Economics Letters 81 (2003) 143–146www.elsevier.com/ locate/econbase

C ollusive price can be lower than equilibrium price if there issearch cost

*Hyung Bae

Department of Economics, Dongguk University, 3-26 Pil-Dong Jung-Gu, Seoul 100-715, South Korea

Received 9 January 2003; received in revised form 7 April 2003; accepted 6 May 2003

Abstract

This paper analyzes a search model in which consumers vary in their demands. We show that the collusiveprice can be lower than the equilibrium price in the market because, more consumers join the market if all firmslower prices than if a single firm lowers its price. This result can explain why sometimes stores around resortareas voluntarily regulate their prices. In our model, such price reducing collusion is better than competition forthe consumers as well as the firms. 2003 Elsevier B.V. All rights reserved.

Keywords: Search; Equilibrium price; Collusive price

JEL classification: D83; L11

1 . Introduction

This paper analyzes a search model in which consumers vary in their demands. In this model, afirm’s small decrease in price from the market price cannot lure consumers of other firms due to thesearch cost. By the same token, a firm’s small increase in price from the market price does not send itsconsumers out to the other firms. But a decrease in the market price increases the number ofconsumers participating in the market. Therefore, the collusive price can be lower than the equilibriumprice in this market. This result explains why sometimes stores around resort areas regulate theirprices voluntarily.

Diamond (1971)shows that under certain conditions the equilibrium price equals the collusive

*Tel.: 182-2-2260-3271; fax:182-2-2260-3684.E-mail address: hbae@dongguk.edu(H. Bae).

0165-1765/03/$ – see front matter 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0165-1765(03)00163-0

144 H. Bae / Economics Letters 81 (2003) 143–146

1price if there is a search cost . His result differs from ours because the consumers are identical in hismodel. If the consumers have identical demand, his result also holds in our model. With identicalconsumers and search costs, independent price setting firms would set the same price as they wouldset collusively. With varying consumers, some consumers may not search in the equilibrium becausesearch cost exceeds their consumer surpluses. In such a case, more consumers join the market if allfirms lower prices than if a single firm lowers its price, and hence the collusive price can be lowerthan the equilibrium price.

There are some cases in which the collusive price can be lower than the equilibrium price. One isthe case in which the consumers have the market-wide network externality. In this case, lowering theprice can increase the utilities of consumers of other firms, and hence can increase the demand ofother firms. The other is the case in which the firms produce complements. In our model as well as inthese two cases, lowering a firm’s price can increase other firms’ demand, at least, in a neighborhoodof the equilibrium price. But, unique from these two cases, in our model, firms produce identicalgoods and there is no externality. In our model, the existence of search cost and heterogeneity inconsumers’ demands make the collusive price lower than the equilibrium price.

The rest of this paper is organized as follows. In Section 2, we introduce the model and derive theequilibrium price. Section 3 derives the collusive price and shows that it can be lower than theequilibrium price. In Section 4, we conclude the paper.

2 . The model and the equilibrium price

In this section, we introduce a search model in which the consumers vary in their demands andderive the equilibrium price. We assume that there areN firms, which produce an indentical productand a continuum of consumers whose measure is 1. We index the consumers witha which isdistributed uniformly on the closed interval [0, 1]. Consumera has a continuously differentiabledownward sloping demand functionD( p, a). There is a search costc for each search. At any time, the

2consumers can recall the buying opportunities, which they forwent . For simplicity, we assume thatthere is no production cost.

`Let CS( p, a);e D(s, a) ds be the consumer’s surplus of consumera at pricep. We assume that,p

¯ ¯ ¯ ¯≠CS( p, a) /≠a$0 for all p; there existsp such thatp . 0, CS(p, 1)5 c, andCS( p, 1).c for p , p;¯ ¯≠CS( p, a) /≠a.0 for all p [ [0, p ]. Then, for allp [ [0, p ], there existsa( p) that is the value ofa,

¯which satisfies thatCS( p, a)5c, wherec is the search cost. Moreover,a9( p).0 for p [ (0, p ).Generally in most search models, the consumers are assumed to know which prices are available in

the market, but they do not know which stores charge which prices. Let us say that a consumer joinsthe market if he ever searches for a product in the market. We assume that there is no income effect inthe demands for the product. Then, it is optimal for a consumer to join (not to join) the market if hisexpected consumer’s surplus is greater (less) than his expected search cost. In this paper, we consider

1Diamond’s result elicited a flurry of responses because it means that there can be no price dispersion in equilibrium. Forsubsequent contributions, which develop models in which there exists dispersion in equilibrium, seeBurdett and Judd (1983)and the studies cited in their research.

