Problem sets for Microeconomics II [110051-0471]

Problem sets for Microeconomics II
[110051-0471]
Spring Semester 2013
Lecture: prof. Jacek Prokop Mon, 13:30-15:10, G - Auditorium B
Seminars: mgr Krzysztof Makowski Mon, 17:10-18:50, G - 105
Office hours: Mon, 16:00-17:00, A - 314
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1 Optimal decisions using marginal analysis
Exercise 1.1. A television station is considering the sale of promotional videos. It
can have the videos produced by one of two suppliers. Supplier A will charge the
station a setup fee of $1,200 plus $2 for each cassette; supplier B has no setup fee
and will charge $4 per cassette. The station estimates its demand for the cassettes
to be given by Q = 1, 600 − 200P, where P is the price in dollars and Q is the
number of cassettes.
a. Suppose the station plans to give away the videos. How many cassettes should
it order From which supplier
b. Suppose instead that the station seeks to maximize its profit from sales of the
cassettes. What price should it charge How many cassettes should it order
from which supplier
Exercise 1.2. As the exclusive carrier on a local air route, a regional airline must
determine the number of flights it will provide per week and the fare it will
charge. Taking into account operating and fuel costs, airport charges, and so
on, the estimated cost per flight is $2,000. It expects to fly full flights (100 passengers),
so its marginal cost on a per passenger basis is $20. Finally, the airline’s
estimated demand curve is P = 120 − .1Q, where P is the fare in dollars and Q is
the number of passengers per week.
a. What is the airline’s profit-maximizing fare How many passengers does it
carry per week, using how many flights What is weekly profit
b. Suppose the airline is offered $4,000 per week to haul freight along the route
for a local firm. This will mean replacing on of the weekly passenger flights
with a freight flight (at same operating cost). Should the airline carry freight
for the local firm Explain.
Exercise 1.3. Under the terms of the current contractual agreement, Burger Queen
(BQ) is entitled to 20 percent of the revenue earned by each of its franchises. BQ’s
best-selling is the Slopper (it slops out of the bun). BQ supplies the ingredients
for the Slopper (bun, mystery meat, etc.) at cost to the franchise. The franchisee’s
average cost per Slopper (including ingredients, labor cost, and so on) is $.80. At
a particular franchise restaurant, weekly demand for Sloppers is given by P =
3.00 − Q/800.
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a. If BQ sets the price and weekly sales quantity of Sloppers, what quantity and
price should it set How much does BQ receive What is the franchisee’s
net profit
b. Suppose the franchise owner sets the price and sales quantity. What price and
quantity will the owner set How does the total profit earned by the two
parties compare to their total profit in part a
c. Now suppose BQ and an individual franchise owner enter into an agreement
in which BQ is entitled to a share of the franchisee’s profit. Will profit
sharing remove the conflict between BQ and the franchise operator Under
profit sharing, what will be the price and quantity of Sloppers (Does
the exact split of the profit affect your answer Explain briefly.) What is the
resulting total profit
d. Profit sharing is not widely practiced in the franchise business. What are the
disadvantages relative to revenue sharing
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2 Demand analysis and optimal pricing
Exercise 2.1. A private-garage owner has identified two distinct market segments:
short-term parkers and all-day parkers with respective demand curves of P S =
3 − Q S /200 and P C = 2 − Q C /200. Here P is the average hourly rate and Q is the
number of cars parked at this price. The garage owner is considering charging
different prices (on a per-hour basis) for short-term parking and all-day parking.
The capacity of the garage is 600 cars, and the cost associated with adding extra
cars in the garage (up to this limit) is negligible.
a. Given this facts, what is the owner’s appropriate objective How can he ensure
that members of each market segment effectively pay a different hourly
price
b. What price should he charge for each type of parker How many of each type
of parker will use the garage at these prices Will the garage be full
c. Answer the questions in part b assuming the garage capacity is 400 cars.
