~„~ ~~ G Discussionfor a ernic Research

' IIII III IItl I I I III I I I II R III II IIIIII I I IIIII IIIII II I

No. 9209

UPSTREAM PRICING AND ADVERTISINGSIGNAL DOWNSTREAM DEMAND

by Svend Albaekand Per Baltzer Overgaard

March 1992

ISSN 0924-7815

March 1992

UPSTREAM PRICING AND ADVERTISINGSIGNAL DOWNSTREAM DEMAND'

Svend AlbakCentER, Tilburg University

and

Per Baltzer OvergaardInstitute of Ernnomics, University of Aarhus

Abstract:

This paper considers a vertical manufacturer-retailer interaction in a content ofasymmetric information. The monopolist manufacturer is privately informed about theintensity of consumer demand which may be high or low. In addition to the state ofnature, consumer demand responds to the retail pria and advertising by themanufacturer. The vertical interaction is modelled as a signalling game, in which themanufacturer initially sets a wholesale price and a level of advertising. The retailer

infers information from the observable choices of the manufacturer and responds bysetting a retail price. By refining retailer inferences, we show that the game óas aunique equilibrium outcome where all information is revealed by the manufacturer'schoices. When consumer demand is high, the manufacturer chooses the full-information pair of wholesale-price and advertising. In contrast, to induce the rightbeliefs, the low-demand manufacturer is forced to distort the price-advertising pair bybehaving as if demand is even lower. As a consequence, the wholesale pria isdistorted downward, whereas demand-enhancing advertising may be distorted in eitherdirection. Dissipative advertising is not distorted since it is never used.

Key words: Manufacturer-Retailer Relations, Demand Uncertainty, Signalling,Wholesale Pricing and Advertising, Equilibrium Refinements.

JEL: C72, D82, L12

1

1. Introduction

Vertical interactions between manufacturers and retailers often take place against abackground of asymmetric information about final demand. In the setting of this paperwe assume that for some exogenous (e.g. legal) reason a monopolist manufacturer isprevented from integrating forwazd into retailing but is forced to operate through anetwork of geographically dispersed retailers with local monopolies. Geographicdispersion is modelled very simply by assuming that consumers in one region areprevented by distance from making their pwchases in another region. Mazket demandin each region is affected by economy-wide shocks, and we assume that themanufacturer has obtained complete and perfect information about demand intensityduring market research. Local retail monopolists, on the other hand, are incompletelyinformed about the state of demand.

We focus on the following simple setting. Since regional retail markets areindependent we analyse a representative market in this paper.' Assume that demandintensiry is either high or low, and that the manufacturer is privately informed aboutthe actual intensity. It follows that the retailer will attempt to infer information fromthe observable actions of the informed manufacturer in order to set the level of itschoice variables optimally. We shall assume that the choice variable of the retaíler isthe retail price, whereas, the manufacturer chooses a wholesale price and (possibly) apositive level of advertising. Final consumer demand depends on the state of nature,the retail price and level of advertising.

At a formal level, we model the following sequence of moves. Before any marketinteractions take place nature chooses the intensity of consumer demand, and themanufacturer is fully informed about nature's choice. Next, the manufacturer sets awholesale price and a level of advertising, and both are fully observed by the retailer.The retailer dces not, however, observe nature's choice, but attempts to inferinformation from the observable actions of the manufacturer. Then, the retailer sets aretail price on the basis of its (possibly updated) beliefs about demand. Finally, theretailer (and, hence, the manufacturer) serves whatever demand is forthcoming at theposted price. The sequence of moves modelled constitutes a signalling game and weinvoke refinements of sequential equilibrium to select outcomes. Formal defuutionsfollow Kreps and Wilson (1982) and Cho and Kreps (1987).

The purpose of this paper is to investigate the extent to which the choice variables ofthe informed manufacturer are distorted compared to the case of full retailer

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information. Now, there are two rypes of manufacturers distinguished by theirobservations of actual demand, and we will refer to them as the high-demand type andthe low-demand type. From the case of full information we can easily conclude that(subject to assumptions about the cost functioas) a manufacturer has an interest inselling as many uníts as possible. So, it is in the interest of the manufacturer that theretailer charges a low retail price. Consequently, when we move to an asymmetricinformation structure, both types of the manufacturer have an incentive to signa! thatactual demand is low since this induces the retailer to charge a low price. Hence, theretailer and a low-demand manufacturer are confronted with an adverse selectionproblem. To alleviate this problem and allow a correct retailer inference, the low-demand manufacturer is forced to choose a combination of wholesale price andadvertising level that it does not pay the high-demand manufacturer to copy.Otherwise manufacturer strategies are non-revealing and the two types aze pooled.

