Oligopoly limit pricing with firm-specific cost uncertainty

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EUI Working Paper ECO No. 92/100

Oligopoly Limit Pricing With Firm-Specific

Cost Uncertainty

Stephen Martin

I

uropean University Institute, Florence

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EUROPEAN UNIVERSITY INSTITUTE, FLORENCE

ECONOMICS DEPARTMENT

EUI Working Paper ECO No. 92/100

Oligopoly Limit Pricing With Firm-Specific

Cost Uncertainty

St e p h e n Ma r t in

BADIA FIESOLANA, SAN DOMENICO (FI)

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All rights reserved.

No part of this paper may be reproduced in any form

without permission of the author.

© Stephen Martin

Printed in Italy in January 1993

European University Institute

Badia Fiesolana

I - 50016 San Domenico (FI)

Italy

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O ligopoly Limit Pricing With Firm -Specific Cost U ncertainty Step h en Martin

Department of Economics European U niversity In stitu te 5 0 0 1 6 San Domenico dì F iesole

Florence, Italy October 1 9 9 2

JEL Codes: D43, L I 3

Abstract:

Bagwell and Ramey ( 1 9 9 1 ) sh ow th a t limit pricing d oes n o t occur as an equilibrium str a te g y in o lig o p o ly if entrants are uncertain about an industry c o s t parameter. I sh ow th a t if entrants are uncertain about firm -sp ecific c o s t param eters of olig o p o ly incumbents, th ere are pooling equilibria in w hich incumbents deter entry by h ig h -c o s t entrants. © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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October 5, 1 9 9 2 1

I. Introduction

Bagwell and Ramey [1 9 9 1 ] model non coop erative en try deterrence duopoly when incumbents' c o s t s depend on an industry c o s t parameter. Because incumbents' c o s t s are b o th high or b o th low, plausible pooling equilibria fail to e x is t: in equilibrium, an entrant can infer th e

true value of th e industry c o s t parameter by observing incumbents' prices.

There are no doubt c a s e s in which one would e x p e c t a common industry ch a ra cteristic to affect all firms' c o s t s in th e sam e way. Examples might be th e e x tra c tio n of oil by different firms from a sin gle oil field or th e e x is te n c e of an ind ustry-w ide union w age agreem ent.1 But th ere m ust be many more c a s e s in w h ich c o s t s , while unknown to an entrant, vary among incumbents on a firm -sp ecific b asis. In particular, th e ca se in which a p oten tial entrant co n tem p la tes an industry w ith m ultiple incumbents, som e of w hich may have high c o s t s w hile o th ers may have low c o s t s , is su rely of in te re st.

In th is paper, I sh ow th a t if an entrant's u n certa in ty about incumbents' c o s t s is firm -sp ecific rather than industry s p ec ific , th en there are conditions under w hich a sequential equilibrium s tr a te g y allow s incumbents to deter som e entry by producing th e o u tp u ts th a t would be produced by lo w - c o s t incumbents.

1. It is more difficult t o think of exam ples in w hich an entrant could n ot form a reliable estim a te of th e nature o f fa c to r s that a ffected th e c o s t s of all firms. A union w age agreem ent would be observable by p oten tial entrants. G eo lo g ists could a s s e s s th e lik ely c o s t of ex tra ctio n from a given oilfield in a p re-en try period.

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October 5. 1 9 9 2

II. Duopoly when rival's c o s t s are uncertain A. A on e-p eriod game

I work w ith a model of q u a n tity -se ttin g o lig o p o ly / A nalysis of duopoly equilibrium in th e presence of u n certain ty about th e rival’s c o s t is a building block for the model of duopoly entry deterrence. In th is se c tio n , I examine equilibrium in a on e-p eriod game.

The inverse demand curve is linear,

( 1) p - a - 0 ,

(where 0 is to ta l outp ut and the assum ption th a t th e slo p e of th e demand curve is -1 is not restrictiv e).

There are tw o incumbents, 1 and 2. Marginal c o s t is con stan t, and ta k es one of tw o values. Marginal c o s t is eith er high (cH) or low (cL). There is n oth in g e sse n tia l about th e assum p tion th a t th e s e values are th e same for both firms.

