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Mixed pricing in oligopoly with consumer switching costs
Journal
of Industrial
Organization
10 (1992) 393411.
North-Holland
Mixed pricing in oligopoly with consumer switching costs A. Jorge Padilla* Nufleld College, Oxford, UK and CEMFI, Final version
received October
Madrid, Spain
1991
In this paper we develop an alternative and more natural two-period model of duopolistic competition with switching costs. We find that overall competition is always less severe when consumers find switching suppliers expensive. However, the relationship between the strength of competition and the importance of switching costs is not monotonic. Interestingly, competition between ex ante identical firms naturally results in firms having asymmetric market shares in equilibrium.
1. Introduction
The study of markets with consumer switching costs has received much attention in recent years. Consumer switching costs make changing suppliers expensive and thus tend to lock consumers into the firms they were previously patronizing. The simultaneous presence of locked-in and uncommited consumers together with the inability of firms to price discriminate among them has important implications for the competitiveness of markets. Our aim is to achieve a better understanding of these implications.’ The existing literature on switching costs presents the unfortunate feature that its various results are obtained in radically different settings. This makes it difficult to understand how these results depend on the particular setting under which they have been derived. The two-period model developed in this paper is able to reproduce in a simple framework many of the existing (but dispersed) results in the literature. It also clarifies the main forces driving competition in one direction or another, i.e., towards fully collusive or more Correspondence to: Dr. A. Jorge Padilla, CEMFI, San Marcos 39, 28004 Madrid, Spain. *This paper is a revised version of the third and fourth chapters of my M.Phil Thesis at the Univesity of Oxford. I am grateful to my supervisors Chris Harris and Paul Klemperer for their helpful suggestions. This paper has benefited from the comments of P. Bertoletti, K.-U. Kuhn, N.H. von der Fehr, J. Fingleton, an anonymous thesis examiner, as well as seminar audiences in Barcelona, L.S.E. and the R.E.S. Conference 1991 at Warwick, two anonymous referees and the Editor of this Journal. Financial support from the Bank of Spain, Fundacion Ramon Areces and the Instituto Valenciano de Investigation Economica is also gratefully acknowledged. ‘A survey of this literature can be found in Padilla (1991a). 0167-7187/92/$05.00
0
1992-Elsevier
Science Publishers
B.V. All rights reserved
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A. Jorge Padilla, Mixed pricing in oligopoly
competitive outcomes. In a market with switching costs, firms are willing to compete quite fiercely to build up a larger customer base. However, they also have an incentive to exploit their current customers. In a two-period model, the first (pro-competitive) force is present in the first period only while the second (anti-competitive) force only influences second-period pricing decisions. Thus, the effect of switching costs on overall competition is potentially ambiguous. Nevertheless, we show that overall competition is less fierce in a market with switching costs than in an otherwise identical market without switching costs. However, the relationship between switching costs and the strength of overall competition is not monotonic. This paper establishes the conditions under which we should expect overall competition to be further relaxed when switching costs become relatively more important. In this model, market structures where ex ante identical firms have ex post asymmetric market shares are found in equilibrium. Overall industry profitability increases with the degree of asymmetry in firms’ market shares. Therefore, the traditional correlation between indices of concentration (such as the Hertindahl index) and industry profitability found in models of Cournot competition is also shown to characterize a model of price competition with switching costs like the one studied here. We also find some support for action-reaction dynamics in the evolution of markets with switching costs, and the possibility of equilibrium stochastic (accidental) price wars. The basic model is slighly modified to analyze growing demand, entry deterrence and finally, the implications of different mechanisms of information gathering on our main results. More precisely, we ask how first and second period competition are affected when demand grows over time. We find that while second period prices are lower, the effect on first period prices is ambiguous. We also show how incumbent firms may deter entry by committing themselves to a lower market share in the pre-entry period. Finally, when the informational setting is modified to consider word-ofmouth externalities, the degree of overall competition is shown to be always decreasing in the proportion of locked-in consumers. Our model differs from previous two-period models of consumer switching costs on several grounds. While most models assume differentiated products [von Weizsacker (1984); Klemperer (1987b)], we consider ex ante homogeneous products which are only differentiated ex post by switching costs. Our model also includes a dynamic consumer turnover which differentiates it from the most similar models with ex ante homogeneous products, Klemperer (1987a). [Klemperer (1987a) cannot cope with new consumers in every period, and Klemperer (1987b) achieves this only by incorporating real (functional) product differentiation.] Also, in order to ensure the existence of equilibrium, previous two-period models with ex ante homogenous products [in particular, Klemperer (1987a)] assumed that switching costs were not the
A. Jorge Padilla, Mixed pricing in oligopoly
395
same for each locked-in consumer. Our model more naturally assumes that all consumers face an identical exogenous switching cost. More importantly, our model is the first to study structures in which asymmetric market shares result in markets with consumer switching costs and ex ante identical firms. This paper is organized as follows. Section 2 presents the model and discusses the assumptions. Sections 3 and 4 are devoted to solving the model for the second period and the overall game respectively. Section 5 provides interpretations of the results of the previous sections. It concentrates on the competitiveness of markets with consumer switching costs, on the existence of price wars and sales, and on the evolution of this type of market. Section 6 extends the model to deal with growing demand, entry and finally, the implications of word-of-mouth externalities. Section 7 concludes.