The signalling game considered generally has a multiplicity of equilibrium outcomessupported by either separating or pooling equilibrium strategies. But, once we rule outequilibrium dominated strategies in the sense of Cho and Kreps (1987), we concludethat a unique separating outcome remains. At this equilibrium outcome, the high-demand manufacturer reverts to the full-information combination of wholesale priceand advertising. The low-demand manufacturer, on the other hand, chooses its signalsso as to induce a demand-reducing distortion (in the terminology of Bagwell andRamey (1990)). By a demand-reducing distortion is understood that the low-demandmanufacturer chooses the opdmal pair of wholesale price and advertising of ahypothetical manufacturer with even lower demand when retailers react as if demandis at the actual level (i.e. low). We show that this ímplies a downward distortion in thewholesale price whereas directly demand-enhancing advertising may be distorted ineither direction. However, if advertising is purely dissipative (as in Milgrom andRoberts (1986)) its level is set to zero, and, hence, is not distorted

The analysis in this paper is related to the rapidly expanding literature on signalling ineconomics originating in the Spence-model (see the seminal paper by Spence (1973)and the game theoretic treatment by Kreps (1984)). Of particular interest are thepapers of Milgrom and Roberts on limit-pricing by a privately informed incumbent inresponse to potential entry (Milgrom and Roberts, 1982) and on price and advertisingas signals of product quality (Milgrom and Roberts, 1986). These papers introducevarious refinements to select equilibria that point to a unique separating equilibriumoutcome (when sepazating equilibria exist). The qualitative features of the marketdistortions resulting from private information are very similar to those spelled out

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above. Most closely related to the present paper are two papers by Bagwell andRamey (1988, 1990) that extend the Milgrom and Roberts (1982) analysis of limitpricing.

Bagwell and Ramey (1988) consider the role of pre-entry price and advertising insignalling whether incumbent-costs are low or high. They conclude that the price-advertising pair of a low-cost incumbent constitutes a cost-reducing distortion. Theyshow that at most one separating outcome as well as a continuum of outcomes wherethe different types of the incumbent are pooled survive the equilibrium refïnements ofCho and Kreps (1987). T'his contrasts the present analysis where separation is alwayspossible and all pooling equilibria are destabilized. Pooling equilibria survive inBagwell and Ramey (1988) because the uninformed player's response is a map frombeliefs into a dichotomous entry choice, which allows the informed player to provokeits unique preferred response (í.e. non-entry) even in a pooling equilibrittm In thepresent paper such a prefened response is ruled out under pooling, and thisestablishes a unique separating outcome.

Bagwell and Ramey (1990) develop a model of entry deterrence (or accommodation)when the competitors of the incumbent are incompletely informed about consumerdemand. In sttvcture their model is very similar to the vertical game considered in thispaper, in the sense that the response map of the uninformed player is smooth, and thequalitative results correspond. In one version of their model the incumbent hasincentives to signal low demand to ensure entry on a small scale.' The incumbent usesthe pre-entry price and advertising level as signals, and Bagwell and Ramey concludethat a low-demand incumbent chooses signals to induce a demand-reducing distortion.As a consequence, the pre-entry price is distorted downward (viz. ~imit-pricing")whereas advertising may be distorted in either direction.'

The outline of the paper is as follows. Section 2 formalizes the vertical interaction.Then, in section 3 the set of separating eyuilibria is analyscd. Pinally, we turn topooling equilibria in section 4. A few concluding remarks are contained in section 5.Proofs are in the appendix.

2. The Model

As outlined in the introduction the monopolist manufacturer supplies his product to amonopolist retailer, who in turn serves the demand of an anonymous mass of pricetaking final consumers. We assume that the final consumer demand is of the form a'

4

t x(p,A) where i e{L,H}, and a` ~ a". We refer to demand as being low or lrigh.The retail price p e!Q, is chosen by the retailer while the manufacturer chooses thelevel of adverdsing A e)e,. x(p,A) is assumed to be continuous, strictly decreasing inp, and non-deaeasing in A. We allow for the possibility that advertising is dissipative,that is, that A has no effect on demand. Besides A the manufacturer chooses awholesale price w e R,. Thus we assume that the manufacturer has all bazgainingpower, since the retailer has no say in determining w. To economize on notation unitcosts of production and retailing are set to zero while the cost of advertising is simplyA.' The profit to the retailer is then (p-w)[a' t x(p,A)] while the profit to the type imanufacturer is w[a' f x(p,A)] - A.

The crucial assumption concerning the structure of information in this paper is thatbefore any market interaction takes place, the manufacturer obtains complete andperfect information about the value of the shift parameter a`, whereas the retailerremains uninformed about the realization of a'. The retailer's prior probabilityassessment about the value of a' is specified as Prob(a") - p' and Prob (a`) - 1-p', p' e(0,1). We assume that p' is rnmmon knowledge. After observing themanufacturer's wholesale and advertising choice (w,A) the retailer forms a posteriorbelief p(w,A) e[0,1] about the probability that demand is high. Using this belief theretailer then maximizes expected profit (both manufacturer and retailer are riskneutral), that is, he solves the problem

max p(P-w)[a"tx(RA)] ' (1-P)(P-w)Ia`tx(P,A)]

with respect to p. We assume that this problem has a unique solution which we callp(w,A,p), p ~ w. It is easy to show that this function is strictly increasing in both wand p. We furthermore assume it to be continuous in all arguments.

