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Utility Misperception in a Vertically Differentiated Duopoly

Sophie Bienenstock EMAIL logo
From the journal Review of Law & Economics

Abstract

When choosing between two goods, consumers anticipate the utility they expect to derive from each product. However, such anticipations are subject to several sources of error, such as quality or price misperception and overoptimism about one’s capacity to use a product. The present paper studies the effect of inaccurate utility anticipations on consumer choice and ultimately on the market outcome in a vertically differentiated duopoly. I come to the conclusion that utility misperception can lead consumers to make suboptimal decisions ex post, although the choice seemed rational at the time of purchase. I show that in a vertically differentiated duopoly, firms are subject to two opposite incentives regarding consumer education. Moreover, the firms’ incentives to educate consumers are not necessarily aligned with the socially efficient outcome. Therefore, this paper also explores several policies aimed at mitigating the negative consequences of consumer misperception.

Acknowledgements:

Ph.D, Centre de Recherche en Economie du Droit (CRED), Université Panthéon-Assas Paris II.

A Appendix

A.1 The market outcome without consumer bias

Assumptions. Consumers are uniformly distributed on the [0,1] interval. Two substitute goods which differ with regards to their quality are offered by firms A and B. The locations a and b on the [0,1] interval represent product quality, with the convention that b>a, which means good B is of higher quality. Consumers make mutually exclusive single purchases. Consumers have the following utility function:

Ux(A)=r+axpAif he buys product AUx(B)=r+bxpBif he buys product B

Firms are assumed to have the same constant marginal cost, independent of quality. Without loss of generality, I assume that this cost is equal to zero.

I assume that prices are set such that all consumers buy one unit of one good. Hence, the market is covered, which holds only if r is sufficiently large. The parameter r needs to verify r>max(pA;pBb).

For given qualities a and b, there is a consumer located at xˆ who is indifferent between the two goods. The indifferent consumer satisfies Ux(A)=Ux(B), which entails :

xˆ=pBpAba

Consumers located at point x on the [0,1] interval such that x<xˆ buy good A, whereas consumers who verify x>xˆ decide on good B. Profit functions depend on the relative prices pA and pB:

  1. if pA>pB, then ΠA=0 and ΠB=pB

  2. if pB(ba)pApB, then ΠA=pA(pBpAba) and ΠB=pB(1pBpAba).

  3. if pA<pB(ba), then ΠA=pA and ΠB=0

I focus on cases where there exists a duopolistic equilibrium, that is to say when both firms make a positive profit. Hence, the analysis is restricted to pB(ba)pApB.

Defining equilibrium prices and profits under the assumption that the market is covered. In what follows, location on the quality axis is given, and firms determine prices according to a and b. In order to define equilibrium prices and profits, I first determine equilibrium prices as functions of locations a and b; and next turn to the profits.

Determining prices as a function of a,b: For any given locations a and b, firms set prices such that pA(a,b) and pB(a,b) constitute a Nash equilibrium.

Firms solve the following program:

maxpiΠi(a,b,pi,pj)=maxpipiDi

According to eq. (4), the expressions of demand are: DA=pBpAba and DB=1pBpAba.

Hence, the maximization constraint above is equivalent to:

maxpApBpAbapAand maxpB1pBpAbapB

The first order conditions are given by:

ΠApA=0pB2pAba=0
ΠBpB=012pBpAba=0

After substitution, one obtains pA(a,b) and pB(a,b):

(22)pA(a,b)=(ba)3and pB(a,b)=2(ba)3

Determining profits as a function of a,b: Once prices are known, one can calculate profits ΠA and ΠB as functions of a, b.

ΠA(a,b)=DApA(a,b)and ΠB(a,b)=DBpB(a,b)

After replacing pA(a,b) and pB(a,b) by the expressions in eq. (22), one obtains:

(34)ΠA(a,b)=DA(ba)3
(25)ΠB(a,b)=DB2(ba)3

Recall that according to eq. (4) DA=pBpAba and DB=1pBpAba.

After substitution, the expressions of the profits Πi yield:

(26)ΠA(a,b)=(ba)9
(27)ΠB(a,b)=2(ba)9

Verifying that there exists a Nash equilibrium under the assumption that the market is covered. In this section, I want to prove that when prices are such that pA=(ba)3 and pB=2(ba)3 as defined in eq. (22) the market is a Nash equilibrium whereby neither firm has incentives to deviate. I first define a condition under which firm A’s best response is to cover the market when pB=pB; and next turn to firm B’s best response when A covers the market.

