On the consistency of consistent conjectures

Economics Letters 16 (1984) 151-157 North-Holland

ON THE CONSISTENCY David J. SALANT Virginia

Polytechnic

151

OF CONSISTENT

CONJECTURES

*

Institute

and State

University,

Blacksburg,

VA 24061,

USA

Received 24 January 1984

A Consistent Conjectures Equilibrium, CCE, requires that firms be correct in their beliefs concerning rival behavior. This paper presents conditions which ensure equivalence of CCE’s in price and quantity models of oligopoly.

1. Introduction Any model of oligopoly includes implicit or explicit assumptions concerning the behavior each firm attributes to its rivals as well as what each firm views to be its decision variables (e.g., either price or quantity). These assumptions may significantly affect the nature of the resulting equilibrium. Much recent work has sought to make endogenous the way firms view each other’s

behavior

or, in other

words,

to find consistent

conjectures. ’ Consistency requires that each firm’s conjectures be in fact correct. A consistent conjectural equilibrium (CCE) is characterized by two conditions: (a) Each firm’s conjecture about rival actions are correct, and (b) the actions chosen by the firms constitute a Nash non-cooperative equilibrium. Conjectures can be expressed in terms of either prices or quantitites. This note examines the relation between price conjectures and quantity conjectures. a consistent conjectural

Conditions equilibrium

are given for duopoly under which (CCE) in prices is also a CCE in

* The author would like to thank James Friedman and the members of the V.P.I. Micro Workshop for their comments. The author, of course, remains responsible for any errors or omissions. i See Bresnahan (1981), Perry (1982), Robson (1982). 0165-1765/84/$3.00

0 1984, Elsevier Science Publishers B.V. (North-Holland)

D.J. Salant / On the consistency of consistent conjectures

152

quantities, assuming equilibrium exists. ’ Thus not only does the CCE provide theoretical grounds for choosing between different possible equilibria,

by requiring

actions

that firms

be correct

and for the right reason,

in their beliefs

about

rivals’

but does so whether firms view price or

quantity as their decision variable. Implicit in both the Cournot and Bertrand versions of oligopoly are the assumptions of zero conjectural variation in quantities and prices respectively. Recent work for the most part has, for reasons of analytical tractability, 3 focused on situations where firms view quantity as the choice variable which necessitates conjectures also be expressed in terms of quantities. In section 2, simple price and quantity models of duopoly are established and consistency of conjectures is defined. 3 assuming that the two firms’ products are not perfect

Then, in section substitutes, it is

shown

into conjectures

about

that conjectures quantities,

about

prices

and vice versa,

can be translated

and in such a way that a consistent

conjectural

equilibrium

Section substitutes

4 concludes with brief discussions of the case of perfect and of the appropriateness of the CCE as a solution to the

oligopoly

in prices (CCE,)

is also a CCE in quantities.

problem.

2. Notation and definitions In what follows pi is used to denote the price firm i receives and quantity of sales, i = 1, 2. Direct demand is given by 4,

=

qi

its

i=lor2

4(Pl~P2)>

and inverse demand is 4 Pi=f,(4iY

C?2)T

Costs

i=lor2.

for firm i are denoted

2 For a discussion Bresnahan (1981), 3 See Laitner (1980) 4 In vector notation inverse demand is

ci(q,)

(i = 1,

2) where ci > 0 and c:’ 2 0.

of the problems of existence (and uniqueness) of equilibrium Laitner (1980) and Robson (1982). who makes this explicit. direct demand is CJ= F(p) where 4 = ( ql, q2) and p = ( p,,p2) p = F(q).

see

and

D.J. Salant / On the consistency

For the price model profits

are

-c,mPN~

dPoP,)=PJxP) And for the quantity

In attempting

i=l,2,

model profits

fl,(%T 4,) =4,.6(4)-c,(q)? about

to maximize

153

of consistent conjectures

(3)

are

i=l,2,

j#i.

its profits,

how its rival will respond

j#i.

(4)

each firm will have conjectures

to change

in its own price or output

which must be taken into account. For the price version, let r, (p,) denote the price firm i believes firmj will choose if its price is p, (which is called firm i’s conjecture about firm j). Then maximization of (3) is equivalent to maximizing rri[ p,, q(p,)] with respect to p, and, for an interior optimum,

the following

first order condition

must hold: 5

[P,-c:(q;)l[F,,(P)+F,,(P)r:(P,)] +4;=0. (5) The term q’( p,) is called firm i’s conjectural variation about firm j. Similarly, for the quantity model, if sI( q,) is firm i’s conjecture about firm j, then maximization of (4) is equivalent to maximizing II,( q,, s,( qi)) with respect

to

q,.