Throughout the paper we shall restrict attention to pure strategy equilibria. Then, thecollection {(w",.~"), (w`,Ê1`), p, p(w,A)} forms a sequential equilibrium ( Kreps andWilson, 1982) if the following conditions are satisfied

i) Optimality for the retailer

P - P(w,A,P(w,A))ii) Optimality for the manufacturer

(~`,t1`) E argmax w[a'tx(P(w,A,á(w.A)),A)] - A

iii) Beliefs are Bayes-consistent;

a) if (w",Á") ~ (w`,Á`)

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then ~(w",A") - 1 and p(w`,Á`) 3 0b) if (w",Á") - (w`,A`)

then V(W",A") - P(w`~A`) - P`c) if (w,A) ff {(w",A"), (w`,A`)}

then any P(w,A) E[0,1] is consistent

The relatively weak requirements on belief systems give rise to the familiarmultiplicity of equilibria in this model. In the foilowing sections we progressivelyrefine these beliefs (i.e. those in iii.c) in order to narrow down the set of sustainableequilibria.

We introduce the notation -rr'(w,A,p) - w[a` t x(p(w,A,p),A)] - A. Let (w",A") be theunique maximizer of ~(w,A,l) and (w`,A`) the unique maximizer of Tr`(w,A,O) wherew' ~ 0, A' z 0. Hence, (v~,A`) is the full information solution when ~- d and theretailer knows this. ~(w",A",1) and ~rr`(w`,A`,0) aze then the full information profits

of, respectively, the high and the low type manufacturer.

3. Separating Equilibria

In this section we first characterize the set of sequential separating equilibria, andthen move on to define and explore the notion of an undominated equilibrium. Weshow that there is an unique undominated sequential separating equilibrium.

Before moving on to the analysis, let us recapitulate the incentives for the high typemanufacturer to mimic the low type. As seen in the previous section, the retail price isstrictly increasing in p, the posterior probability that demand is high. Since demand isstrictly decreasing in p and the manufacturer wants to sell as much as possible given wand A, both types of the manufacturer would like the retailer to believe that demandis low. Hence, for a wide range of wholesale prices and advertising levels the type H

manufacturer will find it profitable to mimic the low type. To avoid this a separatingequilibrium pair (w`,Á`) for type L will be distorted from the full information pair,(w`,A`), as we shall see shortly.

By definition, in any separating equilibrium (w",Á") .(w`,Á`) It follows from Bayes-consistency that p(w",Á") - 1 and p(w`,Á`) - 0. Cleazly, in any sequential sepazatingequilibrium (w",Á") -(w",A"), that is, in rnntrast to the low type, the type H

manufacturer adopts his full information solution. To see this, note that if it was notthe case, deviating to the full information solution would always give greater profits to

6

type H, since ~(w",A",P(w",A")) a~(w",A".1) ~~(w,A,l) for all (w,A) r(~L"~").

To characterize the set of separating equihbrium pairs of type L we follow usualpractice by setting p(w,A) - 1 for all (w,A) r(w`,À`), i.e. any (w,A) different fromthe equilibrium choice of the type L manufacturer triggers the retailer beGef thatdemand is high with certainty. Since this is the least favourable out-of-eyuilibriumbelief from the point of view of the manufacturer it will support the largestconceivable set of separating equilibrium pairs of wholesale prices and advertisinglevels for type L Note that this system of beliefs is Bayes-consistent.

For a pair (w`,À`) to be part of an equilibrium it must be true that the high type dcesnot prefer to mimic the low type and choose (w`,À`) over choosing his full informationsolution. Define the set H as {(w,A)~~(w,A,O) s~(w",A",1)}. Hence, H is the setof (w,A) to which type H would not deviate from (w",A") even if the retailer wouldinfer that a- a`. We see that (w`,À`) e H is a necessary condition for equilibrium.We also need to ensure that the type L manufacturer will adhere to his eyuilibriumstrategy. Now, the profit from the best possible deviation' by type L will be maxTr`(w,A,l) since p(w,A) - 1 for all (w,A) r(w`,À`). We then define the set L-as{(w,A)~~(w,A,O) Z m~x~r`(w,Ê1,1)}. Hence, with (w`,À`) in L, type L will have noincentive to deviate. -

The argumenu above have established that necessary conditions for a separatingequilibrium are (w",À") -(w",A") and (w`,À`) e H fl L To see that these conditions

are also sufficient, note that any such configuration can be sustained as an equilibrium

by setting p(w,A) - 1 for all (w,A) r(w`,À`).

