Applied EconomicsLetters,1994,1, 8-11
Duopsony mode Is with consistent
conjectural variations
Y ANG-MING
CHANG
and VICTOR
~
J. TREMBLA
Y*
Department of Economics, Kansas State University, Waters Hall, Manhattan, Kansas
66506-4001, USA and Oregon State University, Department of Economics, Ballard
Extension Hall 303, Corvallis, OR 97331-3612, USA
In this note we develop a consistent conjectural variation model that generalizes
Bresnahan's (1981) results to a duopoly-duopsony setting.! This is the fIrst duopsony
model in which flfIDS are constrained to have consistent conjectural variations, and two
interesting results emerge! First, Bertrand conjectures occur when inputs are homogeneous and when input demand functions are horizontal. Second, the consistent conjectures equilibrium approaches Cournot when input demand functions are vertical or
when the flfID is a monopolist-monopsonist. This implies that firms will behave more
competitively in a duopsony setting as inputs and outputs become more homogeneous
and as input demand functions become relatively more elastic.
I. THE CONSISTENT
CONJECTURAL
V ARIA TIONS MODEL
geneous, thenpj = Pj and dpjldqj = dpjldqj, and if output is differentiated, thenpj * Pj and dpjldqj * dpj dqj'
Because flfIDS with monopsony power may also have monopoly power, a rather general duopsony-duopoly model with
consistent conjectural variations will be considered.3 In terms
of the output market, two flfIDS (1 and 2) are assumed to compete in a single output market, where qj denotes the output of
firm 1 or 2. Entry barriers are sufficiently high so that entry is
blocked.4 Because firms 1 and 2 are generally assumed to be
symmetrical, we shall focus on the behaviour of firm i and
assume that flfID j is the other flfID when speaking about firm
i. There are a sufficiently large number of buyers who generate the following inverse demand function for firm i:
In the input market, the same two flfIDS are the only flfIDS
that employ input x. For simplicity, these flfIDS are assumed to
use just one inputS and face the same production function
when inputs are assumed to be homogeneous
.= x.
2
ql f(.)
( )
There are many input suppliers who generate the following
input supply function
w.I -w. I ( x.I' x.J')
(3)
where Wi denotes the price of Xi' Note that if the inputs are
homogeneous, then Wj = Wjand dwjldXj = dWi/dXj,and if the
.= .(.
.)
PI PI q.. qJ
(1)
where Pi denotes the price output. Note that if output is homo-
inputs are differentiated, then Wi * Wjand dwjldXj * dWi/dXj'
Firm i' s problem is then to choose x in order to maximize
profit, which is given by
IAlthough this modelling approachhasits critics (Robson(1983),Lindh (1992»,its primary advantageis that,unlike gametheory (Demsetz
(1989),Fisher(1989),and Shepherd(1990», it is easyto empirically testand leadsto severalinterestingandtestableimplications.See
Bresnahan(1989)and Schmalensee(1990)for a discussionof proceduresto estimateoutputflfSt orderconditions (or outputconjectural
variations)and Schroeter(1988)andChangand Tremblay (1991)for proceduresto estimateinput first orderconditions (or input conjectural
variations).
2Schroeter(1988)and Changand Tremblay (1991)developedoligopsonymodelsusingthe conjecturalvariation approachbut did not consider
the casewhereconjecturalvariationsareconsistent.Severalhavedevelopedconsistentconjecturalvariationmodels for outputmarkets.For
example,seeBresnahan(1981),Laitner (1980),Ulph (1983),and Shaffer(1991).According to Paceand Gilley (1990),this modelwas first
developedby Leontief (1936).
3Robinson(1934,p. 227)statesthat, 'The mostimportantcasesof monopsonywill occur in connectionwith monopoly.'
4SeePerry (1982)for a discussionof a consistentconjecturesoutput model with free entry.
5We do not generally assume a fixed proportions technology, which would constrain input and output conjectures to be equal and,simply
reproduce Bresnahan's (1981) results.