Exercise 2.2. A golf-course operator must decide what greens fees (prices) to set
on rounds of golf. Daily demand during week is: P D = 36 − Q D /10 where Q D
is the number of 18-hole rounds and P D is the price per round. Daily demand
on the weekend is P W = 50 − Q W /12. As a practical matter, the capacity of the
course is 240 rounds per day. Wear and tear on the golf course is negligible.
a. Can the operator profit by charging different prices during the week and on
the weekend Explain briefly. What greens fees should the operator set on
weekdays and how many rounds will be played On the weekend
b. When weekend prices skyrocket, some weekend golfers choose to play during
the week instead. The greater the difference between weekday and weekend
prices, the greater are the number of these “defectors”. How might this
factor affect the operator’s pricing policy Give a qualitative answer.
Exercise 2.3. Nearby College Student Recruiting Office has hired a consultant in
an effort to boost enrollment. After detailed surveys are conducted, the consultant
reports that elasticities are as follows: E P = −1.6, E Y − .8, E X
Competition College tuition). Current tuition at Nearby is $4,000.
= .25 (for
a. If tuition at Nearby is set to rise by 7%, income is expected to rise by 1%,
and Competition has announced a tuition increase of 6%, how much will
Nearby’s enrollment change
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. The chief financial officer approaches the Recruiting officer, and asks for a
copy of the report. He says that he would like to see a tuition change because
the school is facing a budget crunch. If the marginal cost of a student
is $2,750, what should tuition be in order to maximize profit
Exercise 2.4. Jonathan Livingstone Yuppie is a prosperous lawyer. Jonathan consumes
three goods, unblended Scotch whiskey ($20 per bottle), designer tennis
shoes ($80 per pair), and meals in French gourmet restaurants ($50 per meal).
After he has paid his taxes and alimony, Jonathan has $400 a week to spend.
a. Write down a budget equation.
b. Draw a three-dimensional diagram to show his budget set.
c. Suppose that he determines that he will buy one pair of designer tennis shoes
per week.
What equation must be satisfied by the combinations of the
restaurant meals and whiskey that he could afford
Exercise 2.5. Nancy Lerner is trying to decide how to allocate her time in studying
for her economics course. There are two examinations in this course. Her overall
score for the course will be minimum of her scores on the two examinations. She
has decided to devote a total of 1,200 minutes to studying for these two exams,
and she wants to get as high overall score as possible. She knows that on the first
examination if she doesn’t study at all, she will get a zero score on it. For every 10
minutes that she spends studying for the first examination, she will increase her
score by one point. if she doesn’t study at all for the second examination, she will
get a zero score on it. For every 20 minutes she spends studying for the second
examination, she will increase her score by one point.
Determine the optimal allocation of the time spent on studying for the examinations.
Show it on the graph.
Exercise 2.6. The utility maximizing consumer has his preferences over goods
x 1 and x 2 described by Cobb-Douglas utility function u(x 1 , x 2 ) = x a 1 x1−a 2
, where
a = 1/3.
a. Find the marginal rate of substitution of good x 1 with good x 2 .
b. Suppose that consumer’s disposable income is m = 20, the price of good x 1 is
p 1 = 5, and the price of good x 2 is p 2 = 4. What is the amount of good x 2
that consumer is willing to consume (Both goods are infinitely divisible).
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3 Production and cost analysis
Exercise 3.1. Suppose that production function of the steelworks is given by Q =
5LK, where L is the number of workers, and K is the amount of capital used to
give Q units of steel daily. Steelwork can hire labor at $10 per unit, and the cost
of capital is $20 per unit.
What is the optimal mix of inputs to produce 40 units of steel
Exercise 3.2. The owner of the “Magic car wash” describes the relation between
number of cars washed and labor input as follows: Q = −.8 + 4.5L − .3L 2 , where
Q is the number of cars washed per hour, and L is the number of employees. For
each car washed the owner gets $5, and he pays $4.5 per hour to his employees.
a. How many persons should the owner employ to maximize profit
b. What is the profit per hour
c. Is the above labor to cars washed relation true for all L Explain.