A sufficient condition which guarantees that firm A’s best response is to cover the market:

Let us assume that the market is covered so that pB=2(ba)3. If firm A does not cover the market, then PA>r. This means that consumers located at point 0 do not purchase good A. Firm A’s profit is then equal to:

ΠA=pApBpAbapAra

A sufficient condition under which firm A never decides not to cover the market is therefore:

pBpAbapAra<0

Recall that pAra is assumed to be positive since the market is not covered. Hence, a sufficient condition is pBpA<0.

Since pA>r, the latter condition implies 2(ba)3<r.

To conclude, firm A covers the market if:

2(ba)3<r

A sufficient condition which guarantees that firm B’s best response is to cover the market:

If B does not cover the market there is a consumer who is indifferent between buying good B and not consuming at all. This consumer is located at point xˉ=pBrb.

When firm B does not cover the market, its profit is equal to:

ΠB=pBxˉpBpAba=pBpBrbpBpAba

A sufficient condition that guarantees that firm B will never decide not to cover the market is pBr<pBpA. Since A covers the market, we know that pA<r. Hence, the condition is always true, provided that A covers the market. In other words, when firm A covers the market, firm B’s best response is always to cover the market.

Conclusion:A sufficient condition which guarantees that the covered market outcome is a Nash equilibrium is:

2(ba)3<r

B The market outcome with consumer bias

Assumptions. Let us define the ex ante anticipated utility U˜x of a consumer located at point x[0,1], such that:

U˜x(A)=r+axpA+ϵAif he buys product AU˜x(B)=r+bxpB+ϵBif he buys product B

The parameters ϵA and ϵB respectively represent consumer bias about goods A and B. We define Δϵ=ϵAϵB. We suppose that r is sufficiently large so that all consumers prefer to buy one unit of either good rather than no good at all. The parameter r needs to verify r>max(pAϵ1;pBϵBb).

The consumer located at point xˆ=pBpA+Δϵba is indifferent between goods A and B. Consumers located at point x such that x<xˆ (respectively x>xˆ) buy good A (respectively B).

Therefore, profit functions are as follows:

  1. if pBpA<ϵBϵA, then ΠA=0 and ΠB=pB

  2. if pBpA(ba)ϵBϵApBpA, then ΠA=pA(pBpA+Δϵba) and ΠB=pB(1pBpA+Δϵba).

  3. if ϵBϵA<pBpA(ba), then ΠA=pA and ΠB=0

We focus on cases where there exists a duopolistic equilibrium, that is to say when both firms make a positive profit. Hence, restrict the analysis to pBpA(ba)ϵBϵApBpA.

Determining prices as a function ofa, b and Δϵ. For any given locations a and b, firms set prices such that pA(a,b) and pB(a,b) constitute a Nash equilibrium.

Firms solve the following program:

maxpiΠi(a,b,pi,pj)=maxpipiDi

According to eq. (4), the expressions of demands are: DA=pBpA+Δϵba and DB=1pBpA+Δϵba.

Hence, the maximization constraint above is equivalent to:

maxpApBpA+ΔϵbapAand maxpB1pBpAΔϵbapB

The first order conditions are given by:

ΠApA=0pB2pA+Δϵba=0
ΠBpB=012pBpAΔϵba=0

After substitution, one obtains pA(a,b) and pB(a,b):

(32)pA(a,b)=(ba)+Δϵ3and pB(a,b)=2(ba)Δϵ3

Determining profits as a function ofa, b and Δϵ. Once prices are known, one can calculate profits ΠA and ΠB as functions of a, b and Δϵ.

ΠA(a,b)=DApA(a,b)and ΠB(a,b)=DBpB(a,b)

After replacing pA(a,b) and pB(a,b) by the expressions in eq. (32), one obtains:

(24)ΠA(a,b)=DA(ba)+Δϵ3
(35)ΠB(a,b)=DB2(ba)Δϵ3

Recall that according to eq. (4) DA=pBpA+Δϵba and DB=1pBpAΔϵba.

After substitution, the expressions of the profits Πi yield:

(36)ΠA(a,b)=(ba)+Δϵ29(ba)
(37)ΠB(a,b)=2(ba)Δϵ29(ba)

The results in eqs. (36) and (37) are only relevant for ϵaϵb. If ϵa=ϵb we are back to the standard situation without biased anticipations.

A sufficient condition which guarantees that the equilibrium with a covered market is a Nash equilibrium. With a similar method as the one used in the absence of consumer bias, we can determine a sufficient condition when neither firm has incentives not to cover the market.

