utilized. Trip
demand is: Q = 10 – (.5)(12.80) = 3.6 million trips with 36,000 taxis (each making
100 trips).
10. a. To produce a fixed amount of output (in this case, 18 units) at minimum total cost, the
firms should set outputs such that MCA = MCB. This implies 6 + 2QA = 18 + QB, or
QB = 2QA - 12. Using this equation together with QA + QB = 18, we find QA = 10 and
QB = 8. The common value of marginal cost is 26.
b. We know that P = 86 - Q, implying MR = 86 - 2Q. Marginal revenue at Q = 18 is 86 (2)(18) = 50. This exceeds either firm’s marginal cost (26); therefore, the cartel can
profit by expanding output.
c. Setting MR = MCA = MCB implies 86 - 2(QA + QB) = 6 + 2QA = 18 + QB. The solution
is QA = 13 and QB = 14. The cartel price is P = $59, and the common value of MR and
the MCs is 32.
*11. a. The bookstore’s profit is: π = (P – AC)Q = (9 – 5)(12) = $48 thousand. Consumer
surplus = ½(15 – 9)(12) = $36 thousand.
b. From the demand curve, the chain sells 6 thousand books online at P = $12 and 10
thousand books in its stores at P = $7. Its total profit is ($12 – $4)(6) + ($7 – $5)(10)
= $68 thousand. Consumer surplus is the sum of triangles A and B. Therefore: CS =
½($15 – $12)(6) + ½($12 – $7)(10) = $34 thousand. From the consumer’s standpoint, online selling has twin countervailing effects. In-store buyers benefit from
lower prices brought by online competition. Online buyers actually pay a higher
price than before (due to the chain’s skillful price discrimination).
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Price
15
A
12
B
9
7
P = 15 - .5Q
6
12
16
Quantity
*12. a. We know that P = 660 - 16Q1 and C = 900 + 60Q1 + 9Q12. Setting MR = MC, we have
660 - 32Q1 = 60 + 18Q1 or Q1 = 12. In turn, we find P1 = $468. The firm’s profit is:
R – C = (468)(12) - [900 + (60)(12) + 9(12)2]
= 5,616 – 2,916 = $2,700.
b. If 10 firms each produce 6 units, total output is 60 and the market price is, indeed:
P = 1,224 - (16)(60) = $264. Setting firm 1’s MR = MC implies: 1,224 - (16)(54) 32Q1 = 60 + 18Q1, implying Q1 = 6 units as claimed. Finally, the firm’s average cost
is: C/Q = [900 + (60)(6) + 9(6)2]/6 = $264. The typical firm earns a zero economic
profit since P = AC.
c. Under perfect competition, Pc = ACMIN. Setting AC = MC, we have 900/QF + 60 +
9QF = 60 + 18QF, implying QF = 10 and ACMIN = 240. Thus, Pc = $240 and Qc = 76.5
- (240)/16 = 61.5. The number of firms is found by dividing total output by each
firm’s output: 61.5/10 = 6.15 firms.
Chapter 9
1. The conventional wisdom points to entry in loose oligopolies for two reasons: i) the
market offers positive economic profits (unlike a perfectly competitive market), and ii)
since the market is not dominated by large firms, a new entrant has the potential to reap
significant market-share gain over time (unlike a tight oligopoly).
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2. a. Before the would-be acquisition, the top four firms accounted for 82% of the market.
After the merger, the former fifth-place firm is included in the top four, increasing the
concentration ration to 87%. The “old” HHI is:
(31.3) 2 + (26.6) 2 + (12.2) 2 + (11.9) 2 + (5.0) 2 + (3.1) 2 + (2.3) 2 + (1.6) 2 = 2,020.
The “new” HHI is:
(31.3) 2 + (38.8) 2 + (11.9) 2 + (5.0) 2 + (3.1) 2 + (2.3) 2 + (1.6) 2 = 2,669.
b. The increase of 649 points in a moderately concentrated market would definitely
trigger close antitrust scrutiny. Alternatively, if T-Mobile were to merge with
TracFone, the new HHI would only increase to 2,142; the more modest 122 HHI
increase would warrant a lesser scrutiny.
c. Mergers frequently generate efficiency benefits (in this case a more seamless call
system and better service). Keep in mind, however, that parties to a merger always
claim efficiency benefits, so the question is how significant they turn out to be.
Because higher industry concentration after the merger means less competition, there
is justifiable concern that AT&T (and possibly rival cellular providers) will raise
prices. If it does try to raise prices, AT&T has to be careful in handling price-sensitive
T-Mobile customers, so as to minimize their defection to alternative low-price plans.
3. a. OPEC’s net demand curve is: QN = QW - QS = (96 – .2P) – (.3P + 26) = 70 – .5P.
Rearranging this, we have: P = 140 – 2QN.
b. Setting MR = MC, we have 140 - 4QN = 20, or QN = 30 million barrels per day. In turn,
P = $80 and QS = (.3)(80) + 26 = 50 million barrels per day. OPEC accounts for 37.5
percent (30/80) of world oil production.
4. a. Each follower maximizes its profit by setting P = MC = 6 + QF. Thus, each firm’s supply
curve is: QF = P - 6. With 4 followers, the total supply curve is: QS = 4QF = 4P - 24.
b. The net demand curve of Firm A (the dominant firm) is:
QA = QD - QS = [48 - 4P] - [4P - 24] = 72 - 8P.
Rearranging the leader’s demand curve, we have: P = 9 - QA/8. The leader maximizes
its profit by setting MR = MC, implying: 9 - QA/4 = 6, or QA = 12 units. In turn, P = 9 12/8 = $7.50 and QS = (4)(7.5) - 24 = 6 units.
5. a. For firm 1, MR1 = MC implies 120 - 5Q2 - 10Q1 = 60, or Q1 = 6 - .5Q2. In equilibrium,
Q1 = Q2 so we can solve the above equation to find Q1 = Q2 = 4.
b. If the firms collude, they maximize total profit by setting MR = MC. Therefore, MR =
120 - 10Q = 60, or Q = 6. With total output split equally, each firm supplies 3 units.
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6. a. The payoff table is
Firm N Advertising
$10 million
$20 million
Firm M
$10 million
50, 50
30, 60
Advertising
$20 million
60, 30
40, 40
b. Each firm's dominant strategy is to spend $20 million on advertising. Yes, a prisoner's
dilemma is present.
c. If the firms could agree to limit spending to $10 million each, their profits would
increase to $50 million each.
7. a. Yes, there is a prisoner’s dilemma in the sense that when all farmers have large crops,
they all make losses. One solution is for farmers to agree to withhold excess supplies
from the market in order to maintain higher prices.
b. If each member’s compensation is based on the team’s overall performance, there is
the incentive to take a “free ride” on the efforts of other members. (If it is a 10member team, one member contributes only 10 percent to the overall performance.)
Countering the prisoner’s dilemma may mean monitoring work effort or increasing the
rewards for individual performance.
8. a. For the union, hiring a lawyer is a dominant strategy – that is, it offers uniformly
higher chances of winning its case. For management, hiring a lawyer is also a
dominant strategy – it minimizes the union’s winning chances.
b. Yes, this is a prisoner’s dilemma. Both sides are inevitably led to hire expensive legal
representation, but these moves are offsetting (having a negligible effect on the
arbitration outcome). The upshot is higher costs for both sides.
9. a. For firm 1, P1 = 75 + .5P2 - Q1. Setting MR1 = MC, we have 75 + .5P2 - 2Q1 = 30,
implying Q1 = 22.5 + .25P2. Substituting this solution for Q1 into the price equation, we
find: P1 = 52.5 + .25P2.
b. A lower P2 shifts firm 1’s demand curve inward, causing firm 1 to set a lower price.
c. Solving P1 = 52.5 + .25P1, we find P1 = P2 = 70. From the demand equations, Q1 = Q2 =
40. Each firm’s profit is 1,600.
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10. a. If Firm 2 cuts price to $67, Firm 1’s best response is: P1 = 52.5 + (.25)(67) = $69.25.
