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On multiplicity of consistent conjectures in free entry oligopoly
Economics Letters North-Holland
ON MULTIPLICITY Yasuhito
109
28 (1988) 109-115
OF CONSISTENT
CONJECTURES
IN FREE ENTRY OLIGOPOLY
TANAKA
Yamagata University, Kojirakawa-machi, Yamagafa, Japan Received Accepted
19 April 1988 1 June 1988
If we allow free entry into oligopoly, we should take into account the fact that the number of firms is variable. Therefore, we should include it in the arguments of the conjectural variation functions, and consider a conjectural variation for the change in the number of firms. We will show that for such conjectural variation functions, there exist multiple consistent conjectures equilibria in free entry oligopoly.
1. Introduction The firms in an oligopolistic market may react to moves of their rival firms, and at the same time have some conjectures for their rivals’ reactions to their own moves. These conjectures are called conjectural variations. The conjectural variations are said to be consistent, if they are consistent with the ‘real’ reactions of the rival firms. The problem of consistency of conjectural variations has been analyzed by several authors such as Bresnahan (1981) Perry (1982), Boyer and Moreaux (1983a, b) and Kamien and Schwartz (1983), and has been applied to analyses in the various economics fields, for example, export subsidy in oligopoly by Eaton and Grossman (1986) tariff equilibrium with retaliation by Turnovsky (1986) and firms’ investment decision by Dixon (1986). The consistent conjecture in free entry oligopoly is analyzed by Perry (1982). He showed that with constant conjectural variation the competitive conjecture (minus one) is the unique consistent conjecture in free entry oligopoly. On the other hand, Boyer and Moreaux (1983a, b) showed that in fixed entry oligopoly there exist multiple consistent conjectures if we consider the conjectural variation function which depends on the outputs of the firms. ’ In an earlier paper, Tanaka (1985), I extended the result of Perry to the case of Boyer and Moreaux type conjectural variation function, and proved that, if we allow free entry into oligopoly, the competitive conjecture is the unique consistent conjecture for any demand, cost and conjectural variation function which depends on the outputs. In this paper we consider a more general conjectural variation function which depends on the number of firms as well as the outputs of the firms, and further we consider a conjectural variation for the change in the number of firms as well as that for the changes in the outputs. If we allow free entry into oligopoly, we should take into account that the number of firms is variable. Therefore, we should include it in the arguments of the conjectural variation function, and consider a conjectural variation for the change in the number of firms. We will show two propositions. Proposition 1 is that the competitive conjecture is always consistent in free entry oligopoly for any demand, cost and ’ They proved oligopoly.
the proposition
0165-1765/X8/$3.50
only in the case of duopoly,
0 1988, Elsevier Science Publishers
but their proposition
B.V. (North-Holland)
may be easily extended
to the case of
Y. Tunakn / Multiplicity o/consistent
110
cort~rcture.~
conjectural variation functions of our type. But, as we will show in Proposition 2, there exists another consistent conjecture, the value of which depends on the form of the conjectural variation functions. This conclusion has a negative implication for the consistent conjectures approach to oligopoly.
2. Free entry oligopoly with conjectural variation We consider the oligopolistic industry is represented as
industry
with a homogeneous
good. The profit of the ith firm in the
where p( .) is the downward sloping ( p’ < 0) inverse demand function, and of the firms which is common to all firms. q, is output of the ith firm and other firms. In choosing q, so as to maximize its profit, firm i forms a conjectural response of the rival firms and that for the change in the number of firms according to
c( .) is the cost function G is total output of the variation for the output to a change in its output
and
where n is the number of firms in the industry. Function f is assumed to be differentiable and common to all firms. If entry into the industry is free, the number of firms, n, is now a variable. Therefore, as stated in the introduction, we should consider the conjectural variation for the number of firms as well as that for the outputs, and should include the number of firms in the arguments of the conjectural variation functions. Such treatment seems to lack economic content. But, considering above general conjectural variation functions, we obtain the results essentially different from the analysis with ordinary simple conjectural variation. Therefore, we think our argument has some significance for the consistent conjectures approach to oligopoly. We denote the derivatives of f as follows:
af -=
a4,
f
”
af == ap fi
and
$
The first- and the second-order
=f3. conditions
P(4,+ e>+(1+f b’(4i + i&-
for profit maximization
c’(a) = 0
of the i th firm are
(4)
and
a1 +f)p'+
(f!+ff*)q,P'+
(1+fhiP”-C”
(5)
Y. Tanaka / Multiplicity
We consider as
only symmetric
P(Q) + (1 +/b'(Q)+
equilibrium.
