Industrial Organization: Markets and Strategies
Paul Belleflamme and Martin Peitz
published by Cambridge University Press
Part VI. Theory of competition policy
Exercises
Exercise 1 Industries with cartels
Briefly describe and analyze a case of your choice concerning a price- or
quantity-fixing cartel (please not OPEC). The following questions may be useful
to bear in mind: What are the relevant characteristics of the industry? What
was the scope of the cartel? How was the cartel enforced? What were the
effects of the cartels? How did the competition authority or court argue and
what was the decision, if any? [suggested length: 1-2 pages, timesnewroman 12
pt, doublespaced]
Exercise 2 Collusion and pricing
Two (advertising-free) newspapers compete in prices for an infinite number
of days. The monopoly profits (per day) in the newspaper market are π M and
the discount rate (per day) is δ. If the newspapers compete in prices, they both
earn zero profits in the static Nash equilibrium. Finally, if the firms set the
same price, they split the market equally and earn the same profits.
1. The newspapers would like to collude on the monopoly price. Write down
the strategies that the newspapers could follow to achieve this outcome.
Find the discount rates for which they are able to sustain the monopoly
price using these strategies.
2. On Sundays, the newspapers sell a weekly magazine (that can be bought
without buying the newspaper). The monopoly (competitive) profits when
selling the magazine are also π M (zero).
3. For which discount rates can the monopoly price be sustained only in the
market for magazines? (Write down the equation that characterizes the
solution.) Compare the solution found in question 1 and 2 and comment
briefly.
4. For which discount rates can the monopoly price be sustained both in the
market for newspapers and in the market for magazines? (Write down the
equation that characterizes the solution.)
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Exercise 3 Collusion and pricing II
Consider a homogeneous-product duopoly. The two firms in the market
are assumed to have constant marginal costs of production equal to c. The
two firms compete possibly over an infinite time horizon. In each period they
simultaneously set price pi , i = 1, 2. After each period the market is closed
down with probability 1 − δ.
Market demand Q(p) is decreasing, where p = min{p1 , p2 }. Suppose, furthermore, that the monopoly problem is well defined, i.e. there is a solution
pM = arg maxp pQ(p). If firms set the same price, they share total demand
with weight λ for firm 1 and 1 − λ for firm 2. Suppose that λ ∈ [1/2, 1).
Suppose that firms use trigger strategies and Nash punishment.
1. Suppose that δ = 0. Derive the equilibrium of the game.
2. Suppose that δ > 0 and λ = 1/2. Derive the condition according to which
firm 1 and firm 2 do not find it profitable to deviate from the collusive
price pM .
3. Suppose that δ > 0 and λ > 1/2. Derive the condition according according
to which pM is played along the equilibrium path. Show that the condition
is the more stringent the higher λ.
4. Show that previous results in (3) also hold for any collusive price pC ∈
(c, pM ).
5. Suppose that δ > 0 and λ > 1/2 and that firm 1 can only adjust its
price every τ periods. Derive the condition according to which pM is
played along the equilibrium path. How does the time span τ influence
the condition?
Exercise 4 Parallel pricing and evidence of collusion
A competition policy authority has noticed that the firms in the Lysine
industry consistently charge very similar prices, and the suspicion is that they
are colluding. Do you think that parallel pricing is proof of collusion? If not,
what kind of evidence would you look for?
Exercise 5 Collusion and quantity competition
Consider the following market: Two firms compete in quantities, i.e., they
are Cournot competitors. The firms produce at constant marginal costs equal
to 20. The inverse demand curve in the market is given by P (q) = 260 − q.
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1. Find the equilibrium quantities under Cournot competition as well as the
quantity that a monopolist would produce. Calculate the equilibrium profits in Cournot duopoly and the monopoly profits. Suppose that the firms
compete in this market for an infinite number of periods. The discount
factor (per period) is δ, δ ∈ (0, 1).
2. The firms would like to collude in order to restrict the total quantity
produced to the monopoly quantity. Write down strategies that the firms
could use to achieve this outcome.