Putting these prices into Firm 2‘s demand curve, we find that Firm 2 sells 42.625
units, generating a profit of (67 – 30)(42.625) = $1,577.13. If instead Firm 2 raises its
price to $73, Firm 1 will respond with: P1 = 52.5 + (.25)(73) = $70.75. Again putting
these prices into Firm 2‘s demand curve, we find that Firm 2 sells 37.375 units,
generating a profit of (73 – 30)(37.375) = $1,607.13. As a price leader, Firm 2’s most
profitable commitment strategy is to raise its price to $73.
b. As price leader, firm 2 is pursuing a soft “puppy dog” strategy. By setting a high price,
the firm induces a higher price from its rival, increasing the profits for both firms.
11. a. Rearranging the price equation shows that raising A increases sales. Advertising
spending is a fixed cost (doesn’t vary with output).
b. Setting MR = MC, we have 50 + A.5 - 2Q = 20 or Q = 15 + .5A.5. Substituting this
solution for Q into the price equation, we find: P = 35 + .5A.5. If advertising is
increased, the firm should plan for increased sales at a higher price.
c. π = (P - 20)Q - A = (15 + .5A.5)(15 + .5A.5) – A = 225 + 15A.5 - .75A. Setting dπ/dA =
0 implies: 7.5/A.5 - .75 = 0. Thus, A = 100. In turn, Q = 20 and P = 40.
12.
a. We can write equation 10.7 as:
[P-MC][Q/A][(dQ/dA)/(Q/A)] = 1,
or
[P-MC][Q/A] = 1/EA,
after dividing each side by the last bracketed term. In the text example, [P-MC]Q/A
= [1 - 0.8][10,000,000/1,000,000] = 2, which exactly equals 1/EA = 1/.5
b. Divide the optimal price equation by the optimal advertising equation and cancel the
common term (P - MC). The resulting expression is (A/Q)/P = (-1/EP)/(1/EA), or
A/(PQ) = -EA/EP. In the example, A/(PQ) is 0.1, which is exactly equal to -EA/EP =
-.5/-5.
c. To justify a much higher advertising ratio, Walt Disney Studio’s sales must be very
elastic with respect to advertising and/or very inelastic with respect to price.
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Chapter 10
1. In a Nash equilibrium, each player’s chosen strategy is optimal, given the strategy of the
other. Thus, neither side can profit by unilaterally deviating. By comparison, a dominant
strategy is optimal against any strategy the other player might choose.
2. It is never advantageous to move first in a zero-sum game. (The best one can do is choose
one’s equilibrium pure strategy, if there is one.) If there are multiple equilibria in a nonzero sum game, the first mover can select her preferred equilibria. A simple setting in
which there is a second-mover advantage involves symmetric price competition. The
second mover can do no worse than the price leader (it can always match the leader’s
price) and can typically do better by undercutting it.
3. a. There are no dominant nor dominated strategies for either player.
b. The equilibrium strategies are R1 and C3; the equilibrium outcome is 10.
4. a. The payoff table is:
Saudi Arabia
Venezuela
3 M barrels 4 M barrels
8 M barrels
480, 180
400, 200
9 M barrels
450, 150
360, 160
b. Saudi Arabia’s dominant strategy is to produce 8 million barrels. Venezuela’s
dominant strategy is to produce 4 million barrels.
c. The basic asymmetry is in the size of the countries’ outputs. By cutting price,
Venezuela can expand output by 33.3 percent. For the same price cut, Saudi Arabia
enjoys only a 12.5 percent increase. Venezuela profits from the extra output; Saudi
Arabia does not. One might call this a “one-sided” prisoner’s dilemma.
5. a. There are two equilibria: firm J develops E and firm K develops D, and vice versa.
Thus, one cannot make a confident prediction as to which outcome will occur.
b. If firm J moves first, it should choose E, knowing firm K will then choose D.
c. Similarly, firm K’s first move should be to choose E.
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6. a. The unique equilibrium outcome has firm A choosing High and firm B choosing
Medium. (Use the method of “circles and squares” to confirm this.)
b. The firms should coordinate their R&D strategies by selecting Medium and Low,
respectively. Here the firms achieve maximum total profit, and each firm’s profit
is greater than it was in the non-cooperative equilibrium of part (a).
7. a. Applying the method of “circles and squares” to the payoff table, we see that
there are two Nash equilibria: i) Both superpowers escalate their weapons
buildup, or ii) Both stop. Strictly speaking this is not a prisoner’s dilemma. (It
is not the case that the play of dominant strategies leads to an inferior outcome
for both sides.)
b. Yes, with the fall of the former Soviet Union, it appears that the superpowers
have switched (at least for the time being) to the Stop-Stop equilibrium.
8. a. Applying the method of “circles and squares” to the payoff table, we see find
that High spending is each side’s dominant strategy. The result is a prisoner’s
dilemma; both sides would benefit if they could maintain Medium spending
instead.
b. With an indefinite number of moves, each should begin and continue to
implement Medium R&D as long as the other does. If either ever
opportunistically pursues High R&D, the other would play tit-for-tat by doing
likewise in its next move. This potential punishment sustains the repeated play
of Medium R&D as a Nash equilibrium.
c. If it’s hard to discern the rival’s R&D moves, it’s more difficult to enforce the
Medium spending equilibrium.
d. No doubt there are a number of diaper producers at the economy, low-price end
of the market. Barriers arising from high advertising and marketing expenditures
(as well as established brand allegiance) preclude new entry at the high end.
9. a. Firm Y has no dominant strategy nor any dominated strategy. For firm Z, C3 is
dominated by C1.
b. Once C3 is eliminated from consideration, R1 is dominated by R2. With R1
eliminated, C2 is dominated by C1. Thus, C1 is firm Z’s optimal choice, and R2 is
firm Y’s optimal response. The resulting value of the game is 0.
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10. a. The payoff table below shows the total sales of the firms. (Carefully check these
entries.) This is a constant-sum game. For any pair of strategies, the sum of the firms’
revenues is $48 million, the total revenue in the two markets.
Firm B
Firm A
2-0
1-1
0-2
3-0
27, 21
22.5, 25.5
30, 18
2-1
33, 15
29, 19
36, 12
1-2
28, 20
27, 21
39,
0-3
18, 30
13.5, 34.5
9
25.8, 22.2
b. For Firm A, the strategies 3-0 and 0-3 are dominated by both 2-1 and 1-2. But neither 21 nor 1-2 dominates the other. For Firm B, the dominant strategy is 1-1. Regardless of
Firm A’s action, 1-1 gives better payoffs than 0-2 or 2-0. For this reason, Firm A
should anticipate B’s play of 1-1. Accordingly, Firm A should play 2-1. The outcome
(by iterated dominance) implies revenues of $29 million and $19 million respectively.
11. a. The town’s dominant strategy is non-enforcement. Anticipating this, the typical motorist
chooses to disobey the law. The outcome is (5, -10).
b. If the town can make the “first move” by committing to 100 percent enforcement, the
situation changes. The typical motorist’s best response is to obey, leading to the
outcome (0, -15). Note, however, that enforcement (because of its high cost) is still not
in the best interest of the town (-15 is worse than -10).
c. Now the town enforces the law with probability p. The typical motorist will obey the
law if and only if his expected payoff from doing so (0) exceeds the payoff if he
doesn’t, -20p + 5(1 - p). Setting these payoffs equal to one another implies p = .2. As
long as the enforcement probability is slightly greater than 20 percent, motorists will
obey the law. The town’s enforcement cost is (.2)(-15) = -3. Probabilistic enforcement,
which successfully deters, is the town’s least costly strategy.
12. a. The buyer does not have a dominant strategy. She buys two units at P = $9, four units at
P = $8, and six units at P = $6. Anticipating this behavior, the seller should set P = $8.
b. With multiple rounds, the buyer could vary its purchases to encourage lower prices
(for instance, by purchasing six units at P = $6, two units otherwise). If this succeeds,
the resulting payoff is (12, 18).
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c. Maximum total profits (32) are achieved at Q = 8 units. A negotiated price of P = $6
(an equal profit split) appears to be equitable.
Chapter 11
1. Although there could be some cost economies from such a merger, the main effect on
consumers likely would be higher soft-drink prices. Aggressive price competition to
claim market share would be a thing of the past. Because the merged entity would
account for over 80 percent of total soft-drink sales, the United States Justice Department
would be likely to fight such a merger on the grounds it would create a monopoly.
2. a. With total output Q = 200,000 units, the resulting equilibrium price is: P = $80.