Exercise 3.3. Firm Z is developing a new product. An early introduction (beating
rivals to market) would greatly enhance the company’s revenues. However, the
intensive development effort needed to expedite the introduction can be very
expensive. Suppose total revenues and costs associated with the new product’s
introduction are given by R = 720 − 8t and C = 600 − 20t + .25t 2 , where t is the
introduction date (in months from now).
a. Some executives have argued for an expedited introduction date 12 months
from now (t = 12). Do you agree
b. How about instant introduction (t = 0) What introduction date is most profitable
Explain.
Exercise 3.4. In a particular region, there are two lakes rich in fish. The quantity
of fish caught in each lake depends on the number of persons who fish in each,
according to Q 1 = 10N 1 − .1N1 2 and Q 2 = 16N 2 − .4N2 2, where N 1 and N 2 denote
the number of fishers at each lake. In all, there are 40 fishers.
a. Suppose N 1 = 16 and N 2 = 24. At which lake is the average catch per fisher
greater In light of this fact, how would you expect the fishers to redeploy
themselves
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. How many fishers will settle at each lake
c. The commissioner of fisheries seeks a division of fisheries that will maximize
the total catch at the two lakes. What division of the 40 fisheries would you
recommend
Exercise 3.5. Let Q = L α K β . Suppose the firm seeks to produce a given output
while minimizing its total input cost: C = P L L + P K K.
Show that the optimal quantities of labor and capital satisfy L/K = (α/β) (P K /P L ).
Provide an intuitive explanation for this result.
Exercise 3.6. Henry is serving lemonade on the corner of the busy street of Philadelphia.
His production function is f (x 1 , x 2 ) = x 1/3
1
x 1/3
2
, where product is measured
in liters, x 1 is the amount of lemons in kilograms, and x 2 is the number of
hours spent on squeezing lemons.
a. What are the returns to scale of the above production function
b. If w 1 is the price of one kilogram of lemons, and w 2 is the hourly wage of
lemon-squeezer, the optimal working method is to work for ............ hours
per one kilogram of lemons.
c. Given that Henry produced y liters of lemonade, he used ............ kilograms of
lemons and worked for ............ hours.
Exercise 3.7. The production function is f (x 1 , x 2 ) = min {x 1 , x 2 }.
a. Suppose x 1 < x 2 . The marginal product of x 1 is ..........., and with small increase
in the amount of x 1 (increases, decreases, remains constant) ........... .
The
marginal product of x 2 is ..........., and with small increase in the amount of
x 2 (increases, decreases, remains constant) ........... .
substitution between x 2 and x 1 is ............ .
(increasing, constant, decreasing) ............ returns to scale.
The technical rate of
This technology demonstrates
b. Let x 1 = x 2 . What is the marginal product of x 1 with small increase in the
amount of x 1 What is the marginal product of x 1 with small increase in
the amount of x 2 The marginal product of x 1 with small increase in the
amount of x 2 (increases, decreases, remains constant) ............ .
Exercise 3.8. A manufacturing firm decided to build a factory. Choice is to be
made between two types. Type A is characterized by following short-term production
costs: C A = 80 + 2Q A + .5Q 2 A ; while type B: C B = 50 + Q 2 B .
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a. If eight units of the product are to be manufactured, which type of the factory
should be chosen How many units need to be produced to justify building
type A factory
b. Suppose the manufacturing company owns two factories (one of each type).
If the total production is big enough it is optimal to maintain production in
both factories. Explain why. Let us assume that planned production is 22
units. How the production should be divided between factories to minimize
cost of production
Exercise 3.9. A multinational firm produces steel at a home facility and at a foreign
subsidiary. The demand for the product is different at home and abroad.
Furthermore, the production costs are different at both facilities. The home demand
is P H = 260 − .1Q H , and foreign is P F = 240. The respective cost curves
are: C H = 1, 000 + .4Q 2 H , and C F = 5, 000 + .25Q 2 F .
a. Due to taxes or import quotas your products are not shipped overseas. Find
firm’s profit-maximizing outputs, sales quantities and prices for both home
and foreign market.
b. Suppose that the fiscal constraints are lifted. (Shipment cost is negligible).
