Econometrica, Vol. 55, No. 5 (September, 19871, 1151-1164
DYNAMIC DUOPOLISTIC COMPETITION WITH
STICKY PRICES
We study duopolistic competition in a homogeneous good through time under the
assumption that its current desirability is a n exponentially. weighted
function of accumulated
past consumption. This implies that-the current price of the good does not decline by as
much t o accommodate any given level of current consumption as it would if its desirability
were a function solely of present consumption. We analyze the consequences of this
"stickiness" of the good's current price through a speed of adjustment parameter that
yields instantaneous adjustment as a limiting case. O u r analysis is conducted in terms of
a differential game. T h e open-loop a n d closed-loop Nash equilibrium strategies are derived,
from which the corresponding asymptotically stable steady-state equilibrium prices for the
good are obtained. These equilibrium prices are then analyzed as the speed of price
adjustment becomes instantaneous. It is found that the equilibrium price corresponding
to the open-loop Nash equilibrium strategies approaches the static Cournot equilibrium
price while the equilibrium price corresponding t o the close-loop Nash equilibrium
strategies, which are subgame perfect, approaches a price below it.
KEYWOKDS:Sticky prices, open-loop Nash equilibrium strategies, closed-loop Nash
equilibrium strategies, asymptotically stable steady-state equilibrium prices, speed of adjustment, limit game.
INTRODUCTION
I N T H I S P A P E R we study duopolistic competition through time under the assumption that the price of a homogeneous product does not adjust instantaneously to
the price indicated by its demand function at the given level of output. Thus, at
each point in time identical duopolists face a constant price that they know will
decline, but not instantaneously, if their joint output exceeds the level of demand
at that price. The evolution of price through time is a function of the difference
between the current price and the price on the demand function for each level
of output. We allow the duopolists to exploit the lag in price adjustment to their
advantage. Previous studies of Cournot duopoly with less than instantaneous
adjustment in quantities have tended to overlook the strategic possibilities provided by lags (see Waterson (1984) for a review of this literature). We employ a
differential game framework to analyze the interaction between the duopolists
through time a n d determine the open-loop and closed-loop (feedback) Nash
equilibria of the duopoly game.
Our main objective is the study of the static Cournot equilibrium as a n
asymptotic limit of competition through time with sticky prices as the speed of
adjustment becomes instantaneous. In order to carry out this investigation we
incorporate a speed of adjustment term in our formulation of the differential
equation governing the evolution of the price. When the speed of adjustment
goes to infinity, price converges instantaneously to its value on the demand
function for the given level of output. Our main conclusion is that the static
' We would like to thank the members of the workshop at the Institute of Advanced Studies in
Jerusalem, Israel, David Kreps, a n d two anonymous referees for many helpful comments.
1151
1152
CHAIM FERSHTMAN A N D MORTON I. K A M I E N
Cournot equilibrium price is the asymptotic limit, as the speed of adjustment
goes to infinity, of the open-loop Nash equilibrium while the stable closed-loop
Nash equilibrium price converges to a value below it. This result is interesting
because the open-loop Nash equilibrium is not subgame perfect while the closedloop Nash equilibrium is.
A differential game framework was employed explicitly by Simaan and Takayama (1978) to study dynamic duopoly with sticky prices and implicitly by Roos
(1925, 1927). Simaan and Takayama (1978) focus on the role of capacity constraints and assume the speed of adjustment to be unity. We allow for an arbitrary
adjustment speed and focus on the limiting case when it is instantaneous. There
have been a number of empirical studies supporting the hypotheses that prices
are sticky-see Rotemberg (1982) for an overview of this literature. An interpretation of open-loop and closed-loop Nash equilibria in terms of the commitments
required of the players in a differential game are provided by Reinganum and
Stokey (1985).Fudenberg and Tirole (1983) and Kreps and Spence (1983) provide
reviews of recent analyses of dynamic oligopoly models.
