This paper shows that even if all consumers face search costs, if these are below a certain level dependent upon the firm numbers and demand elasticity, the Diamond-type equilibrium with all prices at the monopoly level fails to exist.Diamond Paradox ; Search Behaviour ; Oligopoly pricing

Waterson, Michael

English

Research Papers in Economics

 
 
 
 
 
 
 
Consumers and Competition 
 
 
 
Michael Waterson 
 
 
No 679 
 
 
 
 
 
 
WARWICK  ECONOMIC  RESEARCH  PAPERS 
 
 
 
 
 
 
 
DEPARTMENT OF ECONOMICS   1
Consumers and Competition
1 
 
Michael Waterson 
University of Warwick 
May 2003 
 
Abstract 
This paper shows that even if all consumers face search costs, if these are below a certain 
level dependent upon the firm numbers and demand elasticity, the Diamond-type equilibrium 
with all prices at the monopoly level fails to exist. 
 
JEL Classification: D83, D11,  
Keywords: Diamond Paradox, Search Behaviour, Oligopoly pricing. 
 
 
Correspondence to 
Professor Michael Waterson 
Department of Economics 
University of Warwick 
Coventry 
CV4 7AL 
UK 
Telephone +44 2476 523427 
Fax +44 2476 523032 
Email: michael.waterson@warwick.ac.uk 
 
                                                 
1 I would like to thank Norman Ireland, Diane Reyniers and Margaret Slade for helpful discussions and 
comments on this paper.   2
Consumers and Competition 
 
Introduction 
 
The purpose of this paper is to demonstrate in the simplest terms possible, how consumer 
behaviour has a vitally important role to play in the competitiveness of a market.  We first 
note (Proposition 1) that if every consumer searches only once, the optimal price is at 
monopoly level.  We then show a novel proposition (Proposition 2) that relates to a 
significant body of existing work.  It is that once search costs are below a certain level, then 
even if all consumers face positive search costs, it will be worthwhile those consumers 
searching and as a consequence, there will be no equilibrium with price at the monopoly 
level.  A larger number of firms, and a more elastic industry demand, make this outcome 
more likely.  Thus consumer beliefs are important; if consumers believe it is not worthwhile 
searching, it will not be worthwhile, whereas if they consider searching to be worthwhile, it 
may well be so.  This relates to what is commonly called the Diamond Paradox, and the paper 
shows that the paradox is sometimes overstated. 
 
The next section contains the basic analysis, the propositions and a corollary.  Following that, 
short sections contain some discussion of the result and a brief illustrative example. 
 
Analysis 
 
Consider a market in which there is a continuum of risk-neutral potential consumers, indexed 
i, with mass N, each of whom has a different reservation price pi for the product, such that 
their demand is:   3
qi = 1,  p £ pi  
qi = 0,  p > pi   
 
This gives rise to an aggregate demand curve Q = f(p),  f¢ < 0.  Consumers have a constant 
cost per search
2, s. 
 
There are n-1 (>2) firms, with constant marginal costs, c. Firms maximise profit (Pi) and are 
risk neutral.  Consumers allocate themselves randomly to one firm. More formally,  
A1: Each new consumer searches only once for a price and purchases the product if the price 
is below his/ her reservation value.   
 
Then we have 
Proposition 1: In a market with zero/ one demand by individual consumers and constant 
marginal cost of production, if each consumer searches only one firm prior to the purchase 
decision, the pricing outcome is at the monopoly level, regardless of the number of firms in 
the market. 
 
Proof:  
Obvious since max() i p p P and max()/(1) i p pn P- have the same solution, namely that the 
optimal price p* is given by 
 
                           p* = pm = c/(1-1/e)                                                                                       (1) 
                                                 
2 Since a consumer clearly must search at least once, we adopt the convention that the first search is absorbed 
into the price of the product.   4
 
where e is the own price elasticity of demand (written as a positive number), so pm is the 
monopoly price.   
 