A sufficient condition which guarantees that firm A’s best response is to cover the market:

Let us suppose that firm B covers the market so pB=2(ba)Δϵ3. If firm A does not cover the market, its profit is equal to:

ΠA=pApBpA+ΔϵbapArϵAa

A sufficient condition which implies firm A will never decide not to cover the market is pBpA+Δϵ<0, which is equivalent to pA>pB+Δϵ.

Since pA>rϵA, a sufficient condition is:

pB+Δϵ<rϵA

After substitution, one obtains:

r>2(ba)2ϵBϵA3

A sufficient condition which guarantees that firm B’s best response is to cover the market:

Let us suppose that firm A covers the market so pA=(ba)Δϵ3. If firm B does not cover the market, then there is a consumer located at point xˉ=pBrϵBb who is indifferent between buying good B and not buying at all. Firm B’s profit is equal to:

ΠB=pBpBrϵbbpBpA+Δϵba

A sufficient condition which implies firm B will never decide not to cover the market is pBrϵb<pBpA+Δϵ, which is equivalent to pAϵBΔϵ<r.

After substitution, one obtains:

r>(ba)2ϵAϵB3

Conclusion:A sufficient condition which guarantees that the covered market outcome is a Nash equilibrium is:

(38)r>max2(ba)2ϵBϵA3;(ba)2ϵAϵB3

This condition remains valid for ϵA=0 and/or ϵB=0. When ϵA=ϵB=0 we are back to the case without bias mentioned above in Section 8.2. This general condition can be adapted to each case when a firm debiases, either regarding its own product and/or the rival’s good.

C Debiasing conditions

I explain in detail the debiasing conditions for firm A. The same method is used for firm B.

  1. Regarding firm A:

    1. Symmetric education: Firm A educates consumers if and only if:

      (39)ΠACA>ΠA(ϵA;ϵB)

      According to the expression of profits above, eq. (36), the condition (39) is equivalent to:

      [ba]2(ba)+ϵAϵB2>CA
      2(ba)+ϵAϵB[ϵBϵA]>CA
      1. If Δϵ>0:

        2(ba)<CAΔϵΔϵ

        Consequently, A debiases if:

        Δϵ>02(ba)<CAΔϵΔϵ

        This case is impossible since (ba)0.

      2. If Δϵ<0:

        2(ba)>CAΔϵΔϵ
    2. Asymmetric education:

      1. Firm A educates consumers regarding its own good:

        Firm A has incentives to educate consumers concerning good A if and only if:

        (40)ΠA(ϵB)CA>ΠA(ϵA;ϵB)

        Equation (40) is equivalent to:

        [(ba)ϵB]2(ba)+ϵAϵB2>CA

        Two sub-cases can be distinguished:

        ϵA>02(ba)<CAϵAϵA+2ϵB

        and

        ϵA<02(ba)>CAϵAϵA+2ϵB
      2. Firm Ad educates consumers regarding the rival’s good

        Firm A has incentives to educate consumers concerning good B if and only if:

        (41)ΠA(ϵA)CA>ΠA(ϵA;ϵB)

        Equation (41) is equivalent to:

        [(ba)+ϵA]2(ba)+ϵAϵB2>CA

        Two sub-cases can be distinguished:

        ϵB>02(ba)>CAϵB+ϵB2ϵA

        and

        ϵB<02(ba)<CAϵB+ϵB2ϵA
  2. Regarding firm B:

    1. Symmetric education: with an identical method as the one used for firm A, we can show that firm B educates consumers in two situations depending on the sign of Δϵ:

      1. If Δϵ<0:

        Δϵ<04(ba)<CBΔϵ+Δϵ

        This case is impossible.

      2. If Δϵ>0:

        Δϵ>04(ba)>+CBΔϵ+Δϵ
    2. Asymmetric education: Similarly, one finds that, in the case of an asymmetric debiasing policy, firm B can educate consumers regarding good B or regarding good A. One finds the following conditions, with an identical method as detailed above:

      1. Firm B educates consumers regarding its own good:

        Firm B has incentives to educate consumers concerning good A if and only if:

        (44)ΠB(ϵA)CB>ΠB(ϵA;ϵB)

        Two sub-cases can be distinguished:

        ϵB>04(ba)<CBϵBϵB+2ϵA

        and

        ϵB<04(ba)>CBϵBϵB+2ϵA
      2. Firm B has incentives to educate consumers concerning good A if and only if:

        (45)ΠB(ϵB)CB>ΠB(ϵA;ϵB)

        Two sub-cases can be distinguished:

        ϵA>04(ba)>CBϵA+ϵA2ϵB

        and

        ϵA<04(ba)<CBϵA+ϵA2ϵB

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