Alessandro - Idiosyncratic Tastes

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Idiosyncratic Tastes in an Oligopoly Model

Alessandro Bonatti

Yale University

February 2, 2006

Abstract

This paper develops a competitive screening model in which rms face buyers with multi-

dimensional types and private information. Its approach diers from (that of) discrete-choice

models of product dierentiation. It is assumed that buyers have idiosyncratic tastes, i:e: dier-ent marginal utilities for consuming products of dierent brands. The symmetric equilibria for

the duopoly and oligopoly cases are derived and some mechanism design issues that arise in this

context are discussed. The equilibrium allocations are similar to the traditional monopoly case

but they are characterized by the provision of more ecient quality levels. These allocations are

also shown to dier substantially from those of discrete - choice, random participation models.

1 Introduction

This paper analyzes an oligopoly model with adverse selection in which sellers use nonlinear prices

to compete over a buyer with multidimensional characteristics. The buyer has private informationover her preferences, namely on the marginal utilities she derives from the quality of each sellers

product. These marginal utilities are allowed to depend on the products brand. In this sense, the

buyer displays idiosyncratic tastes.

From a more general perspective, this paper describes a game of common agency with an ex-

clusive dealing clause. In the absence of dierentiation in product characteristics or in consumers

demand functions (due for example to idiosyncratic tastes), this is essentially a game of price com-

petition. As such, it displays an equilibrium la Bertrand with zero-prots and fully competitive

(ecient) quality supply. One of the main goals of the literature on competitive price discrim-

ination (surveyed by Stole [10]) is to study the conditions that allow to capture more realistic

aspects of nonlinear taris in strategic environments. A successful approach, in part borrowed

from the empirical literature on industries with market power (surveyed by Bresnahan [3]), is that

of dierentiated products.

In a discrete-choice model of product dierentiation each consumers net utility from choosing

rm j is determined by the (quality, tari) bundle she chooses and by an additive, rm-specic,

For many helpful discussions, I wish to thank (in alphabetical order): Luigi Balletta, Dirk Bergemann, DinoGerardi, Marco Pagnozzi and Maher Said.

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xed component (Stole [10]). The xed component varies across the population of consumers.

The frequently-used functional form for the demand function separates horizontal brand preference

from vertical taste for quality.1 This specication allows to tractably model the consumers choices

from dierent brands product lines. However, it assumes a distribution of preferences under which

the relative value of purchasing similar items from two brands is independent of the products

quality. The automobile industry is often used as an example (see Berry et al. [2]). In this market,

consumers can be thought of having brand preferences (e:g: BMWs over Mercedes) that determine

their choice among products of similar quality and dierent brand. In this setting, the choice of an

item within a brands product line depends only on the buyers taste for quality. This means that

buyers marginal taste for quality does not dier across brands.

However, one may think about dierent complementary aspects to this problem. Again with

reference to the automobile industry, consider the value of optional items in a car (e:g: leather

seats in BMWs as opposed to Fords). This kind of choice is determined by the additional value

the consumer attributes to optional items when she is already (considering) buying a given brands

product. The same could hold true when thinking about upgrading ones choice within a given

brands product line (what is the additional value of a more powerful model?). Therefore this choice

may be better modeled with the introduction of brand-specic taste for quality (marginal utility). 2

This paper characterizes the symmetric equilibria of an oligopoly model when buyers display

idiosyncratic tastes. In equilibrium, sellers oer menus of contracts that share the main qualitative

features of the Mussa-Rosen [7] (henceforth MR) monopoly allocation3. Competition among sellers,

however, reduces quality distortions and increases the agents rents. These eects are even stronger

when the buyers types are positively correlated and when the number of sellers increases. Clearly,

a useful exercise would involve integrating the idiosyncratic tastes approach into a discrete-choice

/ random participation setup, so to better describe the buyers choice within and among menusof contracts. However it is unclear at this stage whether the two approaches can be tractably

combined.

The rest of this paper is organized as follows: section 2 reviews two papers very closely related

to this one; section 3 introduces the model; section 4 discusses the main assumptions; section 5

derives the equilibrium contracts in a duopoly model and compares it to results in the literature;

section 6 provides extensions, e.g. to an oligopoly setting; section 7 concludes.

