Chapter 12
Monopolistic Competition and Oligopoly
n Teaching
Notes
Students viewing this material for
the first time can be overwhelmed because of the number of models presented. Chapter
12 discusses ten models (with some overlap): monopolistic competition,
Cournot-Nash, Stackelberg, Bertrand, price competition with differentiated
products, prisoners’ dilemma, kinked demand, price leadership, dominant firm,
and cartels. The reason for all the models, of course, is that there is no
single oligopoly model. I try to convince students that since oligopoly theory
is still evolving, it is an exciting area to study. Nonetheless, there is a
great deal of material here, and you may want to concentrate on the more basic
models, e.g., monopolistic competition, Cournot-Nash, prisoners’ dilemma, and
cartels. You can otherwise pick and choose among the models as time and
interest dictates.
When introducing the material in this
chapter, start by reminding students that these market structures lie between
perfect competition and monopoly. When presenting monopolistic competition,
focus on why positive profits encourage entry and on the similarities and
differences of this model with competition and monopoly. The example of brand
competition in cola and coffee markets presented at the end of Section 12.1
facilitates a class discussion of the costs and benefits of freedom of choice
among a vast array of brand names and trademarks.
The key to the Cournot-Nash duopoly
model in Section 12.2 is an understanding of the way firms react to each other
and the resulting reaction functions. Stress that reaction functions are
graphed on axes that both represent quantities (see Figure 12.4). Once they
understand reaction functions, students will be better able to follow the
assumptions, reasoning, and results of the Cournot-Nash, Stackelberg, and
Bertrand models. Be sure to compare the competitive, Cournot-Nash, and
collusive (monopoly) equilibria as shown in Figure 12.5. Figure 12.5 gives the
impression that the duopolists always have symmetric reaction curves. Exercise
2 shows that if the cost structures are not identical, the reaction curves are
asymmetric.
An alternative to jumping right into
the Cournot-Nash material is to start with the kinked demand model in Section 12.5. You can use this to discuss the
importance of other firms’ reactions by drawing two complete demand
curves through the current price-quantity point. In fact, there are many
possible demand curves depending on how other firms react to a price change. For
example, other firms may change price by half the amount or double the amount
of the original firm’s change. There may be uncertainty about how other firms
will react, which you could relate back to the material in Chapter 5 if you
covered that chapter. Deriving the kinked demand and associated marginal
revenue curve will be easier if you went through the effect of a price ceiling
on a monopolist in Chapter 10, where there is also a kinked demand.
Section 12.4 on the prisoners’
dilemma is a student favorite. You might add some other examples such as
advertising decisions where the primary effect of advertising is to take sales
away from the other firm. Both firms have “high advertising” as a dominant
strategy even though both would be more profitable if they each chose “low
advertising” instead. Another good example is arms races between nations. Neither
wants to be the one with a low number of weapons, so each stockpiles large
amounts of weapons. Examples 12.2 and 12.3 deal with a pricing decision facing
Procter & Gamble and together are a nice example of a prisoners’ dilemma as
well as an illustration of a pricing problem in foreign markets.
Section 12.5 looks at price signaling
and price leadership. Another type of price signaling you could mention in
class occurs when a firm announces that it will match any competitor’s price. This
is quite common in retail chains selling electronic equipment, office supplies,
food, etc. While such promises sound good to consumers, they may also be a way
to signal willingness to set prices at or near collusive levels. If firm A makes such a pledge, firm B knows that cutting prices will not be
profitable, and it thus has an incentive to keep prices high.
Section 12.6 discusses cartels—an
issue that most students find interesting. You can refer back to the prisoners’
dilemma model to explain why cheating in cartels is often a problem. This
section ends with two topics that invoke opinions from almost every student: a
discussion of OPEC and Example 12.6,
“The Cartelizaton of Intercollegiate Athletics.”
n Review
Questions
1. What are the characteristics of a
monopolistically competitive market? What happens to the equilibrium price and
quantity in such a market if one firm introduces a new, improved product?
The two primary characteristics of a monopolistically
competitive market are that (1) firms compete by selling differentiated
products that are highly, but not perfectly, substitutable and (2) there is
free entry and exit from the market. When a new firm enters a monopolistically
competitive market or one firm introduces an improved product, the demand curve
for each of the other firms shifts inward, reducing the price and quantity
received by those incumbents. Thus, the introduction of a new product by a firm
will reduce the price received and quantity sold of existing products.
2. Why is the firm’s demand curve flatter than
the total market demand curve in monopolistic competition? Suppose a
monopolistically competitive firm is making a profit in the short run. What
will happen to its demand curve in the long run?
The flatness or steepness of the firm’s demand curve is a
function of the elasticity of demand for the firm’s product. The elasticity of
the firm’s demand curve is greater than the elasticity of market demand because
it is easier for consumers to switch to another firm’s highly substitutable
product than to switch consumption to an entirely different product. Profit in
the short run induces other firms to enter. As new firms enter, the incumbent
firm’s demand and marginal revenue curves shift to the left, reducing the
profit-maximizing quantity. In the long run profits fall to zero, leaving no
incentive for more firms to enter.
3. Some experts have argued that too many brands
of breakfast cereal are on the market. Give an argument to support this view. Give
an argument against it.
Pro: Too many brands of any single product signals excess
capacity, implying that each firm is producing an output level smaller than the
level that would minimize average cost. Limiting the number of brands would therefore
enhance overall economic efficiency.
Con: Consumers value the freedom to choose among a wide
variety of competing products. Even if costs are slightly higher as a result of
the large number of brands available, the benefits to consumers outweigh the
extra costs.
(Note: In 1972 the Federal Trade Commission
filed suit against Kellogg, General Mills, and General Foods. It charged that
these firms attempted to suppress entry into the cereal market by introducing
150 heavily advertised brands between 1950 and 1970, crowding competitors off
grocers’ shelves. This case was eventually dismissed in 1982.)
4. Why is the Cournot equilibrium stable? (i.e.,
Why don’t firms have any incentive to change their output levels once in
equilibrium?) Even if they can’t collude, why don’t firms set their outputs at
the joint profit-maximizing levels (i.e., the levels they would have chosen had
they colluded)?
A Cournot equilibrium is stable because each firm is
producing the amount that maximizes its profit, given what its competitors
are producing. If all firms behave this way, no firm has an incentive to
change its output. Without collusion, firms find it difficult to agree tacitly
to reduce output.
If all firms were producing at the joint
profit-maximizing level, each would have an incentive to increase output,
because that would increase each firm’s profit at the expense of the firms that
were limiting sales. But when each firm increases output, they all end up back
at the Cournot equilibrium. Thus it is very difficult to reach the joint
profit-maximizing level without overt collusion, and even then it may be
difficult to prevent cheating among the cartel members.
5. In the Stackelberg model, the firm that sets
output first has an advantage. Explain why.
