The profit-maximizing output of a monopolist is found by solving for the
output at which marginal revenue is equal to marginal cost. Having solved
for this output, you find the monopolist’s price by plugging the profitmaximizing
output into the demand function. In general, the marginal
revenue function can be found by taking the derivative of the total revenue
function with respect to the quantity. But in the special case of linear
demand, it is easy to find the marginal revenue curve graphically. With
a linear inverse demand curve, p(y) = a − by, the marginal revenue curve
always takes the form MR(y) = a − 2by.
24.1 (0) Professor Bong has just written the first textbook in Punk
Economics. It is called Up Your Isoquant. Market research suggests that
the demand curve for this book will be Q = 2, 000 − 100P, where P is
its price. It will cost $1,000 to set the book in type. This setup cost is
necessary before any copies can be printed. In addition to the setup cost,
there is a marginal cost of $4 per book for every book printed.
(a) The total revenue function for Professor Bong’s book is R(Q) =
.
(b) The total cost function for producing Professor Bong’s book is C(Q) =
.
(c) The marginal revenue function is MR(Q) = and the
marginal cost function is MC(Q) = The profit-maximizing
quantity of books for professor Bong to sell is Q∗
= .
24.2 (0) Peter Morgan sells pigeon pies from a pushcart in Central Park.
Morgan is the only supplier of this delicacy in Central Park. His costs are
zero due to the abundant supplies of raw materials available in the park.
(a) When he first started his business, the inverse demand curve for pigeon
pies was p(y) = 100 − y, where the price is measured in cents and y
measures the number of pies sold. Use black ink to plot this curve in
the graph below. On the same graph, use red ink to plot the marginal
revenue curve.
Cents
100
75
50
25
0 25 50 75 100 125 150
Pigeon pies
(b) What level of output will maximize Peter’s profits? What
price will Peter charge per pie? .
(c) After Peter had been in business for several months, he noticed that
the demand curve had shifted to p(y) = 75 − y/2. Use blue ink to plot
this curve in the graph above. Plot the new marginal revenue curve on
the same graph with black ink.
(d) What is his profit-maximizing output at this new price?
What is the new profit-maximizing price? .
24.3 (0) Suppose that the demand function for Japanese cars in the
United States is such that annual sales of cars (in thousands of cars) will
be 250−2P, where P is the price of Japanese cars in thousands of dollars.
(a) If the supply schedule is horizontal at a price of $5,000 what will
be the equilibrium number of Japanese cars sold in the United States?
thousand. How much money will Americans spend in total on
Japanese cars? billion dollars.
(b) Suppose that in response to pressure from American car manufacturers,
the United States imposes an import duty on Japanese cars in such a
way that for every car exported to the United States the Japanese manufacturers
must pay a tax to the U.S. government of $2,000. How many
Japanese automobiles will now be sold in the United States?
thousand. At what price will they be sold? thousand dollars.
(c) How much revenue will the U.S. government collect with this tariff?
million dollars.
(d) On the graph below, the price paid by American consumers is measured
on the vertical axis. Use blue ink to show the demand and supply
schedules before the import duty is imposed. After the import duty is
imposed, the supply schedule shifts and the demand schedule stays as
before. Use red ink to draw the new supply schedule.
Price (thousands)
8
6
4
2
0 50 100 150 200 250 300
Japanese autos (thousands)
(e) Suppose that instead of imposing an import duty, the U.S. government
persuades the Japanese government to impose “voluntary export restrictions”
on their exports of cars to the United States. Suppose that the
Japanese agree to restrain their exports by requiring that every car exported
to the United States must have an export license. Suppose further
that the Japanese government agrees to issue only 236,000 export licenses
and sells these licenses to the Japanese firms. If the Japanese firms know
the American demand curve and if they know that only 236,000 Japanese
cars will be sold in America, what price will they be able to charge in
America for their cars? thousand dollars.
(f) How much will a Japanese firm be willing to pay the Japanese government
for an export license? thousand dollars. (Hint: Think
about what it costs to produce a car and how much it can be sold for if
you have an export license.)
(g) How much will be the Japanese government’s total revenue from the
sale of export licenses? million dollars.
(h) How much money will Americans spend on Japanese cars?
billion dollars.
(i) Why might the Japanese “voluntarily” submit to export controls?
.
24.4 (0) A monopolist has an inverse demand curve given by p(y) =
12 − y and a cost curve given by c(y) = y2
.