* Author to whom correspondence
shouldbe addressed.
1350-5851
@ 1994 Chapman & Hall
Duopsonymodelswith constantconjecturalvariations
9
nj = Pj(qj, qj)qj -Wj(Xj, Xj)Xj
(4)
bl
h
f "'
k
1
h"
0 so ve t IS pro em, owever, Irm I must ma e some
b
h
'
, 1 ' II
h
"
,
assumption a out ow Its nva WI react to a c ange m Its
,
' bl
I th'
d 1 fi
' ,
d
b
own strategic vana es" n IS mo e, Irm I IS assume to e
concerned with j's input response to a change in xi" Thus, firm
i's first order (necessary) condition of profit maximization
requires:
T
same; v = vj = V2" Finally, a consistent conjectures equilibrium
(CCE) exists if all conjectural variations are consistent, denot*"
" "
"
ed by v , and If each firm maximizes profits given the behav,
f ' "
lour 0 ItS nv al "
II, THE CONSISTENT
CONJECTURES
EQUILIBRIUM
onjloxj = [(opjloqj) (oqjloxj) + (opjloqj) (oqjloxj) (oxjlox)]qj
(5)
+ p .(o
.I q .lox.)
I -[Ow.lox.
...J + (owlox.)(ox jox.)]x.
..I -w. = 0
This section will show how differing specifications of output
For notational convenience, this condition will be rewritten as
on. ox. = ! .' .' + .' .'v. .+ ..' -w.'x. -assume
! .PI q. q., PJqJ Iql P.q.
I I
w.J vjX'I -w. I = 0
(6)
demand, input supply and technology affect the CCE. First,
homogeneous inputs and outputs and that each firm
faces the same linear output demand function, linear production function, and linear input supply function. With these
where Pi' = opjloqj, Pj' = opjloqj' qj' = dqjldXj, qj' = oqjlxj, Wi' =
OWjloxj,Wj' = owjloxj' and Vj= oxjOXj. The term v i represents
firm i's input conjectural variation (i,e", firm i's estimate of j's
assumptions, Equation 8 simplifies to:
.= -fp'( 'f -']
vJ
q
W
input reaction to a unit change in Xj)"
Each firm's optimal input quantity cannot be determined
kn . th al
f th
"
1
""
I
"th
p'(q')2(2+vj)-w'(2+v)
(9)
""'
"
.
The true consistent conjectural vanatlons are obtained by rec-
negatively
""
function
d
.
uctlon
sloped
and
.,.
IS posItively
fu
..
nctlon
linear
(at
sloped
and
IS qua
dr
least
locally),
linear
,
..
atlc, pOSItive
the
(locally),
v
al
ue
d
Input
and
the
"
,
, IncreasIng,
pro-
an
d
"""
"" 6"
concave over pOSItive Input quantities, At optimal values,
fi
.,
fi
mn
I s
d
d
Irst or
er con
d
b
"
0 taln:
d ot "'
"
I lerentlate
to
"
"
d
.
IS an I entity
..
Itlon
an
d
can
be
tot
all
y
B
"
,,:\ Q q '.ox,
/ :\ an d q .="
:\ ' / :\
.=
, q .-.I
J -J Q q .ox, J' Y rearrangIn g terms ,
Equation 7 can be expressed as
"",
"v
qj
+
iqj
,
')
' "
Pj qj qj
,
,
"
a
CCE
by
solvIng
".
A C
conditions,
t
.oumo
the
so
1 t'
u Ion
,
b 0 th
that is a CGE; v* = -I"
1 f
t h e numerator
"
an d
Because
t h e d enommator
"
"
equilibrium
0 f E qua t Ion,
"
9
th e
b I"
& B
d
'
Input
nctlon
nee
not
e mear
lor
ertran
conjec
t ures
t0
b
"
G
all h
(1 al) I '
't "
"
e consistent.
ener
y
owever,
oc
mean y IS require d
"
th 1
f th
" f
"
m ord er &
lor e s ope 0 e reaction unction to be a constant
d &.