The Stackelberg leader gains an advantage because the
second firm must accept the leader’s large output as given and produce a
smaller output for itself. If the second firm decided to produce a larger
quantity, this would reduce price and profit for both firms. The first firm
knows that the second firm will have no choice but to produce a smaller output
in order to maximize profit, and thus the first firm is able to capture a
larger share of industry profits.
6. What do the Cournot and Bertrand models have
in common? What is different about the two models?
Both are oligopoly models in which firms produce a
homogeneous good. In the Cournot model, each firm assumes its rivals will not
change the quantity they produce. In the Bertrand model, each firm assumes its
rivals will not change the price they charge. In both models, each firm takes
some aspect of its rivals’ behavior (either quantity or price) as fixed when
making its own decision. The difference between the two is that in the Bertrand
model firms end up producing where price equals marginal cost, whereas in the
Cournot model the firms will produce more than the monopoly output but less
than the competitive output.
7. Explain the meaning of a Nash equilibrium when
firms are competing with respect to price. Why is the equilibrium stable? Why
don’t the firms raise prices to the level that maximizes joint profits?
A Nash equilibrium in price competition occurs when each
firm chooses its price, assuming its competitors’ prices will not change.
In equilibrium, each firm is doing the best it can, conditional
on its competitors’ prices. The equilibrium is stable because each firm is
maximizing its profit, and therefore no firm has an incentive to raise or lower
its price.
No individual firm would raise its price to the level that
maximizes joint profit if the other firms do not do the same, because it would
lose sales to the firms with lower prices. It is also difficult to collude. A
cartel agreement is difficult to enforce because each firm has an incentive to
cheat. By lowering price, the cheating firm can increase its market share and
profits. A second reason that firms do not collude is that such behavior
violates antitrust laws. In particular, price fixing violates Section 1 of the
Sherman Act. Of course, there are attempts to circumvent antitrust laws through
tacit collusion.
8. The kinked demand curve describes price
rigidity. Explain how the model works. What are its limitations? Why does price
rigidity occur in oligopolistic markets?
According to the kinked demand curve model, each firm
faces a demand curve that is kinked at the currently prevailing price. Each
firm believes that if it raises its price, the other firms will not raise their
prices, and thus many of the firm’s customers will shift their purchases to
competitors. This reasoning implies a highly elastic demand for price
increases. On the other hand, each firm believes that if it lowers its price,
its competitors will also lower their prices, and the firm will not increase
sales by much. This implies a demand curve that is less elastic for price
decreases than for price increases. This kink in the demand curve leads to a
discontinuity in the marginal revenue curve, so only large changes in marginal
cost lead to changes in price.
A major limitation is that the
kinked-demand model does not explain how the starting price is
determined. Price rigidity may occur in oligopolistic markets because firms
want to avoid destructive price wars. Managers learn from experience that
cutting prices does not lead to lasting increases in profits. As a result,
firms are reluctant to “rock the boat” by changing prices even when costs
change.
9. Why does price leadership sometimes evolve in
oligopolistic markets? Explain how the price leader determines a
profit-maximizing price.
Since firms cannot explicitly coordinate on setting
price, they use implicit means. One form of implicit collusion is to follow a
price leader. The price leader, often the largest or dominant firm in the
industry, determines its profit-maximizing quantity by calculating the demand
curve it faces as follows: at each price, it subtracts the quantity supplied by
all other firms from the market demand, and the residual is its demand curve. The
leader chooses the quantity that equates its marginal revenue with its marginal
cost and sets price to sell this quantity. The other firms (the followers)
match the leader’s price and supply the remainder of the market.
10. Why
has the OPEC oil cartel succeeded in raising prices substantially
while the CIPEC copper cartel has not? What conditions are necessary for
successful cartelization? What organizational problems must a cartel overcome?
Successful cartelization requires two characteristics:
demand should be relatively inelastic, and the cartel must be able to control most
of the supply. OPEC succeeded in the short run because the
short-run demand and supply of oil were both inelastic. CIPEC has not been
successful because both demand and non-CIPEC supply were highly responsive to
price. A cartel faces two organizational problems: agreement on a price and a
division of the market among cartel members, and it must monitor and enforce
the agreement.
n Exercises
1. Suppose all firms in a monopolistically
competitive industry were merged into one large firm. Would that new firm
produce as many different brands? Would it produce only a single brand? Explain.
Monopolistic competition is
defined by product differentiation. Each firm earns economic profit
by distinguishing its brand from all other
brands. This distinction can arise from underlying differences in the
product or from differences in advertising. If these competitors merge into a
single firm,
the resulting monopolist would not produce as many brands, since too much brand
competition is internecine (mutually destructive). However, it is unlikely that
only one brand would be produced after the merger. Producing several brands
with different prices and characteristics is one method of splitting the market
into sets of customers with different tastes and price elasticities. The
monopolist can sell to more consumers and maximize overall profit by producing
multiple brands and practicing a form of price discrimination.
2. Consider two firms facing the demand curve P
= 50 -
5Q, where Q = Q1
+ Q2. The firms’ cost
functions are C1(Q1) = 20 +
10Q1 and C2(Q2) = 10 +
12Q2.
a. Suppose both firms
have entered the industry. What is the joint profit-maximizing level of output?
How much will each firm produce? How would your answer change if the firms have
not yet entered the industry?
If the firms collude, they face the market demand curve,
so their marginal revenue curve is:
MR = 50 - 10Q.
Set marginal revenue equal to marginal cost (the marginal
cost of Firm 1, since it is lower than that of Firm 2) to determine the
profit-maximizing quantity, Q:
50 - 10Q = 10, or Q = 4.
Substituting Q =
4 into the demand function to determine price:
P = 50 - 5(4) =
$30.
The question now is how the firms will divide the total
output of 4 among themselves. The joint profit-maximizing solution is for Firm
1 to produce all of the output because its marginal cost is less than Firm 2’s
marginal cost. We can ignore fixed costs because both firms are already in the
market and will be saddled with their fixed costs no matter how many units each
produces. If Firm 1 produces all 4 units, its profit will be
p1 =
(30)(4) - (20 + (10)(4)) = $60.
The profit for Firm 2 will be:
p2 = (30)(0) -
(10 + (12)(0)) = -$10.
Total industry profit will be:
pT = p1
+ p2
= 60 -
10 = $50.
Firm 2, of course, will not like this. One solution is for
Firm 1 to pay Firm 2 $35 so that both earn a profit of $25, although they may
well disagree about the amount to be paid. If they split the output evenly
between them, so that each firm produces 2 units, then total profit would be
$46 ($20 for Firm 1 and $26 for Firm 2). This does not maximize total profit,
but Firm 2 would prefer it to the $25 it gets from an even split of the maximum
$50 profit. So there is no clear-cut answer to this question.
If Firm 1 were the only entrant, its profits would be $60
and Firm 2’s would be 0.
