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J Ind Compet Trade (2012) 12:265–272 DOI 10.1007/s10842-011-0097-2

Duopoly Pricing Under ‘Private Knowledge’ of Product Differentiation Carlos A. Ulibarri

Received: 12 July 2010 / Revised: 10 January 2011 Accepted: 4 February 2011 / Published online: 19 February 2011 # Springer Science+Business Media, LLC 2011

Abstract This note studies price decisions in a duopoly industry where firms have private information over the degree of product differentiation (product-type). A Bayesian-Nash price solution is derived assuming firms maximize their ‘certainty-equivalent’ profit levels. The comparative-statics indicate that increased risk aversion over the rival’s product triggers price competition. Consequently, the results of the study suggest revealing information is a higher reward strategy than concealing information in situations where rivals have asymmetric information over product type. These findings contribute to the industrial economic literature by generalizing the Bertrand equilibrium in an asymmetric information game model. Keywords duopoly market . asymmetric information game model . Bayes-Nash equilibrium JEL Classifications D43 . D41

1 Introduction The Bertrand model of price competition has been extensively analyzed in game-theoretic analysis of product differentiation. The necessary and sufficient conditions for Nash (1950) and Bayesian-Nash equilibria are described in Vives (1990, 1999). In brief the Nash equilibrium is guaranteed in pure strategies under complete information but not under asymmetric information. For the latter, a ‘Bayesian Nash solution’ in pure strategies may be established, or possibly mixed strategies with players randomizing over a certain range of prices. The present study uses an asymmetric information game (AIG) model to examine the Bayesian-Nash price equilibrium (BNE prices) assuming firms have private knowledge of the degree of product differentiation. In this setting we show information sharing is a ‘first-best’ strategy for ‘value maximizing’ duopoly firms. For example, the firms could agree to share product design information through a marketing or trade association. C. A. Ulibarri (*) Department of Management, New Mexico Tech, 801 Leroy Place, Socorro, New Mexico e-mail: [email protected]

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Industrial organization models that incorporate private information are particularly useful in examining problems of oligopoly and imperfect competition, including the incentives to reveal cost or demand information (Milgrom and Roberts 1987). In particular, the information sharing problem in oligopoly/duopoly settings has been examined extensively in the trade association literature. Notable cases include two-stage games with Bertrand or Cournot firms competing under uncertainty about the industry’s demand parameter (common knowledge to all participants), or a cost parameter that is firm-specific (private values).1 The present study considers a differentiated product market where duopoly firms are uncertain about each other’s degree of product differentiation, and thus make strategic pricing decisions under asymmetric information over product type. The normal-form representation of Bertrand competition is modeled assuming risk-averse firms move simultaneously in setting prices of differentiated products. The analysis assumes each firm has private information over the degree of product differentiation, and that this is common knowledge. The firms then choose their first-best prices if they are sufficiently compensated for risk-taking, as measured by their ‘certainty-equivalent’ profit E½CEi ¼ p i ri ; that is, the expected profit minus the risk premium. Given the utility from the certainty-equivalent profit ~i Þ; the risk premiums equals the expected utility from the random profit, U ðpi ri Þ ¼ E½U ðp reflect compensation for incomplete information over product type; or in other words, the maximum firms are willing–to-pay to convert random profits into deterministic profits.2 The study proceeds as follows. Section 2 constructs a normal-form representation of the duopoly industry, deriving BNE prices which maximize ‘certainty-equivalent’ profits. Section 3 examines the comparative statics of the model for substitute and complement products. Section 4 interprets the practical implications of the study relative to the literature.