This is the normal monopoly margin equation.  Clearly, no firm has an incentive to raise price 
above the monopoly level. Nor has it an incentive to lower price, since it will not attract 
additional consumers from other firms, under A1. The result is a simplified version of 
Diamond (1971) and also a limiting case of the “sluggish consumer” in Hehenkamp (2002).  
There is an interesting implication, namely that if everyone believes that search is not 
worthwhile because the competitive process works
3, then it does not work. 
 
The assumption that each consumer searches only once is of course a significant limitation.  
Therefore we now extend the game, to three periods, in the following manner.  Firms set 
price in each period, prior to demand being revealed.  The first period is identical to the game 
above.  In the second period, a new firm (E) enters and decides its pricing policy, based upon 
the fact that consumers may indulge in further search as in A2 below. In period 3, based upon 
their experience in period 2, existing firms may react to the new firm’s price; price-cutting is 
punished by retaliatory price-matching (the grim strategy). Consumers make their purchases 
from the lowest- priced supplier they have encountered; if two or more prices are identical, 
they select their supplier randomly.   
 
We now state: 
                                                 
3 That is, everyone believes other people have done the searching for them.   5
A2: In stage 2, consumers of existing firms believe there is a firm that sets a price below pm 
but they do not know the identity of the low-price firm.  Given this assumption, they will 
search for the low-priced firm if it is optimal to do so. 
 
And for completeness: 
A3: The entrant firm will be willing to set a price lower than pm if it increases profit thereby. 
 
We first show: 
Lemma: If it is optimal for firm E to set a price below pm in period 2, that price is given by: 
  
                               pe = pm –(n -1).s/2                                                                                   (2) 
 
Proof: Given assumption A2, the representative consumer will only rationally engage in 
additional search, at a cost s per company searched (beyond the first), if the expected benefit 
is greater than the cost. Therefore, E needs to make pe low enough to encourage search.  
Under assumption A2, the optimal consumer search strategy is to search until the low price is 
found, or not at all.  There are n – 1 additional firms to search, so on average, (n -1)/2 extra 
searches must be carried out.  The expected benefits to an existing consumer are (pm – pe).   
Thus if pe is given by a value less than or equal to that in (2), search by consumers who 
purchased in period 1 is worthwhile, otherwise it is not worthwhile them searching.  It is not 
optimal for E to reduce price in period 2 below the value given by (2); additional consumers 
with reservation prices below pm are attracted to E, but this is equivalent to a monopolist 
reducing its price when it is already below the monopoly level. 
 
We now state our main result:   6
 
Proposition 2: If consumer search costs are below a certain level (but strictly greater than 
zero), it will be worthwhile in expected value terms for consumers to engage in search for a 
lower price than pm and it will be worthwhile for entrant E to price strictly lower than pm.  
This is more likely, the smaller the number of firms in the market. 
 
Proof: 
It remains to be shown that the strategy of E setting a price pe may be optimal, and that this is 
affected as stated by the number of firms in the market. 
 
The first question is whether the firm captures enough additional consumers by this policy to 
make net additional profits, compared with the alternative of setting pm. In considering this, E 
must take into account net profits in periods 2 and 3.  Let Nm (Ne) be the mass of consumers 
who have a reservation price no lower than pm ( pe).  The expected net benefit to firm E of 
lowering price is 
 
()()()/()/()(1)/ emeemeemm EBpcNpcNNnpcNnpcNn dd =-+--+---+                    (3) 
               
Here the first term is the revenue in period 2 from customers with reservation prices above pm 
switching from other suppliers, the second term is the revenue in period 2 from generated 
customers who made their first search at firm E who have reservation values pe < p < pm, the 
third term is revenue from E’s share of customers in period 3 (discounted at the rate d)
4.  The 
                                                 
4 This third term assumes all firms set price pe in period 3 and that consumers search again for a supplier, 
choosing one randomly; this assumption, along with the grim strategy, are the least conducive to price-cutting 
by E in period 2.   7
final term represents E’s revenue under the alternative scenario where it sets price  pm in 
periods 2 and 3.  Substituting in (2) and rearranging slightly yields 
 