If Firm 2 were the only entrant, then it would equate
marginal revenue with its marginal cost to determine its profit-maximizing
quantity:
50 - 10Q2
= 12, or Q2 = 3.8.
Substituting Q2 into the demand equation
to determine price:
P = 50 - 5(3.8) =
$31.
The profits for Firm 2 would be:
p2 = (31)(3.8) -
(10 + (12)(3.8)) = $62.20,
and Firm 1 would earn 0. Thus, Firm 2 would make a larger
profit than Firm 1 if it were the only firm in the market, because Firm 2 has
lower fixed costs.
b. What is each firm’s
equilibrium output and profit if they behave noncooperatively? Use the Cournot
model. Draw the firms’ reaction curves and show the equilibrium.
In the Cournot model, Firm 1 takes Firm 2’s output as
given and maximizes profits. Firm 1’s profit function is
Setting the derivative of the profit function with respect
to Q1 to zero, we find Firm 1’s reaction function:
Similarly, Firm 2’s reaction function is
.
To find the Cournot equilibrium, substitute Firm 2’s
reaction function into Firm 1’s reaction function:
Substituting this value for Q1 into the
reaction function for Firm 2, we find
Q2 =
2.4.
Substituting the
values for Q1 and Q2 into the demand
function to determine the equilibrium price:
P = 50 - 5(2.8 +
2.4) = $24.
The profits for Firms 1 and 2 are equal to
p1
= (24)(2.8) - (20 + (10)(2.8)) = $19.20, and
p2 = (24)(2.4) -
(10 + (12)(2.4)) = $18.80.
The firms’ reaction curves and the Cournot equilibrium are
shown below.
c. How
much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal
but a takeover is not?
To determine how much Firm 1 will be willing to pay to
purchase Firm 2, we must compare Firm 1’s profits in the monopoly situation
versus it profits in an oligopoly. The difference between the two will be what
Firm 1 is willing to pay for Firm 2.
From part a, Firm 1’s profit when it sets marginal revenue
equal to its marginal cost is $60. This is what the firm would earn if it was a
monopolist. From part b, profit is $19.20 for Firm 1 when the firms compete
against each other in a Cournot-type market. Firm 1 should therefore be willing
to pay up to $60 - 19.20 = $40.80 for Firm 2.
3. A monopolist can produce at a constant average
(and marginal) cost of AC = MC
= $5. It faces a market demand curve
given by Q = 53 - P.
a. Calculate
the profit-maximizing price and quantity for this monopolist. Also calculate
its profits.
First solve for the inverse demand curve, P = 53 -
Q. Then the marginal revenue curve has the same intercept and twice the
slope:
MR = 53 - 2Q.
Marginal cost is a constant $5. Setting MR = MC, find the optimal quantity:
53 - 2Q = 5, or Q = 24.
Substitute Q =
24 into the demand function to find price:
P = 53 - 24 =
$29.
Assuming fixed costs are zero, profits are equal to
p = TR -
TC = (29)(24) - (5)(24) =
$576.
b. Suppose a second
firm enters the market. Let Q1 be the output of the first firm
and Q2 be the output of the second. Market demand is now
given by
Q1 + Q2
=
53 -
P.
Assuming that this
second firm has the same costs as the first, write the profits of each firm as
functions of Q1 and Q2.
When the second firm enters, price can be written as a
function of the output of both firms:
P = 53 - Q1 -
Q2. We may write the profit functions for the two firms:
and
c. Suppose (as in the
Cournot model) that each firm chooses its profit-maximizing level of output on
the assumption that its competitor’s output is fixed. Find each firm’s
“reaction curve” (i.e., the rule that gives its desired output in terms of its
competitor’s output).
Under the Cournot assumption, each firm treats the output
of the other firm as a constant in its maximization calculations. Therefore,
Firm 1 chooses Q1 to maximize p1 in part b with Q2
being treated as a constant. The change in p1
with respect to a change in Q1 is
This equation is the reaction function for Firm 1, which
generates the profit- maximizing level of output, given the output of Firm 2. Because
the problem is symmetric, the reaction function for Firm 2 is
.
d. Calculate the
Cournot equilibrium (i.e., the values of Q1 and Q2
for which each firm is doing as well as it can given its competitor’s output). What
are the resulting market price and profits of each firm?
Solve for the values of Q1 and Q2
that satisfy both reaction functions by substituting Firm 2’s reaction function
into the function for Firm 1:
By symmetry, Q2 = 16.
To determine the price, substitute Q1
and Q2 into the demand equation:
P = 53 - 16 -
16 = $21.
Profit for Firm 1 is therefore
pi = PQi - C(Qi) = pi
= (21)(16) - (5)(16) = $256.
Firm 2’s profit is the same, so total industry profit is p1 + p2 = $256 +
$256 = $512.
e. Suppose there are N
firms in the industry, all with the same constant marginal cost,
MC = $5. Find the Cournot
equilibrium. How much will each firm produce, what will be the market price,
and how much profit will each firm earn? Also, show that as N becomes
large, the market price approaches the price that would prevail under perfect
competition.
If there are N identical firms, then the price in
the market will be
Profits for the ith firm are given by
Differentiating to obtain the necessary first-order
condition for profit maximization,
.
Solving for Qi,
If all firms face the same costs, they will all produce
the same level of output, i.e., Qi = Q*. Therefore,
Now substitute Q =
NQ* for total output in the demand function:
Total profits are
pT = PQ -
C(Q) = P(NQ*)
- 5(NQ*)
or
or
Notice that with N firms
and that, as N increases (N ® ¥)
Q = 48.
Similarly, with
as N ®
¥,
P = 53 - 48 =
5.
Finally,
so as N ®
¥,
pT = $0.
In perfect competition, we know that profits are zero and
price equals marginal cost. Here,
pT = $0 and P = MC = 5. Thus,
when N approaches infinity, this market approaches a perfectly
competitive one.
4. This exercise is a continuation of Exercise 3.
We return to two firms with the same constant average and marginal cost, AC
= MC = 5, facing the market demand curve Q1 + Q2 = 53 -
P. Now we will use the Stackelberg model to analyze what will happen if
one of the firms makes
its output decision before the other.
a. Suppose Firm 1 is
the Stackelberg leader (i.e., makes its output decisions before Firm 2). Find
the reaction curves that tell each firm how much to produce in terms of the
output of its competitor.
Firm 2’s reaction curve is the same as determined in part
c of Exercise 3:
Firm 1 does not have a
reaction function because it makes its output decision before Firm 2, so there
is nothing to react to. Instead, Firm 1 uses its knowledge of Firm 2’s reaction
function when determining its optimal output as shown in part b below.
b. How much will each
firm produce, and what will its profit be?