2 Risk-averse Bertrand competition Singh and Vives (1984) propose a convenient demand system to study differentiated product markets in2 which a representative consumer maximizes utility minus expendiP tures: U ðq1 ; q2 Þ pi qi , where pi and qi denote the price and amount of good i. The i¼1 present study assumes U is strictly concave and takes a simplified quadratic form: U ðq1 ; q2 Þ ¼ aðq1 þ q2 Þ

1 2 q þ 2gq1 q2 þ q22 : 2 1

ð1Þ

The ‘gamma’ parameter ðgÞdescribes the nature of product differentiation: the goods are substitutes if γ>0, independent if γ=0 and complements if γ 0. The inverse and direct demand systems are given by expressions (2) and (3): p1 ¼ a q1 gq2 ; p2 ¼ a q2 gq1 ; ð2Þ

q1 ¼

1

½að1 gÞ p1 þ gp2 ½að1 gÞ p2 þ gp1 ; q2 ¼ : 1 g2 1 g2

ð3Þ

See for example Novshek and Sonnenschein (1982), Vives (1984), Gal-Or (1986), Shapiro (1986), and Choi and Jagpal (2004). 2 The seminal contributions to decision theory by Arrow (1970) and Pratt (1964) led to the formulas for risk premia and certainty equivalents applied in the present study. See Milgrom and Roberts (1992) for an intuitive description of the underlying theory and analytical derivations (esp. pp. 246–247).


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The standard model of Bertrand competition assumes complete information over product type—a common gamma parameter observable to both firms. However, suppose perceptions of the rival’s product are based on incomplete information, and that this is common knowledge to both firms. This begs the question: how does private knowledge over product differentiation affect price competition between risk averse firms? To examine this question suppose each firm has private information and sets price under incomplete information over the degree of product differentiation. The price decisions are made assuming the cross-price parameters ð ~g 12 ; ~g 21 Þ in the demand system are i.i.d. random variables, drawn from a joint probability distribution function Fð ~g 12 ; ~g 21 Þ;which is common knowledge to both firms. Essentially the randomized term ~g 12 describes firm 1’s perception of its’ degree of product differentiation and ~g 21 describes firm 2’s perception of its’ degree of product differentiation. Hence substitute products are defined on the positive interval 0 < ~g 12 ; ~g 21 < 1; complement products on the negative interval 1 < ~g 12 ; ~g 21 < 0; and independent products between the two supports ~g 12 ; ~g 21 ¼ 0. Based on the direct demand system (3) the profit equations for the industry are given by ðp1 c1 Þ½að1 ~g 12 Þ p1 þ ~g 12 p2 ~ p1 ¼ ðp1 c1 Þq1 ¼ ; ð4Þ ð1 ~g ~g Þ 12 21

ðp2 c2 Þ½að1 ~g 21 Þ p2 þ ~g 21 p1 p~2 ¼ ðp2 c2 Þq2 ¼ : ð1 ~g 12~g 21 Þ

ð5Þ

The present study focuses on duopolists with constant absolute risk aversion (CARA) defined by parameters r1 and r2, paying risk premiums to convert random profits ~p i to their certainty-equivalent CEi ¼ p i :5ri Varðp i Þ.3 Therefore in compact form the value maximization model for the duopoly market is specified as: # " # " # " #" r1 Varðp 1 Þ CE1 p1 : ð6Þ ¼ :5 CE2 p2 r2 Varðp 2 Þ Two major assumptions are used in specifying the AIG model. Assumption (i): firm 1 takes expectations over its profit Eq. 4 given prior information ~g 21 ¼ o; and likewise, firm 2 takes expectations over its profit Eq. 5 given ~g 12 ¼ o: The logic behind this assumption is that Bayesian agents initially take expectations of variates in their own demand curves, not their rivals. Assumption (ii): firms have ‘common knowledge’ of each other’s degree of risk aversion.4 Clearly ‘common knowledge’ of the rival’s degree of risk aversion is a strong assumption in characterizing strategic choice behavior in the AIG framework. Nonetheless, it yields testable implications regarding duopolistic reactions to uncertainty in Bertrand markets. In this setting the means and variances of the profit equations are defined by p1 ¼ E½ðp1 c1 Þq1 ¼ ðp1 c1 Þ½að1 g 12 Þ p1 þ g 12 p2 ; Varðp 1 Þ ¼ s 2p1 ;