1 To be more precise, a typical formulation of the utility function is

Uij = iqj pj + xij

where i is consumer is taste for (vertical) quality and xij is her additive shock to purchasing from rm j:2 To anticipate the formulation of the utility function:

Uij = ijqj pj :

where ij is consumer i0s taste for the quality of rm j0s products.

3 The distinctive features of the monopoly MR allocation with one-dimensional types can b e summarized as follows:Lowest type (L) receives her reservation utility level (e.g. U(L) = 0).Highest type (H) receives ecient quality provision (q(H) = H) :Quality is distorted downwards everywhere else.

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2 Literature Review

The two papers most closely related to the present one are those by Armstrong and Vickers [1]

(henceforth AV) and by Rochet and Stole [8] (RS). Both AV and RS adopt a formulation for the

buyers utility function that is reminiscent of discrete-choice models. This formulation introduces

randomness in the agents participation decision through type-independent stochastic reservation

utilities. This framework is then applied to both a monopoly and an oligopoly model. Random

participation proves to bear two distinct eects in these two settings. In the monopoly case, it

rationalizes nonlinear tari schemes that assign positive rent (utility) to every consumers (so to

insure positive probability of participation). For the oligopoly case, random participation eectively

introduces spatial dierentiation among sellers, thus relaxing price competition. If dierentiation is

small enough, the equilibrium quality supply proves to be the ecient (rst best) one. Otherwise,

distortions persist, though they prove to be lower under competition than in the monopoly case

(this last result is specic to RS). In either case, all rms obtain positive prots.

The basic technique used in these papers is to dene buyers with multidimensional types indi-cating, roughly speaking, horizontal and vertical taste parameters. The buyers type is assumed to

be a vector ((x1;::xN) ; ) where xj is her reservation utility when dealing with seller j: It then fol-

lows that having a population with a continuum of types (x; ) for a given is equivalent to facing a

single buyer who receives random outside options. The key assumption in both AV and RS is that

the horizontal and vertical taste parameters ((x1;::;xN) ; ) are independently distributed. The

optimal selling mechanisms can then be derived using the traditional tools of single-dimensional

screening via an exclusive dealing assumption and a restriction to deterministic contracts. With

these assumptions, the seller can adopt a direct revelation mechanism in which the buyer is only

asked to report her vertical taste parameter.

This paper adopts a similar simplied approach to the multidimensional screening problem

(section 3) and argues that this type of approach to common agency games with exclusive dealing

clauses is actually general.

3 The Model

This section introduces the main features of an oligopoly model in which buyers have idiosyncratic

taste for quality. More specically, let I = f1;:::;Ig be the set of (identical) sellers. Let there be

a continuum of buyers with types = (1;::;I) 2 Ii=1 [L; H] : Dene the utility type receives

when consuming a good of quality qi produced by rm i is equal to

Ui () = iqi pi

Each component of the buyers type i is identically and independently distributed over [L; H]

according to a continuously dierentiable distribution function F(i) ; with density f(i) :

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3.1 Simple Pricing Game

In order to illustrate the main mechanisms at work in this model, assume that rms can only

produce one quality of the good. For simplicity, let this quality level be qi = 1 for all i; normalize

production costs to zero and let the type space be = [0; 1]I : Firm is strategy consists therefore

of naming a single price pi: This section characterizes the symmetric equilibrium of this game.In this simplied setting, the utility of a type consumer given a strategy prole p 2 RI can

be written as:

U(; p) = maxi2I

(i pi)

Therefore, each sellers expected prot is equal to

E[Vi (p)] = pi Pr

i pi > max

j6=i(j pj)

Notice that a type buyer will prefer the product of rm i over that of rm j ifj i pi +pj : It

follows thatQj6=i F(i pi + pj) is the fraction of types (i; ) that purchase from rm i: Denote

the k-th order statistics of F and f by FIk (i) and fIk (i) : The solutions to programs (1) and (3)

can be then derived as follows.