Firm 1, the Stackelberg leader, will choose its output, Q1,
to maximize its profits, subject to the reaction function of Firm 2:
max p1
= PQ1 - C(Q1),
subject to
Substitute for Q2 in the demand function
and, after solving for P, substitute for P in the profit
function:
To determine the profit-maximizing quantity, we find the
change in the profit function with respect to a change in Q1:
Set this expression equal to 0 to determine the
profit-maximizing quantity:
53 - 2Q1
- 24 +
Q1 - 5 = 0, or Q1 = 24.
Substituting Q1 = 24 into Firm 2’s reaction function gives Q2:
Substitute Q1 and Q2
into the demand equation to find the price:
P = 53 - 24 -
12 = $17.
Profits for each firm are equal to total revenue minus
total costs, or
p1 = (17)(24) -
(5)(24) = $288, and
p2
= (17)(12) - (5)(12) = $144.
Total industry profit, pT
= p1
+ p2
= $288 +
$144 = $432.
Compared to the Cournot equilibrium, total output has
increased from 32 to 36, price has fallen from $21 to $17, and total profits
have fallen from $512 to $432. Profits for Firm 1 have risen from $256 to $288,
while the profits of Firm 2 have declined sharply from $256 to $144.
5. Two firms compete
in selling identical widgets. They choose their output levels Q1
and Q2 simultaneously and face the demand curve
P =
30 -
Q
where Q =
Q1 + Q2.
Until recently, both firms had zero marginal costs. Recent environmental
regulations have increased Firm 2’s marginal cost to $15. Firm 1’s marginal
cost remains constant at zero. True or false: as
a result, the market price will rise to the monopoly level.
Surprisingly, this is true. However, it occurs only
because the marginal cost for Firm 2 is $15 or more. If the market were
monopolized before the environmental regulations, the marginal revenue for the
monopolist would be
MR = 30 - 2Q.
Profit maximization implies MR = MC, or 30 - 2Q = 0. Therefore,
Q = 15, and (using the demand
curve) P = $15.
The situation after the environmental regulations is a
Cournot game where firm 1’s
marginal costs are zero and firm
2’s marginal costs are $15. We need to
find the best response functions:
Firm 1’s revenue is
and its marginal revenue is given by:
Profit maximization implies MR1 = MC1 or
which is firm
1’s best response function.
Firm 2’s revenue function is symmetric to that of Firm 1
and hence
Profit maximization implies MR2 = MC2, or
which is firm
2’s best response function.
Cournot equilibrium occurs at the intersection of the
best response functions. Substituting for Q1 in the response
function for firm 2 yields:
Thus Q2 =
0 and Q1 = 15. P
= 30 -
Q1 - Q2
= $15, which is the monopoly price.
6. Suppose
that two identical firms produce widgets and that they are the only firms in
the market. Their costs are given by C1 = 60Q1 and C2
= 60Q2, where Q1
is the output of Firm 1 and Q2 the output of Firm 2. Price is
determined by the following demand curve:
P =
300 -
Q
where Q =
Q1 + Q2.
a. Find the
Cournot-Nash equilibrium. Calculate the profit of each firm at this
equilibrium.
Profit for Firm 1, TR1 - TC1, is equal to
Therefore,
Setting this equal to zero and solving for Q1
in terms of Q2:
Q1 =
120 - 0.5Q2.
This is Firm 1’s reaction function. Because Firm 2 has the
same cost structure, Firm 2’s reaction function is
Q2 =
120 - 0.5Q1.
Substituting for Q2 in the reaction
function for Firm 1, and solving for Q1, we find
Q1 =
120 - (0.5)(120 - 0.5Q1), or Q1 = 80.
By symmetry, Q2 = 80. Substituting Q1 and Q2
into the demand equation to determine the equilibrium price:
P = 300 - 80 -
80 = $140.
Substituting the values for price and quantity into the
profit functions,
p1
= (140)(80) - (60)(80) = $6400, and
p2 = (140)(80) -
(60)(80) = $6400.
Therefore, profit is $6400 for both firms in the
Cournot-Nash equilibrium.
b. Suppose the two
firms form a cartel to maximize joint profits. How many widgets will be
produced? Calculate each firm’s profit.
Given the demand curve is P = 300 - Q, the
marginal revenue curve is MR =
300 - 2Q. Profit will be
maximized by finding the level of output such that marginal revenue is equal to
marginal cost:
300 - 2Q = 60, or Q = 120.
When total output is 120, price will be $180, based on the
demand curve. Since both firms have the same marginal cost, they will split the
total output equally, so they each produce 60 units.
Profit for each firm is:
p = 180(60) -
60(60) = $7200.
c. Suppose Firm 1 were
the only firm in the industry. How would market output and Firm 1’s profit
differ from that found in part b above?
If Firm 1 were the only firm, it would produce where
marginal revenue is equal to marginal cost, as found in part b. In this case
Firm 1 would produce the entire 120 units of output and earn a profit of
$14,400.
d. Returning to the
duopoly of part b, suppose Firm 1 abides by the agreement, but Firm 2 cheats by
increasing production. How many widgets will Firm 2 produce? What will be each
firm’s profits?
Assuming their agreement is to split the market equally,
Firm 1 produces 60 widgets. Firm 2 cheats by producing its profit-maximizing
level, given Q1 = 60.
Substituting Q1 = 60
into Firm 2’s reaction function:
Total industry output, QT, is equal to Q1
plus Q2:
QT =
60 + 90 =
150.
Substituting QT into the demand equation
to determine price:
P = 300 - 150 =
$150.
Substituting Q1, Q2,
and P into the profit functions:
p1 = (150)(60) -
(60)(60) = $5400, and
p2
= (150)(90) - (60)(90) = $8100.
Firm 2 increases its profits at the expense of Firm 1 by
cheating on the agreement.
7. Suppose that two competing firms, A and
B, produce a homogeneous good. Both firms have a marginal cost of MC
= $50. Describe what would happen to
output and price in each of the following situations if the firms are at (i)
Cournot equilibrium, (ii) collusive equilibrium, and (iii) Bertrand
equilibrium.
a. Because Firm A
must increase wages, its MC increases to $80.
(i) In a Cournot
equilibrium you must think about the effect on the reaction functions, as
illustrated in Figure 12.5 of the text. When Firm A experiences an
increase in marginal cost, its reaction function will shift inward. The
quantity produced by Firm A will decrease and the quantity produced by Firm
B will increase. Total quantity produced will decrease and price will
increase.
(ii) In a
collusive equilibrium, the two firms will collectively act like a monopolist. When
the marginal cost of Firm A increases, Firm A will reduce its
production to zero, because Firm B can produce at a lower marginal cost.
Because Firm B can produce the entire industry output at a marginal cost
of $50, there will be no change in output or price. However, the firms will
have to come to some agreement on how to share the profit earned by B.
(iii) Before the increase in Firm A’s costs, both firms would
charge a price equal to marginal cost (P = $50)
because the good is homogeneous. After Firm A’s marginal cost increases,
Firm B will raise its price to $79.99 (or some price just below $80) and
take all sales away from Firm A. Firm A would lose money on each
unit sold at any price below its marginal cost of $80, so it will produce
nothing.
b. The marginal cost
of both firms increases.