Therefore, total industry profit is: π = (80 - 60)(200,000) = $4,000,000. In turn,
consumer surplus is: ½(120 – 80)(200,000) = $4,000,000. Finally, the total welfare
benefit is $8,000,000.
b. With LAC = LMC = $50, the merged firm maximizes profit by setting MR = LMC.
Therefore, 120 - .4Q = 50, implying Q = 175 thousand units. The corresponding price
is: P = $85.
c. If there is a merger, the new firm’s profit would be: π = (85 - 50)(175,000) =
$6,125,000. In turn, consumer surplus would be: ½(120 – 85)(175,000) = $3,062,500.
Though the slight increase in price (from $80 to $85) has reduced consumer surplus,
total welfare has increased – from $8,000,000 to $9,187,500. On total welfare
grounds, the merger should be approved.
3. a. Setting MR = MC, we have: 500 - 20Q = 150, or QM = 17.5 thousand units and PM =
$325.
b. Under perfect competition, PC = LAC = $150 and QC = 35 thousand.
c. With a $100 tax, the monopolist’s MC is 250. Setting MR = MC, we find QM = 12.5
thousand and PM = $375.
d. The efficient solution calls for a double dose of regulation: promote perfect
competition while taxing the externality. The efficient price is: PC = LMC + MEC =
150 + 100 = $250. The corresponding (efficient) level of output is 25 thousand units.
This is the optimal solution. All of the analysts’ recommended outcomes are
inefficient. (Of the three, the part a outcome, Q = 17.5 thousand is the best. It comes
closest to the efficient outcome, implying the smallest deadweight loss).
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4. a. If management and workers have perfect information about job risks in mining, there
is no externality. The two sides should be able to negotiate working conditions that
provide the efficient amount of safety. However, if information is imperfect and
asymmetric (workers are not aware of all safety risks), there is an externality.
b. The competitors of these large brokerage firms receive a positive externality: more
productive new hires whose training was paid for by someone else. Larger firms are
well aware of these “lost” hires but still profit from these training programs.
c. Spam is an obvious externality. Spammers send out millions of mass emails at an
exceedingly low marginal cost. The idea behind the 5-second requirement is to
increase greatly the cost of sending email by the millions, without imposing any real
cost on normal email users.
d. The couple incurs an adverse effect, but this is not an externality. Rising house values
are a price effect – an effect occurring within a well-functioning market – not a side
effect occurring outside the market.
5. a. The competitive price of studded tires is PC = AC = $60. The price equation P = 170 5Q can be rearranged as Q = 34 - .2P. Thus, one finds the competitive quantity to be
QC = 34 - (.2)(60) = 22 thousand tires.
b. The full MC of an extra tire is 60 + .5Q. Equating industry demand to marginal cost,
we find P = 170 - 5Q = 60 + .5Q. Therefore, the optimal quantity is Q* = 20 thousand
tires. The optimal price is: P = 170 - (5)(20) = $70. Net social benefit is the sum of
consumer surplus and producer profit, net of external costs. Consumer surplus is:
(.5)(170 - 70)(20,000) = $1,000,000. Producer profit is: (70 - 60)(20,000) = $200,000.
External costs are C = .25Q2 = (.25)(20)2 = $100 thousand. Thus, net social benefit is
$1,100,000.
c. At Q* = 20 thousand tires, the marginal external cost is: .5Q* = $10 per studded tire.
Set a tax of $10 per studded tire to obtain the optimal result in part b. The competitive
market price, including tax, becomes: 60 + 10 = $70.
d. At an added cost of $12, low-impact studded tires are not cost effective. At a market
price of $70 (as in part b or c), they cannot compete profitably and should not be
produced.
6. a. Do families make well-informed, rational decisions concerning purchases of toys,
cereal, and so on? If not, government paternalism might justify regulation of
advertising.
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b. One point of view is that workers and employers already structure efficient labor
contracts – ones providing the optimal amount of workplace safety (at the least total
cost). According to this argument, regulating additional safety is unnecessary.
Deregulation is called for. However, if information about work place risks is imperfect,
government regulation might be warranted to ensure an “optimal amount” of safety.
c. By law, there must be access for the disabled. However, the cost of modifying existing
buildings and transport is often very high. In some locals, offering shuttle-bus and taxi
service to the disabled has proved more convenient and cost-effective than
modifications to transit systems.
d. The Department of Agriculture should tradeoff the benefits of pesticides (in increasing
crop yields) against the costs (their risks as possible carcinogens).
e. Maintaining the infrastructure should be a matter of benefits and costs, not dictated by
budget worries.
7. a. The firms’ costs are C1 = 2Q1 + .1Q12 and C2 = .15Q22. It follows that MC1 = 2 + .2Q1;
MC2 = .3Q2. In turn, MB = 9 - .4Q = 9 - .4(Q1 + Q2).
b. Setting MB = MC1 = MC2, we find Q1 = 5 and Q2 = 10, and the common marginal
value is $3. It is economically efficient for firm 2 to clean up more pollution than firm
1 since its marginal cost of cleanup is lower.
c. Each firm cleans up to the point where MC = $4; Using the MC expressions in part a,
we find Q1 = 10 and Q2 = 13.33.
d. The optimal tax is $3.00 (equal to the common value of MB = MC1 = MC2). Facing
this tax, the firms choose Q1 = 5 and Q2 = 10, as in part b.
8. a. Yes, this is a prisoner’s dilemma. It does not pay for any block to unilaterally reduce
emissions. (For instance, for the DNs, the benefit is 10 and the cost is 12.)
b. A multilateral .2 reduction by each block does not quite work; for the U.S., the extra
benefit is 18 and the extra cost is 22. However, if the DN block reduces by .4, while
the other blocks reduce by .2, all blocks benefit: U.S. (24 > 22), Europe (32 > 18), and
DNs (38 > 30).
c. The efficient scheme – found by comparing the marginal benefit and marginal cost of
emission reduction – has EUS = 1.2, EEU = .8, and EDN = .8, implying total emissions
of 2.8. Note that reducing emissions from 3.0 to 2.8 implies a total marginal benefit of
6 + 8 + 8 = $22 billion, and this exceeds the MC incurred by Europe or the DNs. But
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reducing emissions from 2.8 to 2.6 implies a total marginal benefit of only $20 billion,
and this is less than the MC of the additional .2 reduction by US or Europe.
The net benefits (relative to the status quo are: U.S. (24 - 0), Europe (32 - 18), and
DNs (38 - 48). Clearly, the developing nations must be paid monetary compensation
for making the largest emissions cutback. Suppose the U.S. and Europe were to pay
$15 billion and $5 billion respectively per year to the DNs as compensation. Then the
net gains would be $9 billion, $9 billion and $10 billion for the respective blocks.
9. a. To maximize net benefit (i.e. benefit minus cost), RWE should compare MB and MC,
where MC = $150,000 per facility. The optimal number of facilities is: N = 4.
Adopting the program at the fourth facility implies MB = $225,000 (greater than MC)
but adopting at the fifth facility has MB = $100,000 (less than MC). RWE’s
maximum net benefit at N = 4 is: 1,600,000 – (4)(150,000) = $1,000,000.
b. The additional benefit to society means that MB increases by $75,000. Now the
optimal number of facilities is N = 6. Adopting the program at the sixth facility has
MB = $100,000 + $75,000 (greater than MC) but adopting at the seventh facility has
MB = $50,000 + $75,000 (less than MC).
c. Requiring N = 8 reduces total net benefit relative to N = 6 in part b. The marginal
benefits of adopting the program at the seventh and eighth facilities are not worth the
marginal costs.
d. Without any regulatory intervention, RWE would enroll only 4 facilities in the health
and safety program (as in part a). An OSHA subsidy per facility would encourage
RWE to expand the safety program. The optimal subsidy is exactly equal to the
marginal social benefit generated by the program. Thus, the appropriate subsidy is
exactly $75,000 per facility. In response, RWE will extend the program to 6 facilities
as recommended in part b.
10. a. This rule makes sense when deciding whether or not undertake a single program. Note
that B/C > 1 is equivalent to B - C > 0.
b. The benefit-cost ratio does not reveal any information about the scale of either project.