At th e sta r t of th e first period, each incumbent knows it s own marginal c o s t, but d oes not know th e marginal c o s t o f i t s rival. Prior b eliefs are described by th e probabilities

(2 a) u12 = i ' s prior probability th a t 2 *s c o s t s are high; (2b) u21 = 2 's prior probability th a t i ' s c o s t s are high. Prior b eliefs are common knowledge. Following Harsanyi { 1 9 6 7 -6 8 3 , th is makes it p o ssib le to model the game a s one o f com p lete but im perfect information w ith an initial move by nature th a t endows p layers w ith high or low c o s t s w ith the indicated p robab ilities. 2

2. Milgrom and Roberts [1 9 8 2 ] consider en try d eterren ce by a •q u an tity-settin g m onopolist. Bagwell and Ramey [1 9 9 1 ] analyze p rice s e ttin g duopoly w ith product differentiation. The difference betw een Bagwell and Ramey’s r esu lts and th o se p resen ted here d o es n ot depend

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O cto b er 5, 1 9 9 2 3

The natural g en eralization o f Cournot equilibrium to th e p resent of un certain ty is t o require th a t each firm's non coop erative

equilibrium output maximize it s own ex p ected profit, taking th e o u tp u ts of rivals of different c o s t ty p e s as given.3 P ayoffs are (3a)

"1L ‘ {a - CL * - (1 - “lZ^a. * U12q2H]}qiL (3b) _{"lH = <a ' CH}

- lqiH * (1 - ui2^ 2L * U>2q2H])q ik (3c) _{712L ' <a - ct - 1(1 ' U2<Jqil * u2iqiH + q2L^q2L} (3d) _{"2H * <a ' c l - t(l} _{- “ 2i)q1L * u2.q1H * q2H]}q2H}

The first-o rd er con d ition s for m axim ization of th e s e p ayoffs yield th e equations of rea ctio n fun ctions, which vary by firm and by c o s t type: ( 4 a ) 2 q .L ♦ (1 - u >2J q 2L * ui2q2H - a - CL (4b) 2 q .H * (1 - ui2^qa * U12q 2 „ “ 3 - CH (4c) U _{-' U21 ^u . * U21q W * 2 q 2L ' 3 ■- CL} (4d) (1 - u2 1 ^ ,l * _{U2!q iH *} _{2 q 2H = 9 -} CH

Solving th e equations o f th e rea ctio n functions g iv es equilibrium outputs:

(5a) qa - £ [ a - c , - "■» (<;„ - cL) ] ,

(5b) qiH = 1 L 3 * C« ‘ ^ ' U;i(cH - CL) ] ] ,

(5C) q2L ’ 1 C a ' CL * 2Ua 2 U‘2(CH - C0 ] .

3. This is th e approach o f Saloner's [1 9 8 7 ] model o f duopoly w ith o n e-sid ed un certain ty about firm c o s t s .

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October 5, 1 9 9 2

4

[ t 1 P 1 * U|2 ~ 2 u 21r . 1 ~ 1

(5d) - 3 [_ a ~ cH - 2 (cH CLJ J -The e x p ected payoff of each typ e firm is th e square of it s equilibrium output. (This follow s from the equation of th e reaction curve and th e assum ption th a t th e slo p e of th e demand curve is - 1.) B. Pooling equilibrium in a tw o -p er io d game w ith ou t th e p o s sib ility

of entry

Now consider a tw o -p er io d version of th e previous model, and su p p ose th a t entry is n ot p o ssib le. Here I outline th e con d ition s under which th e follow ing is a sequential equilibrium;

(a) b oth firms produce (a - cL) / 3 in period 1-,

(b) each firm produces it s equilibrium output from th e o n e -period game w ith unknown c o s t ty p e s in -period 2; (c) firms carry prior b eliefs forward from period 1 t o period 2 . O ut-of-equilibrium b eliefs, w hich are not r es tr icte d by th e

requirement o f c o n siste n c y w ith Bayes' rule, are su ch th a t if a firm ob serves th a t it s rival produces any output o th er than (a - ct } /3 in period 1, it concludes th a t th e rival has high c o s t and m axim izes it s secon d -p eriod payoff given th is belief.