2Since we consider situations in which there are at most two prices, it is never optimal for a consumer to forgo a buyingopportunity and then recall it later. Thus, whether we allow consumers’ recall or not does not alter the results.

H. Bae / Economics Letters 81 (2003) 143–146 145

the cases in which all the firms set the same price. When firms are expected to set a pricep, it isoptimal for the consumera to join (not to join) the market ifa.(,)a( p).

EQMAs the equilibrium price,p , we refer to the price such that consumers correctly expect the priceand behave optimally, and no firm has an incentive to deviate from setting the price. In order to checkwhether firms have incentives to deviate from setting the equilibrium price, we need an assumption on

EQMhow consumers behave when they find a price other thanp . We assume that when a consumerEQMfinds out that a firm sets pricep and not p , he believes that the other firms still hold the

EQMequilibrium price, p . Then, because there is a search cost, there exists a pricep such thatp isEQMhigher thanp and when consumers find a price, which is lower thanp, they purchase at this price.

Furthermore, we assume that consumers do not know the deviation of a firm from setting theEQMequilibrium price,p , so that such deviation does not alter consumers’ decision of whether or not

EQMto join the market. Therefore, there is a neighborhood ofp such that if a firm set pricep in theEQM 1 EQMneighborhood, its profit isp ( p); (1 /N)p e D( p, a)da. Assume thatp ( p) is strictlyEQMa ( p )

quasiconcave.EQMFor firms not to deviate from settingp , we need:

EQM EQM EQM1 1dp ( p ) ≠D( p , a)1 EQM EQM]]]]] ] ]]]]5 E D( p , a)da 1 p E da 5 0. (1)S D

EQM EQMdp N ≠pa ( p ) a ( p )

Moreover, (1) is sufficient because the profit of a firm when it setsp outside the neighborhood is notEQMgreater thanp ( p).

3 . The collusive price and the results

In this section, we consider whether the firms have incentives to lower their prices from theequilibrium price and advertise the price reduction. If the firms set pricep and advertise it, then the

COL 1 COLfirms obtain the profitp ( p); p e D( p, a)da. Assume thatp ( p) is strictly quasiconcave.a ( p)COL COL COLWe refer to the value ofp which maximizesp ( p) as the collusive price,p . Then, p is

the value ofp, which satisfies that:

COL 1 1dp ( p) ≠D( p, a)]]] ]]]5E D( p, a) da 1 p E da 2 pD( p, a( p))a9( p)50. (2)dp ≠pa ( p) a ( p)

COL EQM COL EQMFrom (1) and (2), we have dp ( p ) /dp,0. Thus, p ,p . Because the consumer’ssurplus increases as the price decreases, in our model, price-reducing collusion is better thanEQM-petition for the consumers as well as the firms.

4 . Conclusions

This paper has analyzed a search model in which the consumers vary in their demands, and showedthat the collusive price can be lower than the equilibrium price. This result explains why at times

146 H. Bae / Economics Letters 81 (2003) 143–146

stores around resort areas voluntarily regulate their prices and advertise such price adjustment. In ourmodel, such price reducing collusion is better than competition for the consumers as well as the firms.

A cknowledgements

The author is grateful to Ill Tae Ahn, Simon P. Anderson, Jai Pil Choi, Jeong-Yoo Kim, Gyu HoWang, Sang-Seung Yi, and an anonymous referee for helpful comments and valuable suggestions.This work is supported by the Dongguk University research fund.

R eferences

B urdett, K., Judd, K., 1983. Equilibrium price dispersion. Econometrica 51, 955–970.D iamond, P., 1971. A model of price adjustment. Journal of Economic Theory 3, 156–168.