Answer the questions in a. once again.
c. Suppose the shipment cost is $16 per ton. Answer the questions in a. once
again.
Exercise 3.10. A firm produces according to the following production function:
Q = K 1/4 L 3/4 , where Q - units of output, K - units of capital, and L - units of
labor. Suppose that the price of K is $4 per unit, and the price of L is $6 per unit.
What is the optimal capital/labor ratio
Exercise 3.11. Ranger construction is preparing to repair potholes, under contract
to the local county road repair agency. Based on past experience, Ranger has
found that output can be described by: Q = K 1/2 L 1/2 , where Q - pot holes filled,
K - units of capital, and L - units of labor. Ranger can hire labor at $12 per unit,
and the cost of capital is $8 per unit. Capacity limitations require that Ranger
accept no more than $96,000 worth of filling this season.
a. What is the optimal mix of inputs for Ranger
b. How many potholes should Ranger agree to fill
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Exercise 3.12. Flo is a university administrator. She has found that happy plants
only need fertilizer and talk. Where f is the number of bags of fertilizer used
and t is the number of hours she talk to her plants, the number of happy plants
produced is exactly h = t + 2 f . Suppose fertilizer costs w f per bag and talk costs
w t per hour.
a. If w t < w f /2, would it be cheaper for Flo to use fertilizer or talk to raise one
happy plant
b. Flo’s cost function is c ( w f , w t , h ) =............ .
c. Flo’s factor demand for talk is t ( w f , w t , h ) = ............ .
Exercise 3.13. Henry is a wealthy entrepreneur, who owns a tropical juice factory.
Its production function is given by F (K, L) = K 2/3 L 1/4 , where K and L are
respectively capital and labor inputs. The price of capital is r, and the cost of labor
is w.
a. Show the production function on the graph (use isoquants).
b. What returns to scale demonstrates the technology used
c. Henry is minimizing costs in his factory and he wants to produce Y units of
juice. Find Henry’s demand for capital
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4 Decision making under uncertainty and value of
information
Exercise 4.1. In 1976, the parents of a 7-year-old boy sued a New York hospital for
$3.5 million. The boy was blinded shortly after he was born two weeks prematurely.
His parents claimed that hospital doctors administered excessive oxygen
to the baby and that caused the blindness. The case went to trial, and just as the
jury announced it hat reached the verdict, the lawyers for the two sides arrived
at an out-of-court settlement of $500,000.
a. If you were the parents, how would you decide whether to accept the settlement
or wait for the jury’s decision What probability assessments would
you need to make Would you have accepted the settlement
b. Answer the questions in part a, taking the hospital’s point of view.
Exercise 4.2. A European consortium has spent a considerable amount of time
and money developing a new supersonic aircraft. The aircraft gets high marks on
all performance measures except noise. In fact, because of the noise, the consortium’s
management is concerned that the U.S. government may impose restrictions
at some of the American airports where the aircraft can land. Management
judges a 50-50 chance that there will be some restrictions. Without restrictions,
management estimates its (present discounted) profit at $125 million; with restrictions,
its profit would be only $25 million. Management must decide now,
before knowing the government’s decision, whether to redesign parts of the aircraft
to solve the noise problem. The cost of the redesign program is $25 million.
There is a .6 chance that the redesign program will solve the noise problem (in
which case, full landing rights are a certainty) and a .4 chance it will fail.
a. Using decision tree , determine the consortium’s best course of action, assuming
management is risk neutral.
b. Find the expected value of perfect information about the redesign program.
