Marketing Exercises Part 6 Double Marginalization Then Increases Price Reduces Quantity Andindustry Prot Also Consumers Would

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6. The stand-alone utility may be affected through negative advertising about the

competitor. Another interpretation can be provided in a setting in which firm

7. Firm Amay overinvest in its own quality to deter entry; even worse for society,

it may overinvest in reducing the stand-alone utility of its competitor to deter

8. maxrA;rB

21 + rA+rB

32c

4(rA)2c

4(rB)2. First-order conditions

are 2

Exercise 25 Upstream merger

Consider an industry with two symmetric upstream firms Aand Band one

downstream firm D. The downstream firm may sell the product of none, one,

or both upstream firms. Total industry profits are a function of the number

of upstream firms selling through D, denoted by V(n),n2 f0;1;2g, which is

assumed to be increasing in n. The outside option for each firm of not selling is

zero. Thus, V(0) = 0.

Rents are the outcome of Nash bargaining between any pair of one upstream

firm and the downstream firm (equal sharing of the surplus within the pair above

the profits that would occur if this pair did not agree).

1. What will be the profits of firms A,Band Dif Dsells both products?

2. Suppose that the two upstream firms merge (and become a two-product

firm) and that function Vcontinues to apply—an interpretation of the

latter property is that prices are set by independent profit centers within

the merged firm. Then, Nash bargaining takes place between the merged

upstream firm and the downstream firm. What will be the profits of the

merged upstream firm AB and the downstream firm D?

3. Provide the exact condition for the merger to be profitable.

4. Explain your findings.

Solution to Exercise 25

1. Suppose that there is an agreement between Band D. If Aand Ddo not reach

an agreement, neither Anor Dobtain any profit from product 1. The additional

2. The industry profit is shared equally between both firms, AB =V(2)=2and

3. The merger is profitable if AB > A+B, which is equivalent to V(2)=2>

4. Prior to the merger, any pair with one upstream and the downstream firm

Exercise 26 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

mis the fixed fee and pis 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 pUand pand comment brie‡y.

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, for c= 1=2, the two-part tariff found

found here with the one in question (2) and comment brie‡y.

Solution to Exercise 26

2. Consumer surplus is R1

p(1 x)dx m= (1 p)2=2m. The problem is now

3. The two-part tariffs implements the first best, as it sets the unit price equal

4. The monopolist can decide to serve one group or both groups. If it only serves

the high-demand group, it faces inverse demand P(q) = 2 2q. Consumer

Exercise 27 Two-part tariffs with downstream competition

Suppose that an upstream monopolist has constant marginal costs of produc-

tion cand sells to symmetric Cournot duopolist in the downstream market at a

two-part tariff (wi; Fi), where i2 f1;2gis the identity of the downstream firm.

Here, wirefers to the linear wholesale price and Fithe fixed fee to be paid by

downstream firm i. Downstream firms have zero marginal costs and face market

demand P(q) = aq. Suppose that a > c > 0. Consider the timing according to

which, at the first stage, the upstream firms set tariffs in the wholesale market

and, at the second stage, downstream firms simultaneously set quantities in the

downstream market.

1. Suppose that the upstream monopolist has to set Fi= 0 and publicly

posts the wholesale prices wi. Calculate the subgame perfect equilibrium

and comment on your findings.

2. Suppose that the upstream monopolist publicly posts two-part tariffs

(wi; Fi). Characterize the subgame perfect equilibria when the upstream

firm is restricted to offer a non-discriminatory offer (w; F )to both down-

stream firms.

3. Suppose that the upstream monopolist cannot publicly post two-part

tariffs—i.e., the two-part tariff that applies to firm iis private information

in the duopoly game at stage 2. Furthermore, suppose that downstream

firms hold passive beliefs—i.e., no matter which contract is offered, a down-

stream firm expects its competitor to be offered the equilibrium tariff. Is

the equilibrium outcome obtained under (2) also an equilibrium outcome

in this case? Explain your finding.

Solution to Exercise 27

1. We solve the game by backward induction. Downstream firms take prices w1,

w2as given. Suppose that these prices are such that both firms are active in

At the first stage, the upstream firm solves

w1;w2

First-order conditions are

2. The upstream firm publicly posts a non-discriminatory tariff (w; F ). One pos-

sibility is offer a tariff with w=cand Fequal to the downstream monopoly

profit at w=c(F= (ac)2=4). In this case the upstream firm extracts the

full monopoly profit in the downstream segment. This is an equilibrium because

3. Public observability is essential to support either of the equilibria above. Under

non-observability and passive beliefs, it is obvious how to destroy the first type

Exercise 28 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

Exercise 29 Vertical contracting [included in 2nd edition of the book]

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 cis uniform. Show pand the expected

payoffs of the parties graphically (put con 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 incum-

bent seller the amount mregardless 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+pand requiring him to pay liquidated damages of mif he switches

to the entrant). If the buyer rejects the contract things are as in part (1).

Given pand 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 pand m.

4. Explain why the incumbent seller will choose pby maximizing the sum of

his expected payoff, I, and the buyer’s expected payoffs, u.