3. For which values of δ is collusion sustainable using the strategies of question (b)? [Hint: Think carefully about what the optimal deviation is.]
Exercise 6 Cournot mergers: profitability and welfare properties
Consider a homogeneous good duopoly with linear demand P (q) = 12 − q,
where q is the total industry output, and constant marginal costs c = 3.
1. Suppose that firms simultaneously set quantities. Determine the equilibrium (price, quantities, profit, welfare).
2. The firms consider to merge although their production costs are not affected. Determine the solution to this problem. Is such a merger profitable? What are the welfare effects of such a merger?
3. Suppose that the merger is efficiency enhancing, leading to marginal costs
cm < c. What are the welfare effects of such a merger. Do you possibly
have to qualify your answer in (2)?
4. Consider the possibility of firm entry after the merger (the entrant produces at marginal costs c = 3 and has entry cost e). Suppose first that the
merger is not efficiency-enhancing. Analyze such a market and comment
on your result (depending on the entry cost e). Suppose next that the
merger is efficiency-enhancing, i.e. cm < 3. Depending on cm and e, when
is a merger profitable? [Hint: Calculate profits for cm = 1/2.]
5. Many countries scrutinize merger and sometimes block them (or impose
remedies)? Discuss which factors should make the courts or the competition authority more inclined to block a merger.
Exercise 7 Cournot mergers and synergies.
Consider a homogeneous-product Cournot oligopoly with 4 firms. Suppose
that the inverse demand function is P (q) = 64 − q.
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1. Suppose that firms incur a constant marginal cost c = 4. Characterize
the Nash equilibrium of the game in which all firms simultaneously choose
quantity.
2. Suppose that firms 1 and 2 consider to merge and that there are synergies
leading to marginal costs cm < c. Characterize the Nash equilibrium. At
which level cm (you may want to give an approximate number) are the
two firms indifferent whether to merge?
3. Is such a merger that just makes the two firms indifferent between merging
and non-merging consumer-welfare increasing?
4. At which level cm would the merger be consumer-welfare neutral?
5. Suppose that instead firms 1, 2, and 3 consider to merge. The new marginal cost of the merged firms is cn < c. At which level cn are the three
firms indifferent whether to merge?
6. Compare your findings in (5) and (2). What can you say about incentives
to merge in this case?
Exercise 8 Cournot mergers and demand
Consider the following Cournot merger game. The inverse demand function
is of the form
P (q) = a − q γ
where γ > 0 and there are n firms with constant marginal costs of production
c.
1. Discuss the shape of the inverse demand function depending on γ.
2. Determine the Cournot equilibrium profits in this set-up.
3. Determine all n for which a single merger, i.e. going from n to n − 1 firms,
is profitable.
Exercise 9 Burning ships
Hernan Cortéz, the Spanish conqueror (”conquistador”), is said to have
burned his ships upon the arrival to Mexico. Why would he do such a thing?
Exercise 10 Quantity commitment
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Up to two firms are in a market in which quantities are the strategic variable.
There are two periods; in the first period firm 1 is a protected monopolist.
In each of the two periods t = 1, 2 the inverse demand function P t is given
by P t (xt ) = 20 − xt . In each period the cost function of firm i is given by
Cit (xti ) = 9 + 4xti . Profits of a firm are the sum of its profits in each period (no
discounting). Firms maximize profits by setting quantities.
1. Determine the monopoly solution.
2. Because of technological restrictions firm 1 has to choose the same quantity
in each period (x11 = x21 ). Observing x11 , firm 2 is considering to enter in
period 2. Determine the profit maximizing x22 given x11 .
3. Assume that firm 2 will enter in period 2. What quantity will firm 1
produce? Determine equilibrium prices, quantities, and profits.
4. Firm 2 only enters in period 2 if it can make positive profits. Determine
the subgame perfect equilibrium of the two-period model.