Later we shall show that sequential separating equilibria always exist. In fact, the set

H fl L will typically be very large so that we get the multiplicity of sequentialequilibrium outcomes usually encountered in signalling games of the kind considered

in this paper. As noted earlier the main reason for this problem is the weakness of the

consistenry requirement embodied in the definition of sequential equilibria. In

particular, only two points in the admissible space of pairs of wholesale prices and

advertising levels are associated with any particular equilibrium. Hence, Bayes-consistenry provides little help in specifying reasonable beliefs at out-of-equilibriumpoints. Most notably, consistency dces not rule out that, following an out-of-equilibrium move, the retailer attaches a high probability to a type for which that

move is dominated by some other move irrespective of the posterior beliefs.

Formally, we say that (w;A') is weakly dominated for type i if

~(w',A-,0) 5 max ~(w,A,l)~A

The notion of weak domination thus states that a pair (w',A') is dominated for type iif the profits associated with (w;A') under the most favorable beliefs, p(w',A') - 0,are no greater than the profits which can be obtained for sure under the leastfavorable beliefs, p(w,A) - 1.

We introduce the following natural restriction on retailer beliefs. After observing apair (w,A) the uninformed retailer should assign zero probability to a type of themanufacturer for whom (w,A) is weakly dominated, as long as (w,A) is not weaklydominated for the other type. If a pair (w,A) is weakly dominated for both types ofthe manufacturer, arbitrary beliefs, p(w,A) e[0,1] can be assigned. This leads us tointroduce the following equilibrium selection criterion:

An equilibrium is called undominated when p(w,A) - 1(0) if (w,A) is weaklydominated for type L(H) but not for type H(L).

From the definition of the set H we know that for all (w,A) e H, ~(w,A,O) s~(w",A",1). Hence, any (w,A) E H is weakly dominated for type H. Then, for allpoints (w,A) in L fl H that are not weakly dominated for type L beliefs must bep(w,A) - 0 according to the restriction above. It follows that in any undominatedseparating equilibrium, (w`,A`) must maximize ~r`(w,A,O) over the set H. To make ourproblem interesting we therefore assume that (w`,A`) ff H, that is, that the low type'sfull information choice is not in H. If it were, it would of course trivially be chosen inany undominated separating equilibrium.

In order to proceed to find (w`,Ê1`) when (w`,A`) ff H it is useful to introduce twoauxiliary profit expressions. First, define for any real number a

~(w,A I a) - ~v[a'x(P(w,A,O),A)] - A

Hence, -rr(w,A ~ a) gives the profit to a manufaMurer with demand parameter a fromchoosing (w,A) when the retailer with probabiliry 1 believes that the true value of a isa`. Let ~j(a) -(w(a),A(a)) be the pair which uniquely maximizes -rr(w,A~a), andassume that ~(a) is continuous. In the appendix we show that w(a) is strictlyincreasing in a. However, in general it is not possible to say that A(a) is monotonic,even when A is strictly demand enhancing. Denote by a(~r(a)~a") the profit to thehigh demand manufacturer from choosing the maximizing pair ~r(a) of the type a, thatis, the pair (w(a),A(a)) which type a would choose if the retailer for sure thinks a-

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a`. Hence

~(~V(a)la") - w(a)Ia"tx(p(w(a),A(a)A)A(a))J - A(a)Note that p- 0, that is the retailer believes with certainty that a - a`. We make thetechnical assumption that there exists an a t a" such that ~(a) a 0 and ~r(~(a)~a")~ ar"(w",A",1). Thus, as a gets sufficiently small the high type manufacturer willprefer his full information solution to choosing ~r(a) even if the retailer thinks a- a`.

We are now ready to state the main result of this section.

Theorem 1: There exists a unique undominated sepazating equilibrium outcome inwhich (w",Á") -(w",A") and ( w`,Á`) -(wL),AL)) for some a ~ a`.

Proof: See Appendix.

The theorem establishes that the sepazating equilibrium wholesale price of the low

demand type manufacturer is biased downwards relative to the full information case,whereas the wholesale price and advertising level of the high demand type azeunbiased. In general, it is not possible to determine the d'uection of the distortion, if

any, on the advertising level of type L However, it is clear that if advertising is

dissipative, that is, has no direct demand-enhancing effects, then A(a) - 0 for all a,and we have that Á` - A` - 0. Furthermore, if A(a) is strictly increasing in a, whichis probably the most natural conjecture, then Á` - A(a) ~ A` - A(a`). Hence,advertising is biased downwards.

4. Pooling Equilibria

We have seen that the vertical mazket game of this paper has a unique, undominated

separating equilibrium outcome. However, in general there remains a multiplicity ofundominated pooling equilibria. In this section we show that invoking a more refinedequilibrium notion than domination eliminates these equilibria.

The set of pooling equilibrium strategies consists of strategies (w,Á) where

~(w,A,P") z ~(w",A",1)

and ~(w,A,p") z max ~(w,A,l)

To support the largest set of equilibria we have again assumed p(w,A) - 1 for all(w,A) r(w,Á). This set may be very large, indeed, although it may aLco be empty ifthe prior p' is sufficiently high.