Therefore, it cannot be used to decide between the two. For instance, suppose the
library’s total benefit is $1,500,000 and its total cost is $1,000,000. The garage’s total
benefit is $1,400,000 and its total cost is $700,000. The garage has the greater net
benefit and the better benefit-cost ratio. Now suppose that the garage were one-half as
large (with correspondingly lower benefits and costs). Its benefit-cost ratio would be
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unchanged (and still better than the library’s), but its net benefit would be only
$350,000, making it an inferior choice compared with the library.
c. This strategy makes great sense. The state should repair bridges where the benefit is
greatest per dollar spent, i.e., where the “bang is biggest for the buck.” In this way, the
state maximizes the total benefit it can achieve with its limited budget.
11. a. Sketching the demand curve, we find the price intercept to be $3.00 and the quantity
intercept to be 900 cars. At a rate of $1.50, 450 cars will park each hour, implying
revenue of $675 per hour. In turn, consumer surplus is (.5)($3 - $1.50)(450) = $337.50
per hour. At a rate of $1.00, 600 cars will park each hour, generating revenue of $600
per hour. Consumer surplus is (.5)($3 - $1)(600) = $600 per hour. The $1 rate generates
the greater total benefit, $1,200 per hour. The annual benefit is (2,600)($1,200) =
$3,120,000. Thus, the net benefit of the garage (in present-value terms) is
(11.9)(3,120,000 - 620,000) - 20,000,000 = $9,750,000.
b. The private developer would use the $1.50/hour rate because it offers the greater
revenue. The annual profit is (2,600)($675) - 620,000 = $1,135,000. The net present
value of the garage is (11.9)(1,135,000) - 20,000,000 = -$6,493,000. The garage is not
profitable.
12. a. The total benefits (B) for the programs (per $1 million spent) are
Program 1. B = (1.0)($4.8 million) + $0 = $4.8 million.
Program 2. B = (.2)($4.8 million) + $3.2 million = $4.16 million.
Program 3. B = (.5)($4.8 million) + $1.5 million = $3.9 million.
Program 4. B = (.75)($4.8 million) + $.2 million = $3.8 million.
Thus, program 1 should be funded up to its limit ($14 million), then program 2 (up to
$12 million), and then the remaining $6 million on program 3.
b. With $7.2 million as the value per life, the program benefits are now
Program 1. B = (1.0)($7.2 million) + $0 = $7.2 million.
Program 2. B = (.2)($7.2 million) + $3.2 million = $4.64 million.
Program 3. B = (.5)($7.2 million) + $1.5 million = $5.1 million.
Program 4. B = (.75)($7.2 million) + $.2 million = $5.6 million.
Again, program 1 should be funded up to its limit ($14 million), then program 4 (up
to $16 million), and the remaining $2 million on program 3. With a greater value for
each life, the programs saving the most lives are fully funded.
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Chapter 12
1. a. The expected values at points E, D, C, B, and A in the decision tree are $15.5, $50,
$30, $19.2, and $19.2.
b. The manager is confused. Point D is a point of decision: The manager simply should
select the top branch (50 is greater than 37). Thus, the value at point D is $50. Putting
probabilities on the branches makes no sense.
2. a. Whether the parents should accept or reject the $1.5 million settlement depends on their
assessment of the expected monetary award (net of legal fees) in court. The parents
would be wise to sketch a decision tree and get input from their lawyers about the
strength of their case, the chances of various court outcomes (including appeals), and
the possible monetary awards. Suppose that the couple (after averaging back the
"litigation tree") estimate an expected value of $2.1 million. If they are risk neutral,
they should reject the settlement. If they are sufficiently risk averse (a plausible
assumption), they would prefer to accept the certain $1.5 million settlement.
b. The defense lawyers should undertake a similar analysis on behalf of the hospital. (In
fact, there may be the involvement of a third party -- the insurance company obligated
to pay part of any malpractice award. This may alter the defendant’s calculations.) If
plaintiff and defendant assess similar decision trees, they will come up with similar
expected values concerning the court outcome. In this case, the sides should be able to
reach a mutually beneficial settlement. (Of course, if both sides are over-optimistic
about their court chances, a settlement may be doomed.)
3. a. The expected value of continuing with its current software strategy is:
(.2)(2) + (.5)(.5) + (.3)(-1) = $.35 million. The expected value of an “open strategy” is:
(.25)(1.5) + (.25)(1.1) + (.25)(.8) + (.25)(.6) = $1.0 million. Thus, the “open” strategy is
preferred.
b. The “open” strategy is less risky in the sense of having a narrower range of possible
outcomes. Managerial risk aversion would be an added reason to pursue this strategy.
4. According to the decision tree below, the consortium’s most profitable decision (on
average) is to redesign the aircraft.
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5. a. The tree lists the six possible outcomes (in thousands of dollars) and the expected value
of each chance circle. Overall expected profit is $1,500.
b. E (revenue) = (.2)(120,000) + (.3)(160,000) + (.5)(175,000) = $159,500. Expected cost
is: (.6)(150,000) + (.4)(170,000) = $158,000. Thus, the expected profit is $159,500 $158,000 = $1,500, the same result as in part a.
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6. a. The film would generate an expected loss so should not be launched.
b. The termination option is worth 4 – 0 = $4 million.
7. a. Let’s compute the expected costs (in $ billions) of the respective safety programs. For
the “standard” program, the expected cost is: .160 + (.01)(10) = $.26 billion. For the
“lax” program, the expected cost is: .040 + (.03)(10) = $.34 billion. For the “ultraconservative” program, the expected cost is: .240 + (.005)(10) = $.29 billion. A risk-neutral
BP would choose the standard program because it delivers the lowest expected cost.
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b. At a judged 2 percent disaster risk, the (apparent) expected cost of the “lax” policy is:
.040 + (.02)(10) = $.24 billion, making it appear to be the least-cost option. A judged
$5 billion liability would reduce the expected cost for all three options. The biggest
apparent reduction would be for the “lax” program, making it the winner at an expected
cost of $.19 billion.
8. The decision tree shows that the best strategy is to attempt a settlement, accept the
$400,000 settlement if this is offered, and otherwise go to court. (Note that the expected
cost of rejecting an offer, $650,000, is the expected value of going to court, plus the
$50,000 cost of trying to settle.) The optimal "try-to-settle" strategy has a $550,000
expected cost, which is less expensive than always settling or always going to court.
9. a. If the customer response is weak, McDonald’s expected profit is:
(.5)(20) + (.5)(-100) = -$40 million. McD’s overall profit (averaging over strong and
weak customer responses) is (.4)(50) + (.6)(-40) = -$4 million. The company should not
have launched the campaign.
b. If the customer response is weak, the company does better by “pulling the plug” − a
$20 million loss is better than an expected $40 million loss from continuing. The
overall expected profit from launching the campaign (and terminating it in the face of
a weak customer response) is (.4)(50) + (.6)(-20) = $8 million. Given the flexibility to
terminate, the company should launch the campaign.
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10. a. The firm should not undertake the software development program. To do so would
mean incurring an expected loss of $13 million.
B Succeeds
.3
A Succeeds
267.5
.4
Develop Software
B Fails
-13
$75
$350
.7
A Fails
-$200
$0
.6
b. The firm should go ahead if it learns that B has failed (expected profit of $20 million);
otherwise it should not invest. Its overall expected profit is: (.7)(20) = $14 million.
c. i) In the joint venture, the chance that both programs fail is (.6)(.7) = .42. Thus, according
to the decision tree, developing both is a losing proposition.
Success
$200
.58
-31
Failure
-$350
.42
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ii) Developing A alone is most profitable: (.4)(350) + (.6)(-200) = $20 million. The
expected profit from B alone is: (.3)(400) + (.7)(-150) = $15 million.
11. The expected utility of pursuing the biochemical approach alone is
E(UChem) = .7U(800) + .3U(400) = (.7)(64) + (.3)(44) = 58.
The accompanying decision tree depicts the strategy of trying the biogenetic approach
first and then pursuing the biochemical approach if necessary. The expected utility of this
strategy is 57.5. Thus, pursuing the biochemical approach alone has a slight edge over
sequential development. Since sequential development has the greater risk (i.e.,
dispersion of possible outcomes), a risk-averse firm chooses the biochemical approach.