ð7Þ

p2 ¼ E½ðp2 c2 Þq2 ¼ ðp2 c2 Þ½að1 g 21 Þ p2 þ g 21 p1 ; Varðp2 Þ ¼ s 2p2

ð8Þ

The Arrow-Pratt coefficient of absolute risk aversion (CARA) is measured at the level of profit p as defined by r ¼ u00 ðpÞ=u0 ðpÞ; The CARA measure determines the firm’s risk premium for small risk: rðpÞ ¼ :5rs 2p . 4 ‘Common knowledge’ is said to exist when two agents know a certain fact, and each agent knows the other agent knows the same fact, etc. (Aumann 1976). 3

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where the bar over the random product-type parameters denotes their expected values. The necessary conditions for value maximizing prices are given by p1 2 arg maxfCE1 g :

@s 2 @CE1 @p 1 ¼0! ¼ :5r1 p1 ; @p1 @p1 @p1

ð9Þ

p2 2 arg maxf CE2 g :

@s 2 @CE2 @p2 ¼0! ¼ :5r2 p2 ; @p2 @p2 @p2

ð10Þ

yielding the price reaction curves: p1 ¼ :5½að1 g 12 Þ þ g 12 p2 þ c1 R1 ;

R1 ¼ :52 r1

@s 2p1 ; @p1

ð11Þ

p2 ¼ :5½að1 g 21 Þ þ g 21 p1 þ c2 R2 ;

R2 ¼ :52 r2

@s 2p2 : @p2

ð12Þ

The price reactions depend on unit cost and CARA parameters and estimates of the expected product type parameters. Note that the intercept and slope of the best reply curves depend on the expected product-type parameter. The Bayes-Nash equilibrium (BNE) prices are obtained by solving (11) and (12) simultaneously, yielding p1 ¼ ð

4 g g Þ :5½að1 g 12 Þ þ c1 þ 12 ½að1 g 21 Þ þ c2 ½R1 þ 12 R2 ; ð13Þ 4 2 4 g 12 g 21

4 g 21 g 21 R1 : Þ :5½að1 g 21 Þ þ c2 þ ½að1 g 12 Þ þ c1 ½R2 þ p2 ¼ ð 4 2 4 g 12 g 21

ð14Þ

The individual BNE prices depend on the cost and CARA parameters of both firms, and the expected product-type parameters of both firms.

3 Comparative statics Figures 1a–b illustrate the impact of risk aversion on the firms’ price reaction curves (11 and 12) and the resulting BNE prices (13 and 14). The curves have positive slopes for differentiated substitute products ð0 < ~g 12 ; ~g 21 < 1Þ; negative slopes for differentiated complement products (1 < ~g 12 ; ~g 21 < 0); and zero slope (not shown) for independent monopoly markets ( ~g 12 ; ~g 21 ¼ 0). Figure 1a illustrates the BNE solution for differentiated substitutes under risk neutral conditions (r1,r2 =0) and risk-averse conditions(r1, r2,>0), denoted by points 0 and 1. The slope and intercept terms of the price reaction curves reflect the expectation of substitutes. Greater risk aversion or lower unit costs shift the curves inward, resulting in the BNE solution at point 1 where the firms set lower prices p11 ; p12 . For stronger substitutes the price reaction curves shift inward and rotate outward. Figure 1b illustrates the BNE solution for differentiated complement products, where the risk neutral and risk-averse solutions are again denoted by points 0 and 1. Here the slope

J Ind Compet Trade (2012) 12:265–272 Fig. 1 a comparative statics of differentiated substitute products b comparative statics of differentiated complement products

269

ap

2

Firm1 r1> 0

O

p20 (r2 = 0)

r1= 0 r2= 0 r2 > 0

Firm2

1

p21 (r2 > 0)

p1 p11 (r1 > 0) p10 (r1 = 0)

b

Firm1 p2

r1> 0

r1= 0

p20 (r2 = 0)