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We now proceed to characterize the set of undominated pooling equilibria. Denote byH"(P') the set {(w,A)~Tr"(w,A,P') a ar"(w",A",1)}. C1earlY (w,A) E H'(P') is anecessary condition for type H to stay in a proposed pool. Denote by LP(p') the set{(w,A)~~(w,A,p') z~r`(w,~0} where (w,A) -(wL), A(a)) as defïned in Section 3.Since (w,A) is weakly dominated for H and not for L we must have p(w,A) - 0 in anyundominated equilibrium.

Hence, type L can always secure himself ~(w,A,O) and any undominated equilibriummust give him at least this profit. We conclude that in any undominated equilibrium(w,Á) e L"(p') n H"(p'). In fact we can prove the following theorem, which uses thatL"(p') is a subset of H'(p').

Theorem 2: Any (w,A) e L"(p') can be supported as an undominated poolingequilibrium strategy.

Proof: See Appendíx.

We see that elimination of dominated strategies does not in general eliminate themultiplicity of equilibria. For low values of p' a large set of undominated equilibriumoutcomes will exist. In fact, we typically end up with an infinity of equilibriumoutcomes. In these cases the predictive power of the model is still limited. Wetherefore turn to consider a more elaborate information processing system on the part

of the retailer.

In the following we argue that the remaining pooling equilibria in L'(p') azedestabilized if the retailer forms beliefs in a slightly more "sophisticated" manner. Toformalize this argument we adapt some of the notions and ideas of Cho and Kreps(1987) to fit our context.

A system of beliefs, p(w,A), that supports an undominated pooling equilibrium (w,Á),

is said to be non-sensible, if there exists a pair (w;A') M(w,Á) such that

1) ~(w',A',0) ~ ~(W,A,P`)2) ~(w;A',0) ~ ~(~,A,P')

Hence, inequality 1) expresses that type H prefers his equilibrium choice (w,Á) to

(w;A') even if (w',A') is followed by the most favourable beliefs p(w',A') - 0,whereas inequality 2) states that type L prefers the out-of-equilibrium "message"

(w,A'), if that message convinces the retailer that demand is actually low.

10If the inequalities 1) and 2) are satisfied simultaneously, the equilibrium associatedwith (w,À) can only be supported by a system of beliefs that assigns p(w;A') ~ 0,which we think non-sensible. We rule out such beliefs, and an equilibrium is only saidto be refined if it can be supported by beliefs that are no-where non-sensible. We canthen prove the following:

Theorem 3: No refined pooling equilibrium exists.

Proof: See Appendix.

Thus, the refinement of out-of-equilibrium beliefs along the lines suggested by Choand Kreps (1987), rules out pooling as a self-enforcing outcome in the signalling gameconsidered here. Intuitively, this result follows from the observation that, relative toany proposed pooling equilibrium, the type L manufacturer can find a potentiallyprofïtable deviation that it would never pay the type H manufacturer to mimic. Facedwith a deviation from an expected equilibrium, a sophisticated retailer should realizethese different incentives and update beliefs accordingly. Hence, type L will in factdeviate, and the pooling equilibrium is destabilized.

S. Concluslon

By inspecting the properties of the unique separating outcome of section 3, it is easilyverified that no potentially profitable deviations are available to type L which wouldnot be copied by type H. It follows that the separating equilibrium component isrefined, and we conclude that the vertical market game has a unique refinedequilibrium outcome. With the outcome is associated the pairs ( w",Êl") - (w",A") and(w`~A`) - (wL),AL)).

We can further conclude that the wholesale price of the type L manufacturer is biaseddownwards relative to the full information price, w`, whereas the wholesale price oftype H is unbiased. Thus, averaging over the states of demand, the effect ofincomplete retailer information is to lower the observed wholesale and retail prices.With respect to advertísing, the picture is less clear, even though we conclude thattype H chooses its full information level, A". Without adding further structure to thedemand side, we cannot in general determine the direction of the advertisingdistortion by type L.

If, however, advertising is purely dissipative, the picture is clear since A` - A" - 0,

iii.e., no distortions of advertising occur. Hence, the welfare effects of asymmetricinformation aze unambiguous if we measure total 'welfaze' as the sum of manufacturerprofits, retailer profits, and consumer surplus. Wholesale and retail prices are loweredon average and total welfare is increased. Tàis net increase in welfaze is composed ofa loss of profits to the type L manufacturer, whereas retailer profits and censumersurplus increase. 7'hus, with dissipative advertising, the unambiguous improvement inretailer profits resulting from asymmetric information provides a tentative explanationof why it is left to the manufacturer alone to gather information in vertical relations ofthe type considered here. In addition, since consumers with a low demand also gain bykeeping the retailer incompletely informed there is little a priori reason to expectconsumers to solve the manufacturer's problem by providing truthful information onthe state of their demand to the retailer.