111
of consistent conJectures
At the symmetric
eqs. (4) and (5) are written
equilibrium,
-c’(z)=O
(6)
and 2(1 +f)p’+
(f, +ff2)$‘+
(1 +l)*+
-c/I
(7)
< 0,
where Q is total output of the industry. At the equilibrium q, = Q/n for all i. With free entry, in addition to eqs. (6) and (7) we have the following zero-profit
condition:
-c(f) =0.
p(Q):
(8)
Eqs. (6) (7) and (8) define the equilibrium oligopoly.
values of the output
and the number
of firms in free entry
3. Multiple consistent conjectures To construct consistent conditions of a rival firm,
conjectures,
we need to consider
the following
first-order
and zero-profit
(9) and
where J is the conjectural f=f
e i n-l’
n?j+q,,
n-l
variation n
function
of the rival firm, and it takes the following
(11)
. 1
Because, for any rival firm of firm i, its output is e/(n industry is (n - 2)/( n - l@ + q, at the equilibrium. Derivatives of f are represented as follows:
af
-= aq,
f 2’
aj 1 --[f,+(n-2)f2] z-n-1
form,
and
-$=
- l), and
-
the output
of the rest of the
’ e(fl-f2)+fx. (n - 1)2
Under free entry, change in e may result from change in the number of firms as well as change in output per firm. Therefore the output response of the rival firms, de/dq,, includes the output change resulting from entry or exit.
112
Y. Tanaka / Multiplicity
Differentiating
of consistent conjectures
eqs. (9) and (lo), we obtain
~((n+i,p.i[~~+(n--z,ril~p.t(l+/)pp..-i/’)~ _(
I
Ql)~[(l+i)p’+(/~-h)~P.-(“-l)fip/-C.’]~
I
n p’+
e (1 +j)---&p”+
(12)
and
-&(p-cf+Qp$g(nfl)*(p-c$p -Apt. Since the values of conjectural
variations
(13)
of all firms are equal at the symmetric
equilibrium,
f=f,
(14)
and from the first-order p-c’=
condition,
-$l+f)p’.
Substituting
(15)
eqs. (14), (15) and the symmetric
condition,
--&(~j(n-2-f)p’~+-&(~j2(l+f)p~~=-~p~. I
Also substituting
@(n
- 1) = Q/H, into eq. (13) we have
(16)
I
(14) and the symmetric
condition
into (12) yields
~{(?7+f)p~+[fI+(“-2)f2]~p~+(l+f)~Qp~~-c~t)~
I
1
---(~j[(l+f)P’+(f~-f2)~P~-(“-l)f3P~-cq~
n-l
= -
n
[
p’+
(1 +f)fp”+
ff*p’
Solving eqs. (16) and (17) simultaneously, conjectural variations: ’
1 we obtain
(17)
the following
conditions
for the consistency
of
The concept of consistency of conjectural variations in this paper is the first order consistency used by Perry (1982) Boyer and Moreaux (1983a, b) and Kamien and Schwartz (1983). It requires that the values of the conjectural variations are equal to the real response ‘at the equilibrium’. On the other hand, Bresnahan (1981) requires the stronger consistency of conjecture. For discussion about the concepts of consistency, see introduction of Boyer and Moreaux (1983a).