(a) What will be its profit-maximizing level of output? .
(b) Suppose the government decides to put a tax on this monopolist so
that for each unit it sells it has to pay the government $2. What will be
its output under this form of taxation? .
(c) Suppose now that the government puts a lump sum tax of $10 on the
profits of the monopolist. What will be its output? .
24.5 (1) In Gomorrah, New Jersey, there is only one newspaper, the
Daily Calumny. The demand for the paper depends on the price and the
amount of scandal reported. The demand function is Q = 15S1/2
P−3
,
where Q is the number of issues sold per day, S is the number of column
inches of scandal reported in the paper, and P is the price. Scandals
are not a scarce commodity in Gomorrah. However, it takes resources to
write, edit, and print stories of scandal. The cost of reporting S units
of scandal is $10S. These costs are independent of the number of papers
sold. In addition it costs money to print and deliver the paper. These
cost $.10 per copy and the cost per unit is independent of the amount
of scandal reported in the paper. Therefore the total cost of printing Q
copies of the paper with S column inches of scandal is $10S + .10Q.
(a) Calculate the price elasticity of demand for the Daily Calumny.
Does the price elasticity depend on the amount of scandal
reported? Is the price elasticity constant over all prices?
.
(b) Remember that MR = P(1 + 1
ǫ ). To maximize profits, the Daily
Calumny will set marginal revenue equal to marginal cost. Solve for the
profit-maximizing price for the Calumny to charge per newspaper.
When the newspaper charges this price, the difference between
the price and the marginal cost of printing and delivering each newspaper
is .
(c) If the Daily Calumny charges the profit-maximizing price and prints
100 column inches of scandal, how many copies would it sell? (Round
to the nearest integer.) Write a general expression for the
number of copies sold as a function of S: Q(S) =
.
(d) Assuming that the paper charges the profit-maximizing price, write
an expression for profits as a function of Q and S.
Using the solution for Q(S) that you found in the last section,
substitute Q(S) for Q to write an expression for profits as a function of
S alone. .
(e) If the Daily Calumny charges its profit-maximizing price, and prints
the profit-maximizing amount of scandal, how many column inches of
scandal should it print? How many copies are sold
and what is the amount of profit for the Daily Calumny if it
maximizes its profits? .
24.6 (0) In the graph below, use black ink to draw the inverse demand
curve, p1(y) = 200 − y.
(a) If the monopolist has zero costs, where on this curve will it choose to
operate? .
(b) Now draw another demand curve that passes through the profitmaximizing
point and is flatter than the original demand curve. Use
a red pen to mark the part of this new demand curve on which the monopolist
would choose to operate. (Hint: Remember the idea of revealed
preference?)
(c) The monopolist would have (larger, smaller) profits at the new demand
curve than it had at the original demand curve. .
Price
200
150
100
50
0 50 100 150 200
Quantity
Problems in this chapter explore the possibilities of price discrimination
by monopolists. There are also problems related to spatial markets, where
transportation costs are accounted for and we show that lessons learned
about spatial models give us a useful way of thinking about competition
under product differentiation in economics and in politics.
Remember that a price discriminator wants the marginal revenue in
each market to be equal to the marginal cost of production. Since he
produces all of his output in one place, his marginal cost of production
is the same for both markets and depends on his total output. The trick
for solving these problems is to write marginal revenue in each market as
a function of quantity sold in that market and to write marginal cost as
a function of the sum of quantities sold in the two markets. The profitmaximizing
conditions then become two equations that you can solve
for the two unknown quantities sold in the two markets. Of course, if
marginal cost is constant, your job is even easier, since all you have to do
is find the quantities in each market for which marginal revenue equals
the constant marginal cost.
A monopolist sells in two markets. The inverse demand curve in market
1 is p1 = 200−q1. The inverse demand curve in market 2 is p2 = 300−q2.
The firm’s total cost function is C(q1 + q2) = (q1 + q2)2
. The firm is able
to price discriminate between the two markets. Let us find the prices
that it will charge in each market. In market 1, the firm’s marginal
revenue is 200 − 2q1. In market 2, marginal revenue is 300 − 2q2. The
firm’s marginal costs are 2(q1 +q2). To maximize its profits, the firm sets
marginal revenue in each market equal to marginal cost. This gives us the
two equations 200 − 2q1 = 2(q1 + q2) and 300 − 2q2 = 2(q1 + q2). Solving
these two equations in two unknowns for q1 and q2, we find q1 = 16.67
and q2 = 66.67. We can find the price charged in each market by plugging
these quantities into the demand functions. The price charged in market
1 will be 183.33. The price charged in market 2 will be 233.33.