7F "
"
d
all
an
lor Uniqueness,
operates
t
m
"
y, no e
,
efficiently
sInce
th
h
v*
-
at w en
the
1
--,
value
of
th
"
t
e mpu
"
the
margInal
Next, we relax the linear technology constraint by allowing
the
production
function
,
8
()
pj qj qi+Pi qi qi +Pj qj viqi +Pi qrtJi +Piqi -Wi -Wj vrwi
This equation has three interpretations: (1) it equals the slope
of firm i's input reaction function; (2) when evaluated at optimal values, it equals the true equilibrium change in Xi in
response to a unit change in Xjand (3) it equals fmnj's consistent input conjectural variation when solved for Vj. Note that
because of symmetry (i,e" fIrmS 1 and 2 are assumed to face
the same demand and technological conditions and behave the
same), each fInn's consistent conjectural variation will be the
to
be
assumption Equation 8 becomes
-(p'q"v*q
v* =
-Wj
"
IS not
and
these
.
h
(p '
"
num
1""
(7)
were
",
is the Bertrand
market
P.' q.' q.' + P .q" -w.'
q.+
.I. P.' q.' q.'
I + P.'
J qJ.' v, q.' +
I
I
I
-Wj'Vj -Wj']dxj + fpj' qj" Vjqj+ Pj' qj' qj' -Wj']OXj = 0
-j
I
under
prod uct lor
x equals the pnce
' 0f Input
"
x fr om the fiIrst
& Input
"
order condition
rn.' q."I
IJ'.
oxjOXj=
'l 'b
symmetry)
that
"'
requires that Vj = VJ' = 0, but when Vj IS
set to 0 m Equation 9,
al 1/2
VjeTh
q~ s -.
IS case par alle1s B resnahan' s (1981) E xamp1e 2 because
fu
supply"
, Note
0
.,
"'
.,
"
"'
, function IS
for simplicity
It ISassumed that the market demand
v
t
.it
First,
as follows.
equi
.,
(given
)
are calculated
oumot
,
J' -v
qjW
c ange m Xj:
,
The consistent conjectures
C
9 for
--*
-V
Vj
d fi d b th
" response
F fi
' tho
al th h
'
e me y at conjecture. or Irm I IS equ s e c ange m
XJ" as indicated by j's reaction function, in response to a unit
h
"
Equation
Vj
(1
"
"f '
IS consistent
I It
th
"l. b "
at
e equi I num
that
"*
+
"".
vanatlon
f ""
al
0 Its nv
" "
ogruzmg
term
al
al
n
con-
th
"
a conjectur
vanatlons,
d
be
e assume to
e
conjectura
"II b
dd
.
e
s
""
slstent.
y
e mltlon,
al
th
"
IS equ
to
e optlm
0
a
fi
ues
""
vanatlons WI
unction
d
v
y
al
supp
e
"
conjectur
Input
owIng
mear
1
non
out
od
our me,
B
,
a
WI
p'q"q
+ pq"
+ p'(q')2(2
concave.
With
this
added
+ P'q'q' -w'
+
v*)
-w'(2
+
v*)
(10)
Again, Equation 10 shows that as the production function
become linear (q" = 0), the CCE approaches the Bertrand
equilibrium" Furthermore, by solving for v* one can see that
as the marginal product and input demand approach the vertical (q" approaches 00),v* approaches 0 and the input market
CCE approaches Cournot,8 This parallels Bresnahan's (1981)
Example 3 and shows that the CCE departs from Bertrand and
becomes less competitive as input demand functions become
steeper.
~ese assumptionssimplify the analysisby ensuringthatthe consistentconjecturalvariationswill be constants"SeeBresnahan(1981)for
furtherdiscussion"
7SeeBresnahan(1981)for further discussionof this point in relationto outputmarketsandthe Appendixfor a more completederivationof the
consistentconjecturalvariationanda discussionof uniquenessandthe secondordercondition.