Y. Tanaka / Multiplicity of consistent conjectures
113
and 2(1 +f)p’ + (fi +fh>(Q/n)~ + (1 +f>‘(Q/+” dn ______=_ da 2(1+f)p’+(f,+ff,)(Q/n>p’-f,p’+(1+f)’(Q/n>p”-c” Rearranging
(1 +f)[2(1 From f=
- cr’] g.
(19)
the terms of (18)
+f)p’+
(fi
+_&,)(Q/~)P’+
(1 +f)‘(Q/+“-
(n - 1 -f)f3~‘-
c”] = 0.
this we obtain
-1
(20)
or
(21) Eqs. (6) (7), (8), (19) and (20) or (21) define the consistent conjectures oligopoly. Now from eq. (20), we can obtain the following proposition. Proposition
1. The competitive variation functions.
conjecture
equilibria
in free entry
( f = - 1) is consistent for any demand, cost and conjectural
f = - 1 is called the competitive conjecture, since, if f = - 1 at the equilibrium, the price is equal to the marginal cost from eq. (6). This proposition says that the competitive conjecture is always consistent. The consistent value of f is obtained by substituting f = -1 into (19). Its value depends on the values of fi, f2 and f3. If f3 = 0, that is, the conjectural variation function does not depend on the number of firms, eq. (21) can be never satisfied because in this case the left-hand side of (21) is the same as that of the second-order condition, eq. (7) and is negative. Then the competitive conjecture is the unique consistent conjecture. But, if f3 is not zero, and f f n - 1, we have another consistent conjectures equilibrium which depends on the form of conjectural variation function (coefficients of the function in the case of linear conjectural variation function considered in the proof of Proposition 2, as the next proposition shows. Proposition 2. For any pair (Q*, n * ) > 0 which satisfies the zero-profit condition eq. (S), we can find conj’ectural variation functions f and g such that (Q *, n * ) is the consistent conjectures equilibrium in free entry oligopoly, provided that the equilibrium value off is not equal to n - 1. Remark. We exclude f = n - 1, because with f = n - 1 the left-hand side of eq. (21) is the same as that of the second-order condition, (7), and (21) can be never satisfied. This implies that we exclude the joint profit maximization conjecture and a pair (Q *, n*) which satisfies (6), (7) and (8) with f = n - 1 from the consistent conjectures equilibrium.
Proof. Consider all firms,
=f(4,,
the following
e>
“>
linear conjectural
variation
functions
of firm i which are common
to
(22)
= a+P9,+Yi2+%
and
(23)
where a, P, Y, and 9 are constants. For these conjectural variation functions, f, = p, f2 = y and f3 = 6. We must show that for any n*) which satisfies eq. (8), there exist (a, /3, y, 6, 0) which makes this pair a consistent pair (Q*, conjectures equilibrium in free entry oligopoly, that is, makes this pair satisfy eqs. (6) (7) (19) and (21). Consider first eq. (6) all the terms are determined, once (Q *, n * ) is chosen, of the conjectural variation is firstly fixed to satisfy (6). Next we rewrite (21) as follows:
s(=f)
=
2(1+f)p’+
(1 +f)*(Q*/n*>p"-c"
(P+vf)(Q*/n*)~'+
3
(n*
-
1 -f)p’
except f. This value
(24)
where the numerator of the right-hand side is the same as the left-hand side of the second-order condition, eq. (7), and is negative. We determine the value of p + yf( =fi + ffi) to satisfy (7). Then the value of 6 is obtained from (24), and it is not zero, because the numerator is negative, p’ < 0 and f# n - 1, and the denominator of eqs. (18) and (19) is not zero. 3 Next, substituting all these values into (19), we can determine the equilibrium value of f3( = g). If we confine f in the range of - 1 sf< n - 1, that is, f lies between the competitive conjecture and the joint profit maximization conjecture, which seems to be reasonable, 6 and 8 are negative. It seems to be natural that the equilibrium consistent conjecture for the change in the number of firms to an increase in the firm’s output, 8, is negative. Finally (Yis determined from eq. (22) as follows:
a=f-/?FY This completes
9,.