12. a. From the given responses, we can conclude that u($500) = 50, u($1,120) = 75,
u($130) = 25, u($1,530) =87.5, u($780) = 62.5, u($280) = 37.5, and (as a second
check) u($500) = 50.
b. The function u = 2.24√y = 2.24y.5, closely tracks the utility values in part a. For
instance, if y = $500, then u = (2.24)(500.5) = 50.1. If y = $1,530, then we find:
u = (2.24)(1,530.5) = 87.6, and so on. In this sense, the function embodies top
management’s attitude toward risk. The firm is moderately risk-averse. For example,
its CE for a 50/50 gamble between $2 billion and $0 is $500 million – well below the
expected value of the risk ($1 billion). For risks involving less variable outcomes, the
gap between the firm’s CE and the expected value is smaller.
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*13. The dealer must commit to ordering and selling some number of yachts (say, Q) before
knowing the course of the economy. Recall that the two price equations are given by:
PG = 20 - .05Q, and PR = 20 - .1Q. Then, the expected price required to sell Q yachts is:
.6PG + .4PR = 20 - .07Q. Expected profit is simply expected revenue minus cost. This is
maximized by setting expected MR equal to MC ($10 thousand). Thus, MR = 20 - .14Q =
10, implying Q* = 71.4. So the optimal (round) number of yachts is Q = 71. This number
is closer to 50 than to 100. This should not be surprising since we found earlier that
ordering 50 was better than ordering 100. Here, we see that the optimal order size (one
that is better than any other quantity) is 71 yachts.
Chapter 13
1. As tough as it may be to do, you should ignore your friend’s story. His experience
represents a single data point. You already have gathered the best available information
on the relative merits of different models. You had a clear choice based on this
information; your friend’s singular experience should not be enough to change your
probabilities or your mind.
2. a. The assessment is subjective in the sense that different “experts” or “odds makers”
would come to different judgments about the team's World Series chances. Prior to the
season, one's assessment rests mainly on determining the strength of the team's hitting,
pitching, and defense relative to their division-, league-, and cross-league rivals.
b. Since there are 30 teams in the Major Leagues, a “naive” assessment would assign each
team an equal, one-in-30 chance of winning the Series. An avid sports fan would
modify this assessment by taking account of the factors in part (a). He would revise his
assessment as teams posted win-loss records over the season, and in light of team
performances, injuries, and so on.
3. a. The chance of a student responding is Pr(R|S) = .08/.24 = 1/3. The chance of a doctor
responding is Pr(R|D) = .05/.18 = .277. The chance of a lawyer responding is Pr(R|L)
= .09/.58 = .155. The promotion is most effective with students.
b. The table identifies the market segments being reached by the promotion. More
important, it measures the effectiveness of the promotion with respect to each
segment.
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4. a. No. The reason this age group accounts for the greatest proportion of accidents is
because they comprise half of all drivers. (Add across the table rows to see this.)
b. The accident rate for Age 17-30 is:
(12,050 + 1,822)/(90,243 + 12,050 + 18,22) = .133.
The accident rates for the other two age groups are .091 and .112. The middle age
group are the best drivers; the youngest age group are the worst.
c. The comparison is misleading because the older group drives many fewer total miles
than the middle group. A more even-handed evaluation would compare accident rates
per miles driven.
5. a. The following decision trees show the consortium’s expected profits from having
perfect information in each instance.
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b. According to Bayes’ Theorem,
Pr(E | S) Pr(S)
Pr(E | S) Pr(S) + Pr(E | F) Pr(F)
Pr(Success | Endorsement) =
= (.9)(.6)/[(.9)(.6) + (.5)(.4)]
= .54/.74 = .73.
Pr(Success | No Endorsement) =
Pr( N | S) Pr(S)
Pr( N | S) Pr(S) + Pr( N | F) Pr(F)
= (.1)(.6)/[(.1)(.6) + (.5)(.4)]
= .06/.26 = .23.
6. The chance is 1/3 that the grand prize is behind your chosen curtain. The fact
that you are shown an empty curtain does not change this prior probability
(although it does eliminate one curtain from consideration). Since your winning
chances are 1/3 if you “stick”, you should switch and gain a 2/3 winning chance.
7. a. Opening directly on Broadway implies an expected profit of:
(.3)(30) + (.5)(10) + (.2)(-50) = $4 million.
Though risky, the musical offers a positive return to investors.
b. The gross profit on average employing out-of-town tryouts is:
(.35)(24) + (.45)(12) + (.2)(0) = $13.8 million.
Accounting for the cost of the previews, the producers’ net profit is $6.8 million. The
preview route is not only more profitable on average. It also limits the downside loss to
$7 million, whereas a Broadway bomb would mean a loss of $50 million.
8. a. Based on the loan officer's assessments reported in the problem, we can construct the
following joint probability table:
Performing
Loan
Defaulted Loan
Total
A (“zero risk”)
.225
.020
.245
B (solid)
.270
.025
.295
C (uncertain)
.360
.045
.405
D (high risk)
.045
.010
.055
Total
.900
.100
1.000
Category
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Therefore:
Pr(Def|A) = .02/.245 = .08
Pr(Def|B) = .025/.295 = .085
Pr(Def|C) = .045/.405 = .11
Pr(Def|D) = .01/.055 = .18
b. The loan officers' assessments of default risk range between 8% and 18%. The scoring
system's assessments range between 5% and 25% and appear to be better at
discriminating between “sound” and “problem” loans. The scoring system provides
more valuable information.
9. a. The firm should not take the R&D program (expected profit = -$4 million).
b. The accompanying decision tree shows that the firm should undertake the R&D program
only if it learns it has exclusive rights ($5 million expected profit). If it must share the
market, investing would imply a $17.5 million expected loss, so in this case, a “no go”
decision ($0) is best. According to the decision tree, the firm’s overall expected profit is:
(.6)(5) + (.5)(0) = $3 million.
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c. According to the joint table, Pr(S|C) = .5/.8 = .625 and Pr(S|H) = 0/.2 = 0.
Prototype
Cool
Hot
Risky Outcomes
Success
Failure
.5
.3
0
.2
.5
.5
Total
.8
.2
d. If the prototype chip runs cool, the expected profit from pursuing the R&D investment
is (.625)(32) + (.375)(-40) = $5 million, so it is worth investing. If the chip runs hot,
the R&D program will fail with certainty, so the firm should walk away. Therefore, the
firm’s overall expected profit from testing the chip is: (.8)(5) + (.2)(0) = $4 million.
Testing makes sense because its expected value ($4 million) is greater than its cost
($2 million).
10. a. According to the decision tree, the government agency should not take the case to
court. It’s expected loss from doing so would be $.2 million.
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b. Here is the completed joint probability table.
Padding
Not
Total
Q
.15
.16
.31
C
.05
.64
.69
.20
.80
1.00
Therefore, Pr(P|Q) = .15/.31 = .484 and Pr(P|C) = .05/.69 = .07.
c. According to the accompanying decision tree, the agency should investigate and go to
court if Q but drop the case if C. Its expected benefit is $70 thousand.
11. a. According to the accompanying decision tree, a bid of $130,000 is the best choice. Its
expected profit is: (.5)(30,000) = $15,000.
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b. Here, the expected cost is $100,000, which is identical to the certain cost in part a.
Thus, there is no change in expected profit. The optimal bid is $130,000 as before.
c. As the decision tree shows, your company’s expected profit with perfect cost
information is $17,500. Thus, the EVI = 17,500 - 15,000 = $2,500.
12. Winning 9 of 10 contracts is not necessarily a sign of good bidding performance. The
main implication of winning this many contracts is that your bids are too low! You should
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probably raise your margins significantly even though this means winning somewhat
fewer contracts. The scarcity of available consultants to service these contracts is an
additional reason for raising prices (and cutting back the number of contracts). In any
case, your bid should include a provision for an opportunity cost -- embodied by the lost
profit that would have been earned if the needed consultants were free to work on other
contracts.
13. a. The bidder might be a much more efficient (i.e. low cost) producer. A more likely
explanation is that the winner (though no more efficient) is overly optimistic about
the actual cost of the job. In this case, the firm risks falling prey to the winner’s
curse, i.e. finding that its actual costs exceed its bid price.
b. The low bid is not all good news for the government. If its contract losses are too
large, the firm may default on the contract, leaving the government in the lurch. Or
the firm may simply lack the management capabilities to complete the contract
satisfactorily. In this sense, a procurement is different from an auction sale. In the
former, the parties have a continuing relationship until the work outlined in the
contract is performed and completed. In the latter, the relationship ends with the sale.