25.1 (0) Ferdinand Sludge has just written a disgusting new book, Orgy
in the Piggery. His publisher, Graw McSwill, estimates that the demand
for this book in the United States is Q1 = 50, 000 − 2, 000P1, where
P1 is the price in the U.S. measured in U.S. dollars. The demand for
Sludge’s opus in England is Q2 = 10, 000 − 500P2, where P2 is its price
in England measured in U. S. dollars. His publisher has a cost function
C(Q) = $50, 000 + $2Q, where Q is the total number of copies of Orgy
that it produces.
(a) If McSwill must charge the same price in both countries, how many
copies should it sell? What price should it charge to maximize
its profits How much will those profits be? .
(b) If McSwill can charge a different price in each country and wants to
maximize profits, how many copies should it sell in the United States?
What price should it charge in the United States?
How many copies should it sell in England? What
price should it charge in England? How much will its total
profits be? .
25.2 (0) A monopoly faces an inverse demand curve, p(y) = 100 − 2y,
and has constant marginal costs of 20.
(a) What is its profit-maximizing level of output?
(b) What is its profit-maximizing price? .
(c) What is the socially optimal price for this firm? .
(d) What is the socially optimal level of output for this firm? .
(e) What is the deadweight loss due to the monopolistic behavior of this
firm? .
(f) Suppose this monopolist could operate as a perfectly discriminating
monopolist and sell each unit of output at the highest price it would fetch.
The deadweight loss in this case would be .
25.3 (1) Banana Computer Company sells Banana computers in both
the domestic and foreign markets. Because of differences in the power
supplies, a Banana purchased in one market cannot be used in the other
market. The demand and marginal revenue curves associated with the
two markets are as follows:
Pd = 20, 000 − 20Q Pf = 25, 000 − 50Q
MRd = 20, 000 − 40Q MRf = 25, 000 − 100Q.
Banana’s production process exhibits constant returns to scale and it
takes $1,000,000 to produce 100 computers.
(a) Banana’s long-run average cost function is AC(Q) = and
its long-run marginal cost function is MC(Q) = (Hint:
If there are constant returns to scale, does long-run average cost change
as output changes?) Draw the average and marginal cost curves on the
graph.
(b) Draw the demand curve for the domestic market in black ink and
the marginal revenue curve for the domestic market in pencil. Draw the
demand curve for the foreign market in red ink and the marginal revenue
curve for the foreign market in blue ink.
Dollars (1,000s)
60
50
40
30
20
10
0 100 200 300 400 500 600 700 800
Banana Computers
(c) If Banana is maximizing its profits, it will sell computers in
the domestic market at dollars each and computers
in the foreign market at dollars each. What are Banana’s
total profits? .
(d) At the profit-maximizing price and quantity, what is the price elasticity
of demand in the domestic market? What is the price
elasticity of demand in the foreign market? Is demand more
or less elastic in the market where the higher price is charged?
.
(e) Suppose that somebody figures out a wiring trick that allows a Banana
computer built for either market to be costlessly converted to work in the
other. (Ignore transportation costs.) On the graph below, draw the new
inverse demand curve (with blue ink) and marginal revenue curve (with
black ink) facing Banana.
Dollars (1,000s)
40
30
20
10
0 100 200 300 400 500 600 700 800
Banana computers
(f) Given that costs haven’t changed, how many Banana computers
should Banana sell? What price will it charge?
How will Banana’s profits change now that it can no longer
practice price discrimination? .
25.4 (0) A monopolist has a cost function given by c(y) = y2
and faces
a demand curve given by P(y) = 120 − y.
(a) What is its profit-maximizing level of output? What price
will the monopolist charge? .
(b) If you put a lump sum tax of $100 on this monopolist, what would its
output be? .
(c) If you wanted to choose a price ceiling for this monopolist so as to
maximize consumer plus producer surplus, what price ceiling should you
choose? .
(d) How much output will the monopolist produce at this price ceiling?
.
(e) Suppose that you put a specific tax on the monopolist of $20 per unit
output. What would its profit-maximizing level of output be? .