12' An earlier version of this paper has been presented at seminars in Hamburg and

Tilburg, and we thank the participants for useful comments.

Notes

1. The extent to which the qualitative results of this paper carry over to a setting withcompetition at the retail stage is left to future reseazch. However, we hazard theconjecture that, with Bertrand compedtion between geographically dispersed andincompletely ínformed retailers, the main qualitative results would continue to hold.The validity of this conjecture clearly depends on the details of the model of retailcompetition, but, with geographical dispersion the goods offered for sale by retailersare imperfect substitutes, and mark-ups and retailer profits can be positive inequilibrium, which is an important feature of the signalling game analyzed.

2. Bagwell and Ramey (1990) also consider a model where the incentives of theincumbent are "turned upside down" so that a high demand is signalled. Thequantitative result are changed, but the qualitative features of equilibria azepreserved.

3. Two remarks should be made at this point. First of all, separating equilibria mayfail to exist in the model of Bagwell and Ramey (1990), whereas existence is provedfor the present model (see the proof of Theorem 1). Secondly, Bagwell and Rameyclaim that both separating and pooling equilibria survive the equilibrium refinements,and they state necessary conditions. However, these conditions are not sufficient, andit turns out that the analysis can be strengthened considerably to prove that, under theassumptions they make, no pooling equilibrium remains (this is shown in Albaek andOvergaazd (1991)).

4. The introduction of constant unit costs of production and retailing would have noeffect on the qualitative results of the paper. The same holds for a general advertisingcost function.

5. We assume that such a best deviation exists.

13Appendiz

Proof of Theorem 1

We first show that w(a) is strictly increasing in a. Pick a, ~ a„ and let (w„A,) -(w(a,), A(a,)), (w„A,) - ( w(a,), Aía,)). Tlten

w,Ia~'x(P(w,,A,,O),A,) - A~ - w:(a~ }x(P(w~~~0),A~] t A3 ~ 0

(Al) w~(a,tx(P(wr~,0),A,)1 - P4 - w~[a,tx(p(w,,~,0),~)1 t~~ 0

Add the inequalities to get

(w, -w~(a~ -a~ ~ 0

Hence, w, ~ w,.

Now, assume a, ~ a, ~ a", and substitute a" into (AI)

w~[a"tx(P(wrA,,O),A,)] -?4 - w~[a"'x(P(w,.A,,O).A,)] t A, ~ 0

Hence, ~rr(i(a) ~ a") is strictly increasing in a for a ~ a". Since, by assumption, there

exists a ~ a" such that ~r(a) Z 0 and a(~j(a)~a") ~~(w",A",1), the above result and

the continuity of ~(a) tells us there exists a unique a ~ a" such that iL) z 0 and~r(~rL)~a") -~(w",A",1). Since we have assumed that (w`,A`) - (w(a`), A(a`)) ff H

we must have -rr(,~(a`) ~a") ~~(w",A",1). Therefore we know that a ~ a`.

We now show that in any separating equilibrium ~(w`,t1`,0) -~(w",A",1). Supposenot, and let (w,A) - (wL),A~a)). Then

~a"tx(P~~O),~) - A- Sv`Ia"tx(P(W`,A`,0),A`)I t Ê1` ~ 0

(A2) ~`Ia`tx(P(W~.,A`,0).Á`)1 - A~ - WIa`tx(Pí~!',~0),~) i A z 0

since (w`,Á`) maximizes -rr`(w,A,O) on H.

ThenL-~`)(a"-a`) ~ 0

andw ~ WL

Substituting a ~ a` into ( A2) gives

w`[atx(P(W`,Á`,0),~`)1 - A` - w[gtx(PL.~O) ~1 t A~ 0

which is a contradiction since trL) maximizes ~rr(w,A~a). Thus, we must have

14~(vY`,A`,p) - ,~(w",p",1).

Now, we still need to show that ( w,A) is the unique pair which maximizes ~r`{w,A,O)under the condition that ~(w,A,O) - ar"(w",A",1). Therefore, pick any (w,A) M(w,,such that Tr"(w,A,O) - ~(w",A",1). Then

wLfx(PL.~~),A)] - A- wL,x(p(w,A,O),A)] t A ~ 0

(A3) w[a"tx(P(w,A,~),A)] - A-~~tx(P(W.~~)~1 t A- 0

Adding gives

(w-w)(a-a") ~ 0

whence

w t wEvaluate

~`(~A,~) - ~`(w,A,~)- W[a`'x(P(W,?~,0).A)] - A- w[a`ax(p(w,A.~),A)] 4 A

Add (A3) to get

w[a`tx(PL,~O),A)] - A- w(a`ix(P(w~A,~)~A)] t At w[a"tx(P(w.A,O),A)] - A-w(a"tx(pL~~~),~] t A- (w-w)(a`-a") ~ 0

Hence, (w,A) -(wL),AL)) is the unique maximizer of ~(w,A,O) given ~(w,A,O) -~(w",A",1).