2. The model We consider a duopolistic market in which two firms produce an identical, non-storable good with identical constant marginal cost without loss of generality equal to zero. Risk-neutral firms set prices simultaneously and independently in each of the two periods, where the second period is discounted with discount factor 0 2 6 5 1. There are N consumers, each with unit inelastic demand with reservation price pr in each period. It is assumed that consumers only care about the price to be paid in the current period. Consumers also have (exogenous) switching costs, that is, once a consumer has bought from either firm, he is locked in. These costs are large enough to deter any switching. Consumers have different information about market supply. Some consumers are informed about the price offers of the two firms while others are only informed about one of these offers. Let 4 E [0, l] be the proportion of fully informed consumers, and (1 - 4)/2 be the proportion of consumers informed only about firm j’s offer, j = 1,2.’ The two periods are linked by a dynamic consumer flow. At the end of the first period each consumer has a positive probability p of leaving the market. Such an outflow is offset by an inflow of new (uncommitted) consumers so that the final number of consumers in the market is in steady state. In period one, firms set prices and then consumers allocate themselves among firms. A consumer will buy a unit of the good as long as the price he faces is lower than or equal to the reservation price. While fully informed consumers buy from the lowest priced firm, partially informed consumers buy from the firm ‘Unfortunately, when 4 = 1 the overall game has no Nash equilibrium at all. In fact, the best reply correspondence is not compact. Two ways of dealing with this problem could be either to look for correlated equilibria (which can be proved to exist) or similarly, to find the equilibrium of a more general game with an endogenous sharing rule as suggested by Simon and Zame (1990).
A. Jorge
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Padilla,
Mixed pricing in oligopoly
about which they are informed. In period two, given first period market shares, firms set prices and subsequently new (uncommitted) consumers choose where to buy according to their respective information. Locked-in consumers who remain in the market continue buying from the firm they were patronizing in period one. Finally, firms cannot price discriminate between uncommitted and locked-in consumers.
3. The second period suhgame In this section, we find and characterize the unique equilibrium of the second period subgame given first-period market shares. For that, we shall use a constructive argument. Let Xjt denote the number of consumers buying from firm j at t, t = 1,2. (Thus, xjt/N defines the market share of firm j at t.) We denote by Pjt the price charged by firm j in period t and by ZIj, the expected profits of firm j at t. There are three different kinds of consumers coexisting in the market in period two: Old or locked-in consumers, new consumers only informed about one firm’s offer, and new fully informed consumers. Given the above definitions of /J and 4 and the fact that the number of consumers is in steady state, the proportions of old consumers, new partially informed consumers, and new fully informed consumers are (1 -/A), (1 - 4),u and &, respectively. Consequently, given xii, xZ1, eq. (1) below represents the current expected profits of firm j in period two. Note that both firms share the market equally when their prices coincide. (It will become evident that for the second period subgame the specific rule used to allocate demand among firms when there is a tie will have no effect on any of our results.)