-an*,
the proof.
(25) Q.E.D.
We have shown that any pair (Q *, n * ) which satisfies (6) and (8) with f f n - 1 can be consistent conjectures equilibrium. 3 Denominator of eqs. (18) and (19):
Y. Tanaka / Multiplicity of consistent conjectures
115
4. Conclusion The conclusion of this paper has negative implications for the consistent conjectures approach to oligopoly, in contrast with the conclusion of our previous paper, Tanaka (1985). Recently Makowski (1987) pointed out the logical weakness of this approach. On the other hand, we have shown that in general, in a model such as the model in this paper this approach fails to pick up a unique or small-number equilibria from multiple equilibria of oligopoly even under free entry.
References Boyer, M. and M. Moreaux, 1983a, Conjectures, rationality and duopoly theory, International Journal of Industrial Organization 1, 23-41. Boyer, M. and M. Moreaux, 1983b, Consistent versus non-consistent conjectures in duopoly theory: Some examples, Journal of Industrial Economics 31, 97-110. Bresnahan, T., 1981, Duopoly models with consistent conjectures, American Economic Review 71, 934-945. Dixon, M., 1986, Strategic investment with consistent conjectures, Oxford Economic Papers. suppl. 111-128. Eaton, J. and G.M. Grossman, 1986, Optimal trade and industrial policy under oligopoly, Quarterly Journal of Economics 101, 383-406. Kamien, M.I. and N.L. Schwartz, 1983, Conjectural variations, Canadian Journal of Economics 16. 191-211. Makowski, L., 1987, Are ‘rotational conjectures’ rational?, Journal of Industrial Economics 36. 35-47. Perry, M.K., 1982, Oligopoly and consistent conjectural variations, Bell Journal of Economics 13, 197-205 Tanaka, Y., 1985, Consistent conjecture and free entry oligopoly: A genera1 analysis, Economics Letters 17, 15-18. Tumovsky, S.J., 1986, Optima1 tariffs in consistent conjectural variations equilibrium, Journal of International Economics 21, 301-312.
ON MULTIPLICITY Yasuhito
109
28 (1988) 109-115
OF CONSISTENT
CONJECTURES
IN FREE ENTRY OLIGOPOLY
TANAKA
Yamagata University, Kojirakawa-machi, Yamagafa, Japan Received Accepted
19 April 1988 1 June 1988
If we allow free entry into oligopoly, we should take into account the fact that the number of firms is variable. Therefore, we should include it in the arguments of the conjectural variation functions, and consider a conjectural variation for the change in the number of firms. We will show that for such conjectural variation functions, there exist multiple consistent conjectures equilibria in free entry oligopoly.
1. Introduction The firms in an oligopolistic market may react to moves of their rival firms, and at the same time have some conjectures for their rivals’ reactions to their own moves. These conjectures are called conjectural variations. The conjectural variations are said to be consistent, if they are consistent with the ‘real’ reactions of the rival firms. The problem of consistency of conjectural variations has been analyzed by several authors such as Bresnahan (1981) Perry (1982), Boyer and Moreaux (1983a, b) and Kamien and Schwartz (1983), and has been applied to analyses in the various economics fields, for example, export subsidy in oligopoly by Eaton and Grossman (1986) tariff equilibrium with retaliation by Turnovsky (1986) and firms’ investment decision by Dixon (1986). The consistent conjecture in free entry oligopoly is analyzed by Perry (1982). He showed that with constant conjectural variation the competitive conjecture (minus one) is the unique consistent conjecture in free entry oligopoly. On the other hand, Boyer and Moreaux (1983a, b) showed that in fixed entry oligopoly there exist multiple consistent conjectures if we consider the conjectural variation function which depends on the outputs of the firms. ’ In an earlier paper, Tanaka (1985), I extended the result of Perry to the case of Boyer and Moreaux type conjectural variation function, and proved that, if we allow free entry into oligopoly, the competitive conjecture is the unique consistent conjecture for any demand, cost and conjectural variation function which depends on the outputs. In this paper we consider a more general conjectural variation function which depends on the number of firms as well as the outputs of the firms, and further we consider a conjectural variation for the change in the number of firms as well as that for the changes in the outputs. If we allow free entry into oligopoly, we should take into account that the number of firms is variable. Therefore, we should include it in the arguments of the conjectural variation function, and consider a conjectural variation for the change in the number of firms. We will show two propositions. Proposition 1 is that the competitive conjecture is always consistent in free entry oligopoly for any demand, cost and ’ They proved oligopoly.