10
Yang-MingChangand ~ J. Tremblay
Finally, we will analyse the CCE when inputs and outputs
are differentiated. For simplicity, we assume a linear production function. In this case, Equation 8 becomes
-(Pj' qj' qi' -Wj')
v*=
2Pi'(qi')2 + Pj'qj'v*qi' -2Wi' -Wj'v*
(11)
Again, the CCE is Bertrand only in the polar case where outputs and inputs are perfect substitutes (i.e., Pi' = Pj' and Wi' =
Wj'). At the other pole, Equation 11 implies that when outputs
and inputs are not substitutable at all (Pj' and Wj' approach 0),
then v* approaches 0 and the model generates a monopolymonopsony solution with just one fInn.
The results of the model have several interesting implications. First, they parallel those of Bresnahan (1981), who
found that under similar conditions the output CCE becomes
less competitive as rival outputs become less substitutable and
as marginal cost functions become steeper. In addition, this
analysis helps to establish situations where Bertrand and
Cournot input conjectures are sensible under the criteria established by Kreps (1990, pp. 443-9). For example, when the
slope input demand functions approaches infinity, Cournot
conjectures are sensible because it is appropriate for a fInn to
ak
m
th
e
xes
t
t
ra
..
eglc
bl
vana
d
e
an
th
assume
t
a
.
nva
I
,.
s
t
mpu
quantities will remain fixed (a 0 input quantity conjectural
vanatlon). AlternatIvely, If mputs are homogeneous and mput
demand functions are horizontal, Bertrand conjectures are
sensible because it would then be appropriate for the firm to
make input price the strategic variable and assume that rivals'
input prices will remain fixed (a 0 input price conjectural variation). Finally, if a firm is a monopolist and a monopsonist
(i.e., the fInn's output and inputs have no substitutes), then a
Cournot conjecture is sensible because it generates the
monopoly-monopsony solution.
III.
CONCLUSION
In this note we develop a consistent conjectures model for a
duopsony
that leads to several
mterestIng
most likely when inputs are homogeneous and when input
demand functions are horizontal, and Cournot conjectures are
most likely when input demand functions are vertical or when
the fInn is a monopolist-monopsonist.
and testable
lmphca-
tions. First,.t~e model pre~icts that an inp~t market will be
less competItIve when the mputs that are hIred by each firm
become more differentiated and when the input demand functions become relatively more inelastic. This result parallels
Bresnahan's (1981) for a duopoly setting and is a rather 'intuitive theory of competition,9 because one might expect product differentiation and the structure of input demand functions
to affect input market performance in this way. In addition,
our results clarify the conditions under which Bertrand and
Cournot conjectures might occur. Bertrand conjectures are
ACKNOWLEDGEMENTS
Comments from Chi-Chur Chao, Carlos Martins, Art
0' Sullivan and Carol Horton Tremblay are gratefully
acknowledged. Any remaining errors are the responsibility of
the authors.
REFERENCES
Bresnahan, T. F. (1981) Duopoly Models with Consistent
Conjectures,AmericanEconomicReview,71,934--45.
Bresnahan, T. F. (1983) Duopoly Models with Consistent
Conjectures:Reply,AmericanEconomicReview,73,457-8.
Bresnahan,T. F. (1989)Empirical Studiesof Industrieswith Market
Power, in Handbook .of Industrial Organization (R.
Schmalensee
USA.
and
R.
D.
WillIg,
eds),
North-Holland,
New
York,
Chang Y .-.an
M
d V ..rem
J T
bl ay (1991) 01 .Igopsony 101.Igopo 1y
Power and Factor Market Performance, Managerial and
Decision Economics,12,405-9.
Demsetz,H. (1989) Industrial Organization at Age Sixty, UCLA
working paper,February9,1989.
Fisher,F. (1989)GamesEconontistsPlay: A NoncooperativeView,
RandJournal ofEconomics,20, 113-24.
