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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 a...