(i) Again refer to Figure 12.5. The increase in the marginal cost of
both firms will shift both reaction functions inward. Both firms will decrease
quantity produced and price will increase.
(ii) When marginal cost increases, both firms will produce less and
price will increase, as in the monopoly case.
(iii) Price will increase to the new level of marginal cost and quantity
will decrease.
c. The demand curve
shifts to the right.
(i) This is the opposite of the case in part b. In this situation,
both reaction functions will shift outward and both will produce a higher
quantity. Price will tend to increase.
(ii) Both firms will increase the quantity produced as demand and
marginal revenue increase. Price will also tend to increase.
(iii) Both firms will supply more output. Given that marginal cost remains
the same, the price will not change.
8. Suppose the airline industry consisted of only
two firms: American and Texas Air Corp. Let the two firms have identical cost
functions, C(q) = 40q.
Assume the demand curve for the industry is given by P = 100 -
Q and that each firm expects the other to behave as a Cournot competitor.
a. Calculate the
Cournot-Nash equilibrium for each firm, assuming that each chooses the output
level that maximizes its profits when taking its rival’s output as given. What
are the profits of each firm?
First, find the reaction function for each firm; then
solve for price, quantity, and profit. Profit for Texas Air, p1, is equal to total revenue
minus total cost:
The change in p1
with respect to Q1 is
Setting the derivative to zero and solving for Q1
gives Texas Air’s reaction function:
Q1 =
30 - 0.5Q2.
Because American has the same cost structure, American’s
reaction function is
Q2 =
30 - 0.5Q1.
Substituting for Q2 in the reaction
function for Texas Air,
Q1 =
30 - 0.5(30 - 0.5Q1), or Q1 = 20.
By symmetry, Q2 = 20. Industry output, QT , is Q1
plus Q2, or
QT =
20 + 20 =
40.
Substituting industry output into the demand equation, we
find P = $60. Substituting Q1,
Q2, and P into the profit function, we find
p1 = p2
= 60(20) -
202 - (20)(20) = $400.
b. What would be the
equilibrium quantity if Texas Air had constant marginal and average costs of
$25 and American had constant marginal and average costs of $40?
By solving for the reaction functions under this new cost
structure, we find that profit for Texas Air is equal to
The change in profit with respect to Q1
is
Set the derivative to zero, and solve for Q1
in terms of Q2,
Q1 =
37.5 - 0.5Q2.
This is Texas Air’s reaction function. Since American has
the same cost structure as in part a, American’s reaction function is the same
as before:
Q2 =
30 - 0.5Q1.
To determine Q1, substitute for Q2
in the reaction function for Texas Air and solve for Q1:
Q1 =
37.5 - (0.5)(30 - 0.5Q1), so Q1 = 30.
Texas Air finds it profitable to increase output in
response to a decline in its cost structure.
To determine Q2, substitute for Q1
in the reaction function for American:
Q2 =
30 - (0.5)(30) = 15.
American has cut back slightly in its output in response
to the increase in output by Texas Air.
Total quantity, QT, is Q1
+ Q2, or
QT =
30 + 15 =
45.
Compared to part a, the equilibrium quantity has risen
slightly.
c. Assuming that both
firms have the original cost function, C(q) = 40q, how much should Texas Air be
willing to invest to lower its marginal cost from 40 to 25, assuming that
American will not follow suit? How much should American be willing to spend to
reduce
its marginal cost to 25, assuming that Texas Air will have marginal costs of 25
regardless
of American’s actions?
Recall that profits for both firms were $400 under the
original cost structure. With constant average and marginal costs of $25, we
determined in part b that Texas Air would produce
30 units and American 15. Industry price would then be P = 100 - 30 - 15 =
$55. Texas Air’s profits would be
(55)(30) - (25)(30) = $900.
The difference in profit is $500. Therefore, Texas Air
should be willing to invest up to $500 to lower costs from 40 to 25 per
unit (assuming American does not follow suit).
To determine how much American would be willing to spend
to reduce its average costs, calculate the difference in American’s profits,
assuming Texas Air’s average cost is $25.
First, without investment, American’s profits would be:
(55)(15) - (40)(15) = $225.
Second, with investment by both firms, the reaction
functions would be:
Q1 =
37.5 - 0.5Q2 and
Q2
= 37.5 -
0.5Q1.
To determine Q1, substitute for Q2
in the first reaction function and solve for Q1:
Q1 =
37.5 - (0.5)(37.5 - 0.5Q1), which implies Q1
= 25.
Since the firms are symmetric, Q2 is
also 25.
Substituting industry output into the demand equation to
determine price:
P = 100 - 50 =
$50.
Therefore, American’s profits when both firms have MC = AC =
25 are
p2 = (50)(25) -
(25)(25) = $625.
The difference in profit with and without the cost-saving
investment for American is $400. American would be willing to invest up to $400
to reduce its marginal cost to 25 if Texas Air also has marginal costs of 25.
9. Demand for light bulbs can be characterized by
Q = 100 - P, where Q is in millions of boxes of lights
sold and P is the price per box. There are two producers of lights,
Everglow and Dimlit. They have identical cost functions:
(i = E, D) Q = QE + QD.
a. Unable to recognize
the potential for collusion, the two firms act as short-run perfect
competitors. What are the equilibrium values of QE, QD,
and P? What are each firm’s profits?
Given that the total cost function is
the marginal cost
curve for each firm is
MCi
= 10 +
Qi. In the short run, perfectly
competitive firms determine the optimal level of output by taking price as
given and setting price equal to marginal cost. There are two ways to solve
this problem. One way is to set price equal to marginal cost for each firm so
that:
Given we now have two equations and two unknowns, we can
solve for Q1 and Q2 simultaneously. Solve
the second equation for Q2 to get
and substitute into the other equation to get
This yields a solution where Q1 = 30, Q2 = 30, and P = $40. You can verify that P = MC for each firm. Profit is total revenue minus total
cost or
40(30)
- [10(30)
+
0.5(30)
2]
= $450 million.
The other way to solve the problem is to find the market
supply curve by summing the marginal cost curves, which yields QM
= 2P - 20. Set supply equal to demand to find P
= $40 and Q = 60 in the market, or 30 per firm since they
are identical.
b. Top management in
both firms is replaced. Each new manager independently recognizes the
oligopolistic nature of the light bulb industry and plays Cournot. What are the
equilibrium values of QE, QD, and P?
What are each firm’s profits?
To determine the Cournot-Nash equilibrium, we first
calculate the reaction function for each firm, then solve for price, quantity,
and profit. Profits for Everglow are equal to TRE - TCE, or
The change in profit with respect to QE
is
To determine Everglow’s reaction function, set the change
in profits with respect to QE equal to
0 and solve for QE:
90 - 3QE
- QD = 0, or
Because Dimlit has the same cost structure, Dimlit’s
reaction function is
Substituting for QD in the reaction
function for Everglow, and solving for QE:
By symmetry, QD = 22.5, and total industry output is 45.