Exercise 11 Strategic quantity choice
Consider a market with two firms, A and B. The firms produce homogenous
goods, compete in quantities, and face a constant marginal cost equal to 1/4.
The timing is the following: First, firm A chooses its quantity qA . Then, after
observing qA , firm B chooses its quantity qB . The price in the market is given
by the inverse demand function P (q) = 1 − q, where q = qA + qB .
1. Find the subgame perfect Nash equilibrium.
2. Assume from now that there is an entry cost of e. Firm A is already
established in the market, and firm B is considering whether to establish
itself in the market or not. The timing is the following: First, firm A
chooses its quantity qA . Then, after observing qA , firm B decides whether
to enter the market and, in case of entry, how much to produce.
3. Write down firm B’s profit function.
4. Assume for now that e = 1/10. Illustrate firm A’s profit as a function of
qA when the reaction of firm B is taken into account.
5. Find the subgame perfect Nash equilibrium for all values of e > 0.
Exercise 12 Taxonomy of entry-related strategies I
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Consider a market with differentiated products. In the first stage firm 1 is
the incumbent firm and can invest an amount K1 ≥ 0 in reducing its marginal
costs, c(K1 ) = c − K1 /10. In stage two firm 2 can decides about entering the
market with constant marginal costs of c and entry costs of e. In stage three if
entry takes place firms engage in price competition and face symmetric demand
functions given by Di (pi , pj ) = A − api + bpj (A > a > b > 0). If no entry takes
place, firm 1 acts as a monopolist with demand, D1 (p1 ) = A − ap1 .
1. Calculate the best response functions for both firms. Draw a graph.
Argue graphically from now on:
2. Does an increase in K1 increase or decrease the profit of the entering firm?
Does an increase in K1 make the incumbent tough or soft?
3. Does a marginal investment K1 increase or decrease the profit of the incumbent? (Assume that e is sufficiently low such that entry takes place
for K1 close to zero. Moreover, assume A ≤ 10)
4. Use your answer of (2): Is entry deterrence via cost reduction possible in
this setting? If your answer is YES, which numbers would you have to
compare to decide whether entry deterrence is optimal? If your answer is
NO, what do we have change in this model to induce entry deterrence?
5. Use your answer of (3): If entry accommodation is optimal how much
should firm 1 invest in cost reduction?
6. How would you answer to (4) change if we consider a Cournot game instead?
7. How would you answer to (5) change if we consider a Cournot game instead?
Exercise 13 Taxonomy of entry-related strategies II
Consider the market from the previous exercise again.
1. Use the best-response functions from the previous exercise to calculate
equilibrium prices for c1 6= c2 in the Nash equilibrium in which firms set
prises.
2. Use your result from (1) to show that p∗i = (A + ac)/(2a − b), for i = 1, 2,
if c1 = c2 = c.
Now, use again that c1 (K1 ) = c − (K1 /10) and c2 = c.
Set c = 4, a = 2, b = 1, A = 10, and F = 7.95.
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3. Show that the critical level of K1D to deter entry is below 0.5.
4. Suppose that investment in cost reduction is restricted to half units, i.e.
K1 ∈ {0, 0.5, 1, 1.5, . . .}. Will firm 1 deter entry in a subgame perfect Nash
equilibrium? State firm 1’s optimal business strategy.
5. Reconsider your answer to (4) if firm 1 as a monopolist faces a demand
of D1 (p1 ) = A − ap1 + bp1 = A − (a − b)p1 .
Exercise 14 Sequential quantity choice and entry
Consider a market for a homogenous good with one incumbent firm (firm
1) and one potential entrant (firm 2). The interaction between the two firms
evolves in two stages. In stage 1, firm 1 chooses its quantity q1 . In stage 2, after
observing q1 , firm 2 decides whether or not to enter the market. If it enters,
it incurs an entry cost e and chooses its own quantity, q2 . If firm 2 does not
enter then q2 = 0 and firm 2 does not pay the entry cost e (firm 1 then is a
monopoly). Assume that the inverse demand for the good is P = a − (q1 + q2 ),
and that the cost of production of each firm i is C(qi ) = qi /2.