This pooling equilibrium corresponds to en try -lim itin g behavior. Exploration o f th e con d ition s under which p ooling w ill occur when entry is n ot p ossib le provides a point of reference for a more general model th a t allow s for th e p o s s ib ility of entry.

If defection leads th e rival to conclude th at th e d efectin g firm has high c o s t, a lo w - c o s t firm would never d efect from su ch a pooling str a te g y . (a - cL) / 3 is th e b e s t-r e s p o n s e output of a lo w - c o s t du opolist if it s rival produces (a - cL) /3 . D efection would reduce th e d efectin g firm's payoff in th e first period and induce th e rival

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October 5, 1 9 9 2 5

t o expand outp ut in th e secon d p e r io d ,r e d u c in g th e d efecto r's secon d -p eriod payoff a s w ell.

If firm 1H fo llo w s th e p ooling str a te g y , it s e x p e cte d f ir s t - period payoff is

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[a - cH - ^-(a - cL)](a - c ^ / 3 2

* ? [ 3 ~ CH " ^ CH ' ClJ ] * ? ( CH _ ClJ2 ’ I t s ex p ected secon d period payoff if it adheres to th e p ooling str a te g y is th e square of equilibrium output from th e on e-p erio d game w ith unknown c o s t ty p e s , given by equation C5b). I t s payoff for th e game, ignoring discoun ting, is th e sum of it s p ayoffs over th e tw o periods.

If firm 1H d e fe c ts , it reveals th at it has high c o s t . A d efectin g firm 1H will th erefore produce a first-p er io d outp ut th a t is it s b e st resp on se to th e equilibrium output (a - cL) / 3 o f firm 2. The resultin g firs t-p er io d payoff is

(7 ) ^ [ a ' CH ^ CH " CL^ ]

When firm 1H s e t s i t s secon d -p eriod outp ut, it d oes n ot know firm 2 's c o s t ty p e. The outp ut th a t m axim izes firm lH ’s e x p e c te d secon d period payoff, found by solvin g th e sy stem o f equations formed by th e reaction fun ctions o f firms 1H, 2L, and 2H, is

4. The difference betw een firm IL's second period p ayoff if it follow s th e pooling s tr a te g y and it s payoff if it d e fe c ts in the first period is

- u21)(cH - clK») .

where (»), th e sum of firm IL's secon d period o u tp u ts if it follow s and if it d e fe c ts from th e equilibrium str a te g y , is p o s itiv e . Unless th e rival's prior b elief is th a t firm 1L has high c o s t w ith

probability 1, firm 1H earns a greater seco n d -p erio d payoff by follow ing th e equilibrium str a teg y .

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O cto b er 5, 1 9 9 2

(8 ) ^ ta * “ (1 hi2)(Cj^ C^)] ,

I t s e x p e cte d seco n d -p erio d payoff is the square of th is output. Comparing adherence and defection p ayoffs, firm 1H w ill follow th e pooling s tr a te g y if

(9} g ( l U2,) ^ _UH 3 - 4Uj₄? -A ch - CL) ] 2 ^ (c H - cL) . Evidently, th is condition fails if u21 is su fficien tly near 1. If firm 2 is convinced th a t firm 1 has high c o s t , firm 1H is b e tter off d efectin g from a p ooling s tr a te g y . But th e left-h a n d sid e o f (9) is larger as u12 is larger. It is more in firm IH's in te r e st t o conceal it s c o s t ty p e if it b eliev es firm 2 has high c o s t s .

A condition corresponding to (9) m ust be m et if firm 2H is to adhere to th e p ooling s tr a te g y . Adherence by firm 2H is more likely, th e smaller is u12 and th e larger is u21, ju st o p p o site th e con d ition s on prior b eliefs th a t make adherence by firm 1H likely. To sh ow th at there is a range o f b eliefs for which b oth firm 1H and firm 2H will follow th is pooling s tr a te g y , I turn to th e sym m etric b e lie fs ca se. If u ., = u , u, (9) becom es

CIO) a - a [ l - U ♦ r - T - u ] f tcH - CL) . As ex p ected from (9), (10) fails if u is su fficien tly near 1. As u g o e s to 0, ( 10) becom es

(1 1 ) a - cH 2 3 (ch - ct ) .