To show the existence of the equilibrium we need to show that ~(a) is in L(bydefinition; ~(a) e H). Suppose not. Define (w,Á) as the unique ma.~mi~pi ofar`(w,A,l) - w(a`fx(p(w,A,l),A)] - A. Then

w[a`tx(P(W,f1,1),~)] -?~ - w[a`,x(PL.~~),~] t A~ 0

w[a"~x(P~~O),~] - A-~[a"tx(P(W,A,1),A)] t Á~ 0By adding we infer

(w-~)(a"-a`) ~ 0

andw~~G

But then

Wjatx(P(W,A,1),A)] - Ê1 - wLtx(P~~~)~] t A ~ 0

which is a contradiction. Hence, ( w,A) e L and a separating equilibrium exists. This

15

Since x(p(~sv,t1,0),A) ~ x(p(J'v,Á,l),A) we get~,Ltx(P(W,A,O),A)1 - A

- w[n.x(PL,~O),~] t A ~ 0concludes the proof of Theorem 1.

Proofof Theorem 2

First we show that any (w,A) in L"(p') must also be in H"(p'). Suppose not, andremember that ~(w,,~0) - ~r"(w",A",1). Then

w[a"tx(P~~O)~] - A- w[a"tx(P(w,A,P~.A)] t A~ 0w[a`tx(p(w,A,P~,A)] - A- w[a`tx(P(~!',~0).~] t A z 0

Adding gives

(w-w)(a"-a`) ~ 0

whence

w~w

But then

wLtx(p(w,A,P~,A)] - A- wLtx(PL.A.O),~] t A ~ 0

Since x(p(w,A,O),A) ~ x(p(w,A,p),A) we getwLtx(P(w,A,O),A)] - A- wLtx(P(~~0),~) t A~ 0

which is a contradiction. The theorem is then proved by noting that the followingbeliefs sustains the undominated pooling equilibrium:

`d(w,A) ff H(P~; P(w,A) - 0`d(w,A) E H(P~`~P~; P(w,A) -1(w,A) - (W,A); P(w.A) - P~d(w,A)El-iP~`{w,A}; P(w,A) - 1.

Proof of Theorem 3

Fix arry undominated pooling equilibrium pair (w,ï~) e L"(p') such that p(w,Á) - p'.Now, the earlier assumptions guarantee the existence of an á ~ a" such that

(Á4) ~[a"tx(P(W,~,P~,A)] - A - w(á)Ia"tx(P(w(á),A(á),0),A(á)]t A(á) - 0

where (w(á ),A(á )) - ~(á ) E B;. By construction we also have

16

(~) w(á)[~tx(P(w(~),A(~) ,0).A(~))] - A(á) -~[~tx(p(~.A,P~,A)] t A~ oAdd (A4) and (AS) to get

(~-w(á))(~„-~) ~ 0

orw ~ w(á)

Now, evaluate

w[a`tx(p(W,A.P~,A)] - A- w(á)[a`tx(p(w(á),A(á),0),A(~))] t A(~)Subtract (A4) to obtain

(w-w(à))(a`-~") ~ 0

Hence, the low type strictly prefers ~(á) to the pooling pair if beliefs are updated top(~r(á )) - 0. By continuity, we can find points ( w,A) in a close neighbourhood of~(á ) that are strictly non-preferred by the high type and strictly preferred by the lowtype if beliefs are updated to p(w,A) - 0. Hence, ( w,Á) collapses. ~

17REFERENCES:

Albaek, S. and P.B. Overgaard, 1991, Advertising and pricing to deter or accommodateentry when demand is unknown: comment, Draft.

Bagwell, K and G. Ramey, 1988, Advertising and limit pricing, Rand Journal ofEconomics 19, no. 1, 59-71.

Bagwell, K and G. Ramey, 1990, Advertising and pricing to deter or accommodateentry when demand is unknown, International Journal of Industrial Organization 8,no. 1, 93-113.

Cho, I.-K. and D.M. Kreps, 1987, Signaling games and stable equilibria, QuarterlyJournal of Economics 102, no. 2, 179-221.

Kreps, D.M. 1984, Signaling games and stable equilibria, Stanford University,Graduate School of Business, Research Paper ~ 758.

Kreps, D.M. and R. Wilson, 1982, Sequential equilibria, Econometrica 50, no. 4, 863-94.

Milgrom, P. and J. Roberts, 1982, Limit pricing and entry under incompleteinformation: An equilibrium analysis, Econometrica 50, no. 2, 443-60.

Milgrom, P. and J. Roberts, 1986, Price and advertising signals of product quality,Journal of Political Economy 94, no. 4, 796-821.

Spence, A.M., 1973, Job market signaling, Quarterly Journal of Economicx 87, no. 3,355-74.

Dtscussion Paper Serles, CentER, TLburQ University, The Netherlands:

(For previous papers please consult previous discussion papers.)