1. Compute the range of e for which entry is blockaded. That is, compute
firm 1’s output when it operates as a monopolist, then given this quantity,
compute the highest profit that firm 2 can earn if it decides to enter, and
finally, compute the range of e for which entry is blockaded.
2. Now, suppose that e is sufficiently low to ensure that entry is not blockaded. Compute the quantities and profits of each firm when entry is
accommodated. That is, compute the outputs that will be selected in
a Stackelberg equilibrium and the resulting profits. (Instruction: first,
compute firm 2’s best response function, br2 (q1 ). Second, substitute for
br2 (q1 ) into firm 1’s profit function and compute firm 1’s profit-maximizing
quantity q1∗ . Third, find firm 2’s best response against q1∗ , using firm 2’s
best response function. Finally, given the pair of quantities you found,
compute the equilibrium profits).
3. Compute the lowest q1 for which entry is deterred. Compute firm 1’s
profits at this output level.
4. Given your answer in (3), show firm 2’s best response function graphically
in the quantities space (recall that firm 2 may wish to stay out of the market when q1 is relatively high). Show on the same graph the Stackelberg
equilibrium you found in Section (3) and the lowest q1 for which entry is
deterred.
5. Given your answers in (2) and (3), find the range of e for which entry
is accommodated, and the range of e for which it is deterred. Explain
in no more than 3 sentences the intuition for the result (i.e., why is it
natural to expect that entry is accommodated/deterred when e is relatively
low/high).
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Exercise 15 Capacity choice and entry
Consider an industry for a homogenous product with a single firm (firm 1)
that can produce at zero cost. The demand function in the industry is given by
Q = a − p. Now suppose that a second firm (firm 2) considers entry into the
industry. Firm 2 can also produce at zero cost. If firm 2 enters, firms 1 and 2
compete by setting prices. Consumers buy from the firm that sets the lowest
price. If both firms charge the same prices, consumers buy from firm 1.
1. Solve for the Nash equilibrium if firm 2 chooses to enter the industry.
Would firm 2 wish to enter if entry required some initial investment?
2. Now suppose that before it enters, firm 2 can choose a capacity, x2 , and a
price p2 (the capacity x2 means that firm 2 can produce no more than q2 =
x2 units). Given q2 and p2 , firm 1 chooses its price and then consumers
decide who to buy from. Compute the Nash equilibrium in the product
market if firm 1 chooses to fight firm 2. What is firm 1’s profit in this
case? Show firm 1’s profit in a graph that has quantity on the horizontal
axis and price on the vertical axis. Would firm 2 choose to produce in
that case?
3. Now suppose that firm 1 decides to accommodate the entry of firm 2.
Compute the residual demand that firm 1 faces after firm 2 sells q2 units,
and then write the maximization problem of firm 1 and solve it for p1 .
What is firm 1’s profit if it decides to accommodate firm 2’s entry? Draw
firm 1’s profit in a graph that has quantity on the horizontal axis and price
on the vertical axis. Would firm 2 wish to enter in this case?
4. Given your answers to (2) and (3), compute for each p2 the largest capacity
that firm 2 can choose without inducing firm 1 to fight it. (Hint: to answer
the question you need to solve a quadratic equation. The solution is given
by the small root).
5. Show that the capacity you computed in (4) is decreasing with p2 . Explain
the intuition for your answer. Given your answer, explain how firm 2 will
choose its price. Computing p2 is too complicated; you are just asked to
explain in words how firm 2 chooses p2 .)
Exercise 16 Investment and incumbency
Consider a differentiated product market. At the first stage firm 1 is the
incumbent firm and can invest an amount I1 ≥ 0 in reducing its marginal
costs, c(I1 ) = c − I1 /10. At stage two firm 2 decides whether to enter the
market with constant marginal costs of c and entry costs of e, which is sunk
at this stage. At stage three, if entry has taken place, firms engage in quantity
competition and face inverse demand functions given by Pi (qi , qj ) = a−bqi −dqj
(a > b > d > 0). If no entry takes place, firm 1 acts as a monopolist with inverse
demand, P m (q1 ) = a − bq1 .