Competitive Case: Under price competition, seller i solves the following program:

maxpi2[0;1]

Z1pi

piYj6=i

F(i pi + pj) f(i) di (1)

The symmetric equilibrium of the competitive pricing game p is characterized by the rst-order

condition

1 FI1 (p)

I

1

IpfI1 (p

) pZ1p

fI11 () f() d = 0 (2)

Collusive Case: If sellers collude to maximize joint prots, then the (collusive) equilibrium

price solves:

maxpi2[0;1]

Z1pi

pi [F(i)]I1 (i) f(i) di (3)

The equilibrium of the collusive pricing game pC is characterized by

1 FI1pC

I

1

IpCfI1

pC

= 0 (4)

In words, condition (2) states that when setting her price, each seller optimally trades o the prot

margin on the units sold1FI

1(p)

I

with loss in market share due to her own price raise

1IpfI1 (p

)

and to competing sellerspR1p

fI11 () f() d

: The latter (strategic) eect is clearly absent in

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the characterization of the collusive price (4). It can be shown that the competitive and collusive

prices respond dierently to an increase in the number of rms.

Proposition 1 The collusive price pC(I) is always increasing in I: The competitive price p (I) is

decreasing in I if the distribution of types satises the condition:

f(p)

F(p)

1

I8p (5)

Proof. See the Appendix

Intuitively, colluding rms will try to capture the rents associated with the buyers who value

their products the most, provided that collusive agreements prevent other rms from challenging

their prices. The sucient condition (5) states that if there is enough probability density at the

upper end of the distribution, then the induced competition among sellers drives the equilibrium

price down as the number of sellers increases. It is worth pointing out that the uniform distribution

over the unit support satises (5) while the standard normal does not. Nevertheless, under bothdistributions, the (competitive) equilibrium prices are decreasing in the number of rms, suggesting

that (5) is perhaps too restrictive a condition. The equilibrium prices for these distributions are

shown on the following graph:

605040302010

1

0.75

0.5

0.25

0

I

p(I )

I

p( I)

Equilibria of Pricing Game for Standard Normal (upper)

and Uniform (lower) distributions

3.2 Nonlinear Pricing

This section introduces competition with nonlinear tari schemes. Consider the same set of I

sellers, each of whom can produce a good of quality qi at a cost of c (qi) = q2i =2. Seller i oers the

buyer a nonlinear tari scheme (menu of contracts) of the form fqi; pi (qi)g : The analysis is carried

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out under three important assumptions. They are stated here and will be discussed in detail in the

following section.

(A1) Restrict attention to deterministic contracts

(A2) Direct revelation mechanisms under which the buyer only reports i to seller i:

(A3) Exclusive dealing: buyers can only buy from one seller.

Under A1 A2; the utility function of a type - buyer when dealing with rm i can be written

in the usual form

Ui

i; i

= iqi

i

pi

i

Therefore, the sucient conditions for global Incentive Compatibility (for seller i) are given by:

dUi (i; i)di

= qi (i)

dqi (i)

di 0

The value of the buyers outside option is normalized to zero. Therefore the Individual Rationality

constraint can be written as:

Ui (i; i) 0

As in the simple pricing game, x a prole fqi (i) ; Ui (i)gIi=1 of incentive-compatible menus and a

seller i. Dene (I 1) indierent types

j (i)j6=i the cuto types j that satisfy the condition4:

Ui (i) Uj

j (i)

= 0 (6)

Therefore, in this framework,Qj6=i F

j (i)

is the fraction of types (i; ) that choose to purchase

from rm i: For a given strategy prole fqi (i) ; Ui (i)gIi=1, seller i

0s expected prots can then be

written as follows:

ZHL

iqi (i)

qi (i)2

2 Ui (i)

!Yj6=i

F

j (i)

f(i) di

4 Discussion of Main Assumptions

The approach adopted in this paper to derive an equilibrium of an oligopoly model with nonlinear

pricing relies heavily on assumptions A1 and A2: In other words, the generality of the results

presented in the following sections is potentially reduced by the restriction to deterministic contracts

4 More precisely, let x be the solution to (6) and dene j := min f1;max fx; 0gg :

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and to partial-revelation direct mechanisms. The main eect of these assumptions is to allow an

adaptation of traditional techniques from single-dimensional screening problems (see e.g. Salani

[9]) to solve for the symmetric equilibrium in a game with competing principals. At least two

mechanism design issues are relevant in this context: (a) the Revelation Principle generally fails

in such common agency games; (b) in a multidimensional types setting, the optimal contract may

require principals to oer lotteries rather than deterministic (p; q) allocations. At this stage, it is

important to discuss these issues and to argue informally that assumptions A1 A2 do not reduce

the generality of the approach.

