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Transaction Utility and Quality Choice

  • Sajeesh Sajeesh EMAIL logo and Sang-Young Song

Abstract

This paper uses a game-theoretic model to examine the role of reference price for firms that vary in their quality positioning in competing for customers. Reference prices provide consumers with additional components of utility. Building on previous research on the impact of consumer decision making on firm strategies, we focus on how firms choose their positioning when consumer utility is driven not only by acquisition utility but also by the transaction utility associated with the purchase and how this, in turn, affects firms’ pricing decisions and profits. Considering a competition between two firms, this paper shows that the firm with higher product quality provides greater discounts to consumers. We also show that when firms are allowed to set a high ‘regular’ price, product differentiation is greater between the firms, and price competition is less intense. Furthermore, under some conditions, the profits of both firms can be higher than the benchmark case (when the effects of transaction utility are ignored).

Appendix

A

Equating consumer utility, the location of the marginal consumer is given by

(A.1)θ˜p1,p2,λ1,λ2=p2p11+λ1+λ2λ1αcs22+λ1αcs12s2s1.

Therefore, the demand observed by the two firms is given by

(A.2)D1=1bap2p11+λ1+λ2λ1αcs22+λ1αcs12s2s1a,
(A.3)D2=1babp2p11+λ1+λ2λ1αcs22+λ1αcs12s2s1.

Now,

(A.4)pˆ1=maxp1p1cs12D1,
(A.5)pˆ2=maxp2p2cs22D2.

Setting the derivative of π1 w.r.t p1 to zero and setting the derivative of π2 w.r.t p2 to zero, the first-order conditions (FOC’s) for profit maximization lead to the following equilibrium prices,

(A.6)pˆ1=b2as2s1+2cs121+λ1+λ2+cs221+λ1+λ2+cαλ1s12s223λ1+3λ2+3,
(A.7)pˆ2=2bas2s1+cs121+λ1+λ2+2cs221+λ1+λ2+cαλ1s22s123λ1+3λ2+3.

The choice of quality level can be found by solving the following maximization problem:

(A.8)s1=maxs1pˆ1cs12D1pˆ1,pˆ2,
(A.9)s2=maxs2pˆ2cs22D2pˆ1,pˆ2.

Setting the derivative of π1 w.r.t s1 to zero and setting the derivative of π2 w.r.t s2 to zero, using the FOC’s for profit maximization and solving, we get eqs (8) and (9) The second-order conditions (SOC’s) are given by d2πidsi2|(s1=s1,s2=s2)=12cλ1λ2+αλ11λ1+λ2+1 for i=1,2. The SOCs are satisfied for α<1+1+λ2λ1.

Substituting the equilibrium locations, the equilibrium prices are given by the following:

p1=16449a21+λ1+λ2+25b21+λ1+λ258ab1+λ1+λ224αλ1ab2λ1+λ2+1cλ1λ2+αλ112,p2=16425a21+λ1+λ2+49b21+λ1+λ258ab1+λ1+λ224αλ1ab2λ1+λ2+1cλ1λ2+αλ112.

Therefore,

p2p1=38b2a2c1+λ1+λ2αλ12,dp2p1dα=34cλ1b2a2λ1+λ2αλ1+13>0.

Not surprisingly, the higher-quality firm charges a higher price. As α increases, the difference in equilibrium prices between the high-quality and the low-quality firm also increases. The equilibrium market share of each firm is 12ba>0. The conditions α<1+1+λ2λ1 and b<5a also ensures that the equilibrium prices are greater than 0 and dp2p1dα>0.

Verifying that firms do not have an incentive to deviate from the equilibrium locations given in equations (8) and (9)

Given s2, when Firm 2 is the higher-quality firm, profits for Firm 1 is given by

(A.10)π1s2=14608a5b+8cs1+8cλ1s1+8cλ2s18cαλ1s117a13b8cs18cλ1s18cλ2s1+8cαλ1s12abcλ1λ2+αλ11λ1+λ2+1.

If Firm 1 deviates to choose a higher-quality than Firm 2, then Firm 2 becomes the lower-quality firm. The profits for Firm 1 under this scenario can be found by interchanging s1 and s2 in π2. In such a deviation, the profits of Firm 1, π1ds2 is given by

(A.11)π1ds2=14608a5b+8cs1+8cλ1s1+8cλ2s18cαλ1s117a13b8cs18cλ1s18cλ2s1+8cαλ1s12abcλ1λ2+αλ11λ1+λ2+1.

Using the first-order condition for profit maximization, the profits in the two settings is given by

(A.12)π1s2=3(ba)216cλ1+λ2+1λ1+λ2+1αλ1,and
(A.13)π1ds2=1144ba2cλ1+λ2+1λ1+λ2+1αλ1.

Since π1s2>π1ds2, firm 1 has no incentive to deviate and offer quality higher than Firm 2. Thus, the local equilibrium is the global product quality equilibrium.

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Published Online: 2017-8-23
Published in Print: 2017-9-26

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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