O

p21 (r2 = 0) 1

r2= 0 Firm2

r2 > 0

p1/ (r1 > 0)

p10 (r1 = 0)

p1

and intercept terms reflect expectations of complement products. As before increased risk aversion or lower unit cost shifts the reaction curves inward, resulting in the BNE solution at point 1, where both firms set lower prices ðp11 ; p12 Þ . The price reaction curves for stronger complements shift outward and rotate inward. Clearly asymmetric information over product differentiation is fundamental in determining price behavior in the industry. If substitute products are expected (g 12 ; g 21 > 0), then lower production costs or increased risk aversion reduce the BNE prices. The lower prices correspond with perceptions that the products are becoming stronger substitutes. On the other hand, expectations of complement products (g 12 ; g 21 < 0) yield mixed effects from changes in unit costs or risk aversion. In this case the BNE price of product 1 also drops if firm 1 lowers its unit cost or becomes more risk averse; however, the price rises if firm 2 lowers unit cost or has increased risk aversion. These relationships are summarized in Table 1, and stated in the following propositions. Propositions (1) and (2) under assumptions (i) and (ii) (1) Given private knowledge of product differentiation for substitutes (g 12 ; g 21 > 0), the BNE price of each firm decreases @pi i when either firm becomes more risk averse, i.e. @p @ri < 0; @rj < 0 i 6¼ j: (2) Given private

270


Table 1 Bayes-Nash equilibrium price reactions

Own risk aversion Rival’s risk aversion Own unit cost Rival’s unit cost

Substitute Products

Independent monopoly

Complement products

@pi @ri @pi @rj @pi @ci @pi @cj



0

0 ¼0

0 >0 0 i ¼ These findings have intuitive explanations. In substitute product markets, a more risk averse firm will mark down its’ price to reduce the risk of competing with the rival firm, which may set an even lower price. Thus in Bertrand markets for differentiated substitutes this immediately explains the negative relationship between a firm’s own price and the rival’s degree of risk aversion. Meanwhile in complementary product markets one firm loses demand if the other firm increases its’ price. Hence, if a firm becomes more risk averse then it will lower its’ price to avoid such losses. Correspondingly, increased risk aversion at one firm lets the other firm profit by raising the price of its’ complement product. Propositions (3) and (4) under assumptions (i) and (ii) (3) Given private knowledge of differentiated substitutes or complements, the BNE prices increase symmetrically in own unit i cost @p @ci > 0 i ¼ 1; 2. (4) However, the BNE prices increase or decrease asymmetrically in i 6 j, depending on whether the rival product is the unit cost of the rival firm @p @cj 0; and @s satisfaction of first-order conditions (9,10) ! @pp11 > 0. Thenproofo of (2) is based on the g @s < i 1 signs of the cross-partial derivatives of (13, 14), i.e. @p 2 @p > 0 i; j ¼ 1; 2 i 6¼ j: @rj ¼ 4g g The mixed signs correspond to the products being substitutes, independent or complements, thereby establishing the proof □ ij

ij ji

2 pj j

@p2 1 Proof of propositions (3) and (4) First, partial differentiation of (13, 14) yields @p @c1 ¼ @c2 ¼ 2 4g 12 g 21 > 0; showing higher own-costs increase the BNE prices symmetrically. Second, when g 12 1 product expectations differ g 12 6¼ g 21 the inequality between partial derivatives, @p @c2 ¼ 4g 12 g 21 g 21 @p2 and @c1 ¼ 4g g , implies the BNE prices adjust asymmetrically to changes in rival-cost: 12 21 increasing for perceived substitutes g 12 ; g 21 > 0 and decreasing for perceived complements g 12 ; g 21 < 0 □

6

Ibid. pp. 302, esp. proposition 2.

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