Substituting industry output into the demand equation
gives P:
45 = 100 - P, or P = $55.
Each firm’s profit equals total revenue minus total cost:
BI = 55(22.5) - [10(22.5) + 0.5(22.5)2] = $759.4 million.
c. Suppose the
Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow
plays Stackelberg. What are the equilibrium values of QE, QD,
and P? What are each firm’s profits?
Recall Everglow’s profit function:
If Everglow sets its quantity first, knowing Dimlit’s
reaction function
we may determine Everglow’s profit by
substituting for
QD in its profit function. We find
To determine the profit-maximizing quantity, differentiate
profit with respect to QE, set the derivative to zero and
solve for QE:
Substituting this into Dimlit’s reaction function,
Total industry output
is therefore 47.1 and
P =
$52.90. Profit for Everglow is
pE = (52.90)(25.7) - [10(25.7) + 0.5(25.7)2]
= $772.3 million.
Profit for Dimlit is
pD = (52.90)(21.4) - [10(21.4) + 0.5(21.4)2]
= $689.1 million.
d. If the managers of
the two companies collude, what are the equilibrium values of QE,
QD, and P? What are each firm’s profits?
Because the firms are identical, they should split the
market equally, so each produces Q/2 units, where Q is the total
industry output. Each firm’s total cost is therefore
,
and total industry cost is
.
Hence, industry marginal cost is
MC = 10 + 0.5Q.
With inverse industry demand given by P = 100 -
Q, industry marginal revenue is
MR = 100 - 2Q.
Setting MR =
MC, we have
100 - 2Q = 10 +
0.5Q, and so Q = 36,
which means QE = QD =
Q/2 = 18.
Substituting Q in the demand equation to determine
price:
P = 100 - 36 =
$64.
The profit for each firm is equal to total revenue minus
total cost:
Note that you can also solve for the optimal quantities by
treating the two firms as a monopolist with two plants. In that case, the optimal
outputs satisfy the condition MR =
MCE = MCD.
Setting marginal revenue equal to each marginal cost function gives the
following two equations:
MR = 100 - 2(QE + QD) = 10 +
QE = MCE
MR = 100 - 2(QE + QD) = 10 +
QD = MCD.
Solving simultaneously, we get the same solution as
before; that is, QE =
QD = 18.
10. Two firms produce luxury sheepskin auto seat covers, Western
Where (WW) and B.B.B. Sheep (BBBS). Each firm has a cost function given by
C (q) =
30q + 1.5q2
The market demand for these seat covers is represented
by the inverse demand equation
P =
300 -
3Q
where Q =
q1 + q2,
total output.
a. If each firm acts
to maximize its profits, taking its rival’s output as given (i.e., the firms
behave as Cournot oligopolists), what will be the equilibrium quantities
selected by each firm? What is total output, and what is the market price? What
are the profits for each firm?
Find the best response functions (the reaction curves) for
both firms by setting marginal revenue equal to marginal cost (alternatively
you can set up the profit function for each firm and differentiate with respect
to the quantity produced for that firm):
R1
= P q1 = (300 -
3(q1 + q2))
q1 = 300q1
- 3q12 - 3q1q2.
MR1 =
300 - 6q1 - 3q2
MC1
= 30 +
3q1
300
- 6q1 - 3q2 = 30 +
3q1
q1 =
30 - (1/3)q2.
By symmetry, BBBS’s best response function will be:
q2 =
30 - (1/3)q1.
Cournot equilibrium occurs at the intersection of these
two best response functions, which is:
q1 =
q2 = 22.5.
Thus,
Q = q1 + q2 = 45
P
= 300 -
3(45) = $165.
Profit for both firms will be equal
and given by:
p = R -
C = (165)(22.5) - [30(22.5) +
1.5(22.52)] = $2278.13.
b. It occurs to the
managers of WW and BBBS that they could do a lot better by colluding.
If the two firms collude, what will be the profit-maximizing choice of output? The
industry price? The output and the profit for each firm in this case?
In this case the firms should each produce half the
quantity that maximizes total industry profits (i.e., half the monopoly
output). Note that if the two firms had different cost functions, then it would
not be optimal for them to split the monopoly output evenly.
Joint profits will be (300 -
3Q)Q - 2[30(Q/2) + 1.5(Q/2)2] = 270Q -
3.75Q2, which will be maximized at Q = 36. You can find this quantity by
differentiating the profit function with respect to Q, setting the
derivative equal to zero, and solving for Q: dp/dQ =
270 - 7.5Q = 0,
so Q = 36.
The optimal output for each firm is q1 = q2 = 36/2 =
18, and the optimal price for the firms to charge is P = 300 -
3(36) = $192.
Profit for each firm will be p = (192)(18) -
[30(18) + 1.5(182)] = $2430.
c. The managers of
these firms realize that explicit agreements to collude are illegal. Each firm must decide on its own whether to produce
the Cournot quantity or the cartel quantity. to aid in making the decision, the manager of WW constructs a
payoff matrix like the one below. Fill in each box with the profit of WW and
the profit of BBBS. Given this payoff matrix, what output strategy is each firm
likely to pursue?
To fill in the payoff matrix, we have to calculate the
profit each firm would make with each of the possible output level
combinations. We already know the profits if both choose the Cournot output or
both choose the cartel output. If WW produces the Cournot level of output
(22.5) and BBBS produces the collusive level (18), then:
Q = q1
+ q2 = 22.5 +
18 = 40.5
P = 300 - 3(40.5) =
$178.50.
Profit for WW =
(178.5)(22.5) - [30(22.5) + 1.5(22.52)] = $2581.88.
Profit for BBBS =
(178.5)(18) - [30(18) + 1.5(182)] = $2187.
If WW chooses the collusive
output level and BBBS chooses the Cournot output, profits will be reversed. Rounding
off profits to whole dollars, the payoff matrix is as follows.
|
Profit Payoff Matrix
(WW profit, BBBS profit)
|
BBBS
|
|
Produce Cournot q
|
Produce Cartel q
|
|
W
|
Produce Cournot q
|
2278, 2278
|
2582, 2187
|
|
W
|
Produce Cartel q
|
2187, 2582
|
2430, 2430
|
For each firm, the
Cournot output dominates the cartel output, because each firm’s profit is
higher when it chooses the Cournot output, regardless of the other
firm’s output. For example, if WW chooses the Cournot output, BBBS earns
$2278 if it chooses the Cournot output but only $2187
if it chooses the cartel output. On the other hand, if WW chooses the cartel
output, BBBS earns $2582 with the Cournot output, which is better than the
$2430 profit it would make with the cartel output. So no matter what WW
chooses, BBBS is always better off choosing the Cournot output. Therefore,
producing at the Cournot output levels will be the Nash Equilibrium in this
industry.