4.1 The Revelation Principle

In most common agency games, restricting attention to truthful equilibria of direct mechanisms

entails loss of generality. The reasons for this failure are eectively summarized in Martimort [5].

For the purpose of this paper, it is enough to point out that the richness of the space of mechanisms

available to the principals depends on the richness of the message space

5

. When oering menus ofcontracts to the same buyer, the latters messages could convey useful market information on the

taris shes being oered by other principals. It can be argued that in this setting the Revelation

Principle holds despite this source of complexity.

In fact, it is sucient to observe that the contracts oered by one principal do not create any

direct or indirect externality on the other principals. The reason for this is that the utility levels

principals oer to the agent only aect her participation decision, not her optimal choice within any

principals menu of oers. More in depth, Martimort and Stoles [6] Delegation Principle6 can be

applied. In the light of this result, the restriction of strategy spaces to nonlinear pricing schedules

is without loss of generality. Then, once a strategy prole for other players has been conjectured,

the Revelation Principle is still available to determine best responses. In general, the need arises at

this point to extend the strategy spaces, so to include out-of-equilibrium menu oers. This would

happen if the agents choice within one principals menu aected her (potential) best option within

menus of other principals. This would yield messages that would not arise in a direct mechanism.

However, in this particular model one can invoke the Delegation Principle, without need to specify

out-of-equilibrium actions. The reason for this is that the agents choice among a rms products

doesnt alter what her choice would have been if she bought some other rms product instead.

4.2 Deterministic Contracts

It seems that the restriction to deterministic contract would still entail loss of generality. However,

consider a direct mechanism. Due to the exclusive dealing assumption, the agents true types i do

not aect her revelation choice of type i to seller i: In fact, conditional on contracting with principal

i; the type component i do not enter the agents utility function. Incidentally, this is another

5 This precise point is made more clearly in Epstein and Peters [4]:6 Replace any indirect mechanism with the decentralized menus of payo-relevant contracting choices: obtain

equivalent equilibrium outcomes.

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way of understanding why competition among principals only aects the participation decisions.

Therefore, screening through random contracts on the agents entire type would imply that

principals oer dierent lotteries to types (i; i). But then, given i; all types would simply choose

the lottery with the highest expected payo. The exclusive dealing assumption (A3) therefore makes

assumptions A1 and A2 without loss of generality7. In fact, if random contracts cannot improve

the principals options, partial revelation of type is sucient to design the optimal contract.

5 Duopoly Equilibrium

Let I = 2: Each seller i solves the following maximization problem:

maxqi();Ui()

ZHL

iqi (i)

qi (i)2

2 Ui (i)

!F

j (i)

dF(i) (7)

s:t: Ui (i) = Uj j (i)

s:t: IC and IR

Notice that the adverse selection problem between a single seller and the buyer is not dierent from

the traditional (Mussa Rosen [7]) case. The only dierences are given by the idiosyncratic tastes and

by competition among sellers. This should lead to expect that the shape of the optimal solution be

closer to the Mussa and Rosen [7] one than to that of Rochet and Stole [8] or other models of price

discrimination under imperfect competition. Note further that, in the absence of dierentiation

among buyers, competition among sellers would allow the agents rent to attain the level of the

entire (ecient) surplus in equilibrium. It is reasonable to conjecture that the equilibrium of

the idiosyncratic tastes model represents an intermediate solution between the monopoly and the

Bertrand (or perfect competition) cases.

Proposition 2 The symmetric equilibrium of the nonlinear pricing game with two rms is char-

acterized by the following rst-order conditions:

( q()) F() f() + () = 0 (8)

F() f()

q()

2

U()

q()

f2 () =

d ()

d(9)

(H) = 0 (10)

Proof. See the Appendix.A few observations from these rst-order-conditions allow to draw insights on the shape of

the optimal taris at the symmetric equilibrium of this game. From the transversality condition

(10), the traditional no distortion at the top result remains true. In the Rochet and Stole [8]

and Armstrong and Vickers [1] models, a further no distortion at the bottom result emerges

7 Of course, the exclusive dealing clause represents a restriction. However, it allows to directly compare this modelsresults with those in the literature on screening (MR) and on discrete-choice models.