Pj2C(l- P)Xj,+ ~-N/21 17j2(Pj2>
Pk2) =
if
Pj2C(l-_)Xjl+~(l+~)N/21 if j,k=l,2,
j#k
and
Pj2
= Pk2? (1)
Pj2
PjzSp,,
We deal with the non-trivial case where .D,4 ~(0, l), so that there are consumers from each of the three types we mentioned above.3 Under these conditions, it is easy to observe that no equilibrium in pure strategies exists. 31t is easy to see that when &=O, there is a NE in pure strategies where both firms charge the reservation price. On the other hand, when PC) = 1, the unique equilibrium of the game is characterized by each firm charging the marginal cost. Note in addition that the case where p = 1 and C#J #O can be understood as the benchmark case of no switching costs.
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391
However, a mixed strategy equilibrium does exist and has very intuitive properties.” Therefore, we shall find two distribution functions 5r2(p) and 5&p) representing the equilibrium pricing strategies of firms 1 and 2, respectively. tjz gives the probability that firm j charges a price smaller than p. (Recall that the support of a probability distribution is the smallest closed set with probability one). Let supp tjz be the support of tj2 with pjz,jjjijz, being its lower and upper bounds, respectively. In equilibrium, firm j has to be indifferent between all prices in its support. Finally, ijt =a(/+ (almost everywhere) is the corresponding density function. In what follows we assume xl1 zxzl without loss of generality. Current expected profits can be rewritten as,
nj2(p)=p[(l-~)~j1+~((1-_)/2+~(1-_5k2(~))Nlr
k.i=LZ
kfj. (2)
Lemmas 1-3 below tell us what are the basic properties that have to be satisfied by the respective supports of any pair of equilibrium strategies. They also give us the expected profits for each firm in such an equilibrium.’ Lemma 1.
Lemma2. p,, j=L2,
pjzspI,
j=1,2.
Pj22i)j2=PrC(1-PL)Xj1
+~(1-~)N/21/C(1-~)u)xj,+P(1+(P)N/21<
where ~jz is the lowest price firm j is willing to pay in order to be the lowest priced firm. In other words, given that firm j cannot price discriminate between locked-in and uncommitted consumers, fijz is the price that makes the firm indifferent between only selling to its locked-in consumers at the reservation price pr, and selling to them and all informed consumers at p. Lemma 3. Suppose that either p12>&, or that p12=p22 and pz2 is not named with positive probability. Then, in any equilibrium: and the equilibrium profits of firm 1 are given by (4 P12=pr B12c(1-/4~l,
(b) pIz=p22=p
+M
+dWPl;
and neither of them was named with positive probability;
4The existence of a mixed strategy equilibrium for the second period subgame can be easily guaranteed using the Dasgupta and Maskin (1986) existence theorems for games with discontinuous payoffs. ‘All proofs can be found in the appendices at the end of the text.
398
A. Jorge
Padilla, Mixed pricing in oligopoly
(e) The equilibrium level of profits of firm 2 is uniquely determined (x119xZ1) and it is equal to Q[(1-~)xz1+~(1+~)N/2].
by
Therefore. from Lemmas l-3
+A1 +41N121Y
nj2=nj2(P)=pC(1-_I")xjl
(3)
where (1 -P)X11 +p(l -@N/2 P=~r(1-~)x,,+~(1+~)N/2. In equilibrium nj2(p) =
71j*
for all p E SUpp
tj2,
for all j,
(4)
where njz(p) and rcj2 are given by eqs. (2) and (3), respectively. It is easy to see that 7t12zrrn22 for all p, that is, current expected profits in the second period are larger for the firm with larger market share in period one, which makes it clear why market share is of importance. Notice that total industry profits, rc12+rr22, are increasing in the degree of asymmetry in market share or, xl1 -x21. Solving the system of equations in (4) we obtain, 5j2(P)=(p-P)[(1-~)xkl+~(1+~)N/21
P&V where tlz(~)St22(p)
when ~11~x21
2
k, j=l,2,
and ~12(~)=522(~)
k# j,
when xll=xzl.