the proposition
0165-1765/X8/$3.50
only in the case of duopoly,
0 1988, Elsevier Science Publishers
but their proposition
B.V. (North-Holland)
may be easily extended
to the case of
Y. Tunakn / Multiplicity o/consistent
110
cort~rcture.~
conjectural variation functions of our type. But, as we will show in Proposition 2, there exists another consistent conjecture, the value of which depends on the form of the conjectural variation functions. This conclusion has a negative implication for the consistent conjectures approach to oligopoly.
2. Free entry oligopoly with conjectural variation We consider the oligopolistic industry is represented as
industry
with a homogeneous
good. The profit of the ith firm in the
where p( .) is the downward sloping ( p’ < 0) inverse demand function, and of the firms which is common to all firms. q, is output of the ith firm and other firms. In choosing q, so as to maximize its profit, firm i forms a conjectural response of the rival firms and that for the change in the number of firms according to
c( .) is the cost function G is total output of the variation for the output to a change in its output
and
where n is the number of firms in the industry. Function f is assumed to be differentiable and common to all firms. If entry into the industry is free, the number of firms, n, is now a variable. Therefore, as stated in the introduction, we should consider the conjectural variation for the number of firms as well as that for the outputs, and should include the number of firms in the arguments of the conjectural variation functions. Such treatment seems to lack economic content. But, considering above general conjectural variation functions, we obtain the results essentially different from the analysis with ordinary simple conjectural variation. Therefore, we think our argument has some significance for the consistent conjectures approach to oligopoly. We denote the derivatives of f as follows:
af -=
a4,
f
”
af == ap fi
and
$
The first- and the second-order
=f3. conditions
P(4,+ e>+(1+f b’(4i + i&-
for profit maximization
c’(a) = 0
of the i th firm are
(4)
and
a1 +f)p'+
(f!+ff*)q,P'+
(1+fhiP”-C”
(5)
Y. Tanaka / Multiplicity
We consider as
only symmetric
P(Q) + (1 +/b'(Q)+
equilibrium.
111
of consistent conJectures
At the symmetric
eqs. (4) and (5) are written
equilibrium,
-c’(z)=O
(6)
and 2(1 +f)p’+
(f, +ff2)$‘+
(1 +l)*+
-c/I
(7)
< 0,
where Q is total output of the industry. At the equilibrium q, = Q/n for all i. With free entry, in addition to eqs. (6) and (7) we have the following zero-profit
condition:
-c(f) =0.
p(Q):
(8)
Eqs. (6) (7) and (8) define the equilibrium oligopoly.
values of the output
and the number
of firms in free entry
3. Multiple consistent conjectures To construct consistent conditions of a rival firm,
conjectures,
we need to consider
the following
first-order
and zero-profit
(9) and
where J is the conjectural f=f
e i n-l’
n?j+q,,
n-l
variation n
function
of the rival firm, and it takes the following
(11)
. 1
Because, for any rival firm of firm i, its output is e/(n industry is (n - 2)/( n - l@ + q, at the equilibrium. Derivatives of f are represented as follows:
af
-= aq,
f 2’
aj 1 --[f,+(n-2)f2] z-n-1
form,
and
-$=
- l), and
-
the output
of the rest of the
’ e(fl-f2)+fx. (n - 1)2
Under free entry, change in e may result from change in the number of firms as well as change in output per firm. Therefore the output response of the rival firms, de/dq,, includes the output change resulting from entry or exit.