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quite generally. This result does not necessarily extend to the idiosyncratic taste model. In the

equilibrium of this model the quality level q(L) that each seller associates to the buyer with lowest

valuation for her good is unconstrained (evaluate (8) at L). The reason for this indeterminacy is

that in equilibrium each rm will not serve such types with positive probability8.

This is arguably the main dierence between the equilibrium of this model and that of Rochet

and Stole. In RS, the random participation assumption induces sellers to serve a positive fraction

of all types9. Contrarily to this scenario, the symmetric equilibrium of the model with idiosyncratic

tastes reects the fact that competition will more likely attract away from a rm those customers

who like its products the least. The next (conjectured) result formalizes the dierence between the

two models.

Conjecture 1 In the duopoly symmetric equilibrium, in a neighborhood around L; distortions

persist, i:e: q() < :

Proof. From (8) (9), if q(L) = L then _ (L) = 0 (dierentiate (8)). From (9) then

U(L) =12 (L)

2 which is the value of the entire ecient surplus at L: This shouldnt have an

impact since in equilibrium no type L buys from the rm. However, iff(L) > 0 then since () is

continuous, the (near-)ecient quality provision will have to extend to a neighborhood of L. This

means that the agent obtains rents equal to the entire surplus over this neighborhood. Thus the

rm will make zero prots over a positive measure of customers. It could therefore make strictly

positive gains by introducing distortions, reducing rents accordingly and giving up market shares

in exchange for positive prots.

5.1 Analytical Solution with Uniform Distribution - an Example

Consider the duopoly model in which consumer types are independently, identically and uniformly

distributed on [L; H]2 : The following result can be proved:

Proposition 3 When H =52L; there is a solution with linear quality supply schedule in which

the lowest type receives zero quality (hence zero utility). This solution is characterized by

q() =1

3(5 2H)

U() =5

62

2

3H +

2

152H

8

This result can be generalized. At the symmetric duopoly equilibrium, (see f.o.c. (8)) each rm will meet a buyerof type i with probability F(i).

9 The reason for this dierence is related to the fact that in both models the condition (L) = 0 is obtained aspart of the characterization of an equilibrium with U(L) > 0: In RS, this condition and the f.o.c.s imply q(L) = L(eciency at the bottom). In the model with idiosyncratic taste, when (L) = 0; the f.o.c. at L is satised for anyvalue of q.

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Proof. For the uniform distribution, the rst order conditions simplify to

( q()) L

(H L)2 = () (11)

q()

2

+U()

q()

L1

(H L)

2 =d ()

d

(12)

q(H) = H (13)

Look for an equilibrium with a linear quality supply function q() = a + b: The transversality

condition (13) immediately imposes the restriction b = H (1 a) : The utility function is there-

fore U() = H aH +12a

2 + c: Plugging in these functional forms and using the fact thatdd

[LHS(11)] = LHS(12) one obtains the restriction on the support and the characterization of

q() and U() :

Let L = 1 and H =52 ; then the equilibrium quality schedule can be represented in the following

graph (the dashed lines indicate the monopoly and ecient quality levels).

2.52.2521.751.51.251

2.5

2

1.5

1

0.5

0

Theta

q(Theta)

Theta

q(Theta)

Ecient, Duopoly and Monopoly quality provision.

The solution here obtained conrms the intuitive prediction of a quality provision path that lies

between the competitive (Bertrand) case and the MR monopoly solution. The result for the case

of a uniform distribution of types, this result can hopefully be generalized. It can be conjectured

that the solution involves shutdown of low types whenever H >52L (in the MR case this happens

when H > 2L).

5.2 Numerical Simulations

Numerical solutions are available for the simple case in which rms oer fully-separating menus of

contracts that do not shut down any type : Using the IC constraint dUd

= q, conditions (8) (10)

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can be reduced to the following boundary value problem:

_U() = + ()

F () f()

_ () = 0@F() 0@f()2

()

2F()

U() f()

+ ()F()f()1A1A f()

(L) = (H) = 0

One diculty associated with solving this problem numerically is that the functions

_U; _

are

not well-dened at L (where F() = 0). The following graph describes another example using

the uniform distribution. In this simulation, parameter values were chosen so to highlight the

dierences between the fully competitive, monopoly, idiosyncratic taste and random participation

models:

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 53

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5Monopoly, Duopoly and 1st Best Qualities

Theta

q(theta)

Types i Uniformly Distributed on [4; 5]

The dashed line represents the RS allocation (eciency at top and bottom) whereas the central

straight line is the idiosyncratic tastes quality provision. The top and bottom lines represent the

ecient and the monopoly qualities respectively.Incidentally, it may be useful to consider cases with dierent distribution functions.

