2023年6月24日发(作者：)

1

Market equilibrium with price-beating firms under imperfect

knowledge

Pedro Cosme Costa Vieira (pcosme@)

Faculdade de Economia do Porto

R. Dr. Roberto Frias, s/n

4200-464 PORTO, PORTUGAL

Tel +351.225571215 Fax +351.225505050

Presented at the

X Summer Meeting of Economic Analysis

at Santiago de Compostela, Spain

23-24 Sept 2004

Abstract

It is unquestionable both theoretically and empirically that trade in competitive

markets is the best way to promote the efficient allocation of goods. Nevertheless, there

could be some traders‟ strategies that interfere with competition, collusion among

traders, which results in losses to society.

In this work, I study the implication that guarantees have on the competitiveness

of the market when information is imperfect and costly. In formal terms, I study the

„price matching‟ and „price beating‟ guarantees that may be extended to other types of

guarantee.

From the theoretical model it results that both strategies are anti-competitive

strategy but, contrary to „price matching‟, „price beating‟ is not in the best interest of

sellers (in a Nash equilibrium perspective).

JEL codes: D82, D83, L12

Keywords: Collusion; Price beating guarantees; Imperfect information

2

1. Introduction

Guarantees are declarations from seller that its products have a certain quality

standard. So, guarantees only make sense if they are a commitment on a characteristic

that buyers do not known perfectly and if there is a compensation in case the claim is

not accurate. Otherwise, it is “cheap talk” (e.g., Farrell, 1987).

The imperfect knowledge results from the existence of a search cost (e.g. the

quantification of the effort to acquire the information) or resulting from the value to be

realized in future (Lippman and McCall, 1981).

Firm‟s guarantees may be in absolute or relative terms. The most common in

absolute terms is the “good functioning” which encompass a legal obligation imposed

on sellers that artifacts will be “good functioning” for at least a certain period of time.

Other common guarantee is the “return if not pleased” by which a seller sustains that a

buyer will not regret from the purchase.

Guarantees in relative terms are based on a comparison between goods, e.g., a

cleaning product seller assuring that “this product cleans better that others”.

The question I discuss in this work is whether guaranteeing is a strategy that

increases competition between traders, which conduces to better products with smaller

prices or, on the contrary, it turns possible sellers to collude, behaving as if they were

monopolists, decreasing quality and increasing prices.

In this work I present a market equilibrium model with sellers guaranteeing in

relative terms that their prices are no higher than other firms‟ prices. Then, I investigate

whether it is, a priori, optimal sellers to promise paying the price difference or adding to

that a compensation sum in case that the buyer discovers the claim is false.

Contrary to Corts (1995), I assume that buyers do not have perfect knowledge of

prices.

2. First strategy: firms matching other firms’ prices

There are N identical firms that have constant and null marginal cost.

A firm sets the price and guarantees that it matches other firms‟ prices.

Firms do not know prices affixed by other firms. 3

There are Q buyers, being  the fraction of uninformed buyers (that only known

the price of one seller) and being (1–) the fraction of informed buyers (that known the

price of two sellers).

The assumption that there are a percentage of buyers that know the price of only

a seller, tourists, is used, among others, in Salop and Stiglitz (1977), Wilde and

Schwartz (1979) and Burdett and Judd (1983). If all buyers are identical, it is known

from the literature that informed buyers only have to know the prices set by two firms

(Burdett and Judd, 1983, p. 962, cl. 1).

Since Simon (1955), imperfect knowledge is modeled assuming that the value is

an extraction from a known aleatory function. In this vein, each firm assumes that other

firms‟ strategy is to randomly pick a price from the distribution function F(p).

Property 1: If firms guarantee price matching, the equilibrium market

Proof: If a seller guarantee that matches other firms‟ price, all buyers that ask

him the price, will buy. That result, on average, the quantity Q/N. In this way, a firm

that affix price P have the expected profit formalized by next expression:

E(P)PQQP(1)Pxf(x)dx0NN

PQP(1)xf(x)dx0Nprice will be unique and equal to the monopoly price

(1)

In a Nash equilibrium situation, the F(x) function will be such that the expected

profit function is horizontal (expected profit is identical to all prices observed in the

market). In particular, considering there is a maximum price, Pmax, and a minimum

price, Pmin, for both extreme prices, it will result the same expected profit:

E(Pmin)E(Pmax)Pmin(1)

PminPminxf(x)dxPmax(1)PmaxPminxf(x)dx (2) 4

The average price is in-between Pmin and Pmax. Being so, one may assume k 

[0,1] and rewriting this expression results that Pmin must be equal to Pmax:

PmaxPminPmax(1)Pminxf(x)dx (3)

PminPmax(1)kPmax(1k)Pmin

PminPmax,,k

Assuming that all firms set price P, except one firm that sets p, if p is smaller

than P, that firm‟s expected profit is

E(p)Q/Np. If p is higher than P, that firm‟s

expected profit is

E(p)Q/Np(1)P. Thus, the firm‟s profit is increasing

with the price affixed so the firm will affix the monopoly price (since there is at least

one uninformed buyer). By symmetry, all firms set the monopoly price.