Calculate separately the expected value of perfect information about the
U.S. government’s decision.
c. Suppose the management of the consortium questions its engineers about the
success or failure of the redesign program prior to committing to it. Management
recognizes that its engineers are likely to be biased in favor of the
program. It judges that if the program truly will succeed, the engineers will
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endorse it 90 percent of the time, but even if the program will fail, they will
endorse it 50 percent of the time. What is the likelihood of success in light
of an endorsement What if the engineers do not endorse the program
Exercise 4.3. Firm A is facing a possible lawsuit by legal firm B. Firm B represents
the family of Mr. Smith, who was killed in a motel fire (allegedly caused by faulty
wiring). Firm A was the builder of the motel. Firm A has asked its legal team
to estimate the likely jury award it will be ordered to pay in court. Expert legal
counsel anticipates three possible court outcomes: awards of $1,000,000, $600,000,
or $0, with probabilities .2, .5, and .3, respectively. In addition to any awards,
firm A’s legal expenses associated with fighting the court case are estimated to be
$100,000. Firm A also has considered the alternative of entering out-of-court settlement
negotiations with firm B. Based on the assessments of its lawyers, firm A
envisions the other side holding out for one of two settlement amounts: $900,000
(a high amount) or $400,000 (a more reasonable amount). Each demand is considered
equally likely. If presented with one of these settlement demands, firm A
is free to accept it (in which case firm B agrees to waive any future right to sue)
or reject it and take its chances in court. The legal cost of pursuing a settlement
(whether or not one is reached) is $50,000.
Determine the settlement or litigation strategy that minimizes firm A’s expected
total cost (any payment plus legal fees).
Exercise 4.4. Consider the following simplified version of the television game
show Let’s Make a Deal. There is a grand prize behind one of three curtains;
the other two curtains are empty. As a contestant, you get to choose a curtain
at random. Let’s say you choose curtain 3. Before revealing what’s behind the
curtain, the game show host always offers to show you what one of the other
curtains contains. She shows you that curtain 2 is empty; in fact, she always
shows you an empty curtain. (You know that’s how the game works; so does the
audience and everybody else.)
Now you must decide: Do you stick with your original choice, curtain 3, or
switch to curtain 1 Which action gives you the better chance of finding the grand
prize
Exercise 4.5. Filene’s Basement, a Boston-based department store, has a policy of
marking down the price of sale items each week that they go unsold. You covet
an expensive brand of winter coat that is on sale for $100. In fact, you would be
willing to pay as much as $120 for it. Thus, you can buy it now (for a profit of
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$120 - $100 = $20) or wait until next week, when the price will be reduced to $75
if the coat is still available. The chances of its being available next week are 2/3.
If it is available in week 2, you can buy or wait until week 3. There is a 1/2 chance
it will be sold between weeks 2 and 3 and a a 1/2 chance it will be available at a
reduced price of $60. Finally, if it is available in week 3, you can buy or wait until
week 4. There is a 1/4 chance it still will be available, at a price of $50 (and a 3/4
chance it will be sold in the meantime). Week 4 is your last chance to buy before
the coat is withdrawn.
a. How long should you wait before buying Illustrate via a decision tree.
b. Filene’s has 120 of these winter coats for sale. What is expected total revenue
from the pricing scheme in part a
c. Alternatively, Filene’s can set a single price for all coats. Its demand curve
is P = 180 − Q. Would it prefer a common-price method or the pricereduction
method in part b Explain.
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5 Perfect competition and monopoly
Exercise 5.1. In a competitive market, the industry demand and supply curves
are P = 200 − .2Q D and P = 100 + .3Q S , respectively.
a. Find the market’s equilibrium price and output.
b. Suppose the government imposes a tax of $20 per unit of output on all firms
in the industry. What effect does this have on the industry supply curve
Find the new competitive price and output. What portion of the tax has
been passed on to consumers via a higher price
c. Suppose a $20-per-unit sales tax is imposed on consumers. What effect does
this have on the industry demand curve Find the new competitive price
and output. Compare this answer to your findings in part b.
Exercise 5.2. Firm Z, operating in a perfectly competitive market, can sell as much
or as little as it wants of a good at a price of $16 per unit. Its total cost function
is TC = 50 + 4Q + 2Q 2 . The associated marginal cost is MC = 4 + 4Q and the
point of minimum average cost is Q min = 5.
a. Determine the firm’s profit-maximizing level of output. Compute its profit.
b. The industry demand curve is Q = 200 − 5P. What is the total market demand
at the current $16 price If all firms in the industry have cost structures
identical to that of firm Z, how many firms will supply the market
c. The outcomes in part a and b cannot persist in the long run. Explain why. Find
the market’s price, total output per firm in the long run.
d. Comparing the short-run and long-run results, explain changes in the price
and in the number of firms.