Hence if th e c o s t disadvantage of h ig h -c o s t firms is n ot to o great and if firms are n ot to o certain th a t their rivals have high c o s t s , th e pooling s tr a te g y described above will be a sequ en tial equilibrium.

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An Example

Suppose a - 10, cL - 1, cH - 2, u(2 - 0 .4 , and u21 - 0 .6 . Then Firm 1H earns a payoff o f 12.41 if it follow s th e p ooling s tr a te g y , w h ile it s d efection payoff is 1 2 .3 3 . Firm 2H earns a p ayoff 1 2 .9 3 if it p o o ls, and 1 2 .6 6 if it d e fe cts. The condition (9) is 0 .3 3 > 0 .2 5 for firm 1H and 0 .5 2 > 0 .2 5 for firm 2H. For th e s e param eter values, h ig h -c o s t firms earn g reater p ayoffs by pooling in th e first period. III. Pooling equilibrium in a tw o -p erio d game w ith entry

A. Structure of the game

Now modify th e tw o -p er io d game considered above by su p p osin g t th ere is a sin gle entrant (E) who ob serves first period o u tp u ts before deciding w h eth er or not t o enter, who must pay a sunk en try c o s t K > to come into th e market, and who breaks even by sta y in g out o f th e market. The p oten tial entrant's c o s t s are e ith er high or low and are known t o th e entrant but n ot to th e incumbents.

Prior b eliefs are given by th e probabilities

( 1 2 a) u * incumbent i‘s prior probability th at incumbent j's c o s t s are high, i, J « 1,2, i f j;

Cl2 b) v - incumbents' prior probability th at entrant's c o s t s are high; ( 1 2 c) w = entrant's prior probability th a t incumbent i's c o s t s are

high, i - 1. 2 .

For sim p licity I have im posed a lim ited sym m etry of b eliefs. This part of th e sp ec ific a tio n can be relaxed w ithou t alterin g th e nature of th e resu lts. Prior b eliefs are common knowledge.

B. A precondition for entry lim itation

To determine con d ition s under w hich it is p o ssib le for incumbents t o pool output in th e first period and limit th e entry o f h ig h -c o s t en tran ts, it is first n e c essa ry to determine equilibrium p a y o ffs from

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October 5, 1 9 9 2 8

th e game played in th e secon d period if entry occu rs but each firm knows only it s own c o s t typ e. In such a game, each firm o f ea ch c o s t ty p e m axim izes it s e x p e cte d profit. That of a lo w - c o s t firm 1. for exam ple, is

(1 3 ) 711L = {a - cL - [qiL * (1 - u)qa * u q ^ - (1 - v)q£L * v q ^ D q ^ The first-o rd er conditions for th e m axim ization o f ex p e cte d profit by three firms, of tw o c o s t ty p e s each, can be so lv ed for equilibrium ou tp u ts. The equilibrium o u tp u ts of lo w - c o s t and h ig h - c o s t en tran ts are

(1 4 a ) q£L = 2j-la - cL - (u + v - 3w )(cH - c L)]

(14b) qEH = ^-[a - c H - ( l * u + v - 3w )(cH - cL)] r esp ectiv ely . E xpected p ayoffs are the squares o f equilibrium o u tp u ts, le s s sunk c o s t s of entry K.