And for an interior

solution

this implies

(6) The term s,!( qi) is called firm i ‘s conjectural uariation. Note that (5) implicitly defines pi as a function of p,, denoted p,( p,), and (6) implicitly defines q, as a function of q,, denoted a,( q,). The functions p;(p,) and u,(q,) are best replies for firm i in the price and quantity versions of the model, respectively. A consistent conjectural equilibrium (CCE) requires that (i) each firm’s choice be a best reply to its rival’s decision, given its conjectures about the rival, and (ii) these conjectures are in fact consistent with optimal response on the part of the rival - at least in a neighborhood equilibrium. Whence:

of

’ For a function y, = y,(x,, xk), the notation y,,(x) r = J’ or k.

for

is used to denote ay,(x,,

xk)/6’x,

154

D.J. Salant / On the consrstency of consistent conjectures

Definition

A CCE in prices (CCE,)

I.

is a pair of prices (p:,

p:)

and

conjectures [rS*(pl), rT(pz)] such that (a) p,* = p,( p,*) for i = 1, 2 and j # i, and (b) in some neighborhood of equilibrium, p*, c*( p,) = p,( p,) for both firms.

Definition 2.

A CCE in quantities (CCE,)

is a pair of quantities

(q;,

q;),

and conjectures [s;( ql), sT( q2)] such that (a) q: = a,( q,*) for i = 1, 2 and j # i, and (b) in some neighborhood of q*, s:(q,) = u,(q,) for i = 1, 2 andj#i. Note parts (b) of Definitions 1 and 2 imply that the conjectural variations are consistent as well, as r,*’ =p; and s,*’ = u,‘.

3. The relation between In order conjectures,

to make

price and quantity the translation

conjectures

s,(q,),

[i.e., s,( p,) or ‘J( p,), respectively]

f[w,G7,)]

of quantity

conjectures

into price

and vice versa, one takes either the price conjectures

or the quantity

F[P,,~,(P,)]

conjectures

= [4,~s,(q,)],

as given and then defines

c( p,),

the other

by setting

or (7)

= [Pd,(Pi)].

Given q( p,), the conjectures s,( q,) are well defined in a neighborhood of some (equilibrium) p” provided

(8) and similarly,

the

conjectures 5 ( p,), given S, ( p, ), are well defined in some neighborhood some (equilibrium) q”, provided the terms

is always strictly of the same sign in that neighborhood,

of

(4;

(9)

+~j';)/(fit+Lj';)

are each always strictly of the same sign in that neighborhood. Note, that (7) implies that given price conjectural variations, the quantity

conjectural

variations

TJ’(p,),

are given by (10)

D.J. Salant / On rhe consistency of consistent conjecams

Similarly, (7) implies that given quantity conjectural the price conjectural variations are given by

155

variations,

s,‘( q,),

(aP,/aP,)c=~‘(P,)=(f,,+tf,sJ)/(f,,+f,,s~). Whence,as (C,),.,=I.2 = (fr,);,L1.2, the maximization

(11)

first order condition

for profit

in the price model

i P, - c:] ( 4, + 4,$)

(12)

+ q, = 0

is equivalent to the first order condition quantity model:

for profit

maximization

in the

(13)

P,-c:-q,(&+f,,sl)=O. Thus, given conjectures of only its own decision pressed

equivalently

which variable choices

about its rival, each firm’,s profit is a function variable. Conjectures and profits can be ex-

in terms

of price

is viewed as the decision

and response

or quantity variable

and it matters

in determining

not

optimal

rule:

Theorem.

Suppose the two firms’ products are not perfect substitutes and the terms (8) and (9) are always of the same sign and finite in some neighborhood of an equilibrium. Then the vector ( pT, p;, rT( p2), r2*( pl)) is a CCE,

if and only if (q:,

q;,

s:(q2),

s;(ql))

is a CCE,,

where

q* =F(P*)

(14)

and for each i = 1 or 2,

for allp Proof

in a neighborhood Note

well defined, If (q:,

of p* and all q in a neighborhood

first that the assumptions

of the theorem

of q*. imply s,*(q,)

is

by (13), given r~*( p, ) and vice versa.