Our analysis is conducted under the assumptions that demand is a linear
function of price, total cost is a quadratic function of output, and there is no
uncertainty. We show that the Nash equilibria of the differential game are
symmetric. In the next section we introduce our formulation of the differential
game and derive the open-loop Nash equilibrium and the closed-loop Nash
equilibrium in the subsequent two sections, respectively. In Section 4 we define
the "limit game" and derive our basic results regarding the asymptotic properties,
as price adjusts instantaneously, of the open-loop and closed-loop equilibria.
The last section contains a summary of our analysis.
1 . FORMULATION A N D NOTATIONS
Consider a duopoly in which both firms have the cost function
(1.1)
~ ( u=
; )cui +$u:
(i=1,2)
where ui 3 0 is the ith firm's output rate.
In static analysis an inverse demand function is given, in its linear version, by
p = a - b ( u , + u,), where for simplicity we will set b equal to one. Under the
assumption of "sticky prices," the price does not adjust instantaneously to the
level given by the demand function for a given level of output. This adjustment
takes time and the rate of change of price is a function of the gap between the
current market price and the price indicated by the demand function for the
currently produced quantities. Adopting Simaan and Takayama's (1978) formulation of a market with sticky prices we assume that the change in price is governed
by the differential equation
where O < s G co denotes the speed in which the price converges to its level on
DYNAMIC DUOPOLISTIC COMPETITION
1153
the demand function and a - [ u l ( t )+ u , ( t ) ] is the price on the demand function
for the given level of output.
Under the above assumptions the objective of each firm is to maximize
subject to (1.2) and u i ( t )3 0.
The role of the speed of adjustment can be seen by solving for p ( t ) in (1.2)
and substituting into (1.3) to get
(1.4)
Ji =
lom
e-"[(a - u1- u,)ui -
d ~ i -/ ~CU; - ~ : / 2 dl]
( i = 1,2).
It is evident from (1.4) that firms face a downward sloping linear inverse demand
function but that the decline in price along it, as a firm's level of output increases,
is retarded when s is finite. As s + a,the term p u i / s vanishes and price adjusts
instantaneously along the demand function.
The source of the stickiness in prices can be traced to the inverse demand
function implied by (1.2). Integration of (1.2), with p ( 0 ) = a and u , + u, = u yields
(1.5)
p ( t ) = a -s
lo'
e-s('-T)U ( T ) dr.
This inverse demand function is consistent with consumers' utility functions that
depend on both current consumption and past consumption of a good, as in
Ryder and Heal (1973). Specifically, Ryder and Heal posit a utility function of
the form W' W ( C ( t ) ,Z ( t ) ) ,where C ( t ) represents current consumption and
Z ( t ) represents exponentially weighted accumulated past consumption. A special
case of this utility function is of the form W = C f ( Z ) ,from which it follows that
the marginal utility of current consumption a W I d C =f ( Z ) . Identification of f ( Z )
with the right side of (1.5) yields the result that (1.5) is an inverse demand
function stemming from the posited utility function. According to ( I S ) , recent
consumption of the good has a more depressing effect on its current desirability
than earlier consumption does.
The demand function (1.5) yields a linear inverse demand function in current
consumption as can be seen by integrating its right side by parts, and assuming
differentiability of u ( r ) ,
and letting s + a to get
The linear inverse demand function (1.6) is a special case of (1.5) in which the
marginal utility of current consumption is concentrated entirely on present consumption. This relationship between (1.5) and (1.6) through s provides the key
1154
CHAIM FERSHTMAN A N D MORTON I. K A M I E N
to the meaning of sticky prices in our model. For with s finite, the marginal utility
of current consumption is not entirely concentrated on consumption at the present
instant of time. Thus, with s finite the marginal utility of current consumption
does not decline with an increase in present consumption by as much, and
therefore price does not have to decline by as much to induce this level of present
consumption, as it does when s + co.
The appropriate framework for analyzing the problems posed in (1.2) and (1.3)
is as differential games. In analyzing a differential game two major strategy spaces
are commonly examined in the literature-the open-loop and the closed-loop
(feedback) strategy spaces.