The e x p e cte d profit o f a lo w -c o s t entrant e x ce ed s th e e x p e cte d profit of a h ig h -c o s t entrant. If a lo w -c o s t entrant's e x p e cte d profit from en try is n eg a tiv e, then incumbents can preclude all entry by pooling in th e first period. If a h ig h -c o s t entrant's e x p e cte d profit is p o sitiv e , th en incumbents cannot limit entry. The interm ediate c a se is th a t in w hich a lo w -c o s t entrant e x p e c t s a p o s itiv e profit, but a h ig h -c o s t entrant a n egative profit. This occu rs if

a - cL - (u * v - 3w )(cH - cL) z 4 *JK

(1 5)

> a - c L - ( 2 f U + v - 3w) (ch - cL) , a condition I h en ceforth assum e is met. It is then p o ssib le for incumbents to limit th e en try by h ig h -c o s t entrants. I now outline con d ition s under w hich incumbents will engage in su ch behavior as an equilibrium str a teg y . © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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O cto b er 5, 1 9 9 2 9

C. Equilibrium olig o p o ly limit pricing

Structure of th e equilibrium pooling s tr a te g y

I in v e stig a te a pooling en try -lim itin g sequ en tial equilibrium w ith th e follow ing ch a ra cteristics:

(a) incumbent d u o p o lists each produce (a - cL) / 3 in period 1; (b) a h ig h -c o s t entrant s t a y s out in period 2 :

(c) a lo w -c o s t entrant com es in in period 2 ;

(d) if entry d oes n ot occur, incumbents produce th e appropriate ou tp u ts from (5);

(e) if entry occu rs, th e entrant is revealed a s having lo w - c o s t , w hile incumbents' c o s t ty p e s are unknown e x c e p t to th em selves. Each player produces th e seco n d -p erio d output th at m axim izes it s e x p e cte d payoff, given prior b eliefs: (f) incumbents and th e lo w - c o s t entrant carry prior b eliefs

forward from th e first to th e secon d period.

Out-of-equilibrium b eliefs are n ot r estr icte d by th e requirem ents of sequential equilibrium. I assum e th a t if an incumbent produces any output oth er than (a - cL) / 3 in period 1, i t s rivals conclude th a t th e d efectin g firm has high c o s t. Incumbents' p a y o ffs are th e sum of th eir p ayoffs in th e tw o periods.

Second period equilibrium p ayoffs

If th e pooling s tr a te g y is follow ed, entry will occur only if th e entrant has low c o s t. Incumbents' prior probability for th is is (1 - v). Hence if th e pooling s tr a te g y is follow ed, incumbents e x p e c t en try to occur w ith probability 1 - v.

If entry occurs, th e entrant is revealed a s having low c o s t , but an incumbent’s c o s t ty p e is known only to its e lf. Equilibrium o u tp u ts are found by solving th e s y s te m o f 5 equations given by th e reaction functions of th e entrant and incumbents 1L, 1H, 2L, 2H, and EL. They are Cl6 a) - cL - (u - w )(cH - c L)) CIO») qiH ” q2H * i la " CH • Cl - U + W))(CH - CL)1 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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O c to b er 5, 1 9 9 2 1 0

(1 6 c] q£L = ^-[a - cL + (u - 3w ](cH - cL)] .

An incumbent's e x p ected payoff is th e square o f i t s outp ut. The lo w - c o s t entrant's e x p e cte d payoff is th e square o f i t s ou tp u t, le s s sunk entry c o s t K.

The entrant's output, and therefore payoff, in th is 5 -p la y e r game e x ce ed s equilibrium output (1 4 a ) of th e lo w - c o s t entrant in th e 6 - player game trea ted above. Condition (15) th erefore im plies th a t output (1 6 c) yield s th e lo w - c o s t entrant a p o s itiv e e x p e cte d profit.

If entry does not occur, an event w hich incumbents regard as having probability v, incumbents' ex p ected seco n d -p erio d p a y o ffs are th o s e of th e on e-p eriod game w ith unknown c o s t ty p e s (using (5) and (1 2 ))

(1 7 a ) _{n IL =} _™2L_{* g [ a} _{CL + 2^CH} cL)l2

(1 7 b ) _{n !H} _{n 2H " g t a} _CH ₂ _{^CH - CL) ) 2}

Behavior of th e entrant

In equilibrium, the entrant acquires no inform ation about incumbents' c o s t ty p e s by observing firs t-p er io d outp ut. The alleged equilibrium s tr a te g y ca lls for th e p oten tial entrant t o come in if ex p ected profit from en try is p o sitiv e , and o th erw ise to s ta y out. Given it s b eliefs, th e entrant m axim izes it s e x p e cte d payoff by behaving in th is way.