25.5 (1) The Grand Theater is a movie house in a medium-sized college
town. This theater shows unusual films and treats early-arriving movie
goers to live organ music and Bugs Bunny cartoons. If the theater is
open, the owners have to pay a fixed nightly amount of $500 for films,
ushers, and so on, regardless of how many people come to the movie.
For simplicity, assume that if the theater is closed, its costs are zero. The
nightly demand for Grand Theater movies by students is QS = 220−40PS,
where QS is the number of movie tickets demanded by students at price
PS. The nightly demand for nonstudent moviegoers is QN = 140−20PN .
(a) If the Grand Theater charges a single price, PT , to everybody, then
at prices between 0 and $5.50, the aggregate demand function for movie
tickets is QT (PT ) = Over this range of prices, the inverse
demand function is then PT (QT ) = .
(b) What is the profit-maximizing number of tickets for the Grand Theater
to sell if it charges one price to everybody? At what
price would this number of tickets be sold? How much profits
would the Grand make? How many tickets would be sold to
students? To nonstudents? .
(c) Suppose that the cashier can accurately separate the students from
the nonstudents at the door by making students show their school ID
cards. Students cannot resell their tickets and nonstudents do not have
access to student ID cards. Then the Grand can increase its profits by
charging students and nonstudents different prices. What price will be
charged to students? How many student tickets will be sold?
What price will be charged to nonstudents? How
many nonstudent tickets will be sold? How much profit will the
Grand Theater make? .
(d) If you know calculus, see if you can do this part. Suppose that the
Grand Theater can hold only 150 people and that the manager wants
to maximize profits by charging separate prices to students and to nonstudents.
If the capacity of the theater is 150 seats and QS tickets are
sold to students, what is the maximum number of tickets that can be
sold to nonstudents? QN = Write an expression for the
price of nonstudent tickets as a function of the number of student tickets
sold. (Hint: First find the inverse nonstudent demand function.)
Write an expression for Grand Theater profits
as a function of the number QS only. (Hint: Make substitutions using
your previous answers.)
How many student tickets should the Grand
sell to maximize profits? What price is charged to students?
How many nonstudent tickets are sold? What price
is charged to nonstudents? How much profit does the Grand
make under this arrangement? .
25.6 (2) The Mall Street Journal is considering offering a new service
which will send news articles to readers by email. Their market research
indicates that there are two types of potential users, impecunious students
and high-level executives. Let x be the number of articles that
a user requests per year. The executives have an inverse demand function
PE(x) = 100 − x and the students have an inverse demand function
PU (x) = 80 − x. (Prices are measured in cents.) The Journal has a zero
marginal cost of sending articles via email. Draw these demand functions
in the graph below and label them.
Price in cents
120
100
80
60
40
20
0 20 40 60 80 100 120
Quantity
(a) Suppose that the Journal can identify which users are students and
which are executives. It offers each type of user a different all or nothing
deal. A student can either buy access to 80 articles per year or to none
at all. What is the maximum price a student will be willing to pay for
access to 80 articles?
(Hint: Recall the lesson on consumer’s surplus and the
area under the demand curve.) An executive can either buy access to
100 articles per year or to none at all. What is the maximum price an
executive would be willing to pay for access to 100 articles?
.
(b) Suppose that the Journal can’t tell which users are executives and
which are undergraduates. Thus it can’t be sure that executives wouldn’t
buy the student package if they found it to be a better deal for them.
In this case, the Journal can still offer two packages, but it will have to
let the users self-select the one that is optimal for them. Suppose that it
offers two packages: one that allows up to 80 articles per year the other
that allows up to 100 articles per year. What’s the highest price that the
undergraduates will pay for the 80-article subscription? .
(c) What is the total value to the executives of reading 80 articles per
year? (Hint: Look at the area under their demand curve and
to the right of a vertical line at 80 articles.)
(d) What is the the maximum price that the Journal can charge for 100
articles per year if it wants executives to prefer this deal to buying 80
articles a year at the highest price the undergraduates are willing to pay
for 80 articles? .
(e) Suppose that the Mall Street Journal decides to include only 60 articles
in the student package. What is the most it could charge and still
get student to buy this package? .
(f) If the Mall Street Journal offers a “student package” of 60 articles
at this price, how much net consumer surplus would executives get from
buying the student package? .
(g) What is the most that the Mall Street Journal could charge for a 100article
package and expect executives to buy this package rather than the
student package? .
(h) If the number of executives in the population equals the number of
students, would the Mall Street Journal make higher profits by offering a
student package of 80 articles or a student package of 60 articles? .