DEFINITION
1: The open-loop strategy space for player i is
s:' = {ui(t)lui(t)is piecewise continuous and ui(t)2 0 for every t}.
The open-loop strategies can be characterized as path strategies. Each player
chooses a path of action ui(t) to which he commits himself at the outset of the
game. Neither has the option to reconsider this strategy and change it. A Nash
equilibrium in such strategies (Definition 2) is a pair of paths (or n-tuple in the
case of n players) such that each player's path is the best response to its rival's
path.
DEFINITION
2: An open-loop Nash equilibrium for the above game is a pair of
open-loop strategies (uT, u;) such that for every ui E s:',
J~(U:, u;)
2 J'(u,,
u;),
J ~ ( u T u;)
,
2 J ~ ( u T UJ.
,
Following Reinganum (1982, pp. 674-675), we define the closed-loop strategy
space as follows.
DEFINITION
3: The closed-loop (feedback) strategy space for player i is the set
Si= {ui(t,p)lui(t,p ) is continuous in (t, p); ui(t, p ) 2 0 and
Iui(t,p) - ui(t,p')l s m(t)lp -p'l for some integrable m(t) 2 0 ) .
The closed-loop strategies describe decision rules that prescribe an output rate
as a function of time and the observed market price. Players in this case do not
commit themselves to a particular path at the outset and can respond to different
prices they observe.
DEFINITION
4: A closed-loop Nash equilibrium is a pair of closed-loop strategies
(u?, u;) E S1x S2 such that for every possible initial condition (p,, to):
J ' ( u r , u):
2~
' ( uu:),
~,
for every ui E Si
(i, j = 1, 2; i # j).
D Y N A M I C DUOPOLISTIC COMPETITION
1155
For later reference note that in the static game when the demand function is
given by p = a - ( u , u,) and the cost function is specified by (1.1), the Cournot
equilibrium price is
+
and the "competitive" price when both firms behave according to the rule
"marginal cost equals price" is
2. OPEN-LOOP NASH E Q U I L I B R I U M
THEOREM1: There is a unique stationary open-loop Nash equilibrium for the
above game. The price at this equilibrium is:
and theJirms' strategies are given by
PROOF: For every given path u j ( t ) of firm j, firm i faces the problem of
maximizing (1.3) subject to (1.2) and given u j ( t ) . A similar problem of course
faces player j. An equilibrium in the market is a pair of open-loop strategies that
solves the two optimization problems simultaneously. Forming the current value
Hamiltonian in the standard way, the necessary conditions for an open-loop
equilibrium are
(2.3)
p ( t ) - c - u i ( t )- A i ( t ) s = 0
(2.4)
- A i ( t ) = u i ( t )- A i ( t ) ( s+ r ) , and
( i = 1 , 21,
lim eKrrA,(t)= 0
r-m
Differentiating (2.3) with respect to t and substituting
Ai
( i = 1,2).
and h i in (2.4) yields
In Appendii 1 we show that the open-loop equilibrium strategies are symmetric.
Keeping in mind that at the stationary point p ( t ) = u , ( t ) = u , ( t ) = O we can
use equations (1.2) and (2.5) to find the stationary equilibrium given by (2.1)
and (2.2).
Q.E.D.
REMARK:Notice that equation (2.3) implies that at every instance each player
will follow the policy u i ( t )+ c = p ( l ) - A,(t)s. This rule is the well-known MC =
MR, but in this case marginal revenue consists of two elements. The price at
1156
CHAIM FERSHTMAN A N D MORTON I. K A M I E N
time t is the instantaneous marginal revenue and -Ai(t)s is the long run effect
of an incremental change in the output rate. Thus, this condition equates marginal
cost with long run marginal revenue.