Assuming as Burdett and Judd (1983) that buyers have a positive search cost,

the introduction of firms‟ price-matching cause the disappearing of equilibrium points

with search and price dispersion so the unique equilibrium consists in sellers behaving

as if they where monopolists.

3. Second strategy: firms beat other firms’ prices

Now I investigate, in a market equilibrium situation, if it is optimal a firm to

promise a priori to pay the price difference plus a compensation sum to everyone that

finds a price lower than the seller guarantees and if that increases competition.

Claim 1: Price beat with compensation is an anti competitive strategy.

Proof: All buyers that search a firm buy there at price P. But a fraction (1 –

) of

them will ask the price to another seller and if they find a smaller price, they recover the

price difference plus a percentage compensation (1 –

), 1





 0. Being so, the

expected profit of that firm when it affixes price P is:

E(P)PQP(1)Pminxf(x)dx

N QED

This result is identical to that of Salop (1986) and Corts (1995) who assume

perfect knowledge.

(4) 5

profit:

PmaxPmin(1)xf(x)dxPmax(1)xf(x)dx

PminPminPminIn equilibrium, the expected profit function is horizontal. Therefore, from the

minimum price, Pmin, and maximum price, Pmax, it must result the same expected

(5)

As explained, the average price is in-between Pmin and Pmax (0 < k < 1)

PminPmax(1)kPmax(1k)Pmin

1(1)(1k)PmaxPmin(1)k

(6)

Comparing this value with the standard model of Wilde and Schwartz (1979),

where

PmaxPmin(2)/, the price beating strategy decreases competition

because

1(1)(1k)2.

(1)k QED

Nevertheless, price-beating guarantees increase the competitiveness in relation to

the price-matching strategy because the difference between maximum price and

minimum price observed in the market is increasing with compensation, 1 –

.

Property 2: In market equilibrium, price beating strategy is dominated by

Proof: It is known that if

 = 1, there is no price dispersion so the market average

price is equal to the maximum price (Property 1). If  < 1, whatever is the price

dispersion distribution, F(x), the market average price,

results from expression (4) is increasing with

:

PmaxPminprice matching strategy, so no firm will adopt such strategy.

xf(x)dx, is not higher than

the maximum price. Then, if a seller affixes the maximum price, his expected profit that 6

dE(Pmax)QdP(1)P

dNd

(7)

Being so, in a situation of market equilibrium, sellers do not have incentives to

price beat but only to price matching ( = 1) other sellers, situation already studied

above.

QED

Note that Corts (1995)‟s result that price beating guarantee is pro-competitive and

that push the market to the perfect competition situation (Pmax = Pmin = 0) is a

particular result of my model for  = 0.

4. Conclusion

I derive theoretically that, under imperfect knowledge, price-beating strategy is

anti competitive. Nevertheless, this strategy is not in the interest of sellers because it is

dominated by price matching strategy. This theoretical result is in conflict with Corts

(1995)‟s but it is in accordance with the very low occurrence of such phenomena in real

world.

The price matching strategy is a limit situation of the price beating strategy and it is an

anti-competitive strategy, resulting that sellers behave as if they where monopolist.

Therefor, the price matching strategy interferes with competition promoting collusion

among traders, which results in losses to society.

Bibliographic References

Burdett, Kenneth, e Kenneth L. Judd (1983), “Equilibrium Price Dispersion”,

Econometrica, vol. 51, pp. 955-69.

Corts, Kenneth S. (1995), “On the Robustness of the Argument that Price Matching is

Anti-competitive”, Economic Letters, vol. 47, pp. 417-421.

Farrell, Joseph (1987),“Cheap Talk, Coordination, and Entry”, Rand Journal of

Economics, vol. 18, pp. 34-9. 7

Lippman, Steven A., e John J. McCall (1981), “The Economics of Belated Information”,

International Economics Review, vol. 22(1), pp. 135-54.

Salop, Steven (1986) “Practices that (credibly) facilitate oligopoly coordination”, in

George Stigler and Frank Mathewson, eds, New Developments in the Analysis of

Market Structure, MIT Press, Cambridge. MA

Salop, Steven, e Joseph E. Stiglitz (1977), “Bargains and Ripoffs: A Model of

Monopolistically Competitive Price Dispersion”, Review of Economic Studies, vol.

44, pp. 193-510.

Simon, Herbert A. (1955), “A Behavioural Model of Rational Choice”, The Quarterly

Journal of Economics, vol. 64, pp. 99-118.

Stigler, George (1964), A Theory of Oligopoly, Journal of Political Economy, vol. 72,

pp.44-61.

Wilde, Louis L. and Alan Schwartz (1979), “Equilibrium Comparison Shopping”,

Review of Economic Studies, vol. 46, pp. 543-54.