()[(2)(1)]/(1).[(1)(1)]/ mmeme EBpcNnNnnsNnNn ddd =---++---++                     (4) 
 
The firm will benefit from price-cutting if (4) is positive.  After some rearrangement, using 
(1), it can be shown that a sufficient condition for search costs to be low enough to make 
price cutting worthwhile is the following 
 
                              2/[(2).] m spn de <--                                                                              (5) 
 
Given that firm E benefits by undercutting the existing players when this condition holds, an 
equilibrium where all firms set  pm cannot exist.  Thus consumer beliefs regarding the 
existence of a low price firm are justified. 
QED 
 
Corollary: In markets where consumers’ search costs are significant, the monopoly price can 
be the Nash equilibrium outcome.  This is the more likely, the larger the number of firms in 
the industry.  
Proof: A sufficient condition for (4) to be negative, under which it is in no firm’s interest to 
price below pm, is given by (6) below.  In these circumstances, there is a Nash equilibrium at 
the monopoly price. 
 
                                                 2/[(1).] m spn e >-                                                                  (6) 
   8
To gain some idea of magnitude, if n is 20 and e is 1.7, search costs greater than around 6% 
of monopoly price will render price-cutting undesirable from the viewpoint of any particular 
firm.   
 
Discussion 
 
Proposition 2 looks superficially like Stahl’s (1989) proposition 4(a), but the mechanism 
involved is completely different.  In Stahl, a proportion of the consumers are “shoppers” and 
have zero search costs, so they seek out the cheapest store, whilst the others do have positive 
search costs.  As the proportion varies, the price distribution changes.  By contrast in the 
present model, all consumers have search costs.  Also, the main point is not to identify an 
equilibrium, in either pure or mixed strategies, but rather to show that the Diamond (1971) 
equilibrium need not exist, where it is commonly viewed as doing so.  This point deserves 
amplification. 
 
The intuition underlying Diamond’s (1971) result is deceptively simple.  Consider for 
example the following exposition in Anderson and Renault (1999, pp. 719-20) “Suppose 
there were an equilibrium at which some firm set a price below the monopoly price and 
(weakly) below those set by all other firms.  Then all consumers who go to the low-price firm 
would still buy there even if the firm raised its price by an amount less than the cost of 
searching another firm, since a consumer who searched again could not expect to gain 
enough to offset the search cost.  This incentive to raise price means that all firms must 
charge the monopoly price in equilibrium.”  (emphasis added).  Our point is that under   9
certain circumstances, this equilibrium does not exist and that these circumstances are well-
characterised by (5) in terms of search costs, demand elasticity and firm numbers
5.  
 
The analysis above is reminiscent of discussions of tacit collusion, in which a collusive 
agreement may be maintained if there is a punishment that can be inflicted on the price-
cutting firm that is credible and makes collusion sustainable (see e.g. Rees, 1993).  The 
example above assumes such a punishment may be imposed.  However, the difficulty in this 
line of argument is that simply matching the price-cutter’s price may not be sufficient as a 
punishment, given consumers’ limited information.  If it does not lead to firms deserting the 
price-cutter, there would appear to be no credible punishment.  In these circumstances, the 
argument above would be modified somewhat, but proposition 2 would remain valid.  It is 
also worth noting that collusion is generally viewed as being easier to maintain if firm 
numbers are small.  In the model above, the result is the opposite. 
 
A Brief Illustration 
 
Finally, there is a question of whether this model bears a resemblance to reality.  The market 
for many professional services, prior to around 1980, provides an illustration.  For example, 
around much of the world, legal services were provided by a large number of relatively small 
firms that were organised as partnerships.  These were commonly prevented by their 
professional bodies from advertising except in the most trivial of ways. In particular they 
                                                 
5 In this sense it makes rigorous the intuitive discussion in Carlton and Perloff (1999, p 433).  Note that their 
discussion captures the importance of firm numbers but not of search costs or demand elasticity.   10
were not allowed to advertise the price of their services
6.  In these circumstances, it is clear 
that the Corollary is likely to hold, with high prices being maintained.    
 