PROPOSITION
1: For everyjinite r > 0, and s > 0, the stationary open-loop equilibrium price is below the static Cournot equilibrium price. Only in the limit as r+ 0
or s + do the two prices coincide. Moreover, as the interest rate increases or the
speed of adjustment decreases, the equilibrium price declines. A s r + co or s + 0 the
stationary open-loop equilibrium price converges to the static competitive price.
PROOF: Letting r + 0 or s + co in (2.1) yields that the stationary open-loop
equilibrium price is p* = ( a + c ) / 2 = P*,. Differentiating (2.1) yields
(2.6)
ap* 12s(P*, - P*,)
-- <o,
ar
(4~+3r)~
and
ap* - 12r(P*, - P*,) > 0.
-as
(4~+3r)~
For r+ co or s + 0 it is straightforward to check that the stationary equilibrium
Q.E. D
price converges to p* = P*,.
The intuitive explanation of Proposition 1 is that, if we start at the Cournot
equilibrium price, increasing output has little short-run impact on prices, and so
given impatience, firms expand production.
The above necessary conditions can be used to find the open-loop equilibrium
price trajectory for a game that starts at po Z p*.
PROPOSITION
2: The open-loop Nash equilibrium price trajectory is given by
(2.7)
p e ( t )= p * + ( p o - p * ) ekl'
where p* is the stationary equilibrium price given by (2.1), po is the initial price at
t = 0, and k , is a negative constant that depends on the parameters of the problem.
PROOF:See Appendix 2.
An immediate corollary from this proposition is that the open-loop Nash
equilibrium is globally asymptotically stable. The equilibrium price trajectory
converges to the stationary equilibrium price, which does not depend on the
initial price.2
* For more detailed discussion about existence, stationarity and global asymptotic stability of
open-loop equilibria, see Fershtman and Muller (1984, 1986).
DYNAMIC DUOPOLISTIC COMPETITION
1157
3. T H E SUBGAME PERFECT CLOSED-LOOP NASH EQUILIBRIUM
THEOREM
2: Let
Then ( u T ( p ) , u q ( p ) ) constitutes a symmetric global asymptotically stable closedloop subgame perfect Nash equilibrium for the infinite horizon dynamic game
under consideration.
PROOF:The proof will be carried out in two steps. First we consider the case
in which p , a $ . In this case we have an interior solution. Then we will consider
the case po <$.
CASE1: Assume that p, j?. Using the value functions approach the closed-loop
equilibrium strategy ( u ? , u q ) must satisfy the following Hamilton-JacobiBellman equation (see Starr and Ho ( 1 9 6 9 ) ) .
(3.5)
rvi(p)
ax { ( p - c ) u i - ; u : + s ~ b ( p ) [ a- p - ( u i + u j ) ] )
u,
(i= 1,2),
where ~ ' ( pis)the value for player i of the game that starts at price p. Note that
although in the general case V' is also a function of t and not just of the state
variable, it can be shown that if time enters the objective function ( 1 . 3 ) only
through the discount term that the value functions do not depend on t (see
Kamien and Schwartz (1981, p. 2 3 8 ) ) .
Since the right side of the above equation is concave, the ui that maximizes it
is given by
(3.6)
u*(p)= p -c -s v ; ( p )
(i=1 ~ 2 ) .
Substituting ( 3 . 6 ) into ( 3 . 5 ) yields
(3.7)
r v i ( p )= ( p - c ) ( -~c - V ~ S- +)( p - c - ~ 6 s ) ~
+v~s[~-~-(~~-~c-sv~-sv',)]
(i = 1,2).
Expression ( 3 . 7 ) presents a system of two partial differential equations. By
solving this system and finding the value functions ( ~ ' ( p )v ,2 ( p ) ) ,we can use
( 3 . 6 ) to find the equilibrium strategies. For every p a $ we propose the following
quadratic value function:
(3.8)
v i ( p )= ; K i p 2 - ~ , p + g
which implies that v ; ( p ) = K i p - Ei.