To guard against these ongoing contract risks, the government’s selection procedure
should scrutinize contractors’ capabilities and cost estimates, in addition to their
price bids. Incentive contracts allowing for cost sharing are also recommended.
Chapter 14
1. a. We know that Pr(L) = .04, Pr(R | L) = .5, and Pr(R|N) = 1/16, where L denotes lemon,
R denotes return, and N denotes normal car. The joint
table is
Lemon (L)
Normal (N)
Total
Return (R)
.02
.06
.08
Keep (K)
.02
.90
.92
Total
.04
.96
1.00
For instance, one computes the first row: Pr(L & R) = (.5)(.04) = .02, and Pr(N & R) =
(1/16)(.96) = .06. Thus, we find Pr(L | R) = .02/(.08) = .25. Of all cars returned, 25
percent are lemons. In turn, Pr(L | K) = .02/.92 = .021.
b. We see that the return policy screens out half of the lemons (a substantial benefit to
customers), but at the cost that about 6 percent of normal-quality cars will be returned
as well.
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2. a. Commissions can lead to both adverse selection and moral hazard problems. The Sears
compensation system may have attracted dishonest employees who saw a chance to
make money by defrauding customers. This is the adverse selection problem. Once
employed, all employees had the incentive to convince customers that they needed
repairs. This is the moral hazard problem. By removing commissions, some employees
could be expected to quit and others to change their behavior.
b. The management of Sears may not have much incentive to monitor its employees and
discover the abuses. After all, the employees were generating strong profits for the
company and thus there may have been an incentive to turn a blind eye.
c. The downside of changing the compensation structure is that employees may have an
incentive to shirk. Without the commission, fewer customers mean less work.
3. a. With equal chances of both types of workers, the firm offers a wage of $37,500 (equal
to the workers’ average productivity).
b. By attending college, HP workers can distinguish themselves from LP workers (i.e.,
signal their higher productivity). Consider an equilibrium in which workers with
college educations are paid $45,000, and all others are paid $30,000. By going to
college, HP workers increase their incomes by $15,000 per year or $75,000 over their
expected five-year job tenure. Since these added earnings exceed the cost of a college
education ($60,000), it pays HP workers to go to college. Not so for LP workers
whose college costs are $90,000. Thus, the signaling outcome is, indeed, an
equilibrium. However, if the average job stay is only three years, this signaling
equilibrium breaks down.
4. First, from the university’s point of view, it is probably relatively inexpensive to offer
these educational benefits. (The marginal cost to the system of accommodating additional
employee students may be quite low.) Second, the more subtle point is that the university
may be able to attract better employees by offering a fringe benefit that only ambitious,
highly-motivated workers would value.
5. If the bill is split five ways, each time a couple orders an extra menu item (say, an
expensive shrimp cocktail or a baked Alaska dessert), its share of the extra cost is only 20
percent. The other couples pay for 80 percent of the cost. Moral hazard occurs because
couples will tend to overindulge themselves in expensive items because they bear only a
fraction of the costs. The couple who mistakenly expects separate checks is in double
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jeopardy. By economizing, it forgoes a lavish meal, yet it pays for the others’
extravagance.
6. a. Potential advantages include” better coordination between the physician and the
radiologist performing the test, perhaps greater efficiencies, not to mention greater
patient convenience.
b. Here, there is a potential moral hazard problem; the urology practice gains extra
income from additional tests. This might cause urologists, consciously or
unconsciously, to order more tests. The result is overtesting, overtreatment, and
escalating costs.
7. a. Guarantied deliver is not efficient, because it forces Firm X to deliver even when its
cost of doing so is greater than Firm Y’s benefit ($100,000).
b. Setting the penalty at $50,000 is also NOT efficient. For instance, Firm Y would
default with a cost such as c = $70,000 (it’s cheaper to pay the penalty), even though
Firm X’s value is much higher ($100,000). Setting the penalty at exactly $100,000 is
efficient. This contingent contract acts like a “tax” set exactly equal to the harm nondelivery inflicts on Firm Y. So Firm X delivers if and only if its cost is less than the
benefit Firm Y stands to gain. Of course, the penalty serves to fully insure Firm Y.
8. The corporation favors the arrangement because it obtains a guaranteed price. (This price
is slightly below the price the underwriter will set for the shares on the market. The
difference is the underwriter’s profit.) The company also knows that the underwriter will
have a strong incentive to sell the securities. However, there is one obvious area of
conflict between the two parties. There will be hard bargaining over the guaranteed price.
Clearly, the underwriter will argue for a low price in order to be as certain as possible not
to be stuck with unsold securities. In addition, the underwriters will need cooperation
from the company in order to satisfy disclosure regulations. Typically the price is not set
until the morning of the offering.
9. a. Having only imperfect information, the winning bidder may have been overly
optimistic about the player’s “true” long-run ability. (For instance, the winning team
may not have known that the pitcher had a sore arm, a bad attitude, and so on.) The
winning bidder might ask itself, “If this pitcher is so great, why didn’t his original team
retain him?”
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b. If a ball player is guaranteed exorbitant sums for the duration of his contract, he may
have a reduced incentive to give a 100 percent effort on the field (and therefore
perform poorly). Obviously, his incentive increases in the last year of his contract if he
expects to become a free agent.
c. An owner should estimate what a player is worth based on the best available
information, and place a bid somewhat below this estimate in order to acquire the ball
player at a profit.
10. A seniority system avoids influence costs, that is, the maneuvering, negotiation, and
office politics that inevitably occur when one person has the power to dole out
perquisites. On the other hand, perquisites could be a way of motivating employees. By
using a seniority system the company forgoes these potentially powerful incentives.
11. Although team decision making can generate valuable information and promote problem
solving (five heads are better than one), it is also costly (enlisting additional human
resources) and time-consuming. In addition, team decision making may suffer from freerider problems, that is, team members may shirk and expect other members to pick up the
slack. For these reasons, it is important to limit the size of workable teams.
12. a. No, the plant manager will report a much lower output, so that his actual output will
greatly exceed the understated report, thereby claiming a large bonus payment for
himself.
b. Under the new bonus schedule, the manager will always strive for maximum output
(because the bonus depends positively on actual output) and will also have the
incentive to report the plant’s true capacity, QT = 10,000. To confirm this last result,
suppose that the manager knows that maximum output is 10,000. If this is his report
and his actual output matches the report (Q = QT = 10,000), then his bonus is .4QT =
$4,000. If he were to report a higher target (say QT = 12,000), his bonus would fall to:
(.4)(12,000) + .5(10,000 – 12,000) = $3,800,
using the second equation of the bonus payment. Alternatively, if he were to report a
lower target (say QT = 8,000), his bonus would fall to:
(.4)(8,000) + .3(10,000 – 8,000) = $3,800,
using the first equation of the bonus payment. Any overstatement or understatement
reduces his bonus; his incentive is to report truthfully.
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Chapter 15
1. a. The plaintiff’s expected court receipt (net of legal costs) is 50,000 - 15,000 = $35,000.
The defendant’s expected court payment (including legal costs) is 50,000 + 15,000 =
$65,000. The zone of agreement lies between these two amounts. If each side believes
its winning chances are 60 percent, then the plaintiff’s expected court receipt is
$45,000 and the defendant’s expected court payment is $55,000. The parties’
optimistic (and conflicting) opinions have reduced the zone of agreement.
b. When the potential damages are $200,000, the expected court outcomes of the
disputants become $105,000 and $95,000. Now there is no zone of agreement. The
plaintiff’s minimally acceptable settlement exceeds the defendant’s maximum
acceptable payment.
c. Facing a nuisance suit, the defendant knows it will win its court case but still faces an
expected cost equal to its legal fees. Thus, it rationally might settle out of court for
any amount smaller than this. For example, it might well settle a nuisance suit for
$5,000 if it knows that defending the suit will cost $10,000. The most immediate way
to deter nuisance suits is to make the losing party pay the other side’s legal (i.e., court)
costs.