5. Write the first-order condition for pand show that the profit-maximizing

price of the incumbent seller, p, is such that p <1=2. Also show that

if g(0) >0then 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 cis uniform, and show the

expected payoffs of the parties and the social loss graphically (again, put

con the horizontal axis and the equilibrium price function on the vertical

axis).

8. Compute p under the assumption that G(c) = c, where  > 0. How

does p vary with ? Give an intuition for this result.

Solution to Exercise 29

1. In equilibrium, the price is p = 1=2if c1=2(hence, the buyer buys from the

entrant), and p =c, otherwise (hence, the buyer buys from the incumbent).

Given this price, the buyer’s expected benefit is

2)g(c)dc;

E=Z1

3. Given pand c, the buyer will buy from the entrant if cpand from the incum–

bent seller otherwise and will pay p. The buyer’s expected payoffs is therefore

4. First, note that the contract must be designed so as to ensure the buyer the

same expected payoff as in Part 1, otherwise the buyer will reject the contract.

5. Differentiating I+u, the first-order condition for p is given by

6. Since the buyer always buys the good, efficiency is achieved if and only if the

lowest cost seller produces the good. In Part 1 competition ensured that this is

7. When cis distributed uniformly on the unit interval, the first-order condition

for p becomes

8. When G(c) = c, the first-order condition for p becomes

Exercise 30 Franchising

A monopolistic manufacturer produces a good that is sold to three retailers.

The manufacturer has constant marginal cost equal to c < 1=2. 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

1. Let w < 1=2and take fas 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 bin all three cities and that z= 2. What

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 wand foptimally?

Exercise 31 Exclusive dealing [included in 2nd edition of the book]

Suppose that two firms produce at constant marginal costs c. There are two

periods and mbuyers. Each buyer has an inverse demand curve: P(qI+qE) =

1(qI+qE)where qIis the quantity sold by the incumbent and qEis 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 ï¿ 1

2la 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.

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

cIwill exclusive dealing arise in equilibrium?

Solution to Exercise 31 The value an exclusive dealing is worth VI= 1=4(1 

cI)21=9(1 2cI)2to the incumbent. A buyer accepts if she is offered a payment of,

Exercise 32 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 1can be active in periods

1 and 2, firm 2can 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 Vper consumer in each period.

An upgrade by firm 1leads to a surplus of r+1, while an upgrade by firm

2 would lead to a surplus of r+2. Firm 2 is assumed to be more efficient,

2> 1. The upgrading cost is C. Suppose furthermore that 1> 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. Dis-

cuss your result.

Solutions to Exercise 32

1. Bertrand competition in period 2 implies that consumers make a net surplus of

2. Alternatively, firm 1 could offer a long-term contract requiring the consumer not

to buy from the rival in period 2. Consumers are willing to take this option if

Exercise 33 Vertical integration. [included in 2nd edition of the book]

1. Suppose that two downstream retailers sell a homogeneous product in a

downstream market. They have to pay w < 1for each unit of the product

that they sell on to final consumers and do not incur any further variable

costs. The inverse downsteam market demand P(q) = 1 q, where q=

q1+q2. Determine the Nash equilibrium when both firms set quantities

simultaneously.

2. Suppose that there are many such downstream markets (to be precise,

a continuum of mass 1) and that prior to the quantity setting in those

downstream markets two upstream firms simultaneously set quantities xi.

For each unit they incur marginal costs c. Determine the subgame perfect

equilibrium of the two-stage game.

3. What would change if there is only one instead of a continuum of down-

stream markets? Is there any conceptual difference between those two

settings? Explain in at most three sentences.

4. Suppose that upstream firm 1 merges with downstream retailer 1 in each of

the many downstream markets. Suppose furthermore that firm 1 commits

neither to sell to any downstream retailer 2 nor to buy any units from

upstream firm 2. The timing of the game is that, at stage 1, upstream

firms simultaneously set xiand that, at stage 2, retailers set qi. Determine

the subgame-perfect equilibrium of this two-stage game.

5. Compare your results in (2) and (4). Does the vertical merger increase the

input price for non-integrated downstream firms? Does the vertical merger

make consumers better off? Explain in one or two sentences.

Solutions to Exercise 33

1. The profits of each downstream firm is D

i= (1 qiqjw)qi. Solving the

2. Since q=x, the inverse demand in the upstream market is w= 1 3x=2.

3. With a continuum of downstream markets the demand by each downstream firm

4. At stage 2, downstream firm 2 obtains the input at wholesale price w2, whereas

downstream firm 1 obtains the input through internal production and, thus, at

5. The wholesale price paid by the non-integrated firm is less than without the

vertical merger. This may sound surprising as upstream firm 2 exerts monopoly

Exercise 34 Exclusive dealing and vertical integration.

Consider a vertical duopoly with exclusive dealing contracts in place, i.e.,

upstream firm ionly sells to downstream firm {,i= 1;2. Suppose that, at stage

1. Characterize equilibrium upstream and downstream prices.

0).

3. Are there incentives for vertical integration (for b=d= 1)? Discuss your