(i =1,2),
1158
CHAIM FERSHTMAN A N D MORTON I. K A M I E N
Substituting v i ( p ) and v ; ( p ) into ( 3 . 7 ) yields a condition that must hold for
every p. We show in Appendix 3 that only a symmetric equilibrium can be
asymptotically stable. Using straightforward algebraic manipulation we can conclude that the symmetric equilibrium K and E are given by
and
Now substitute
(3.11)
v;(~
into
) (3.6) to yield that the strategies
uT=(l-sK)p+(sE-c)
(i = 1,2),
where K and E as defined by ( 3 . 9 ) and (3.10), respectively, constitute a subgame
perfect closed-loop Nash equilibrium for the dynamic game under consideration.
However, since (3.9) is a quadratic equation it has two possible solutions for K .
These two solutions define two possible equilibrium strategies. In Appendix 3
we prove that only the symmetric solution with the minus sign, i.e., ( 3 . 2 ) , defines
a globally asymptotically stable Nash equilibrium.
CASE2 : po<p^. In this case the constraint ui 0 is binding and we need to
modify the proof. In maximizing the right side of ( 3 . 5 ) for p<p^ no interior
solution can be reached and the optimal output policy in this case is u? = O .
However, as ( 1 . 2 ) indicates when u , = u 2 = 0 the price goes up. If a <p^, price
will go up until it will be equal to a, but since a <p^ no production will take
place. If a >p^, price will go up until p 3; and then the equilibrium is the one
discussed in Case 1 . In order to establish the subgame perfectness of this
equilibrium we need to also define a value function for prices below p^ and then
to show that condition ( 3 . 5 )is satisfied for this value function. When ui = 0 , i = 1 , 2 ,
the price path p( t ) is given by
(3.12) p ( t ) = p o e-"' + a ( 1 - e-"').
Let i ( p )denote the time that it takes for the price to reach the level p^ from the
level p. Now for every p <p^, let us define the value function
(3.13)
v ' ( ~=
) e-r"P' vi(P^),
where vi(p^)
is defined by ( 3 . 8 ) .The economic explanation of this value function
is straightforward. For every p <p^ optimal output is zero and thus profits are
zero. The first time the firms deviate from the zero production level is when price
reaches p^; the value of a game starting at p^ is already discussed and defined by
(3.8). Thus, the value of the game that :tarts at p <p^ is the discounted value of
vi(p^).
Using (3.12),the condition that t ( p ) must satisfy is p^ = p e-"'+ a ( 1 - e-"')
which implies that d i / d p = - l / s ( a - p ) . Differentiating the value function (3.13)
with respect to p yields
DYNAMIC DUOPOLISTIC COMPETITION
1159
Thus, it is straightforward to verify that the suggested value function satisfies
(3.5).
Q.E. D.
COROLLARY
1 : There is a stationary closed-loop Nash equilibrium price given by
+
Moreover, if a game starts at p, p* the closed-loop subgame perfect equilibrium
price path converges to this stationary price.
PROOF:AS is demonstrated in Appendix 3, the equilibrium price path is
p ( t ) = p * + ( p o - p * ) eDT where p* is specified by (3.15) and D is a negative
constant that depends on the parameters of the problem.
Q.E.D.
COROLLARY
2: A S the price in the market increases, the jirms increase their
output rate.
PROOF:The equilibrium strategies are given by (3.1) which are linear functions
Q.E. D.
of price. The coefficient that multiplies p is given by 1 - sK > 0.
An immediate corollary of the above discussion is that as r + w the equilibrium
price converges to the competitive price as defined in (1.8). This is true since
(3.2) and (3.3) imply by 1'Hospital's rule that lim,,, K = 0 and lim,,, E = O
which implies that VL = 0, i = 1 , 2 . The equilibrium strategies in this case (see
( 3 . 1 ) )are u* = p - c, i = 1 , 2 . This policy implies that u* + c = p which is identical
to the well-known rule MC = M R , when M R is taken to be instantaneous marginal
revenue and not long run marginal revenue. The stationary equilibrium price is
given in this case by p* = ( 1 / 3 ) ( a+ 2 c ) which is exactly the competitive equilibrium price of the static game in which firms set marginal cost equal to price.