That
is, t12 first-order stochastically dominates r22 whenever firm 1 is the largest firm. This is an intuitive result because it indicates that the largest firm charges high prices with a greater probability than the smallest firm. It is interesting to notice the presence of an atom of positive probability in the distribution function of the larger firm, i.e., c12(pr) < 1. Notice in particular that the probability mass at such an atom is an increasing function of the degree of asymmetry in market share in the first period. This is because the more asymmetric the market, the more tempted the largest firm is to exploit its locked-in consumers giving up competition for uncommitted consumers. Finally, we check in Appendix A that t12, 522 are proper cumulative distribution functions satisfying Lemma 3. The following proposition summarizes our results (see also fig. 1). Proposition
I.
Zf x1 1)=xzl
for all p, C#J ~(0, l), there is a unique equilibrium
A. Jorge
Padilla, Mixed pricing in oligopoly
399
Fig. 1
for the second-period subgame. Moreover the equilibrium is in mixed strategies and has the following properties: Each firm set prices according to continuous and strictly increasing distribution functions over a coincident interval, except that firm I names the uppermost price with positive probability whenever i.e., firm l’s strategy stochastically xl1 >xzl, and finally, 512(~)5522(~), dominates the strategy of firm 2, with strict inequality when xI1 >x,,.
4. Equilibrium in the full game
In the previous section we found and characterized the unique equilibrium of the second-period subgame given the market shares of the first period. In this section we concentrate on the first period when market shares are determined. In this period there are no locked-in consumers. We focus on the case in which consumers have different information about market supply, 4~(0,1). Firm j names pjI to maximize ~~~~~~~~~~~~~~~~~~~~~~~~~ ~njz(xjl(p11,pzl)) subject to njz0 and pjl Q, for all j. AS in the second period subgame there is no equilibrium in pure strategies. However, there is a unique equilibrium in mixed strategies. We first show the existence of a symmetric equilibrium by means of a constructive argument. Looking for
400
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symmetric equilibrium seems the natural procedure given the symmetric structure of the full game. Then, we prove that this is indeed the unique equilibrium of this game. The number of consumers buying from firm j in period 1, Xjr can only take three different values, (1 - 4)N/2, N/2, (1 + 4)N/2 depending on whether firm j fails to be the lowest priced firm, whether there is a tie with both firms naming the same price, or whether firm j succeeds being the lowest priced firm. Let x* be the maximum of xll,xZ1. Then, x* is either N/2 or (1 + 4) N/2. Total discounted profits in 17, are given by
=Pjl N/2 + 6P’( 1 + ,~4)N/2
Pjl
=Pkl,
=pj1(1+~)N/2+6p”(l+~)N/2
Pjl
(6)
where p’ and p” are the values taken by p, the intimum of the support of the second-period equilibrium strategy,
-WP))=(l
-s2(P))(n,(P)--n,(P))+n,(P), (7)
where Q(p) represents the pricing strategy of each firm in a symmetric equilibrium. (Q(p) is a cumulative distribution function.) The corresponding density function is represented by o(p) (o(p)=ll’(p))) almost everywhere. Notice that there is a zero probability of a tie, or in other words that no symmetric equilibrium can be characterized by pricing strategies involving mass points (a formal proof of this statement can be found in Lemma 6.1 in the appendix). Lemma
4.
W(P) = 0
forP > pr.
Zf Q(p)
1-p+
6Hence, P'=P~~,
is a Nash
I
P =P~
(1+4)-W, (I+91
equilibrium
and
pn>p’
NE
distibution
function,
then
A. Jorge
Padilla, Mixed pricing in oligopoly
401
Lemma 5. If Q(p) is a NE distribution function, then p the supremum of the support of Q(p) is equal to the reservation price pr.