Exercise 5.3. In a competitive market, the industry demand and supply curves
are P = 70 − Q D and P = 40 + 2Q S .
a. Find the market’s equilibrium price and output.
b. Suppose that the government provides a subsidy to producers of $15 per unit
of the good. Since the subsidy reduces each supplier’s marginal cost by
15, the new supply curve is P = 25 + 2Q S . Find the market’s new equilibrium
price and output. Provide an explanation for the change in price and
quantity.
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c. A public interest group supports the subsidy, arguing that it helps consumers
and producers alike. Economists oppose the subsidy, declaring that it leads
to an inefficient level of output. In your opinion, which side is correct
Explain carefully.
Exercise 5.4. Until recently, the market for air travel within Europe was highly
regulated. Entry of new airlines was severely restricted, and air fares were set
by regulation. Partly as a result, European air fares were and continue to be
higher than U.S. fares for routes of comparable distance. Suppose that, for a given
European air route (say, London to Rome), annual air travel demand is estimated
to be Q = 1, 500 − 3P, where Q is the number of trips in thousands and P is
the one-way fare in dollars. In addition, the long-run average (one-way) cost per
passenger along this route is estimated to be $200.
a. Some economists have suggested that during the 1980s and 1990s there was
an implicit cartel among European air carriers whereby the airlines charged
monopoly fares under the shield of regulation. Given the preceding facts,
find the profit-maximizing fare and the annual number of passenger trips.
b. In the last five years, deregulation has been the norm in the European market,
and this has spurred new entry and competition from discount air carriers
such as Ryanair and easyJet. Find the price and quantity for the European
air rout if perfect competition became the norm.
Exercise 5.5. Demand for microprocessors is given by P = 35 − 5Q, where Q is
the quantity of microchips (in millions). The typical firm’s total cost of producing
a chip is C i = 5q i , where q i is the output of firm i.
a. Under perfect competition, what are the equilibrium price and quantity
b. Under perfect competition, find total industry profit and consumer surplus.
c. Suppose that one company acquires all the suppliers in the industry and thereby
creates a monopoly. What are the monopolist’s profit-maximizing price and
total output
d. Compute the monopolist’s profit and the total consumer surplus of purchasers.
e. Discuss changes between your answers in parts b. and d.
Exercise 5.6. Suppose that, over the short run (say, the next five years), demand
for OPEC oil is given by Q = 52.5 − 1.25P. (Here Q is measured in millions of
barrels per day.) OPEC’s marginal cost per barrel is $10.
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a. What is OPEC’s optimal level of production What is the prevailing price of
oil at this level
b. Many experts contend that maximizing short-run profit is counterproductive
for OPEC in the long run because high prices induce buyers to conserve
energy and seek supplies elsewhere. Suppose the demand curve just described
will remain unchanged only if oil prices stabilize at $20 per barrel or
below. If oil price exceeds this threshold, long-run demand (over a second
five-year period) will be curtailed to Q = 60 − 2P. OPEC seeks to maximize
its total profit over the next decade. What is its optimal output and price
policy (Assume all values are present values.)
Exercise 5.7. Firm 1 is a member of a monopolistically competitive market. Its
total cost function is C = 900 + 60Q 1 + 9Q 2 1
. The demand curve for the firm’s
differentiated product is given by P = 660 − 16Q 1 .
a. Determine the firm’s profit maximizing output, price and profit.
b. Attracted by potential profits, new firms enter the market. A typical firm’s
demand curve (say firm’s 1) is given by P = [1, 224–16(Q 2 + Q 3 + ... +
Q N )–16Q 1 ], where N is the total number of firms.
(If competitors’ outputs
or numbers increase, firm 1’s demand curve shifts inward.) The longrun
equilibrium under monopolistic competition is claimed to consist of 10
firms, each producing 6 units at a price of $264. Is this claim correct
c. Based on the cost function given, what would be the outcome if the market
were perfectly competitive (Presume market demand is P = 1, 224 − 16Q,
where Q is total output.) Compare this outcome to the outcome in part b.