This result is intuitively appealing. A very high interest rate implies that the
importance of the future declines. As r approaches infinity firms stop taking into
consideration the future effects of their current actions. Thus, the policy that they
follow is marginal cost equals short run marginal revenue which is equal, under
the assumption of this model, to price. The resultant equilibrium price in this
case is, of course, the competitive price.
4. T H E "LIMIT GAME": T H E D Y N A M I C G A M E WITH INSTANTANEOUS PRICE
ADJUSTMENT
In previous sections we based our dynamic game on the assumption that the
speed of adjustment is finite. It is this assumption that makes the game we consider
different from a repeated Cournot game in its continuous time version. When s
goes to infinity the dynamic structure disappears and the price jumps instantaneously to its level on the demand function for each level of output. Thus, for
s + co the game can be viewed as a repeated Cournot game in its continuous time
version. In this section we let s -,co and examine the limits of the open-loop and
closed-loop equilibria.
1160
C H A I M FERSHTMAN A N D MORTON I . K A M l E N
THEOREM
3: The Cournot equilibrium price of the static game is the limit of the
stationary open-loop equilibrium price.
PROOF:This was established in Proposition 1.
THEOREM
4: The stationary subgame perfect closed-loop equilibrium price converges to a price,
which is a convex combination of the static Cournot equilibrium price and the static
competitive price and, therefore, is below the static Cournot equilibrium price.
PROOF:The stationary closed-loop equilibrium price is given by (3.15). Let
sK and y = lim,,, sE. Using equation (3.2) yields that /3 = 1
0. Similarly, from (3.3) we obtain that y = ( c - ap -2cp)/(3 - 3 p ) . Note that from
(3.2) and (3.3) that lim,.,o sK = P and lim,,, sE = y. Thus, as s + co or r + 0, the
equilibrium price approaches
/3
-a>
= lim,,,
Substituting for y and rearranging yields
upon recalling that
P = 1 -a,
and
(1.7) and (1.8).
Q.E. D.
From (3.1) it is evident that the limit as s + a or r+O of the closed-loop
equilibrium strategies is
The intuitive reason for the difference between the open-loop and closed-loop
equilibrium price is that in the formulation of the closed-loop strategy each player
takes into account the optimal reaction of its rival to a change in the state variable
while in the formulation of an open-loop strategy it does not. The stable closedloop equilibrium strategy, in the problem we analyze, is an increasing linear
function of the state variable, price. Thus, each firm will decrease its output when
price decreases. Let us see what taking the rival's reaction to a decrease in price
into account means in terms of a firm's output decision. If a firm ignores this
reaction by its rival and simply makes the Cournot assumption that its rival's
output will remain at its present level, then it will make its output decision on
the basis of the residual demand curve it faces. If, on the other hand, it takes its
rival's reaction to a price change into account, it will know that as it expands its
output and causes prices to fall, its rival will contract his output. Thus, its
movement down its residual demand curve will be offset somewhat by an outward
D Y N A M I C DUOPOLISTIC COMPETITION
1161
shift of the residual demand curve as its rival contracts his output. This, of course,
will cause the-firm to optimally expand its output beyond the optimal level when
its rival's reaction to a price change is ignored. As both rivals will take each
others' optimal reaction to a change in price into account in the formulation of
their optimal closed-loop strategies, the equilibrium output will be greater than
in the equilibrium of the open-loop strategies where rivals' reactions to price
changes are ignored.
It is not difficult to show that profits are higher at the stationary open-loop
equilibrium than at the stationary closed-loop equilibrium. Since the open-loop
equilibrium is the case in which each player commits himself to an output path
at the outset, and does not condition his output rate on the observed price, it is
clear that the players can benefit from such commitments.