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1. Calculate the best response functions for both firms. Draw a graph.
2. Does an increase in I1 increase or decrease the profit of the entering firm?
Does an increase in I1 make the incumbent tough or soft?
3. Does a marginal investment I1 > 0 increase or decrease the profit of the
incumbent? (Assume that e is sufficiently low such that entry takes place
for I1 close to zero.)
4. If entry accommodation is optimal how much should firm 1 invest in cost
reduction? (Use your answer of (3).)
5. Is entry deterrence via cost reduction possible and profitable in this setting? Discuss your results in the light of the taxonomy developed in the
book.
Exercise 17 Competition and entry
Consider a homogeneous good duopoly with linear demand P (q) = 1 − q,
where q is the total industry output. Suppose that firms are quantity setters
and firms incur constant marginal costs of production ci .
1. Suppose that firms have constant marginal costs of production c. Determine the Nash equilibrium in quantities (report prices, quantities, profit,
welfare)
2. Reconsider your answer in (1) because of the following: A tabloid runs a
series on consumers paying “excessive” prices. The government considers
introducing a non-negative special sales tax t ≥ 0 per unit on this product (and plans to use the revenues for some project from which nobody
benefits). Determine the welfare-maximizing tax rate (the government is
assumed to be able to commit to the tax; welfare is total surplus which includes tax revenues). Discuss your result. What would be your conclusion
if the government was considering subsidizing the firm?
3. Return to the case without taxes. Consider now the duopoly with c1 = 0
and c2 = c ∈ [0, 1]. Determine the equilibrium (price, quantities, profit,
welfare).
4. Consider now an extended model in which only firm 1 is necessarily
present. At stage 1, firm 1 can make an investment I after which firm
2’s marginal costs is c2 = 1/2 instead of c2 = 0. Afterwards, firm 2 observes the investment decision of firm 1 and, at stage 2, decides whether
to enter at a negligible entry cost e > 0. At stage 3, active firms set
quantities simultaneously. Determine the subgame perfect equilibrium of
this game. Discuss your result in the light of what you have learnt reading
about entry-related strategies (max 3 sentences).
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5. Consider now a different entry model. Both firms have zero marginal costs
of production but consumers have become accustomed to product 1 (even
if they did not consume it themselves). Therefore, consumers are willing
to pay 1/2 money units less for product 2 than for product 1. The inverse
demand function P (q) = 1 − q gives demand for product 1. At stage 1,
the potential entrant, firm 2, considers to enter the market at an entry
cost e > 0. (There is no investment stage in this game.) At stage 2,
firms set quantities simultaneously. Report the profit function for each
firm. Determine the equilibrium in case firm 2 has entered (report prices,
quantities, profit, welfare). Determine the subgame perfect equilibrium
and comment on your result. You may want to reuse some of the results
derived above.
Exercise 18 Non-linear pricing in the supply chain
A monopolist produces a good with constant marginal cost equal to c, c <
1. Assume for now that all consumers have the demand Q(p) = 1 − p. The
population is of size 1.
1. Suppose that the monopolist cannot discriminate in any way among the
consumers and has to charge a uniform price, pU . Calculate both the price
that maximizes profits and the profits that correspond to this price.
2. Suppose now that the monopolist can charge a two-part tariff (m, p) where
m is the fixed fee and p is the price per unit. Expenditure then is m + pq.
Calculate the two-part tariff that maximizes profits and the profits that
correspond to this tariff. Compare pU and p and comment briefly.
3. Compare the situation with a uniform price and a two-part tariff in terms
of welfare (a verbal argument is sufficient).