Behavior of lo w - c o s t incumbents

If a lo w - c o s t incumbent produces any outp ut other than (a - cL) / 3 in period 1, it reduces it s e x p ected first-p er io d p ayoff and p ortrays it s e lf as a h ig h -c o s t firm to th e other incumbent and t o th e entrant. This induces th e o th er incumbent to produce more ou tp u t in period 2

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October 5, 1 9 9 2 11

than if th e equilibrium s tr a te g y w ere follow ed. I t d o es n o t reduce, and may increase, th e probability of entry. F ir st-p er io d d efection therefore reduces th e e x p e cte d secon d -p eriod payoff as w ell as th e ex p ected first-p er io d payoff. It fo llo w s th a t a lo w - c o s t incumbent would never d efect from th e s tr a te g y outlined above.

Behavior of h ig h -c o s t incumbents

A h ig h -c o s t incumbent’s e x p e cte d payoff from adhering to th e alleged sequential equilibrium s tr a te g y is

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g ta CH " CL^ 2 " 4 ^CH " CL^2 +

| [ a ' ch - 1 2 U(c« - ClJ ] + T r C 8 * ch - « u * w ^(cH

-where th e first tw o term s give th e first-p er io d p ayoff and th e final tw o term s are th e e x p e cte d secon d -p eriod payoff.

Given th e nature of out-of-equ ilibriu m b e lie fs, if firm 1H d e fe c ts it will produce an outp ut th a t m axim izes i t s firs t-p er io d payoff. The resultin g d efection payoff in the first period is

cL)

(1 9 ) g(a - Cjj - 2 ^ch ~ CL^

By defecting, firm 1H reveals its e lf as a h ig h -c o s t firm. It is no longer certain th a t a h ig h -c o s t entrant w ill s t a y out in th e secon d period. To evalu ate seco n d -p erio d payoffs, tw o c a s e s must be

considered.

H incumbent d efection d oes n ot make EH en try profitable Suppose first th a t a sin gle h ig h -c o s t firm could d efect w ith o u t making it profitable for a h ig h -c o s t entrant t o come in to th e market. A condition for th is t o occur is given in th e follow in g se c tio n . At th e time firm 1H decid es w h eth er or n o t it w ill d e fe ct from th e equilibrium s tr a te g y , it has before it 2 altern ative seco n d -p erio d

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October 5, 1 9 9 2 1 2

s c e n a r io s . If i t d e f e c t s , i t r e v e a l s i t s e l f a s h a v in g h ig h c o s t .

With probability v th e p o ten tia l entrant has high c o s t and s t a y s out of th e market in the secon d -p eriod . Second-p eriod o u tp u ts are determined by solving th e equations of the reaction fu n ction s for firms 1H, 2L, and 2H; (2 0 a) _{2q,H * Cl - u)q2L + Uq2H = 3 ' CH} (20 b) * 2 ^2L= a - cL (2 0 c) * 2 <,2H = a - cH •

Solving th is s y s tem of equations, firm lH's seco n d -p erio d output if entry does n ot occur a fter d efectio n is

(2 1 ) ^-[a - cH - (1 - u)(cH - cL)] .

I t s secon d -p eriod payoff in th is ca se is th e square of i t s output. With probability 1 - v, th e entrant has low c o s t and will come into th e market in th e secon d period. Second period o u tp u ts are determined by th e reaction fun ctions for firms 1H, 2L, 2H. and EL:

(2 2 a ) 2 q + (1 4ih ' - u )q a * Uq2H * qEL = (2 2 b ) * 2 q 2L + qEL - a * CL ( 2 2 c ) * 2 q 2H + qEL ^ a - c H (2 2 d ) q (1 - MH w i q ^ - Wq2H * 2 q Et "

In entry occu rs, firm lH 's p o s t-e n tr y output is

(2 3 ) i | > - c„ - 4 ~ 32U ^ W(c„ - c L) ] ,

and it s secon d -p eriod e x p e cte d payoff is th e square of th is output. Firm lH ’s ex p e cte d payoff if it d e fe cts from th e s tr a te g y is

g"£ a — Cj^ — 2"(Ch “ c^) J + ~ — (2 — u)(c^ — cL)] (24) * T O - CL * 4 ~ ?2U * W(CH - cL) ] 2 • © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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O cto b er 5, 1 9 9 2 13