REMARK:Besides the stable stationary equilibrium price (4.2), there is another
stationary equilibrium price,
which is not asymptotically stable. This equilibrium price is higher than the
Cournot price. It arises from the larger root in (A.3.10) and is an unstable
equilibrium since starting from any initial price Po # P there will not be convergence to P (see (A.3.9)).
The above two theorems imply the following corollary:
3: The static Cournot equilibrium price is the limit, as price adjusts
COROLLARY
instantaneously, of the open-loop equilibrium price which is not subgame perfect
while the subgame perfect equilibrium converges to an equilibrium price below the
static Cournot equilibrium price.
We note that in defining the closed-loop strategy space we restricted ourselves
to Markovian decision rules that are continuous in the state variable. Thus, we
ruled out discontinuous strategies that allow the use of threats and trigger
strategies. It can be shown by standard methods that if discontinuous history
dependent strategies are allowed, then a collusive equilibrium can be supported
in our model. The stable closed-loop equilibrium that we consider is, however,
subgame perfect even in the game with discontinuous history dependent strategies.
SUMMARY
We have analyzed a differential game of duopolistic competition through time
under the supposition that price does not adjust instantaneously to its level on
the demand function for each level of output. Our focus has been on the difference
between the stationary state open-loop and closed-loop Nash equilibria of this
game, in the limit, as price adjusts instantaneously. We find that the stationary
state open-loop equilibrium price converges to the static Cournot equilibrium
1162
C H A l M F E R S H T M A N A N D MORTON I . K A M I E N
price, as price adjusts instantaneously, while the stationary state closed-loop
equilibrium price converges to a price below it. An intuitive explanation for this
difference is that in the closed-loop strategy, which is subgame perfect, each
duopolist knows that the loss in future profits from expansion of current output
will be shared by his rival, who will attempt to at least partially offset it by
contracting his output. Thus each duopolist will expand his current output beyond
the level that he would if he alone bore the full loss in future profits. On the
other hand, when the duopolists follow open-loop strategies their ability to shift
some of the loss in future profits from expansion of current output on their rival
is limited by a commitment to an output path at the outset. This shifting of the
loss in future profits on the rival persists when the price adjusts instantaneously
for the stationary state closed-loop strategies but vanishes for the stationary state
open-loop strategies.
Department of Economics, Hebrew University, Jerusalem 91905, Israel,
and
Graduate School of Management, Northwestern University, Evanston, Illinois
60201, U.S.A.
Manuscript received August, 1984; jinal revision received September, 1986.
APPENDIX 1
SYMMETRY O F T H E OPEN-LOOP EQUILIBRIUM
To show that the open-loop equilibrium must be symmetric we solve for u, in ( 2 . 3 ) and substitute
into ( 2 . 4 ) ,to get
Solving this differential equation and employing the transversality condition to evaluate the constant
of integration yields
Thus, A , ( t ) = A 2 ( t ) as they both equal the right side of the above equation. It follows from (2.3) and
the strict concavity of the Hamiltonian in u , , that u , ( t )= u 2 ( t ) .
APPENDIX 2
P R O O F O F PROPOSITION
2
A detailed proof is long but standard. A proof will merely be out!ined here. Differentiating the
kinematic equation (1.2) with respect to t and substituting in it A, A and u from equations (2.31,
( 2 . 4 ) , and ( 1 . 2 ) , respectively, yield that the equilibrium price trajectory must satisfy the following
second order linear differential equation:
when P = ( d 2 p ) / d t 2and
,
(A.2.2)
A
= s - r,
(A.2.3)
B
=-s2-3s(s+
(A.2.4)
R
= -[s2a
r),
+ s ( 2 c + a ) ( s+ r ) ] .