6 Extensions

The duopoly model can be extended to a more general setting, namely by introducing correlation

among types and by considering an arbitrary number of rms.

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6.1 Correlated Types

Up to this point it has been assumed that the buyers types are identically and independently

distributed. The independence assumption does not seem particularly realistic. There may be

more than one reason to believe that a buyers marginal valuation of one brands product should

not dier substantially from her valuation of the same product of a dierent brand. This leads tothe conclusion that the marginal utilities from consuming products of dierent brands should be

positively correlated. To take this possibility into account, let types be distributed according to

the joint density:

f(1; 2) = k (")

1 " (1 2)2

= [1; 2]2

This particular form chosen in this example can be viewed as a modied uniform distribution in

which the distance between the two type components 1 and 2 reduces the probability density

at type : The constant k (") just ensures that the density integrates to one over the support.

The intuitive prediction is that positive correlation should increase competition. The following

numerical example that for some parameter values this is indeed the case: quality provision (and

the agents rent) increase as compared to the independence case (the dashed line). Both allocations

lie in between the monopoly and the ecient levels of quality.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Monopoly, Oligopoly and 1st Best Qualities

Theta

q(theta)

Equilibrium with Correlated Types [" = 1; L = 1; H = 2]

It is important to point out that other simulations show radically dierent results. Simply changing

the support induces the following quality provision that is reminiscent of Rochet-Stole [8] (the other

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lines are as in the previous graph).

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 53

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8


Theta

q(theta)

Equilibrium with Correlated Types [" = 1; L = 4; H = 5]

This could be a promising way of establishing further conditions under which rms provide ecient

quality supply "at the bottom" of the distribution.

6.2 Oligopoly Equilibrium and Bargaining

The main qualitative features of the duopoly symmetric equilibrium can be shown (by way of

numerical simulations) to extend to the generic Irms case. When looking for a symmetric

equilibrium in the I sellers model, program (7) is virtually unchanged. The Hamiltonian can be

written as follows

H(q;U;) =

q

q2

2 U

[F( (U))]I1 f() + q

Computational diculties allow extension only to I = f2; 3; 4g ; but the numerical simulations

suggest that the shape of the optimal quality provision does not change. As the number of rmsincreases, quality provision shifts towards the ecient levels and the agents rent increases. In this

setting, however, it does not seem possible to extend a result like the one in Conjecture 1. As a

matter of fact, the rst step in the conjectured proof would not be replicated if I 3: Intuitively,

ecient quality provision cannot be ruled out even for the lowest types as competition among rms

becomes more intense. The evolution of the quality provision for the uniform case and I = f2; 3; 4g

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can be seen from the following graph:

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Theta

q(theta)

Symmetric Equilibrium when I = 2; 3; 4

The formulation of the principals maximization program in the Irms case can also be interpreted

as a bargaining problem. Note that for each type i, the Hamiltonian is very similar to a Nash

product. In this ctitious bargaining game, the utility functions are given by the rms prot

margin on type i and an increasing function of type is informational rent. The latter component

is weighted by the number of outside options (I 1) available to the buyer. As I increases, the Nashbargaining solution to this problem must favor the buyer (higher utility, lower quality distortions).

The numerical simulations conrm this prediction.

7 Conclusions

This paper has attempted to characterize the symmetric equilibria of several pricing games when

the buyers valuation of the quality of dierent brands products may vary across the population.

The main nding is that this source of product dierentiation restores the MR quality distortions

that instead often disappear in random participation models. One exception is represented by the

case of correlated types.

The approach introduced in this paper adds realism to a simple model of competition with

nonlinear prices and is to be considered complementary to the random participation approach of

AV and RS. On the one hand, a "mixed model", combining the two approaches would be worthwhile

pursuing, although there is little scope for imagining that the results (in terms of quality provision)

would dier substantially from those already obtained in the literature. On the other hand, the

method developed in this paper is hopefully general and useful enough to study non-zero prot

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equilibria of competitive nonlinear pricing games in a dynamic setting.