25.7 (2) Bill Barriers, CEO of MightySoft software, is contemplating
a new marketing strategy: bundling their best-selling wordprocessor and
their spreadsheet together and selling the pair of software products for
one price.
From the viewpoint of the company, bundling software and selling it
at a discounted price has two effects on sales: (1) revenues go up due to
to additional sales of the bundle; and (2) revenues go down since there is
less of a demand for the individual components of the bundle.
The profitability of bundling depends on which of these two effects
dominates. Suppose that MightySoft sells the wordprocessor for $200 and
the spreadsheet for $250. A marketing survey of 100 people who purchased
either of these packages in the last year turned up the following facts:
1) 20 people bought both.
2) 40 people bought only the wordprocessor. They would be willing to
spend up to $120 more for the spreadsheet.
3) 40 people bought only the spreadsheet. They would be willing to
spend up to $100 more for the wordprocessor.
In answering the following questions you may assume the following:
1) New purchasers of MightySoft products will have the same characteristics
as this group.
2) There is a zero marginal cost to producing extra copies of either
software package.
3) There is a zero marginal cost to creating a bundle.
(a) Let us assume that MightySoft also offers the products separately
as well as bundled. In order to determine how to price the bundle, Bill
Barriers asks himself the following questions. In order to sell the bundle
to the wordprocessor purchasers, the price would have to be less than
.
(b) In order to sell the bundle to the spreadsheet users, the price would
have to be less than .
(c) What would MightySoft’s profits be on a group of 100 users if
it priced the bundle at $320?
.
(d) What would MightySoft’s profits be on a group of 100 users if it priced
the bundle at $350?
.
(e) If MightySoft offers the bundle, what price should it set?
.
(f) What would profits be without offering the bundle?
.
(g) What would be the profits with the bundle? .
(h) Is it more profitable to bundle or not bundle? .
(i) Suppose that MightySoft worries about the reliability of their market
survey and decides that they believe that without bundling t of the 100
people will buy both products, and (100−t)/2 will buy the wordprocessor
only and (100 − t)/2 will buy the spreadsheet only. Calculate profits as a
function of t if there is no bundling. .
(j) What are profits with the bundle? .
(k) At what values of t would it be unprofitable to offer the bundle?
.
(l) This analysis so far has been concerned only with customers who
would purchase at least one of the programs at the original set of
prices. Is there any additional source of demand for the bundle? What
does this say about the calculations we have made about the profitability
of bundling?
.
25.8 (0) Colonel Tom Barker is about to open his newest amusement
park, Elvis World. Elvis World features a number of exciting attractions:
you can ride the rapids in the Blue Suede Chutes, climb the Jailhouse
Rock and eat dinner in the Heartburn Hotel. Colonel Tom figures that
Elvis World will attract 1,000 people per day, and each person will take
x = 50 − 50p rides, where p is the price of a ride. Everyone who visits
Elvis World is pretty much the same and negative rides are not allowed.
The marginal cost of a ride is essentially zero.
(a) What is each person’s inverse demand function for rides?
.
(b) If Colonel Tom sets the price to maximize profit, how many rides will
be taken per day by a typical visitor? .
(c) What will the price of a ride be? .
(d) What will Colonel Tom’s profits be per person? .
(e) What is the Pareto efficient price of a ride? .
(f) If Colonel Tom charged the Pareto efficient price for a ride, how many
rides would be purchased? .
(g) How much consumers’ surplus would be generated at this price and
quantity? .
(h) If Colonel Tom decided to use a two-part tariff, he would set an
admission fee of and charge a price per ride of .
25.9 (1) The city of String Valley is squeezed between two mountains
and is 36 miles long, running from north to south, and only about 1
block wide. Within the town, the population has a uniform density of
100 people per mile. Because of the rocky terrain, nobody lives outside
the city limits on either the north or the south edge of town. Because of
strict zoning regulations, the city has only three bowling alleys. One of
these is located at the city limits on the north edge of town, one of them is
located at the city limits on the south edge of town, and one is located at
the exact center of town. Travel costs including time and gasoline are $1
per mile. All of the citizens of the town have the same preferences. They
are willing to bowl once a week if the cost of bowling including travel
costs and the price charged by the bowling alley does not exceed $15.
(a) Consider one of the bowling alleys at either edge of town. If it charges
$10 for a night of bowling, how far will a citizen of String Valley be willing
to travel to bowl there? How many customers would
this bowling alley have per week if it charged $10 per night of bowling?