112
Y. Tanaka / Multiplicity
Differentiating
of consistent conjectures
eqs. (9) and (lo), we obtain
~((n+i,p.i[~~+(n--z,ril~p.t(l+/)pp..-i/’)~ _(
I
Ql)~[(l+i)p’+(/~-h)~P.-(“-l)fip/-C.’]~
I
n p’+
e (1 +j)---&p”+
(12)
and
-&(p-cf+Qp$g(nfl)*(p-c$p -Apt. Since the values of conjectural
variations
(13)
of all firms are equal at the symmetric
equilibrium,
f=f,
(14)
and from the first-order p-c’=
condition,
-$l+f)p’.
Substituting
(15)
eqs. (14), (15) and the symmetric
condition,
--&(~j(n-2-f)p’~+-&(~j2(l+f)p~~=-~p~. I
Also substituting
@(n
- 1) = Q/H, into eq. (13) we have
(16)
I
(14) and the symmetric
condition
into (12) yields
~{(?7+f)p~+[fI+(“-2)f2]~p~+(l+f)~Qp~~-c~t)~
I
1
---(~j[(l+f)P’+(f~-f2)~P~-(“-l)f3P~-cq~
n-l
= -
n
[
p’+
(1 +f)fp”+
ff*p’
Solving eqs. (16) and (17) simultaneously, conjectural variations: ’
1 we obtain
(17)
the following
conditions
for the consistency
of
The concept of consistency of conjectural variations in this paper is the first order consistency used by Perry (1982) Boyer and Moreaux (1983a, b) and Kamien and Schwartz (1983). It requires that the values of the conjectural variations are equal to the real response ‘at the equilibrium’. On the other hand, Bresnahan (1981) requires the stronger consistency of conjecture. For discussion about the concepts of consistency, see introduction of Boyer and Moreaux (1983a).
Y. Tanaka / Multiplicity of consistent conjectures
113
and 2(1 +f)p’ + (fi +fh>(Q/n)~ + (1 +f>‘(Q/+” dn ______=_ da 2(1+f)p’+(f,+ff,)(Q/n>p’-
(1 +f)[2(1 From f=
- cr’] g.
(19)
the terms of (18)
+f)p’+
(fi
+_&,)(Q/~)P’+
(1 +f)‘(Q/+“-
(n - 1 -f)f3~‘-
c”] = 0.
this we obtain
-1
(20)
or
(21) Eqs. (6) (7), (8), (19) and (20) or (21) define the consistent conjectures oligopoly. Now from eq. (20), we can obtain the following proposition. Proposition
1. The competitive variation functions.
conjecture
equilibria
in free entry
( f = - 1) is consistent for any demand, cost and conjectural
f = - 1 is called the competitive conjecture, since, if f = - 1 at the equilibrium, the price is equal to the marginal cost from eq. (6). This proposition says that the competitive conjecture is always consistent. The consistent value of f is obtained by substituting f = -1 into (19). Its value depends on the values of fi, f2 and f3. If f3 = 0, that is, the conjectural variation function does not depend on the number of firms, eq. (21) can be never satisfied because in this case the left-hand side of (21) is the same as that of the second-order condition, eq. (7) and is negative. Then the competitive conjecture is the unique consistent conjecture. But, if f3 is not zero, and f f n - 1, we have another consistent conjectures equilibrium which depends on the form of conjectural variation function (coefficients of the function in the case of linear conjectural variation function considered in the proof of Proposition 2, as the next proposition shows. Proposition 2. For any pair (Q*, n * ) > 0 which satisfies the zero-profit condition eq. (S), we can find conj’ectural variation functions f and g such that (Q *, n * ) is the consistent conjectures equilibrium in free entry oligopoly, provided that the equilibrium value off is not equal to n - 1. Remark. We exclude f = n - 1, because with f = n - 1 the left-hand side of eq. (21) is the same as that of the second-order condition, (7), and (21) can be never satisfied. This implies that we exclude the joint profit maximization conjecture and a pair (Q *, n*) which satisfies (6), (7) and (8) with f = n - 1 from the consistent conjectures equilibrium.