This is a prisoners’ dilemma game, because both firms
would make greater profits if they both produced the cartel output. The cartel
profit of $2430 is greater than the Cournot profit of $2278. The problem is
that each firm has an incentive to cheat and produce the Cournot output instead
of the cartel output. For example, if the firms are colluding and WW continues
to produce the cartel output but BBBS increases output to the Cournot level,
BBBS increases its profit from $2430 to $2582. When both firms do this,
however, they wind up back at the Nash-Cournot equilibrium where each produces
the Cournot output level and each makes a profit of only $2278.
d. Suppose WW can set
its output level before BBBS does. How much will WW choose to produce in
this case? How much will BBBS produce? What is the market price, and what
is the profit for each firm? Is WW better off by choosing its output first?
Explain why or why not.
WW will use the Stackelberg strategy. WW knows that BBBS
will choose a quantity q2, which will be its best response to q1
or:
.
WW’s profits will be:
Profit maximization implies:
.
This results in q1 = 25.7 and q2 = 21.4. The equilibrium price and profits
will be:
P = 300 -
3(q1 + q2)
= 300 -
3(25.7 + 21.4) = $158.70
p1
= (158.70)(25.7) - [(30)(25.7) +
1.5(25.7)2] = $2316.86
p2
= (158.70)(21.4) - [(30)(21.4) +
1.5(21.4)2] = $2067.24.
WW is able to benefit from its first-mover advantage by
committing to a high level of output. Since BBBS moves after WW has selected
its output, BBBS can only react to the output decision of WW. If WW produces its
Cournot output as a leader, BBBS produces its Cournot output as a follower. Hence
WW cannot do worse as a leader than it does in the Cournot game. When WW
produces more, BBBS produces less, raising WW’s profits.
11. Two firms compete by choosing price. Their demand
functions are
Q1 =
20 -
P1 + P2 and
Q2 = 20 + P1 - P2
where P1 and P2
are the prices charged by each firm, respectively, and Q1 and
Q2 are the resulting demands. Note that the demand for each
good depends only on the difference in prices; if the two firms colluded and
set the same price, they could make that price as high as they wanted, and earn
infinite profits. Marginal costs are zero.
a. Suppose the two
firms set their prices at the same time. Find the resulting Nash
equilibrium. What price will each firm charge, how much will it sell, and what
will its profit be? (Hint: Maximize the profit of each firm with respect
to its price.)
To determine the Nash equilibrium in prices, first
calculate the reaction function for each firm, then solve for price. With zero
marginal cost, profit for Firm 1 is:
Marginal revenue is the slope of the total revenue
function (here it is the derivative of the profit function with respect to P1
because total cost is zero):
MR1 =
20 - 2P1 + P2.
At the profit-maximizing price, MR1 = 0. Therefore,
This is Firm 1’s reaction function. Because Firm 2 is
symmetric to Firm 1, its reaction function is
Substituting Firm 2’s
reaction function into that of Firm 1:
By symmetry, P2 = $20.
To determine the quantity produced by each firm,
substitute P1 and P2 into the demand
functions:
Q1 =
20 - 20 +
20 = 20, and
Q2 = 20 + 20 - 20 =
20.
Profits for Firm 1 are P1Q1
= $400, and, by symmetry, profits for
Firm 2 are also $400.
b. Suppose Firm 1 sets
its price first and then Firm 2 sets its price. What price will each
firm charge, how much will it sell, and what will its profit be?
If Firm 1 sets its price first, it takes Firm 2’s reaction
function into account. Firm 1’s profit function is:
To determine the profit-maximizing price, find the change
in profit with respect to a change in price:
Set this expression equal to zero to find the profit-maximizing
price:
30 - P1
= 0, or P1 = $30.
Substitute P1 in Firm 2’s reaction
function to find P2:
At these prices,
Q1 =
20 - 30 +
25 = 15, and
Q2 = 20 + 30 - 25 =
25.
Profits are
p1 = (30)(15) =
$450 and
p2
= (25)(25) = $625.
If Firm 1 must set its price first, Firm 2 is able to
undercut Firm 1 and gain a larger market share. However, both firms make
greater profits than they did in part a, where they chose prices
simultaneously.
c. Suppose you are one
of these firms and that there are three ways you could play the game: (i) Both
firms set price at the same time; (ii) You set price first; or (iii) Your
competitor sets price first. If you could choose among these options, which
would you prefer? Explain why.
Compare the Nash profits in part a, $400, with the profits
in part b, $450 for the firm that sets price first and $625 for the follower. Clearly
it is best to be the follower, so you should choose option (iii). From the
reaction functions, we know that the price leader raises price and provokes a
price increase by the follower. By being able to move second, however, the
follower increases price by less than the leader, and hence undercuts the
leader. Both firms enjoy increased profits, but the follower does better.
12. The dominant firm
model can help us understand the behavior of some cartels. Let’s apply this
model to the OPEC oil cartel. We will use isoelastic curves to describe world
demand W and noncartel (competitive) supply S. Reasonable numbers
for the price elasticities of world demand and noncartel supply are -1/2 and 1/2, respectively. Then, expressing W
and S in millions of barrels per day (mb/d), we could
write
and 
Note that OPEC’s net demand is D = W - S.
a. Draw the world
demand curve W, the non-OPEC supply curve S, OPEC’s net demand
curve D, and OPEC’s marginal revenue curve. For purposes of
approximation, assume OPEC’s production cost
is zero. Indicate OPEC’s optimal price, OPEC’s optimal production, and
non-OPEC production on the diagram. Now, show on the diagram how the various
curves will shift and how OPEC’s optimal price will change if non-OPEC supply
becomes more expensive because reserves of oil start running out.
OPEC’s initial net demand curve is
Marginal revenue is quite difficult to
find. If you were going to determine it analytically, you would have to solve
OPEC’s net demand curve for
P. Then take that expression and multiply by
Q (
=D) to get total
revenue as a function of output. Finally, you would take the derivative of
revenue with respect to
Q. The
MR curve would look approximately
like that shown in the figure below.
OPEC’s optimal production, Q*, occurs where MR = 0 (since production cost is assumed to be
zero), and OPEC’s optimal price, P*, is found from the net demand curve
at Q*. Non-OPEC production, QN, can be read off the
non-OPEC supply curve, S, at price P*.
Now, if non-OPEC oil becomes more expensive, the supply
curve S shifts to S¢.
This shifts OPEC’s net demand curve outward from D to D ¢,
which in turn creates a new marginal revenue curve, MR¢,
and a new optimal OPEC production level of Q¢, yielding a new higher price
of P ¢.
At this new price, non-OPEC production is QN¢. The
new S, D, and MR curves are dashed lines. Unfortunately,
the diagram is difficult to sort out, but OPEC’s new optimal output has
increased to around 30, non-OPEC supply has dropped to about 10, and the
optimal price has increased slightly.
b. Calculate
OPEC’s optimal (profit-maximizing) price. (Hint: Because OPEC’s cost is
zero, just write the expression for OPEC revenue and find the price that
maximizes it.)