.
(b) Write a formula for the number of customers that a bowling alley at
the edge of town will have if it charges $p per night of bowling.
.
(c) Write a formula for this bowling alley’s inverse demand function.
.
(d) Suppose that the bowling alleys at the end of town have a marginal
cost of $3 per customer and set their prices to maximize profits. (For
the time being assume that these bowling alleys face no competition from
the other bowling alleys in town.) How many customers will they have?
What price will they charge? How far away from the
edge of town does their most distant customer live? .
(e) Now consider the bowling alley in the center of town. If it charges a
price of $p, how many customers will it have per week? .
(f) If the bowling alley in the center of town also has marginal costs
of $3 per customer and maximizes its profits, what price will it charge?
How many customers will it have per week? How
far away from the center of town will its most distant customers live?
.
(g) Suppose that the city relaxes its zoning restrictions on where the
bowling alleys can locate, but continues to issue operating licenses to
only 3 bowling alleys. Both of the bowling alleys at the end of town
are about to lose their leases and can locate anywhere in town that they
like at about the same cost. The bowling alley in the center of town is
committed to stay where it is. Would either of the alleys at the edge of
town improve its profits by locating next to the existing bowling alley in
the center of town? What would be a profit-maximizing location
for each of these two bowling alleys?
.
25.10 (1) In a congressional district somewhere in the U.S. West a
new representative is being elected. The voters all have one-dimensional
political views that can be neatly arrayed on a left-right spectrum. We
can define the “location” of a citizen’s political views in the following way.
The citizen with the most extreme left-wing views is said to be at point
0 and the citizen with the most extreme right-wing views is said to be at
point 1. If a citizen has views that are to the right of the views of the
fraction x of the state’s population, that citizen’s views are said to be
located at the point x. Candidates for office are forced to publically state
their own political position on the zero-one left-right scale. Voters always
vote for the candidate whose stated position is nearest to their own views.
(If there is a tie for nearest candidate, voters flip a coin to decide which
to vote for.)
(a) There are two candidates for the congressional seat. Suppose
that each candidate cares only about getting as many votes as possible.
Is there an equilibrium in which each candidate chooses the
best position given the position of the other candidate? If so,
describe this equilibrium.
.
25.11 (2) In the congressional district described by the previous problem,
let us investigate what will happen if the two candidates do not care
about the number of votes that they get but only about the amount
of campaign contributions that they receive. Therefore each candidate
chooses his ideological location in such a way as to maximize the amount
of campaign contributions he receives, given the position of the other.*
Let us define a left-wing extremist as a voter whose political views
lie to the left of the leftmost candidate, a right-wing extremist as a voter
whose political views lie to the right of the rightmost candidate, and a
moderate voter as one whose political views lie between the positions
of the two candidates. Assume that each extremist voter contributes to
the candidate whose position is closest to his or her own views and that
moderate voters make no campaign contributions. The number of dollars
that an extremist voter contributes to his or her favorite candidate
is proportional to the distance between the two candidates. Specifically,
we assume that there is some constant C such that if the left-wing candidate
is located at x and the right-wing candidate is located at y, then
total campaign contributions received by the left-wing candidate will be
$Cx(y − x) and total campaign contributions received by the right-wing
candidate will be $C(1 − y)(y − x).
(a) If the right-wing candidate is located at y, the contributionmaximizing
position for the left-wing candidate is x = If the
left-wing candidate is located at x, the contribution-maximizing position
for the right-wing candidate is y = . (Hint: Take a derivative
and set it equal to zero.)
(b) Solve for the unique pair of ideological positions for the two candidates
such that each takes the position that maximizes his campaign
contributions given the position of the other. .
* This assumption is a bit extreme. Candidates typically spend at least
some of their campaign contributions on advertising for votes, and this
advertising affects the voting outcomes.
(c) Suppose that in addition to collecting contributions from extremists
on their side, candidates can also collect campaign contributions from
moderates whose views are closer to their position than to that of their
rivals’ position. Suppose that moderates, like extremists, contribute to
their preferred candidate and that they contribute in proportion to the
difference between their own ideological distance from their less-preferred
candidate and their ideological distance from their more-preferred candidate.
Show that in this case the unique positions in which the left- and
right-wing candidates are each maximizing their campaign contributions,
given the position of the other candidate, occurs where x = 1/4 and
y = 3/4.
.