If firm 1H adheres to th e p ooling s tr a te g y , on th e oth er hand, it s ex p ected payoff is given by (1 8 ). Comparing (1 8 ) and (2 4 ), firm 1H will earn at lea s t as great a payoff by adhering to th e pooling str a te g y a s by d efectin g if

If v = 1, the first term on th e left van ish es. If firm 1H b eliev es th e entrant has high c o s t, it b eliev es th a t entry will not occur. The lo s s of p rofit on d efectio n is e x p e cte d to come from revealing to firm 2 th a t firm 1 has high c o s t. This explain s the sim ilarity betw een th e left-h a n d sid e o f (9) and th e seco n d term in (25).

If u - 1, th e secon d term on th e left v an ish es. If firm 2 b eliev es th a t firm 1 has high c o s t , then firm lH 's d efectio n from th e equilibrium str a te g y provides firm 2 w ith no new information. The lo s s of profit on d efectio n is e x p e cte d to come from revealing to th e entrant th at firm 1 has high c o s t.

H incumbent d efectio n m akes EH entry profitable

If a h ig h -c o s t entrant would come in to th e market, knowing th at one incumbent has high c o s t , th en th e fact o f en try d o es n o t reveal th e entrant's c o s t typ e. S econd-p eriod o u tp u ts are found by solvin g th e equations of reaction curves for firms 1H, 2L, 2H. EL, and EH. The resultin g equilibrium o u tp u ts are

6 - V 3y-^H - cl ) ] ( 2 - u - w) (25) (26a) ₂ (26b) . 2 - u ♦ 2v 2 (2 6 c) u - 2v + w ,_₂ _lCH © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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October 5, 1 9 9 2 14

(26(1) = ^-[a - cL * - — 2 2v * 3w (cH - cL)]

(2 6 e) = i [ a - cH - u * ? 2 3W(ch " CL^ • The condition for d efectio n to induce entry by a h ig h -c o s t p o ten tia l entrant is th a t su ch an entrant e x p e ct a nonnegative payoff a fter entry. From (2 6 e ), th is w ill be th e c a se if

(27) a - cH - 14 * 2V2 ~ 3w(ch - cL) > 4^|K . If inequality (2 7 ) is n ot m et, then the d efection payoff is given by (24) and th e an aly sis of th e previous se c tio n app lies. If (2 7 ) holds, firm lH's d efection payoff is

(2 8 ) ^ [ a - cH - i ( c „ - cL) ] * j ^ [ a - c H - 4 ~ > 3u (cH - Comparing (1 8 ) and (2 8 ), when d efection leads a h ig h -c o s t entrant to come into th e market, firm 1H w ill prefer to follow th e sequential equilibrium s tr a te g y if v ( i - T V ) [ > - cH - 4 - 3u - ?v . w(Ch .

c0y

* - CH - - - - ^ -" - - - — (Ch - c0 ] ( 2 - U - 2 v - W ) ( C H - | [ a - cH - 5...- . 4U . - 2 v w ^ . ^ ] (3 2 u 2 v * w ) ( c „ -(2 9 ) 2 ? ( CH - Cl /

The first term on th e le ft r eflec ts th e e x p e cte d lo s s of profit if th e entrant has high c o s t and com es into th e market in th e knowledge th a t a t le a s t one of th e incumbents has high c o s t. This term makes it more lik ely th a t h ig h -c o s t incumbents w ill be w illing to pool in th e first period, and it appears p recisely b ecause th ere is some probability th a t pooling in th e first period will d eter en try in th e secon d period. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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O cto b er 5, 1 9 9 2 15

Observe further th a t if v = 0, conditions (2 5 ) and (2 9 ) are identical. If incumbents believe th a t the p o ten tia l entrant has low c o s t w ith probability 1, their in cen tiv es to pool are unrelated t o the d ecision s th a t a h ig h -c o s t entrant would make. If v - 1, h ig h -c o s t incumbents will pool if th e y earn a su fficien tly g rea ter p ayoff in th e second period by keeping their c o s t ty p e s hidden.