4. Assume now instead that there are two types of consumers. The consumers
of type 1 have the demand Q1 (p) = 1 − p, and the consumers of type 2
have the demand Q2 (p) = 1 − p/2. The population is of size 1 and there
are equally many consumers of the two types. Finally, it is assumed in this
question that c = 1/2. Calculate the two-part tariff that maximizes the
profits of the monopolist. Compare the two-part tariffs found in questions
(2) and (3) for c = 1/2 and comment briefly.
Exercise 19 RPM
RPM was common in a number of industries. In particular, it could be
observed in the clothing, consumer electronics, and food industry. What is the
probable motivation for firms to use RPM in these industries. Discuss the likely
welfare consequences in these industries.
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Exercise 20 Vertical contracting
A buyer wants to buy one unit of a good from an incumbent seller. The
buyer’s valuation of the good is 1, while the seller’s cost of producing it is
1/2. Before the parties trade, a rival seller enters the market and his cost, c, is
distributed on the unit interval according to a distribution function with density
g(c). The two sellers then simultaneously make price offers and the buyer trades
with the seller who offers the lowest price. If the two sellers offer the same price
the buyer buys from the seller whose cost is lower.
1. Determine the price that the buyer pays in equilibrium, p∗ , as a function
of c. Given p∗ , write the payoffs of the expected payoffs of the buyer and
the two sellers.
2. Suppose that the distribution of c is uniform. Show p∗ and the expected
payoffs of the parties graphically (put c on the horizontal axis and the
equilibrium price function on the vertical axis and show the payoffs by
pointing out the appropriate areas in the graph).
3. Now suppose that the incumbent seller offers the buyer a contract before
the entrant shows up. The contract requires the buyer to pay the incumbent seller the amount m regardless of whether he buys from him or from
the entrant, and gives the buyer an option to buy from the incumbent at a
price of p (this is equivalent to giving the buyer an option to buy at a price
m + p and requiring him to pay liquidated damages of m if he switches to
the entrant). If the buyer rejects the contract things are as in part (1).
Given p and c, what is the price that the buyer will end up paying for the
good? Using your answer, write the expected payoffs of the buyer and the
two sellers as a function of p and m.
4. Explain why the incumbent seller will choose p by maximizing the sum of
his expected payoff, π I , and the buyer’s expected payoffs, UB .
5. Write the first-order condition for p and show that the profit-maximizing
price of the incumbent seller, p∗∗ , is such that p∗∗ < 1/2. Also show that
if g(0) > 0 then p∗∗ > 0.
6. Explain why the contract is socially inefficient. Is the outcome in part (1)
socially efficient? Explain the intuition for your answer.
7. Compute p∗∗ assuming that the distribution of c is uniform, and show the
expected payoffs of the parties and the social loss graphically (again, put
c on the horizontal axis and the equilibrium price function on the vertical
axis).
8. Compute p∗∗ under the assumption that G(c) = cα , where α > 0. How
does p∗∗ vary with α? Give an intuition for this result.
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Exercise 21 Franchising
A monopolistic manufacturer produces a good that is sold to three retailers.
The manufacturer has constant marginal cost equal to c < 12 . The retailers are
monopolists in three different cities. The marginal cost of the retailers is equal
to the wholesale price of the good. Each of the retailers faces a demand function
Qb = 1 − b p where p is the retail and b is a variable characterizing the local
demand function. Each retailer knows the value of b in its city. Furthermore, it
is common knowledge that b = 1 in one city and b = z in two cities, z ∈ (1, 2].
The manufacturer offers each of the retailers a two-part tariff consisting of a
wholesale price w and a franchise fee f . Let us assume that the retailers accept
any contract that results in nonnegative profits. The retailers choose the retail
price p in their market (i.e., there is no resale price maintenance).
1. Let w < 12 and take f as given. Find the price that a retailer sets as a
function of b. Who earns the highest profit, retailers with b = 1 or b = z?
