Dynamic limit pricing

• DOI: http://doi.org/10.1111/1756-2171.12176
• Author: Flavio Toxvaerd
• Published date: 01 March 2017
• Date: 01 March 2017

RAND Journal of Economics

Vol.48, No. 1, Spring 2017

pp. 281–306

Dynamic limit pricing

Flavio Toxvaerd∗

I study a multiperiod model of limit pricing under one-sided incomplete information. I characterize

pooling and separating equilibria and their existence and determine when these involve limit

pricing. Forsome parameter constellations, the unique equilibrium surviving a D1 type refinement

involves immediate separation on monopoly prices. For others, there are limit price equilibria

surviving the refinement in which different types may initially pool and then (possibly) separate.

Separation involves setting prices such that the inefficient incumbent’s profits from mimicking

are negative. As the horizon increases or as firms become more patient, limit pricing becomes

increasingly difficult to sustain in equilibrium.

1. Introduction

Since Bain’s (1949) pioneering work, limit pricing has been a staple of industrial economics.

In a nutshell, limit pricing is the practice by which an incumbent firm (or cartel) deters potential

entry to an industry by pricing below the profit maximizing price level. Early workon the subject

took its cue from the casual observation that in some industries, firms price below the myopic

profit maximizing price level on a persistent basis. This observation lead to the notion that by

doing so, incumbent firms could somehow discourage potential entry which would otherwise

have occurred, in effect by sacrificing profits in the short run in return for a maintenance of the

monopoly position in the long run.

Bain (1949) noted that “[ . . . ] established sellers persistently or ‘in the long run’ forego

prices high enough to maximize the industry profit for fear of thereby attracting new entry to the

industry and thus reducing the demands for their outputs and their own profits.”

The present work revisits receivedwisdom on equilibrium limit pricing in dynamic contexts,

by way of a dynamic extension of a simple static model of one-sided incomplete information

∗University of Cambridge; fmot2@cam.ac.uk.

I thank YairTauman for constructive conversations and exchanges of ideas at an early stage of this project. I also thank

Heski Bar-Isaac, Jean-Pierre Benoit, Alex Gershkov,Paul Heidhues, GeorgeMailath, Joao Montez, Romans Pancs, In-Uck

Park, Patrick Rey,Charles Roddie, and seminar participants at the Hebrew University of Jerusalem, Universidad Torcuato

Di Tella, Universidad de San Andres, the University of Bonn, Universidad del CEMA, the University of Cambridge,

and the First London IO Day for useful comments. Next, I wish to thank participants at the following conferences

for feedback and comments: the Econometric Society European Meetings, Vienna (2006), the annual meetings of the

American Economic Association, Chicago (2007), the Royal Economic Society,Warwick (2008), the annual meetings of

the European Association for Research in Industrial Economics, Toulouse (2008), the Latin American Meetings of the

Econometric Society, Buenos Aires (2009), and the Conference on Research on Economic Theory and Econometrics,

Milos (2011). Last, I wish to thank the Editor and two anonymous referees for constructive suggestions that helped

improve this work.

C2017, The RAND Corporation. 281

282 / THE RAND JOURNAL OF ECONOMICS

in the spirit of Milgrom and Roberts (1982). I demonstrate that although some aspects of the

standard (static) analysis may be preserved qualitatively when moving to dynamic contexts, the

quantitative results may radically differ. The analysis shows that when the horizon is sufficiently

long and the players sufficiently patient, limit pricing becomes infeasible altogether.

In this article, I analyze a model of limit pricing with one-sided incomplete information in

which a simple entry game is repeated as long as entry has not occurred. In this model, I identify

two distinct regimes, a monopoly price regime and a limit price regime. In the monopoly price

regime, limit price equilibria may existb ut all such equilibria are ruled out byusing a combination

of equilibrium dominance and Cho and Kreps’ (1987) criterion D1 at all information sets off

the equilibrium path, as compared to a natural benchmark equilibrium in which the uninformed

player makes use of all available information (in a sense that will be made precise). The unique

equilibrium, using this notion, is one of immediate separation on monopoly prices. In the limit

price regime, both pooling and separating equilibria may exist and all these involve limit pricing.

I find that in the limit price regime, the basic logic of separating equilibria of a static single-round

setting carries over to the separating equilibria of the dynamic setting. In particular, I find that

by sacrificing enough at some (single) stage of the game, the efficient incumbent may credibly

convey his identity to the entrant. Whether this signalling effectively precludes entry, and is thus

worthwhile from the perspectiveof the incumbent fir m, in turn depends on the entrant’s incentives

to enter. In the dynamic setting, as the future becomes more important, the relevant conditions

needed to deter entry are increasingly unlikely to be satisfied. Specifically, I show that as the

horizon becomes longer, it becomes more difficult to deter entry simply because the entrant’s

one-off cost of entry may not outweigh a long sequence of postentry profits, even if discounted.

Similarly, I show that for a sufficiently patient entrant firm, an infinite sequence of discounted

future profits will outweigh any bounded entry fee and thus, make entry inevitable. In both cases,

adding dynamics to a static limit pricing model makes entry deterrence through limit pricing

more difficult (or impossible) to sustain as an equilibrium outcome. Thus, immediate entry is

likely to result, with each firm setting its respective monopoly price (regardless of the prevailing

regime).

Although these results cast serious doubt on the viability of limit pricing, it should be

mentioned that the basic trade-off found in the static analysis can be recovered in the dynamic

setting, if one disregards the caveats above and assumes all incentive constraints to be sat-

isfied. Even in this case, the dynamics of the model make somewhat unrealistic predictions.

Specifically, one important difference with a static setting is that in the static setting, the ben-

efits from deterring entry are bounded, whereas this is no longer the case in the dynamic set-

ting, if the players are sufficiently patient. For a large enough discount factor and a sufficiently

long horizon, the efficient incumbent needs to press the inefficient incumbent to make strictly

negative profits from mimicking (e.g., by pricing below marginal cost). When the players are

very patient, the short-term losses necessary to credibly signal to be a low-cost incumbent are

unbounded.

Assuming that all the relevant feasibility constraints are satisfied, in the limit price regime,

all equilibria satisfying the D1 type refinement (anchored D1) belong to a single class, consisting

of (i) a (possibly nonzero and possibly infinite) number of periods during which the two types of

incumbent pool; (ii) a period in which the efficient type engages in costly signalling, whereas the

inefficient type reveals himself and invitesentr y; and (iii) continuation playin which the efficient

type charges monopoly prices in all subsequent periods and deters entry, whereas the inefficient

type competes against the entrant.

The welfare properties of these equilibria are not straightforward. It is true, as is the case in the

static benchmark model, that in the period where separation takes place, welfarei sunambiguously

higher than it would be under symmetric information. This is because entry occurs under the

same states of nature as under symmetric information, but the efficient type sets lower prices than

would a monopolist. However, if separation is preceded by periods with pooling, the conclusion

is less clear-cut. This is because pooling deters entry, which counterweights the benefits of lower

C

The RAND Corporation 2017.