From Lemma 5, profits in a NE are equal to
Therefore Q(p)= I_ WPJ-G(P) G(P)-G(P)
= I_
(I-@(P,-P) WP+6P”(l-4)’
Finally, we need to check that Q(p) is a proper distribution function and that pr is effectively the supremum of the support. We proceed in three steps: (a) !S(p,)= i as we can easily observe by substituting pr into (9); (b) o(p) 20 for all p in the support of Q(p). Hence Q(p) is increasing, and it is also easy to see that it is concave; (c) Q(p*) = 0 where p* is the inlimum of the support of Q(p), or in other words is the largest price such that Q(p) is equal to zero. Hence, p* = [(l - ~P)p,-26p”$$l -p)]/(l + 4). Notice that p* is also the minimum price each firm is willing to pay in order to serve all informed consumers in the market. This price may be negative, that is, firms may be willing to charge prices below marginal cost in the first period of the game. Proposition 2. For C/I~(0, l), 0 j 6 5 1, there is a unique equilibrium for the full game. In equilibirum each firm sets prices according to a common distribution function Q(p) with support [p*, pr] (see jig. 2).
5. The nature of competition In this section we give some interpretations for our previous results. In particular, we focus on the degree of competitiveness, on the existence of stochastic price wars and sales, and on the evolution of market structure. The importance of market share in markets with consumer switching costs is made clear by the fact that second-period expected profits are larger the larger the first period market share. The unique equilibrium of the second period subgame is generically asymmetric (it would be symmetric only if both firms had identical initial market shares which happens with probability zero). The largest firm sets high prices with larger probability than its rival (i.e., the distribution function representing the pricing strategy of the largest firm stochastically dominates that of the smallest firm). This implies higher expected prices for the largest firm. 7 Therefore, we should expect the market 7Firms’second period expected prices are E pz2 = A,p In (p,/p) > 0 and E p12 = Alp In (p,/p) + _ j, k = 1,2, j #I. p,)) > E pz2 z 0, respectively, where Aj = [( 1- p)xkl + p( 1 c4)N/27&N
402
A. Jorge Padilla, Mixed pricing in oligopoly
Fig. 2
share of the largest firm to fall so that action-reaction, the lack of persistence of dominant positions in the market, describes in expected value terms the equilibrium evolution of the market.’ Competition in the second period yields outcomes between those that would obtain under full collusion and Bertrand competition depending on the proportion of price-sensitive consumers in the market, ~4. When & approaches zero, outcomes that look collusive are obtained, when it is equal to one, the result is that of Bertrand competition. In between, prices are random but in expected terms competitiveness is reduced [see Varian (1980) for a related result]. To evaluate the effect of switching costs on second period competition, take 4 E (0,l) as given. Note that the larger is p the less important are the switching costs in this market. In particular, the case of p= 1 can be understood as the absence of switching costs. From (5) we can see that a fall in p clearly reduces the intensity of second period competition. Switching costs have the effect of making the demand faced by each firm more inelastic thus increasing each firm’s monopolistic power. It is interesting to observe that overall industry profits are larger the larger *It is obvious that the nature of our model makes towards more asymmetric outcomes impossible.
increasing
dominance,
that is, the tendency
A. Jorge Padilla, Mixed pricing in oligopoly
403
the asymmetry between firms’ market shares. This observation provides support for the existence of a positive correlation between the Herfindahl index, H (which is equal to the sum of the squares of each firm’s market share) and overall industry protitability.g A greater H in period 1 implies higher industry profits in period 2. The result extends to price competition the traditional correlation between concentration and pro~tability found in Cournot models. In the first period, competition can be shown to be fiercer, in general, than in the second period. For q5 constant over both periods, price competition is necessarily more severe in the first period. Firms tight to obtain a larger market share today and thus greater expected profits in the next period. An increase in the proportion of locked-in consumers in the second period (a fail in cl) leads then to lower expected prices. In fact, as p approaches zero expected prices fall below marginal cost. Notice, however, that the presence of uninformed consumers may moderate competitiveness so that no definitive conclusion can be established when (f, is allowed to vary. In conclusion, while switching costs lead to less intense competition in the second period, they encourage first-period competition. The overall competitive effect has to be evaluated through the analysis of the net effect of switching costs on total profits. Total profits, 7c, are given in eq. (8). Note that rc(,~)2 rc(1) for all h ~(0,l). Therefore, starting from a situation of no switching costs (p= l), these costs necessarily relax overall competition. However, the relationship between the proportion of locked-in consumers and competition is not monotonic. From (g), dn/~~uO (to) if and only if P-C l/2 (> l/2). An increase in the proportion of locked-in consumers both makes first period competition fiercer and second period competition milder. The second-period anti-competitive effect only dominates for ,u not too small. Finally, price wars are found to be pervasive in this type of market. They are represented by low realizations of the equilibrium mixed strategies of the duopolistic firms and therefore may be thought to constitute an accidental but recurrent phenomenon in markets with switching costs. Such low realizations of the mixed strategy equilibrium of our model could be alternatively interpreted as endogenous sales, or price promotions.”