2. Assume in the rest of the exercise that z = 2. Suppose first that the
manufacturer knows the value of b in all three cities and that z = 2. What
contract will the manufacturer offer to the retailers? Is the franchise fee
the same for all retailers? Is it possible for the retailer to extract all profits
in the vertical chain? Assume from now on that the manufacturer does
not know the retailers’ individual b. However, the manufacturer knows
that two of the retailers have b = z = 2 and that one retailer has b = 1.
Assume also that the manufacturer wishes to serve all retailers.
3. Find the optimal franchise fee as a function of the wholesale price w.
4. Find the optimal wholesale price as a function of c. Is the wholesale price
greater or less than c?
5. Can the manufacturer extract all profits by setting w and f optimally?
6. Assume now instead that c = 1/4. Show that it is optimal for the manufacturer only to sell to the retailer with b = 1. What happens if c → 0?
Exercise 22 Exclusive dealing
Suppose that two firms produce at constant marginal costs c. There are two
periods and m buyers. Each buyer has an inverse demand curve: P (qI + qE ) =
1 − (qI + qE ) where qI is the quantity sold by the incumbent and qE is the
quantity sold by the entrant. In the first period, there is only the incumbent
in the market. Thus, the incumbent produces the monopoly quantity. The
incumbent has marginal cost equal to cI . In the second period, an entrant
with constant marginal cost equal to zero enters into the market. The entry is
foreseen by the buyers. After entry, the two firms compete à la Cournot. In
the first period, the incumbent offers the buyers a fee for an exclusive dealing
agreement (take-it-or-leave-it). If a buyer accepts the offer, she cannot buy from
the entrant in the second period.
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1. Suppose that the incumbent and entrant are equally efficient, i.e. cI =
0. What is the maximal fee for an exclusive dealing agreement that the
incumbent is willing to offer to a buyer? What is the minimum fee that a
buyer is willing to accept for signing an exclusive dealing agreement? Will
there be exclusive dealing in equilibrium?
2. Consider a general marginal cost of the incumbent cI . For which values
of cI will exclusive dealing arise in equilibrium?
Exercise 23 Long-term contracts, upgrades, and exclusion
Consider a market for a base good that is sold over two periods. In period 2,
also an upgrade may become available. For simplicity, set the mass of consumers
equal to 1 and marginal costs equal to zero. Firm 1 can be active in periods
1 and 2, firm 2 can only be active in period 2. At the beginning of period 2,
firms decide simultaneously whether to upgrade. Suppose that firms maximize
the sum of profits in periods 1 and 2.
The willingness-to-pay without upgrades is V per consumer in each period.
An upgrade by firm 1 leads to a surplus of r + λ1 , while an upgrade by firm
2 would lead to a surplus of r + λ2 . Firm 2 is assumed to be more efficient,
λ2 > λ1 . The upgrading cost is C. Suppose furthermore that λ1 > C. This
assumption means that upgrading is socially superior to not upgrading even if
it is done by the less efficient firm.
1. Characterize the equilibrium if firms can only offer short-term contracts,
i.e., firm 1, when selling to consumers in period 1, cannot make them sign
a contract that binds consumers to buy from it in period 2.
2. Characterize the equilibrium if firm 1 can offer a long-term contract that
does not allow consumers or prevents them from buying from firm 2. Discuss your result.
Exercise 24 Vertical duopoly and vertical integration.
Consider a vertical duopoly with exclusive dealing contracts in place, i.e.,
upstream firm i only sells to downstream firm ı́ , i = 1, 2. Suppose that, at
stage 1, upstream firms and then, at stage 2, downstream firms set prices.
Downstream demand is of the form Qi (pi , pj ) = 1 − bpi + dpj . Upstream firms
have zero marginal costs of production and set their wholesale price. Consider
subgame perfect equilibria.
1. Characterize equilibrium upstream and downstream prices.
2. Suppose that b = d = 1. Characterize the equilibrium if firms indexed by
1 have vertically integrated (so that the integrated firm’s transfer price is
0).
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3. Are there incentives for vertical integration (for b = d = 1)? Discuss your
results.
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