Exercise 5.8. A single buyer who wields monopoly power in its purchase of an
item is called a monopsonist. Suppose that a large firm is the sole buyer of parts
from 10 small suppliers. The cost of a typical supplier is given by C = 20 + 4Q +
Q 2 .
a. Suppose that the large firm sets the market price at some level P. Each supplier
acts competitively (i.e., sets output to maximize profit given P). What is the
supply curve of the typical supplier Of the industry
b. The monopsonist values the part at $10. This is the firm’s break-even price, but
intends to offer a price much less than this and purchase all parts offered. If
it sets price P, its profit is simply: π = (10 − P)Q S , where Q S is the industry
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supply curve found in part a. (Of course, Q S is a function of P.) Write down
the profit expression and maximize profit with respect to P. Find the firm’s
optimal price. Give a brief explanation for this price.
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6 Oligopolistic competition
Exercise 6.1. The OPEC cartel is trying to determine the total amount of oil to sell
on the world market. It estimates world demand for oil to be Q W = 60 − .5P,
Q W denotes the quantity of oil (in millions of barrels per day) and P is price per
barrel. The cartel’s marginal cost is approximately $20 per barrel.
a. Determine OPEC’s profit-maximizing output and price.
b. OPEC’s economists recognize the importance of non-OPEC oil supplies. These
can be described by the estimated supply curve Q S = 1.5P − 20. Write
OPEC’s net demand curve. Find its optimal output and price. What portion
of the world’s total oil supply comes from OPEC
Exercise 6.2. Firm A is the dominant firm in a market where industry demand
is given by Q D = 48 − 4P. There are four “follower” firms, each with long-run
marginal cost given by MC = 6 + Q F . Firm A’s long run marginal cost is 6.
a. Write the expression for the total supply curve of the followers (Q S ) as this
depends on price.
b. Find the net demand curve facing firm A. Determine A’s optimal price and
output. How much output do the other firms supply in total
Exercise 6.3. In the small town there are two confectioneries: one of the Mr Long
and one of the Mr Short. Nobody can tell the cakes from those confectioneries
apart. The marginal costs of Mr Long are constant - $1 per cake. Mr Short has
constant marginal costs of $2 per cake. There are no fixed costs. The inverse
demand function is p(q) = 6–.01q , where q is the sum of the cakes sold by both
confectionery owners.
a. Find the reaction functions, quantities and profits in the equilibrium, assuming
that confectionery owners competition is of Cournot type.
b. Find the quantities and profits in the equilibrium, assuming that confectionery
owners competition is of Bertrand type.
c. Assume that Mr Long is a leader of Stackelberg type competition (Probably
due to fact, that he starts working an hour earlier than Mr Short.) Find the
quantities and profits in the equilibrium.
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7 Game theory
Exercise 7.1. Firms M and N compete for a market and must independently decide
how much to advertise. Each can spend either $10 million or $20 million on
advertising. If the firms spend equal amounts, they split the $120 million market
equally. (For instance, if both choose to spend $20 million, each firm’s net profit
is 60 – 20 = $40 million.) If one firm spends $20 million and other $10 million, the
former claims two thirds of the market and the latter one-third.
a. If the firms act independently, what advertising level should each choose Explain.
Is a prisoner’s dilemma present
b. Could the firms profit by entering into an industry-wide agreement concerning
the extent of advertising Explain.
Exercise 7.2. A firm sells two goods in a market consisting of three types of consumers.
The table shows the values consumers place on the goods. The unit cost
of producing each good is $5.
Good
Consumers X Y
A 13 6
B 7 11
C 15 2
Find the optimal prices for (a) selling the goods separately, (b) pure bundling,
and (c) mixed bundling. Which pricing strategy is most profitable
Exercise 7.3. Find all Nash equilibriums in the game of two banks, whose management
considers selling to debtors their liabilities for the discounted price.