The case of Bates (Supreme Court, 1976) provides an interesting example.  Bates and 
O’Steen were attorneys licensed to practice law in Arizona.  They started a “legal clinic” in 
Phoenix, providing straightforward services (uncontested divorces and the like) at low prices.  
However, “After conducting their practice in this manner for two years, appellants concluded 
that their practice and clinical concept could not survive unless the availability of legal 
services at low cost was advertised and, in particular, fees were advertised.” (Supreme Court, 
1976, p. 354).  It was at this stage, when they advertised “very reasonable fees”, that they 
came into conflict with the Bar and the legal proceedings commenced.  Here we see that an 
attempt to cut price did not create sufficient business to enable the lower price to be 
sustained, in the absence of active information provided to consumers.   
 
An interesting contrast is provided by the example of solicitors’ house conveyancing services 
in the England and Wales.  As Domberger and Sherr (1989) recount, a “Private Member’s 
Bill” was introduced into the House of Commons in 1984, proposing to end the solicitors’ 
monopoly on conveyancing (legal transfer of title in real estate).  This was withdrawn, 
following an undertaking by the Government to introduce its own legislation, and indeed the 
first (non-solicitor) licensed conveyancers came into practice in 1987.  However, the 
interesting thing is that “Reports of reductions in conveyancing charges first appeared in 
1984.” (Domberger and Sherr, 1989, p. 41), that is much before the competition from 
                                                 
6 Monopolies and Mergers Commission (1976) contains surveys of restrictions coming from solicitors and 
similar professional bodies from most of the major North American, West European and Australasian 
jurisdictions.  All severely restricted advertising (as of 1974/5) or banned it outright.   11
licensed conveyancers appeared, and indeed before the Law Society’s restrictions on 
advertising by lawyers had been relaxed, or advertisements had appeared.  Thus it appears 
that consumers had been sensitised to the possibility of finding cheaper provision of this 
standardised service through search, and that as a result some firms of solicitors had 
responded by reducing charges in order to generate extra business for themselves, as 
suggested by Proposition 2. 
 
 
   12
References 
 
Anderson, S.P. and Renault, R., 1999, Pricing, product diversity and search costs: a Bertrand-
Chamberlin-Diamond model, RAND Journal of Economics, 30, 719-735. 
 
Carlton, D.W and J.M. Perloff, 2000, Modern Industrial Organization (3rd ed.) (Addison 
Wesley: Reading MA). 
 
Diamond, P., 1971, A model of price adjustment, Journal of Economic Theory, 3, 156-68. 
 
Domberger, S. and Sherr, A., 1989, The Impact of Competition on Pricing and Quality of 
Legal Services, International Review of Law and Economics, 9, 41-56. 
 
Hehenkamp, B, 2002, Sluggish consumers: An evolutionary solution to the Bertrand Paradox, 
Games and Economic Behavior, 40, 44-76. 
 
Monopolies and Mergers Commission, 1976, Services of Solicitors in England and Wales, 
HCP 557, (HMSO: London). 
Rees, R., 1993, Collusive equilibrium in the great salt duopoly, Economic Journal, 103, 833-
848. 
Stahl, D., 1989, Oligopolistic pricing with sequential consumer search, American Economic 
Review, 79, 700-712. 
 
Supreme Court, 1976, Bates v. State Bar of Arizona, No. 76- 316, 433 US, pp. 350-405. 

A model of price adjustment,

Bates v. State Bar of Arizona,

Collusive equilibrium in the great salt duopoly,

Oligopolistic pricing with sequential consumer search,

Pricing, product diversity and search costs: a BertrandChamberlin-Diamond model,

Sluggish consumers: An evolutionary solution to the Bertrand Paradox,

Consumers and Competition

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