No. Author(s) Title

9061 F. van der Ploeg Macroeoonomic Policy Coordination during the Various

9062 E. Bennett andE. van Damme

9063 S. Chib, J. Osiewalskiand M. Steel

9064 M. Verbeek andTh. Nijman

9065 F. van der Ploegand A. de Zeeuw

9066 F.C. Drost andTh. E. Nijman

9067 Y. Dai andD. Talman

9068 Th. Nijman andR. Beetsma

9069 F. van der Plceg

9070 E. van Damme

9071 J. Eichberger,H. Haller and F. Milne

9072 G. AlogoskouGs andF. van der Ploeg

9104 G. Alogoskoufis andF. van der Plceg

9105 R.M.WJ. Beetsma

9106 C.N. Teulings

9107 E. van Damme

9108 E. van Damme

9073 KC. Fung

Phases of Economic and Monetary Integration in Europe

Demand Commitment Bargaining: - TheCase of Apex Games

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Empirical Tests of a Simple Pricing Model for Sugar Futures

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Fair Division under Asymmetric Information

Naive Bayesian Learning in 2 x 2 Matrix Games

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9111 F. van der Ploeg andTh. van de Klundert

9112 Th. Nijman, F. Palmand C. Wolff

9113 H. Bester

9114 R.P. Gilles, G. Owenand R. van den Brink

9115 F. van der Ploeg

9116 N. Rankin

9117 E. Bomhoff

9118 E. Bomhoff

9119 J. Osiewalski andM. Steel

9120 S. Bhattacharya,J. Glazer andD. Sappington

9121 J.W. Friedman andL Samuelson

9122 S. Chib, J. Osiewalskiand M. Steel

9123 Th. van de Klundertand L. Meijdam

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EMS Credibility

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A Bayesian Note on Competing Correlation Structures in theDynamic Linear Regression Model

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9124 S. Bhattacharya Banking Theory: The Main Ideas

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9125 J. Thomas

9126 J. Thomasand T. Worcall

9127 T. Gao, AJ.J. Talmanand Z. Wang

9128 S. Altug andR.A. MIDer

9129 H. Keuzenkamp andA.P. Barten

9130 G. Mailath, L. Samuelsonand J. Swinkels

9131 K. Binmore andL. Samuelson

9132 L. SamueLson andJ. Zhang

9133 J. Greenberg andS. Weber

9134 F. de Jong andF. van der Plceg

9135 E. Bomhoff

9136 H. Bester and E. Petrakis

9137 L. Mirman,L. Samuelson andE. Schlee

9138 C. Dang

9139 A. de Zeetnv

9140 B. Lockwood

9141 C. Fershtman andA. de Zeeuw

9142 J.D. Angrist andG.W. Imbens

9143 A.K. Bera andA. Ullah

Title

Non-Computable Rational E~tations EquIIibria

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The D';Triangulation for Continuous Deformation Algorithmsto Compute Solutions of Nonlinear Equations

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Rao's Score Test in Econometrics

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9144 B. Melenberg andA. van Soest

9145 G. Imbens andT. Lancaster

9146 Th. van de Klundertand S. Smulders

9147 J. Greenberg

9148 S. van Wijnbergen

9149 S. van Wijnbergen

9150 G. Koop andM.FJ. Steel

9151 A.P. Barten

9152 R.T. Baillie,T. Bollerslev andM.R. Redfearn

9153 M.FJ. Steel

9154 A.K. Bera andS. Lee

9155 F. de Jong

9156 B. le Blanc

9157 AJJ. Talman

9158 H. Bester

9159 A. t7zcam, G. Judge,A. Bera and T. Yancey

9160 R.M.WJ. Beetsma

9161 A.M. L,ejour andH.AA. Verbon

9162 S. Bhattacharya

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9165 Q.H. Vuong and W. Wang

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9167 D.O. Stahl [I

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The Missing Equilibria in Hotelling's Location Game

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Strategic Advertising and Pricing with Sequential Buyer Search

9168 T.E. Nijman and F.C. Palm Recent Developments in Modeling Volatiliry in Financial Data

9169 G. Asheim

9170 H. Carlsson andE. van Damme

9201 M. Verbeek andTh. Nijman

9202 E. Bomhoff

4203 J. Quiggin and P. Wakker

9204 Th. van de Klundertand S. Smulders

9205 E. Siandra

9206 W. HBrdle

9207 M. Verbeek andTh. Nijman

9208 W. Hïrdle andA.B. Tsybakov

9209 S. Alb~k andP.B. Overgaard

Individual and Collective Time Consistency

Equilibrium Selection in Stag Hunt Games

Minimum MSE Estimation of a Regression Model with Fi~tedEffects from a Series of Cross Sections

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Incomplete Panels and Selection Bias: A Survey

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PO RC1X ~M~.~ .~,Mfl I F TII RI IR(; THF NFTHFRLANDBibliotheek K. U. Brabant~ ii h ~i u ~ u uuuNiui i~in i h~~ u i

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