6. Extensions 6.1. Growing demand
We proceed
to investigate
the effects of growing demand
‘1 wish to thank one of the referees for suggesting the existence “Another reason for price wars in markets with consumer (1989).
on first and
of such a correlation. switching costs is Klemperer
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A. Jorge Padilla, Mixed pricing in oligopoly
second period competition. In the second period, although the equilibrium is qualitatively similar to the case of stationary demand, prices are lower and profits higher. In the first period, the impact of demand growth is ambiguous. Consider the model of previous sections but assume now that the total number of consumers is not in steady state but instead is growing at a positive rate p(y - 1) where y > 1. While the expected outflow of consumers remains at pAr, the corresponding inflow is now multiplied by y and hence the total number of consumers in the second period is given by N( 1 + ~(y - 1)). Again we have three types of consumers in the second period but their relative proportions are changed. The expected proportions of old consumers, new fully informed consumers and, partially informed consumers are (1 -p), yp4 and yp(1 - 4), respectively. Notably, both the ratio of new to old consumers and the ratio of price sensitive to price insensitive consumers increase with y, The mere introduction of growing demand does not modify the main characteristics of the equilibrium of the second period subgame. The solution of this problem is identical to that of the original one, where demand was in steady state, if the discount factor were 6* = 6y/3,> 6, and the proportion of new consumers p is multiplied by ;1 (A= y/Cl+p(y - l)]).‘l Second period prices are lower because demand growth reduces the proportion of locked-in consumers. There are two effects of an increase in y on second period expected profits. First, for given prices, profits increase because of the absolute increase in total demand. Second, as we have just been discussing, competition is more intense which has a negative effect on profits. Both effects can be found in eq. (10) below. The first term between brackets constitutes precisely the increase in profits due to the market expansion. The second term in the brackets reflects the decrease in profits due to increased competition. For p,+ ~(0, l), the first effect always dominates since imperfect information about firms’ prices limits the extent to which competition is strengthened when the market expands.
“In this section, the proportions of old locked-in and new consumers are (1 -p)/( 1+ /.~(y- 1)) and py/(l +p(yl)), respectively. In addition second period total demand is 1 +p(y - 1) times larger than first period demand. Therefore, the second period is now more valuable, i.e., the effective discount factor increases by 1 +/.~(y - 1).
A. Jorge Padilla, Mixed pricing in oligopoly
40.5
Whereas market expansion generates, in general terms, fiercer competition in the second period, its effect on first period competition is ambiguous. First, lower prices (and thus larger market shares) today means capturing a lower percentage of new consumers in the next period. However, since the future is more important (6’ > S) the value of a given market share increases. While the first effect leads to less intense competition, the second works in the opposite direction. The final effect depends upon the elasticity of demand to price cuts, that is, upon the particular values of p, y, and $. For instance, when 4 is close to zero the second effect is expected to dominate since second period profits are higher. However, when 4 is equal to one, the first effect is likely to dominate since firms will generally value their market shares less.
6.2. Entry deterrence To consider the possibility of entry deterrence our game is modified as follows. In period one, only firm one enters the market. This firm decides the number of consumers to be served, that is x1%. In the second period a potential entrant considers whether to enter the market or not. If it does, it faces a positive cost of entry, k. For ,u=O, entry is clearly blockaded since all consumers are locked into the firms they were previously patronizing. On the contrary, in a market without switching costs, p= 1, entry can never be profitably deterred. In those two cases the incumbent firm will sell to all consumers, xi1 =N. For 0
k*, there exists Z1 1 such that, for all xl1 smaller than XII, entry is never profitable.
Finally, provided the discount factor 6 is not too small, the incumbent prefers to commit itself to a low market share. It sells to a smaller number of
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A. Jorge Padilla, Mixed pricing in oligopofy
consumers as given.”