Proof. Consider all firms,
=f(4,,
the following
e>
“>
linear conjectural
variation
functions
of firm i which are common
to
(22)
= a+P9,+Yi2+%
and
(23)
where a, P, Y, and 9 are constants. For these conjectural variation functions, f, = p, f2 = y and f3 = 6. We must show that for any n*) which satisfies eq. (8), there exist (a, /3, y, 6, 0) which makes this pair a consistent pair (Q*, conjectures equilibrium in free entry oligopoly, that is, makes this pair satisfy eqs. (6) (7) (19) and (21). Consider first eq. (6) all the terms are determined, once (Q *, n * ) is chosen, of the conjectural variation is firstly fixed to satisfy (6). Next we rewrite (21) as follows:
s(=f)
=
2(1+f)p’+
(1 +f)*(Q*/n*>p"-c"
(P+vf)(Q*/n*)~'+
3
(n*
-
1 -f)p’
except f. This value
(24)
where the numerator of the right-hand side is the same as the left-hand side of the second-order condition, eq. (7), and is negative. We determine the value of p + yf( =fi + ffi) to satisfy (7). Then the value of 6 is obtained from (24), and it is not zero, because the numerator is negative, p’ < 0 and f# n - 1, and the denominator of eqs. (18) and (19) is not zero. 3 Next, substituting all these values into (19), we can determine the equilibrium value of f3( = g). If we confine f in the range of - 1 sf< n - 1, that is, f lies between the competitive conjecture and the joint profit maximization conjecture, which seems to be reasonable, 6 and 8 are negative. It seems to be natural that the equilibrium consistent conjecture for the change in the number of firms to an increase in the firm’s output, 8, is negative. Finally (Yis determined from eq. (22) as follows:
a=f-/?FY This completes
9,.
-an*,
the proof.
(25) Q.E.D.
We have shown that any pair (Q *, n * ) which satisfies (6) and (8) with f f n - 1 can be consistent conjectures equilibrium. 3 Denominator of eqs. (18) and (19):
Y. Tanaka / Multiplicity of consistent conjectures
115
4. Conclusion The conclusion of this paper has negative implications for the consistent conjectures approach to oligopoly, in contrast with the conclusion of our previous paper, Tanaka (1985). Recently Makowski (1987) pointed out the logical weakness of this approach. On the other hand, we have shown that in general, in a model such as the model in this paper this approach fails to pick up a unique or small-number equilibria from multiple equilibria of oligopoly even under free entry.
References Boyer, M. and M. Moreaux, 1983a, Conjectures, rationality and duopoly theory, International Journal of Industrial Organization 1, 23-41. Boyer, M. and M. Moreaux, 1983b, Consistent versus non-consistent conjectures in duopoly theory: Some examples, Journal of Industrial Economics 31, 97-110. Bresnahan, T., 1981, Duopoly models with consistent conjectures, American Economic Review 71, 934-945. Dixon, M., 1986, Strategic investment with consistent conjectures, Oxford Economic Papers. suppl. 111-128. Eaton, J. and G.M. Grossman, 1986, Optimal trade and industrial policy under oligopoly, Quarterly Journal of Economics 101, 383-406. Kamien, M.I. and N.L. Schwartz, 1983, Conjectural variations, Canadian Journal of Economics 16. 191-211. Makowski, L., 1987, Are ‘rotational conjectures’ rational?, Journal of Industrial Economics 36. 35-47. Perry, M.K., 1982, Oligopoly and consistent conjectural variations, Bell Journal of Economics 13, 197-205 Tanaka, Y., 1985, Consistent conjecture and free entry oligopoly: A genera1 analysis, Economics Letters 17, 15-18. Tumovsky, S.J., 1986, Optima1 tariffs in consistent conjectural variations equilibrium, Journal of International Economics 21, 301-312.