2. a. When the firm’s profits are booming, management is vulnerable to a costly strike by
labor. Labor’s bargaining power is strengthened.
b. Again, labor is in a strong position when it has ample reserves in its strike fund.
c. With high unemployment, labor has more to lose from a prolonged strike.
Management’s bargaining power is strengthened.
3. Paying the developer 1 percent of the store’s first year’s revenue might be beneficial for
two reasons. First, if the parties are risk averse, this arrangement is one way to share the
risk of uncertain revenues. Second, the arrangement might depend on different
probability assessments of the parties. For instance, the store may be relatively
pessimistic (and the developer may be optimistic) about the volume of shoppers coming
to the new mall.
4. a. No. Achieving zero percent defects is very costly for firm S who will insist on an
accordingly high price.
b. The value-maximizing agreement calls for 4 percent defects. Here the players’ total
trading gain is: Vb - Cs = 72 - 37 = $35 thousand.
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5. a. Since the mill has the right to pollute, the fishery must pay it to clean up. With 50
percent cleanup, the benefit to the fishery is 100,000 - 30,000 = $70,000. The mill’s
cost is $50,000, so the total net benefit (relative to no cleanup) is $20,000. A 100
percent cleanup, however, costs more than it is worth: $120,000 > $100,000. Thus a 50
percent cleanup (at a price between $50,000 and $70,000) is mutually beneficial.
b. The same 50 percent reduction would be negotiated if the fishery held the legal right to
clean water. Moving from 100 percent cleanup to 50 percent cleanup costs the fishery
$30,000 in reduced profit, but saves the mill $50,000 in abatement costs. Since the total
net benefit from this change is positive ($20,000), the parties can benefit mutually from
the cleanup. Here, the mill will pay the fishery an amount between $30,000 and
$50,000. A further move to zero percent cleanup is not warranted. (The fishery’s
reduction in profit exceeds the mill’s cost saving.)
6. a. As a general rule, an outcome on an issue should be adopted if and only if doing so
increases the value of the contract to the parties together. The table below summarizes
these impacts (in $ millions). The UMW’s values, originally expressed in wage
equivalents, have been converted into total dollars (in millions).
UMW
Producers
Parties Together
Right-to-Strike
+30
-50
-20
Open up Job
-20
+60
+40
Impact of Both
Clauses
+10
+10
+20
Introducing the right-to-strike clause has a negative (-$20 million) impact on the parties
together. Since it reduces total value, it should not be introduced. Opening non-mine
jobs increases total value (management’s gain exceeds labor’s loss) and therefore
should he introduced. Thus, only the second clause should be part of an efficient
contract. Of course, with the introduction of the second clause, the UMW will insist
upon a wage concession from the producers. Any wage concession worth between $20
million and $40 million will generate a mutually beneficial improvement in the
contract terms.
b. Now it is impossible to introduce a single clause, since a compensating wage change is
ruled out. Besides the status quo, the only other option is to introduce both clauses. Is
this mutually beneficial? The table shows that each side gains $10 million, so the
answer is yes. Introducing both clauses is an example of negotiation quid pro quo.
However, notice that the total profit of the parties is now only $20 million (less than
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the $40 million of part a). Both parties would be better off if wages were unfrozen
allowing adoption of the job clause alone (along with appropriate wage compensation).
7. a. The eight possible agreements (and associated payoffs) are:
1. 95%, 3yr, w/o Bio.: 180, -140
2. 95%, 5yr, w/o Bio.: 100, -80
3. 80%, 3yr, w/o Bio.: 160, -90
4. 80%, 5yr, w/o Bio.: 60, -50
5. 95%, 3yr, w/Bio.: 150, -100
6. 95%, 5yr, w/Bio.: 70, -60
7. 80%, 3yr, w/Bio.: 130, -50
8. 80%, 5yr, w/Bio.: 30, -30
Only agreements 1, 3, 7, and 8 are efficient. Agreements 2, 4, and 6 are dominated by
agreement 7. Agreement 5 is dominated by agreement 3.
b. Agreement 7 is optimal since the parties’ total gains, (130 - 50), are maximized.
8. a. Company A’s expected value for company is (.5)(50) + (.5)(20) = $35 million.
Company T’s expected value for company is (.8)(40) + (.2)(30) = $38 million. Thus,
there is no zone of agreement, making a 100% cash transaction impossible.
b. A 100% stock transaction (at the right terms) works because it allows Company A to
pay a 50% depreciated price in the event of liability. Effectively, Company A pays a
contingent price. For instance, suppose Company A pays a price of $44 million in
stock (based on the current stock price). Therefore, Company A’s expected payment is
(.5)(44) + (.5)(22) = $33 million which is less than its $35 million value. In turn,
Company T’s expected receipt is (.8)(44) + (.2)(22) = $39.6 million which is more
than its $38 million value. The stock deal is mutually beneficial.
c. A buyback provision makes good sense because if Company T is liable, the firm will
be worth more under T’s management ($30 million) than under Company A’s ($20
million). The buyback is mutually beneficial. Including such a provision (at a
mutually beneficial price) upfront in the terms of the acquisition increases both sides’
values.
*9. The buyer’s expected profit is π b = (vb - P)F(P). The buyer determines the optimal price P
that maximizes this expression by setting marginal profit equal to zero. Thus, Mπ b = dπ b/dP
= (vb - P)dF(P)/dP – F(P) = 0. This can be rewritten as (vb - P)f(P) – F(P) = 0, where f(p) =
dF(P)/dP is the density function of F(P). Solving for P we confirm that P = vb – F(P)/f(P).
10. If firm A claims 55%, its expected profit is (.9)($110,000) = $99,000. If it claims 60%,
its expected profit is (.85)($120,000) = $102,000. If it claims 65%, its expected share is
(.8)($130,000) = $104,000. Thus, a 65% claim is optimal.
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*11. The value of the target under current management ranges between $60 and $80 per share,
with an expected value of $70 (since all values are equally likely). What if firm A offered
a price of $70? Current management accepts this price when vT is between $60 and $70.
(Obviously, if vT > 70, firm T will not sell.) Thus, when its offer is accepted, the
acquisition value to firm A ranges between $60 and $75. (Remember that vA = 1.5vT 30.) This means that firm A’s expected acquisition value is $67.5. On average, it obtains
a company worth less than the price it pays! The trick is to realize that companies that
accept its offer are likely to be low-value companies. One can check that firm A cannot
earn a positive profit at any price between $60 and $80. No deal is possible.
12. a. Presumably, Icahn has the first move (whether to spend $50 million to launch a
dissident campaign). Such a campaign costs AOL $200 million, so the zone of
agreement has AOL paying an amount of greenmail somewhere between these two
values. The best prediction is that Icahn will initiate an opposition movement and
AOL will pay greenmail. Things change if AOL must pay all “large” holders the same
stock price premium as Icahn receives. This will likely raise AOL’s greenmail cost to
an amount exceeding $200 million. Therefore, AOL will have no incentive to pay
greenmail and in turn Icahn will have no incentive to mount an opposition (unless he
does so for other motives).
b. The efficient outcome is an AOL-Microsoft partnership (with total value $500 - $50 =
$450 million) which exceeds the value of the current AOL-Google deal ($250 + $70 =
$320).
c. The firms’ different views about the source of value can definitely impede a deal. For
instance, it is perfectly possible that each side thinks that it is responsible for 70% of
the $250 total value and, therefore, deserves such a share. (Of course, 70% and 70% do
not fit into 100%, so there could well be a stalemate.) The way around this problem is
to structure the deal so that the parties’ shares depend on Microsoft’s “measured
performance.” Thus, Microsoft’s dollar share must be based on both the number of site
visits and the number of searches (the two variables about which the firms have very
different predictions).
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Chapter 16
1. a. Increasing or decreasing returns to scale implies that either the objective function or
some constraint is nonlinear. Thus, the LP formulation cannot be used.
b. The LP method can handle any number of decision variables. The earlier problem of
producing a maximum level of output contained more variables (3) than constraints (2).
c. A downward-sloping demand curve implies a nonlinear revenue function. (The revenue
function is linear only if the demand curve is horizontal, i.e., the price is constant.)
Thus, the LP formulation cannot be used.
d. Here, the constraints are Q1/Q2 ≥ .4 and Q1/Q2 ≤ .6. These can be rewritten as Q1 - .4Q2
≥ 0 and Q1 - .6Q2 ≤ 0, respectively. Since these are both linear, the LP formulation
applies.