92, &+Xq2), $(q,))

is a CCE,,

then (16)

156

D.J. Salant / On the consistency of consistent conjeciures

for all qi 2 0. Whence

II:

= vi( P,*, ‘/*(P,* )) and

for all pi. For if not, there is a p: + p:

with

But

T( Pi, rj*(Pl))= fl,(d 5%)) for (q:,S,(ql))=F(P:,$*(P:)) which contradicts (16). And therefore, (17) holds or rr*( p,) = p,( p,), i.e., ri*( p,) is firm i’s best reply to q*( p,). An analogous argument shows that if (P?,P:, CCE,.

GYM Q.E.D.

%Ypt)

is a CCE,

then (q:,

&,

G’Yq2), G(q,))

is a

4. Conclusions What the above result establishes is that when duopolists produce differentiated products, then whether firms view price or quantity as the decision

variable the equilibrium

consistent. criterion

Consistency for selecting

oly problem

is the same provided

of conjectures an appropriate

equilibrium

in that only when conjectures

about their rivals’ actions

be correct

the conjectures

has been proposed concept

are correct

are

as a theoretical for the oligop-

will firms’ beliefs

and for the right reason.

The result

of the above theorem lends further support to this view. Note, however that the above analysis pertained only to the case where the duopolists’ products are not perfect substitutes. With homogeneous products, it must be the case that both firms receive the same price. Thus, dp, = dp, = P’(Q)(l + sj’(q,))dq,, where P = P(Q) is inverse demand and Q = q1 + q2 is total output. Thus, no matter what is sJI, (dp,/dp,). = $(p,) must be unity, and then quantity conjectural variations equivalent to these price conjectural variations are not uniquely defined. But it is true that the ‘competitive conjectures’ 6 in prices, 6 Bresnahan’s Bertrand conjecture is what is called here and by Perry (1982) the competitive conjecture.

D.J. Snlant

157

On the consistency of consistent conjectures

/

TJ’(p,) = 0, and in quantities, J,!( q,) = - 1, are equivalent, and, where both firms have constant and identical average costs, c, they imply that [see Bresnahan 1/2Q(p*), equilibrium.

(19Sl)] where

the Q(p)

CCE has p* =pT is direct demand

The CCE provides an equilibrium ties which are unique if equilibrium

=pT

-

and q; = q; = is the Bertrand

= c

which

with determinant prices and quantiis too. ‘x8 Ruled out, however, is the

possibility that any firm will choose any non-equilibrium strategy, such a choice would contradict consistency. Once a firm has found

for the

correct conjecture, choice of a non-equilibrium strategy can only decrease its profits. This means that at a CCE no firm would wish to change its strategy

knowing

its rival(s)

optimal

response.

It also indicates

that an

exogenous change in the economic environment displacing equilibrium should result in all firms immediately choosing their new CCE strategies should equilibrium to describe

still exist. An essential

how

firms

revise

consistent

ones. This question

oligopoly

which fully accounts

question

conjectures

yet to be addressed

so as to possibly

seems intrinsic

arrive

‘to any dynamic

is at

model of

for firms’ conjectures.

References Bresnahan,

Timothy

Economic Bresnahan,

Timothy

can Economic Daniel,

Coldwell

American Fellner,

1981,

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F., 1983,

James

Duopoly

Review 73, March, III,

Economic

William,

Friedman,

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models

with consistent

conjectures,

American

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1960,

1983,

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Duopoly

models

Review 73, March, Competition

W., 1977,

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Reply, Ameri-

240-241.

among

Oligopoly

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Comment,

238-239. tlie few (August

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John,

1980,

Nancy XVI,

‘Rational’

Schwartz,

1983,

Conjectural

variations,

The

Canadian

May, 191-211. duopoly

equilibrium,

Quarterly

Journal

of Economics

95,

Sept., 947-966. Perry,

Martin

K., 1982,

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Arthur,

duopoly,

’ For

Oligopoly

and consistent

conjectural

variations,

Bell Journal

of

13, Spring, 197-205. T., 1982, On the existence

Discussion

a more elaborate

s See Bresnahan

(1981)

paper (University

of consistent of Western

defense of the notion or Laitner

(1980)

conjectural

Ontario,

of a CCE

for a discussion

equilibria in models of

London).

see Bresnahan

(1981)

and (1983).

of the issue of uniqueness.