6.3.
than
would
maximize
total profits
taking
the entrant’s
behaviour
Word-of-mouth externalities
In this subsection we modify the informational setting we have been using throughout the paper to account for word-of-mouth externalities. Traditionally [see Diamond (1989)] four different mechanisms of information gathering are distinguished: Sequential search, price guides, advertising, and, word-of-mouth. The latter reflects the fact that some information is transmitted freely through the random contacts of informed and uninformed agents. In previous sections we assumed an information mechanism where 4 consumers freely acquired a price guide to lowest cost shopping, while the remaining l-4 found the same price guide prohibitively costly. These uninformed consumers were then assumed to split evenly across competing firms. Now the rate of arrival of uninformed consumers to a given firm is assumed to be a direct function of the firm’s market share. It should be uncontroversial that information transmission through random contacts between informed and uninformed consumers is more likely to be beneficial for the firm with a larger customer base. With this assumption we try to capture the (previously discussed) fact that some information passes between consumers in a costless way [see Sutton (1980)]. This is, we believe, a natural way of modelling word-of-mouth externalities.13 Let Bj be the proportion of all uninformed consumers who purchase from l4 For simplicity, we assume that t3, equals firm j’s firm j, 0,+8,=1. first-period market share xjl/N. Under this assumption, for any given price, the proportion of total demand captive in the largest firm is relatively increased which generates a greater incentive to exploit price-insensitive consumers. Consequently, the largest firm names a higher (expected) price which makes it possible for the smallest firm to raise its (expected) price without losing demand. Therefore, second-period prices increase on average. However, first-period competition is encouraged. First-period prices fall
iZIn our model, given the inelastic character of consumer demand, accumulation of market share above the profit maximizing level in the absence of entry will never be a protitable strategy. For a general analysis of the conditions under which overinvestment and/or underinvestment may take place in markets with consumer switching costs; see Klemperer (1987~). Our result cannot be considered as a special case of this latter paper though. In fact, Klemperer’s model is a model of Cournot competition while in ours firms compete in prices. r3An alternative modelling strategy could be, following Diamond’s (1989) suggestion, to include price reputations. In our model the two strategies are somewhat equivalent because the firm with the larger market share was the firm with the lowest prices in the previous period. ‘%ee Appendix C for proofs of the results discussed in this section.
A. Jorge Padilla, Mixed pricing in oligopoly
407
because word-of-mouth externalities introduce an extra incentive to obtain market share.r’ As in the absence of word-of-mouth externalities, an increase in the proportion of locked-in consumers has opposite effects on first and second period competition. However, in the presence of word-of-mouth externalities, it can be shown (see Appendix C) that an increase in the proportion of locked-in consumers makes the market unequivocally less competitive. 7. Conclusions In this paper we have developed a simple specification of consumer switching costs that is perhaps more natural than existing models, and also results naturally in ex ante identical firms having ex post asymmetric market sharse. Most of the important previous results in two-period models of switching costs have been derived in our model. The model clarifies the overall competitive effect of switching costs in a two-period model with homogeneous goods. We have also found some clear support for actionreaction in the evolution of market share, accidental price wars, price promotions and the possibility of underinvesting in market share to deter entry when markets are characterized by the presence of consumer switching costs. Our model can be extended in several directions. In particular, it is not difficult in this model to allow for firms with different marginal costs. Padilla (1991b) also draws on this model to develop a multiperiod analysis of competition with switching costs.16 Appendix A Proof of Lemma 3.
(a) Given our assumptions about the upper bounds of is strictly increasing in ~UPP<,Z and supp 522, 5&L)= 1. H ence n,,(pl,) its argument and it attains a maximum at pr. Then the level of profits in equilibrium has to be equal to 7112(~r)=~12[(1-11)~11+~(1+~)N/2] as it can be easily checked by substituting fii2 by its value. (b) Suppose pi2
A. Jorge Padilla, Mixed pricing in oligopoly
408
increase its profits. smaller price.) Thus firm 2 names p with equilibrium no firm ZIj2(p) =zjz constant
(It could sell ~c$N more consumers at only a slightly firm 1 will name p with zero probability. Similarly, if positive probabilityTn,,(p-a) > nlZ(p+ E). Therefore, in names p with positive probability. -(c) In equilibrium for al p in supp tj2, j= 1,2. From (a) fl,, =
~1*C(l-~)~~~+~(l+~)N/21. Hence p=h.