2. a. This maximization problem cannot be solved since the feasible region is unbounded
above (x and y can be made indefinitely large).
b. This is not an LP problem since the objective function (xy) is nonlinear.
c. This is an LP problem. The solution is x = 1.5, y = .5.
d. This maximization problem has no solution since the inequalities are contradictory.
There is no feasible region.
e. The second inequality (though it looks strange) can be rearranged to fit the LP
requirements. The expression x/(x + y) ≤ .7 can be rewritten x ≤ .7(x + y), or
.3x - .7y ≤ 0. The LP solution is y = 2, x = 0.
3. a. The slope of the objective function (-10/15) lies between the slopes of the two
constraints (-2/5 and -6/3). Therefore, the optimal solution has both constraints
binding: 2x + 5y = 40 and 6x + 3y = 48. The solution is x = 5 and y = 6. The value of
the objective function is 140.
b. The slope of the objective function (-.75) lies outside the slopes of the two constraints
(-1/.5 and -1/1). Therefore, the optimal solution has y = 0 and only the second
constraint is binding: x + y = 16. Thus, x = 16 and the minimum value of the objective
function is 12.
4. Let the initial resource constraints be:
1A + 2B ≤ X
2A + 2B ≤ Y
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and assume that both are binding in the optimal solution. Note that the LP solution is:
A = Y - X.
The innovation means that producing A units requires fewer units of good X. For
instance, the new constraint might be: .5A + 2B ≤ X, so that 50% fewer units of input X
are needed to produce good A. The new LP solution is: A = (Y - X)/1.5. The result of the
innovation is that the optimal output of A has fallen!
5. a. The formulation is
Minimize:
.1M + .15C
Subject to:
2M + 2C ≥ 50 (calcium)
2M + 6C ≥ 90 (protein)
6M + 2C ≥ 66 (calories),
where M and C are the nonnegative quantities of milk and cereal. A graph shows that
the lowest contour touches the feasible region at the corner formed by the protein and
calcium constraints. (The slopes of these constraints are -1/3 and -1, respectively; the
slope of the typical cost contour is -.1/.15 = -2/3.) Solving 2M + 2C = 50 and 2M + 6C
= 90, we find C = 10 and M = 15. The minimum cost of a healthy diet is $3.
b. If we increase the calcium requirement by a small amount (say, by 4 units to 54), the
new solution becomes C = 9 and M = 18. The cost of meeting this higher health
requirement is $3.15. Therefore, the shadow price of an extra unit of calcium is .15/4 =
$.0375.
6. a. Let x and y be the numbers of tires produced via processes 1 and 2 respectively. The
LP formulation is:
Maximize:
π = 4x + 6y
subject to:
x + y ≤ 10
(capital)
4x + 2y ≤ 32
(labor)
2x + 4y ≤ 32
(raw materials)
After graphing the feasible region and comparing slopes, we find the first and third
constraints to be binding. Solving the two equations, x + y = 10 and 2x + 4y = 32, we
find: x = 4, y = 6, and π = $52.
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b. Since the labor constraint is not binding, labor's shadow price is zero. If we raise the
supply of raw materials to 33, the new solution becomes x = 3.5, y = 6.5, and π = $53.
Therefore, the shadow price of materials is: ∆π/∆M = (53-52)/(33-32) = $1.
7. a. The formulation is
Maximize:
4B + 6T
Subject to:
5B + 5T ≥ 3.5
.4B + 4T ≥ 1.5
.4B + 4T ≤ 2.5
B + T = 1.0
Since bonds have better returns, the investor would like to make T as large as possible.
Clearly, the first two constraints never are binding. However, the last two constraints
do bind the proportion of bonds. Solving .4B + 4T = 2.5 and B + T = 1, we find B =
.417 and T = .583. The expected return of this portfolio is 5.17 percent.
b. The formulation is
Maximize:
4B + 6T + 4.4C + 5.6M + 8J
Subject to:
5B + 5T + 3.5C + 3M + 1J ≥ 3.5
B + T + C + M + J = 1.0.
Notice that Treasury bonds dominate (are more profitable and safer) than Treasury
bills, corporate bonds, and municipal bonds. Eliminating these three securities reduces
the binding constraints to 5T + J = 3.5 and T + J = 1. The solution is T = .625 and J =
.375. The portfolio’s expected return is 6.75 percent.
c. If risk is not an issue, the manager should invest 100 percent of the portfolio in junk
bonds (J = 1), earning a maximum rate of return and just meeting the maturity constraint.
8. Let T and M be the numbers of TV and magazine ads respectively. The LP formulation is:
Minimize:
Cost = 120,000T + 40,000M
subject to:
10,000T + 5,000M ≥ 600,000
5,000T + 1,000M ≥ 150,000
250T + 500M ≥ 30,000
(total)
(prime)
(sweepstakes)
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Note that this is a minimization problem involving “greater to or equal” constraints.
After graphing the feasible region and comparing slopes, we find the first and second
constraints to be binding. Therefore, the solution is T = 10, M = 100, and the producer's
minimum advertising cost is $5,200,000.
9. a. Let x1 and x2 denote the levels of the two processes. At a unit level, process 1 produces 2
units of H and 1 unit of P for a total contribution of ($2)(2) + ($1)(1) = $5. The
contribution of process 2 is: ($2)(2) + ($1)(4) = $8 at the unit level. Thus, the LP
formulation is
Maximize:
Subject to:
5x1 + 8x2
x1 + 2x2 ≤ 110
2x1 + 2x2 ≤ 160.
In the graphic solution, both constraints are binding. The optimal solution is x1 = 50
and x2 = 30. Total contribution is $490.
b. Let the supply of labor increase to 120. The new solution is x1 = 40 and x2 = 40, and
total contribution increases to $520. Labor’s shadow price is 30/10 = $3.
c. If the contribution of plywood rises to $3, the new objective function is to maximize 7x1
+ 16x2. The slope of the objective function (-7/16) no longer lies between the slopes of
the input constraints (-1/2 and -1). Therefore, only the labor constraint is binding and
the firm only uses the second process (i.e., x1 = 0). Solving the binding labor constraint,
we have x2 = 55. The firm’s maximum contribution is (16)(55) = $880.
10. a. The LP formulation is
Minimize:
6H + 5S + 8F + 20R
Subject to:
100H + 70S + 20F + 50R ≥ 70
100H + 70S + 40F + 70R ≥ 80,
H + S + F + R = 1,
where H, S, F, and R (are all non-negative) and denote the proportion of weekly meals
in the respective meal categories.
b. The minimum cost weekly meal plan calls for the proportions, H = .333 and S = .667,
implying an average cost per meal of $5.33. The analyst should avoid expensive and
unhealthy fast-food and restaurant meals.
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c. Now the LP formulation is
Maximize:
3H + 8S + 12F + 28R
Subject to:
6H + 5S + 8F + 20R ≤ 10
100H + 70S + 20F + 50R ≥ 70
100H + 70S + 40F + 70R ≥ 80,
H + S + F + R = 1.
The maximum value meal plan (with an average value of $12.56) calls for the
proportions, H = .33, S = .36, F = 0, and R = .31. The analyst should avoid all fast
food and split his meals roughly equally between the other categories. (Note that
high-value restaurant meals are now represented.) If his average meal budget
increases to $15, the new value-maximizing plan is: H = .4 and R = .6, implying an
average value of $18.
11. a. The LP formulation is
Minimize:
Subject to:
425L + 300M + 200D
2L + 2M + 2D ≥ 12
5,200L + 2,520M + 1,224D ≥ 20,000
where L, M, and D (are all nonnegative integers) and denote the number of
roundtrips to Los Angeles, Miami, and Durham, respectively. Using a spread-sheet
optimizer, one finds the solution, L = 3, M = 1, and D = 2. The total cost of these
six round trips (comprising 20,568 total miles and 12 segments) is $1,975.
b. If the challenge is to fly 25,000 miles, the best solution is: L = 4, M = 2, and D = 0.
The total cost of these trips (covering 25,840 miles) increases to $2,300. Finally, if the
requirement is 20,000 miles and only 10 segments, the optimal solution is: L = 3, M =
2, and D = 0 (20,640 miles flown) at a cost of $1,875.
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