Since costs are zero, OPEC will choose a price that
maximizes total revenue:
To determine the profit-maximizing price, take the
derivative of profit with respect to price and set it equal to zero:
Solving for P,
At this price, W =
40, S = 13.33, and D = 26.67 as shown in the first diagram.
c. Suppose the
oil-consuming countries were to unite and form a “buyers’ cartel” to gain
monopsony power. What can we say, and what can’t we say, about the impact this
action would have on price?
If the oil-consuming countries unite to form a buyers’
cartel, then we have a monopoly (OPEC) facing a monopsony (the buyers’ cartel).
As a result, there are no well-defined demand or supply curves. We expect that
the price will fall below the monopoly price when the buyers also collude,
because monopsony power offsets some monopoly power. However, economic theory
cannot determine the exact price that results from this bilateral monopoly
because the price depends on the bargaining skills of the two parties, as well
as on other factors such as the elasticities of supply and demand.
13. Suppose the market
for tennis shoes has one dominant firm and five fringe firms. The market demand
is Q = 400 - 2P. The dominant firm has a constant
marginal cost of 20. The fringe firms each have a marginal cost of MC = 20 +
5q.
a. Verify that the
total supply curve for the five fringe firms is
.
The total supply curve for the five firms is found by
horizontally summing the five marginal cost curves, or in other words, adding
up the quantity supplied by each firm for any given price. Rewrite each fringe
firm’s marginal cost curve as follows:
Since each firm is identical, the supply curve is five
times the supply of one firm for any given price:
.
b. Find the dominant
firm’s demand curve.
The dominant firm’s demand curve is given by the
difference between the market demand and the fringe total supply curve:
.
c. Find the
profit-maximizing quantity produced and price charged by the dominant firm, and
the quantity produced and price charged by each of the fringe firms.
The dominant firm will set marginal revenue equal to
marginal cost. The marginal revenue curve has the same intercept and twice the
slope of the linear inverse demand curve, which is shown below:
Now set marginal revenue equal to marginal cost to find
the profit-maximizing quantity for the dominant firm, and the price charged by
the dominant firm:
Each fringe firm will charge the same $80 price as the
dominant firm, and the total output produced by the five fringe firms will be
Each fringe firm will
therefore produce 12 units.
d. Suppose there are ten
fringe firms instead of five. How does this change your results?
We need to find the new fringe supply curve, dominant firm
demand curve, and dominant firm marginal revenue curve as above. The new total
fringe supply curve is
The new dominant firm
demand curve is
The new dominant firm
marginal revenue curve is
The dominant firm will
produce where marginal revenue is equal to marginal cost which occurs at 180
units. Substituting a quantity of 180 into the demand curve faced by the
dominant firm results in a price of $65. Substituting the price of $65 into the
total fringe supply curve results in a total fringe quantity supplied of 90, so
that each fringe firm will produce 9 units. Increasing the number of fringe
firms reduces market price from $80 to $65, increases total market output from
240 to 270 units, and reduces the market share of the dominant firm from 75% to
67% (although the dominant firm continues to sell 180 units).
e. Suppose there
continue to be five fringe firms but that each manages to reduce its marginal
cost to MC = 20 + 2q. How does this change your
results?
Follow the same method as in earlier parts of this
problem. Rewrite the fringe marginal cost curve as
The new total fringe supply curve is five times the
individual fringe supply curve, which is also the fringe marginal cost curve:
The new dominant firm demand curve is found by subtracting
the fringe supply curve from the market demand curve to get
The new inverse demand curve for the dominant firm is
therefore,
.
The dominant firm’s new marginal revenue curve is
Set MR = MC
= 20. The dominant firm will produce
180 units and will charge a price of
Therefore, price drops from $80 to $60. The fringe firms
will produce a total of
units, so total
industry output increases from 240 to 280. The market share of the dominant
firm drops from 75% to 64%.
14. A lemon-growing
cartel consists of four orchards. Their total cost functions are:
TC is in hundreds of dollars, and Q is
in cartons per month picked and shipped.
a. Tabulate total,
average, and marginal costs for each firm for output levels between 1 and 5
cartons per month (i.e., for 1, 2, 3, 4, and 5 cartons).
The following tables give
total, average, and marginal costs for each firm.
|
|
Firm
1
|
Firm 2
|
|
Units
|
TC
|
AC
|
MC
|
TC
|
AC
|
MC
|
|
0
|
20
|
__
|
__
|
25
|
__
|
__
|
|
1
|
25
|
25
|
5
|
28
|
28
|
3
|
|
2
|
40
|
20
|
15
|
37
|
18.5
|
9
|
|
3
|
65
|
21.67
|
25
|
52
|
17.33
|
15
|
|
4
|
100
|
25
|
35
|
73
|
18.25
|
21
|
|
5
|
145
|
29
|
45
|
100
|
20
|
27
|
|
|
Firm
3
|
Firm 4
|
|
Units
|
TC
|
AC
|
MC
|
TC
|
AC
|
MC
|
|
0
|
15
|
__
|
__
|
20
|
__
|
__
|
|
1
|
19
|
19
|
4
|
26
|
26
|
6
|
|
2
|
31
|
15.5
|
12
|
44
|
22
|
18
|
|
3
|
51
|
17
|
20
|
74
|
24.67
|
30
|
|
4
|
79
|
19.75
|
28
|
116
|
29
|
42
|
|
5
|
115
|
23
|
36
|
170
|
34
|
54
|
b. If the cartel
decided to ship 10 cartons per month and set a price of $25 per carton, how
should output be allocated among the firms?
The cartel should assign
production such that the lowest marginal cost is achieved for each
unit, i.e.,
|
Cartel
Unit Assigned
|
Firm
Assigned
|
MC
|
|
1
|
2
|
3
|
|
2
|
3
|
4
|
|
3
|
1
|
5
|
|
4
|
4
|
6
|
|
5
|
2
|
9
|
|
6
|
3
|
12
|
|
7
|
1
|
15
|
|
8
|
2
|
15
|
|
9
|
4
|
18
|
|
10
|
3
|
20
|
Therefore, Firms 1 and 4 produce
two units each and Firms 2 and 3 produce three units each.
c. At this shipping
level, which firm has the most incentive to cheat? Does any firm not
have an incentive to cheat?
At this level of output, Firm 2 has the lowest marginal
cost for producing one more unit beyond its allocation, i.e., MC = 21 for the fourth unit for Firm 2. In
addition, MC = 21 is less than
the price of $25. For all other firms, the next unit has a marginal cost equal
to or greater than $25. Firm 2 has the most incentive to cheat, while Firms 3
and 4 have no incentive to cheat, and
Firm 1 is indifferent.