Appendix

Proof of Proposition 1

Show rst that prot functions are concave in pi: Then sign ofdp(I)dI

is entirely determined by the

second partial cross-derivative of the prot function.

Collusive Case: Dierentiating the f.o.c. (3) wrt I at the optimal solution pC one obtains

that:

dpC

dI= I2

FI (p) ln FI (p) 1

Now let FI (p) = x and note that (x ln x 1) 0 8x 2 [0; 1] :

Competitive Case: The cross derivative of the prot function can be written and simplied

as follows:

@2

@p@IV =

1

I2

1 FI (p)

@

@I

p

Z1p

(I 1) f2 () FI2 () d

1

I(ln F(p)) FI (p) pf(p) FI1 (p) ln F(p) (14)

= 1

I2

1 FI (p)

1

I(ln F(p)) FI (p) pf(p) FI1 (p) ln F(p)

p

Z1p

f2 () FI2 () + (I 1) f2 () FI2 () ln F()

d (15)

1 + FI

ln FI

FI pd

dp FI ln FI Z1p

ddp FI

d

dp ln FI

1 + ln FI ()

d

1 + FI

ln FI

FI pd

dpFI ln FI

Z1p

dFI

dp

d ln FI

dpd

1 + FI 2

ln FI

FI

Z1p

dFI

dp

d ln FI

dpd

=

1 FI + 2

ln FI

FI +

Z1p

d ln FI

dp

dFI

dpd

(16)

Now, if

d ln FIdp

1 8p (17)

then Z1p

d ln FI

dp

dFI

dpd

Z1p

dFI

dpd = 1 FI (p)

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16/17

and (16) can be simplied to

2

1 FI + FI ln FI

(18)

Now let x = FI (p) and note that:

(1 x + x ln x) 0 8x 2 [0; 1]

therefore (18) is negative everywhere. Finally, rewriting (17) ; one obtains the sucient condition

(5) :

f(p)

F(p)

1

I

Proof of Proposition 2

The Hamiltonian for this problem is:

Hi (qi; Ui; i; i) =

iqi

q2i2

Ui

F

j (Ui)

f(i) + iqi

Remember again that j (i) solves Ui (i) Uj

j (i)

= 0: Therefore, applying the Implicit

Function Theorem at each point i;

@j (i)

@Ui=

1

dUj=dj=

1

qj

j

the f.o.c.s can be written as follows:

Hq : = (i qi) F

j

f(i) + i = 0

HU : = F

j

f(i) iqi q

2i =2 Ui

qj

j f2 (i) = di

di

i (H) = 0

The symmetric equilibrium, is characterized by:

( q()) F() f() + () = 0

F() f()

q()

2 U()

q()

f2

() =d ()

d

(H) = 0

References

[1] Armstrong, M. and Vickers, J. (2001) Competitive Price Discrimination, The RAND Journal

of Economics, 32: 579-605.

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[2] Berry, Steve, Levinsohn, James, and Pakes, Ariel (1995) Automobile Prices in Market Equi-

librium Econometrica 63(4), 841-890.

[3] Bresnahan, Timothy F. (1989) : Empirical studies of industries with market power. Handbook

of Industrial Organization, vol.2, eds R. Schmalensee and R. Willig: North Holland.

[4] Epstein, L. and M. Peters (1999): A Revelation Principle for Competing Mechanisms, Jour-

nal of Economic Theory, 88, 119-160.

[5] Martimort, David. (2006): Multi-Contracting Mechanism Design, mimeo, Univrsit de

Toulouse.

[6] Martimort, D. and L. Stole (2002): The Revelation and Delegation Principles in Common

Agency Games, Econometrica, 70, 1659-1674

[7] Mussa, Michael, and Rosen, Sherwin, (1978). "Monopoly and Product Quality" Journal of

Economic Theory, 18: 301-17.

[8] Rochet, Jean-Charles, and Stole, Lars, (2002). "Nonlinear Pricing with Random Participation"

Review of Economic Studies. 69, 277-311.

[9] Salani, B. (1997). The Economics of Contracts. The MIT Press.

[10] Stole, Lars (2005). Price Discrimination in Competitive Environments, in Handbook of In-

dustrial Organization, eds. M. Armstrong and R. Porter, Amsterdam: North Holland.

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