1163
D Y N A M I C DUOPOLISTIC COMPETITION
Clearly, a particular solution of (A.2.1) is p ( t ) = R I B which is exactly the stationary equilibrium
price (2.1). Note that the roots of the characteristic equation associated with the homogeneous part
of (A.2.1) are both real-one positive and the other negative. This is true since B is negative. Then,
if we take the stable solution and use the initial price p ( 0 ) = p o , the following trajectory
is the open-loop Nash equilibrium price trajectory where
APPENDIX 3
G L O B A L ASYMPTOTIC STABILITY O F T H E S Y M M E T R I C C L O S E D - L O O P E Q U I L I B R I U M
To establish that only a symmetric solution, u , ( t ) = u,(t), provides an asymptotically stable
closed-loop equilibrium, we substitute from (3.8) into (3.7) to get
(A.3.1)
f r ~ , ~ ' - r ~ #rg,+= ( f - 3 s K i +s,K,K,
+ f ~ ~ ~ f ) ~ ~
+ [ 3 s E , - s 2 ~ , E-2s2K,EJ
,
-c+sK,(a+2c)]p
The requirement that these equations be satisfied for all values of p implies that
(A.3.2)
( i ,j = 1 , 2 , i # j ) .
S~K:+(~S~K~-~S-~)K,+~=O
A similar expression can be obtained for E , by equating the coefficients of p, from which it can be
seen that E, is a function of K , , K,, and c. The g, in turn depend on E l and E,.
Now by subtracting the expression for K , given by (A.3.2) from the expression for K , given by
(A.3.2) we get
(A.3.3)
( K , - K , ) [ s 2 ( K ,+ K,)
- ( r + 6 s ) ] = 0.
This expression implies that either K , = K,, in which case E l = E,, g , = g,, and, by substitution into
(3.6), that uT = u?. The alternative is that K , # K , and uT # u?, but
(A.3.4)
s'(K,
+ K,)
=
r+6s.
In this case we have by substitution from (3.6) for u: and u? into (1.2)
(A.3.5)
p
-sp[s(K,
+ K,) -31
=s[a+2c-s(E,
+ E,)].
The particular solution to this first order differential equation is
(A.3.6)
=
+
a + 2 c - s ( E , E,)
3 - s ( K , K,)
+
The solution to the homogeneous part of (A.3.5) is
(A.3.7)
p ( / )= ~ ~ ' ! " ~ i + ~ : ' - ~ ] '
where C is the constant of integration. Finally, the complete solution of (A.3.5) is
(A.3.8)
p(t) =p+(p,-P) e'[""~+"~'-~]'
where p ( 0 ) =p,, is employed to determine the constant of integration C. Now in order for (A.3.8) to
converge to as t + co, to be asymptotically stable, we need that s ( K , + K,) - 3 < 0 . However, by
substituting for s ( K , K,) from (A.3.4) this means that r+3s < 0 , which is impossible as r and s
are both nonnegative. Thus, an asymmetric closed-loop equilibrium uT # u z cannot be asymptotically
stable.
We turn then to the symmetric equilibrium, where K , = K , = K, E l = E , = E, and g , = g,. The
explicit values of K and E are given by (3.9) and (3.10), respectively. The complete solution of the
kinematic equation ( 1 . 2 ) ,the counterpart of (A.3.8) in the symmetric equilibrium, is
+
(A.3.9)
where p
p ( t ) = j i + ( p , , - P ) en'
= p*
is given by (3.15) and D = s [ 2 ( s K - 1 ) - 11.
1164
C H A I M FERSHTMAN A N D MORTON I . K A M I E N
Thus, lim,,,p(t) = p iff D<O. This means that a necessary condition for asymptotic stability is
that s[2(sK - 1) - 11< 0, or that K < 3/2s. Note that the explicit solution to (3.9) is
Taking first the positive sign in (A.3.10) implies that K > 513s > 312s which contradicts the condition
for asymptotic stability. Taking the negative sign implies that K < 1/3s <3/2s. In this case K is
sufficiently small which implies that if K is given by (A.3.10) with the negative sign, the equilibrium
is asymptotically stable.
Since the choke off price, a, is never below p, we can conclude that a >$. This condition is
sufficient for p (equation (3.15)) to be above p*. Using (A.3.9) we can conclude now that p(t)>p* for
every t. Thus, the equilibrium strategy u: is always positive.
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