Fisher, I. (1989) Cournot and Mathematical Econontics, Quarterly
Journal ofEconomics,12, 117-38.
Kamien, M. I. and Schwartz,N. I. (1983) Conjectural Variations,
CanadianJournal ofEconomics,16, 191-211.
Kreps,D. M. (1990)A Coursein Microeconomic Theory,Princeton
University Press,Princeton,USA.
Laititer,J. (1980) 'Rational' Duopoly Equilibria, QuarterlyJournal
ofEconomics,95,641-62.
Leo . f W S k lbe
M
1. ti. C
t.t "
,
ntie ,
., tac e
rg on
onopo IS c
Political Economy, August 1936,554-9.
ompe I Ion, Journa I 0if
Lindh, T. (1992) The Inconsistency of Consistent Conjectures,
Journal ofEconomicBehaviorand Organization,18,69-90.
Pace,R. K. and Gilley, W. (1990)A Note on ConsistentConjectures:
The LeontiefLegacy.SouthernEconomicJournal,56,788-92.
Perry, M. K. (1982) Oligopoly and Consistent Conjectural
Variations,Bell Journal ofEconomics,13,197-205.
Robinson, J. (1934) The Economics of Imperfect Competition,
Macntillan, London,UK.
Robson,A. (1983)Existence of ConsistentConjectures:Comment,
AmericanEconomicReview,73,454-6.
81nthis case,the secondorderconditionof profit maxintizationrequiresthat:
q"(p'q + p) + p'(q')2(2 + VJ -w'(2 + vJ < O.
This conditionis satisfiedif vi < 2 [which is reasonablesincea fIrm's conjecturalvariationis likely to lie betweenBertrand (Vi = -1) and cartel (Vi = 1)], the outputdemandfunctionis negativelysloped(p' < 0), the input supplyfunctionis positively sloped(w'>O),andp' q + P > O.
Note thatthis last term canbe rewrittenasp[1 -(If)] [where n= (dqldp (plq) > 0] and will bepositive if the equilibriumprice is greaterthan
zero andthe firm operatesin the elastic regionof demand.
~s is the phraseusedby Bresnahan(1981,p. 943)to describethe implications of his duopoly with consistentconjectures.
Duopsonymodelswith constantconjecturalvariations
11
Schrnalensee,
R. (1990)Empirical Studiesof Rivalrous Behaviour,in
(G. Bonannoand D. Brandolini, eds),Industrial Structurein the
New Industrial Economics,Cl~endon ~ess, Oxford, u~. ..Vj
which simplifies to
Shaffer, S. (1991) ConsistentConjectures m a Value-Maxlmlzmg
Duopoly, SouthernEconomicJournal,57,993-1009.
Shepherd, W. G. (1990) Publishing Performance in Industrial
Organization: A Comment,Review of Industrial Organization,
5,139-46.
..
Setting vi = Vj = v (by symmetry), EquatIon Al can be rewntten as
2
V +2v+ 1 =0.
(A2)
Ulph, D. (1983) Rational
International
Journal
The roots of this quadratic
. rural
t
Conjectu~s
of Industrial
in the .Th~ory of Oligopoly,
Organization, 131-54.
equation
(AI)
produce
single
consistent
. .
vana Ion equal t 0 -. 1
conJec
Under the assumptions given above, the sac of profit
maximization is satisfied. To see this, note that profit maximization requires that
APPENDIX
This appendix demonstrates that there is a unique consistent
conjecture that satisfies the second order sufficient (necessary)
condition (SaC) when inputs and outputs are homogeneous
and when output demand, input supply, and the production
function are linear. Uniqueness can be established as follows.
Recall Equation 9
y-j -p'(q')2(2
-1
= -(~)
_fp'(q')2 -w1
+ vJ -w'(2 = vJ
p'(q')2(2 + VJ -w'(2 + vJ < O.
(A3)
This condition is satisfied when Vi = -1 and the output
demand and input supply functions are regular (i.e., p' < 0 and
w' > 0, respectively).