Bank II
S
NS
Bank I S 6,6 6,11
NS 11,6 6,6
Exercise 7.4. There are three payoff tables given.
a. Identify the equilibrium outcome(s) in each of the three payoff tables.
I. C 1 C 2
R 1 12,10 10,4
R 2 4,8 9,6
II. C 1 C 2
R 1 12,10 4,4
R 2 4,4 9,6
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III. C 1 C 2
R 1 12,10 4,4
R 2 4,-100 9,6

. In each table, predict the exact outcome that will occur and explain your reasoning
c. In table III, suppose the column player is worried that the row player might
choose R 2 (perhaps a one-in-ten chance). Given the risk, how should the
column player act Anticipating the column player’s thinking, how should
the row player act
Exercise 7.5. Two firms dominate the market for surgical sutures and compete
aggressively with respect to research and development. The following payoff
table depicts the profit implications of their different R&D strategies.
a. Suppose that no communication is possible between the firms; each must choose
its R&D strategy independently of the other. What actions will the firms
take, and what is the outcome
b. If the firms can communicate before setting their R&D strategies, what outcome
will occur Explain.
Firm B’s R&D Spending
Low Medium High
Firm A’s R&D Spending Low 8,11 6,12 4,14
Medium 12,9 8,10 6,8
High 11,6 10,8 4,6
Exercise 7.6. One way to lower the rate of auto accidents is strict enforcement of
motor vehicle laws (speeding, drunk driving, and so on). However, maximum
enforcement is very costly. The following payoff table lists payoffs of a typical
motorist and a town government. The motorist can obey or disobey motor vehicle
laws, which the town can enforce or not.
Town
Enforce Don’t Enforce
Motorist Obey 0,-15 0,0
Don’t Obey -20,-20 5,-10
a. What is the town’s optimal strategy What is the typical motorist’s behavior
in response
b. What if the town could commit to a strict enforcement policy and motorists
believed that this policy would be used Would the town wish to do so
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c. Now suppose the town could commit to enforcing the law part of the time.
(The typical motorist cannot predict exactly when the town’s traffic police
will be monitoring the roadways.) What is the town’s optimal degree (i.e.,
percentage) of enforcement Explain.
Exercise 7.7. In the following game, players A and B alternate moves. At each
turn, a player can terminate the game or pass move to the next player. By passing,
the player increases the rival’s potential payoff by five units and reduces her own
by one unit. Thus, as long as both players pass the move on to one another, their
payoffs increase. If the game terminates immediately - the payoff is (2,2). Player
A starts the game.
a. Draw a game tree for this extensive form game.
b. Suppose you are paired with another student with whom you will play the
game. Just based on your judgment (no analysis), how would you play
c. Now analyze the game tree by working backward. What actions should the
players take What is the outcome Briefly explain this result.
d. The experiments with the above described game show that many of the players
A decides to pass the move. How could you explain this result
Exercise 7.8. Over the last decade, the Delta Shuttle and the U.S. Air Shuttle have
battled for market share on the Boston – New York and Washington, DC – New
York routes. In addition to service quality and dependability (claimed or real),
the airlines compete on price via periodic fare changes. The hypothetical payoff
table lists each airline’s estimated profit (expressed on a per-seat basis) for various
combinations of one-way fares.
U.S. Air Shuttle
$139 $119 $99
Delta Shuttle $139 34,38 15,42 6,32
$119 42,20 22,22 10,25
$99 35,7 27,9 18,16
a. Suppose that the two airlines select their fares independently and “once and
for all.” (The airlines’ fares cannot be changed.) What fares should the airlines
set
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. Suppose, instead, that the airlines will set fares over the next 18 months. In
any month, each airline is free to change its fare if it wishes. What pattern
of fares would you predict for the airlines over the 18 months
c. Pair yourself with another student form the class. The two of you will play
the roles of Delta and U.S. Air and set prices for the next 18 months. You
will exchange written prices for each month. You then can determine your
profit (and your partner’s profit) from the payoff table. The competition
continues in this way for 18 months, after which time you should compute
your total profit (the sum of your monthly payoffs). Summarize the results
of your competition. What lessons can you draw from it
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