(4 If x21>x11, then t22>h2
which contradicts either (b) or (c). (e) Immediate and (b) and (c) above. 0
from the definition
of NE
Proof of Proposition 1. Lemmas l-3 above guarantee the existence of an equilibrium satisfying all conditions in Lemma 3, when firms use tr2, 522 [as defined in eq. (5)], respectively, provided 5j2 is a proper distribution function, j= 1,2. However, it is easy to check that this is in fact the case since lj2 is such that (a) 5jz(p)=O Vj, (b) dc,,/dp>O Vj, and (c) tj2(p,)j1 Vj. In indifferent particular, 5,,(p,)< 1 when x2r
There are no mass points in the symmetric equilibrium strategies.
Proof. In any symmetric equilibrium both firms use the same pricing strategy defined by an identical distribution function over a coincident support. For any price p in that common support, no firm will name p with positive probability. Otherwise since the number of mass points has to be countable, there exists E>O such that P--E is named with zero probability. However, this cannot be an equilibrium strategy because the other firm will deviate, naming p-c with positive probability and p with zero probability. 0 The deviant firm increases its profits for E>O sufficiently small.
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A. Jorge Padilla, Mixed pricing in oligopoly
Lemma 6.2. The (common) support of any symmetric (i.e., there are no gaps in the support.)
equilibrium
is connected
Proof. Recall that a set is said to be connected if it is not the union of two non-empty, disjoint closed subsets. In particular a subset of [wis connected iff it is an interval. Let (p,,p,) be the largest gap in the support of the equilibrium strategy Q(p). Let {p,>, {p,} be two sequences of prices converging to pa,pb, respectively. Since we already know from Lemma 6.1 that Q(p) is atomless, lim,,,a Q(p,) = limB+pbQ(p,) which in turn implies pa=p,,. Otherwise, each firm moves density from prices below pa to prices in the gap since this does not involve the loss of any consumer. 0
To show that there are no asymmetric equilibria, following Osborne and Pitchick (1986), it is sufticient that both U,(p), II,(p) differ from rci for some pf, pS E [p,p], respectively, to guarantee that no degeneracies (mass points and/or gaps in the support) exist in equilibrium. This is true for our game as it can be observed in eqs. (6), (7) and the definition of rri. 0 Proof
of Proposition
and that ~ll=max{N,
3.
It is enough to see that k* =p,,u(l -4)‘N/(1+4) [~~/2(1-~)lC(k(l+4)-~,~~(1-4~))/(p,~~-k)l).
Cl
Appendix B When the demand grows over time at a positive rate .~(y- l), y > 1 (see section 6.1), second period profits are given by
The equilibrium pricing strategy of firm j is equal to
where ~=~,C(1-~)x,,+y~(l-_)N/21/C(1-~)x,,+y~(1+~)~/21. that aplay < 0. Similarly, first period profits are equal to -
Notice
where 6* = 6y/i, p* = PII and 1= y/[ 1 + p + (y - l)]. Finally, the symmetric equilibrium pricing strategy in period one is equal to
410
A. Jorge Padilla, Mixed pricing in oligopoly
where P”=P,((~-~*)(~+~)+~*Y(~-~~I/C(~-~*)(~+~)+~Y(~+~)I~~ creasing in y. Notice that &2(p)/LG > 0 and dQ(p)/dp
de<
0 for all
p.
Appendix C
Second period profits are now equal to,
where A(P,@)= 1-~4. obtain
Then, using the same procedure
of section 3, we
It is then possible to see by direct comparison that for all p, for all k = 1,2,
where 17-~=p*[(~,~--x~~)A(~,~)]. Hence, 52*(p) Vp. Total profits are n* = p,{( 1- 4) +>(A’( 1+ 4)/P$)}N/2. Hence, &c*/aP < 0 for all P E (0,l).
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A. Jorge Padilla, Mixed pricing in oligopoly
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