Examples

Let a = 10, u = 0.4, v = 0 .5 , and w = 0 .6 . In a 6 -p la y e r game w ith unknown c o s t ty p e s, entrants ou tp u ts would be qEL « 2 .4 7 5 and qEH = 1 .9 7 5 . For any entry c o s t le s s than 3.9, a h ig h -c o s t entrant would come into th e market again st incumbents of unknown c o s t ty p e s. Entry lim itation would th erefore be im possible. For en try c o s t s g reater than 6 .1 2 5 , even a io w - c o s t entry would s t a y out again st incumbents of unknown c o s t ty p e s.

Suppose entry c o s t K » 4. If a h ig h -c o s t firm d e fe c ts in the first period, and a h ig h -c o s t entrant would produce outp ut 2 .3 in th e secon d period and e x p e ct a profit 1.29. D efection by a h ig h -c o s t incumbent would th erefore induce entry by a h ig h -c o s t entrant. A h ig h -c o s t incumbent’s payoff if it p o o ls in th e first period is 1 0 .7 3 , w hile d efection brings it only 9 .1 4 .

If on th e other hand K = 6 , a h ig h -c o s t entrant would s ta y out of th e market even if d efectio n were to reveal th a t one o f th e incumbents had high c o s t. A h ig h -c o s t incumbent's p ayoff if fo llo w s th e pooling s tr a te g y is again 1 0 .7 3 , a gain st 1 0 .5 3 if it d e fe c ts.

IV. Conclusion

Limit pricing fa ils as an equilibrium s tr a te g y in Bagwell and Ramey (1 9 9 1 ) b ecause th e entry decision depends on an ind ustry-w ide c o s t param eter and firms cannot n on coop eratively coord in ate decep tion .

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O cto b er 5, 1 9 9 2 16

In the model developed here, the entry decision depends on firm- sp ecific c o s t param eters, and limit pricing may em erge a s a sequ en tial equilibrium o lig o p o ly str a te g y . Part of th e incen tive to adhere to su ch a s tr a te g y is th e secon d -p eriod profit th a t is e x p e cte d to be preserved if entry is deterred.

Two con clu sion s may be drawn. First, where u n certa in ty about c o s t s is firm -sp ecific rather than in d u stry -sp ecific, o lig o p o lis tic entry deterrence is a p o ssib ility . Second, th e likelihood th a t su ch a s tr a te g y will emerge as a noncooperative equilibrium fa lls a s th e number o f incumbents r ises, sin ce the ex p ected saving in profit from deterring entry falls, th e larger the number of incum bents.5

5. This can be show n formally be generalizing th e model p resen ted

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October 5, 1 9 9 2 17

References

Bagwell, Kyle and Ramey, Garey "Oligopoly limit pricing," Rand Journal of Econom ics. Volume 22, Number 2, Summer 1 9 9 1 , pp. 1 5 5 - 1 7 2 . Harsanyi, John C. "Games w ith Incom plete Inform ation Played by 'Bayesian' Players, I-III," Management Science Volume 14. Number 3, November 1 9 6 7 , pp. 1 5 9 - 1 8 2 : Volume 14, Number 5, January 1 9 6 8 , pp. 3 2 0 - 3 3 4 : Volume 14, Number 7 , March 1 9 6 8 , pp. 4 8 6 - 5 0 2 : reprinted in Harsanyi, John C. Papers in Game Theory. Dordrecht: D. Reidel Publishing Company, 1 9 8 2 , pp. 1 1 5 - 1 7 0 .

Milgrom, Paul and Roberts, J. "Limit pricing and en try under incom plete information," E conom etrics Volume 50, Number 2. March 1 9 8 2 , pp. 4 4 3 - 4 6 6 .

Saloner, Garth "Predation, Mergers, and Incom plete Information," Rand Journal of Econom ics. Volume 18, Number 2, Summer 1 9 8 7 , pp. 1 6 5 - 1 8 6 . © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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