Strategic Adjustment Asymmetries *)

Transcription

1 Strategic Adjustment Asymmetries *) Per Svejstrup Hansen Department of Economics and Natural Resources, Royal Veterinary and Agricultural University H Peter Møllgaard Department of Economics, Copenhagen Business School Centre for Industrial Economics, University of Copenhagen Per Baltzer Overgaard Department of Economics, University of Aarhus Centre for Industrial Economics, University of Copenhagen Jan Rose Sørensen Department of Economics, University of Southern Denmark January 001 Keywords: JEL: Symmetric duopoly, Cournot and Bertrand, asymmetric equilibrium adjustment, multiple equilibria C7, D43, L13 Abstract: In symmetric Cournot and Bertrand duopoly models, we show how responses to unforeseen demand shocks may be asymmetric in two ways: First, there can be (multiple) equilibria in which only one firm adjusts after the shock Secondly, adjustments following positive and negative shocks differ, in the sense that both firms adjust fully to a positive shock, whereas one of the firms may choose not to adjust to a negative shock Starting from a state of symmetry, it follows that strategies are more flexible upwards than downwards These qualitative results on adjustment asymmetries hold generally and irrespective of whether firm choices are strategic substitutes or complements * Comments from Peter Skott, Svend Albæk and Torben M Andersen are gratefully acknowledged Correspondence: H Peter Møllgaard, Department of Economics, Copenhagen Business School, Solbjerg Plads 3, DK- 000 Frederiksberg, phone , fax , hpmeco@cbsdk

2 1 Introduction and Outline The aim of this paper is to study the scope for asymmetric adjustment patterns, hence, asymmetric equilibria, in symmetric duopoly models ) To do this we consider a class of two-period duopoly games where, at the end of the first period, the market is assumed to have settled into a symmetric Nash equilibrium Before the market interaction in period two, consumer demand is hit by a completely unanticipated shock In period two, the firms make their decisions in two stages Initially, they decide whether or not to adjust their choice variables, eg prices or quantities These decisions are publicly observed and irreversible Then, contingent on a decision to adjust, a firm will choose its second period action optimally If a firm has decided not to adjust, its second period action is by default equal to its first period action The primary role of the first period is thus to fix an initial action profile to serve as a reference in period two Our motivation is twofold First, there is by now a rather substantial literature that attempts to endogenously explain sequential move structures in duopoly games (see eg Gal-Or (1985), Dowrick (1986), Boyer & Moreaux (1987), Hamilton & Slutsky (1990), Robson (1990), and Albæk (1990, 199)) Rather than imposing leader and follower roles exogenously, this literature tries to develop models to explain how agents might allocate the roles endogenously between them While the literature is far from mature, the paper by Hamilton & Slutsky (1990) provides an overview of the possible endogenous role formations by embedding the various, standard one-shot duopoly games in a larger extensive form game, where the actual market interaction is enlarged by a pre-play stage With observable delay, for a sequential move structure, akin to the Stackelberg model, to evolve endogenously, two things are required: The (subgame perfect) equilibrium outcome of the sequential move game must be preferred by both players to the simultaneous move outcome, and the action of the first-mover has to be irrevocable, ie a partial commitment technology has to be introduced Our second period game is structurally similar to Hamilton & Slutsky's extensive form with observable delay, in the sense that a player can decide to be an early mover (by choosing not to adjust) but not a leader, since becoming a leader depends on the other player deciding to be a late mover (by choosing to adjust) Hence, there are three possible structures; both players moving early, one moving early and one late, and both moving late However, our model is significantly different from those covered by Hamilton & Slutsky, since moving first in our sequential structure does not imply an ordinary leadership role, but rather a role of inactivity We study the circumstances in which a player may have an interest in taking on such a role when the market is hit by unforeseen shocks Finally, the strategic adjustment to demand shocks is asymmetric in the sense that both firms always adjust fully to positive shocks, whereas one (and only one) firm may decide not to adjust to a negative shock Secondly, we take our cue from the literature on the role of imperfect competition in explaining price rigidity (see eg the survey in Martin, 1993, ch 15) In particular, our model is somewhat related to the model of the kinked oligopoly demand curve (see eg Sweezy, 1939, Hall and Hitch, 1939, Maskin and Tirole, 1988) In this model, the initial (focal) price becomes a likely equilibrium because each firm has a conjecture about its rivals reaction to a price cut which makes such a cut unattractive Similarly, in our model, the initial price (or quantity) will, under some circumstances, remain unchanged in the face of a demand shock because the firms conjecture that, if they both choose to change the price (or quantity), the outcome will be less attractive than if one of them chooses to remain passive and keep the price (or quantity) unchanged Although, there are some similarities between the two models, there are also some important differences First of all, our model is sufficiently general to analyse price rigidities as well as quantity rigidities, whereas, the model of the kinked demand curve mainly has something to say about price rigidities Moreover, the model of the kinked demand curve focuses on symme- Our focus will be on duopoly models, but below it will be argued that our results generalize to other two-player games

3 3 tric equilibria whereas one of our main points is that we may obtain asymmetric equilibria in a completely symmetric duopoly To complete this introduction, we should also note that related papers by Spencer and Brander (199) and Sadanand and Sadanand (1996) have studied the endogenous formation of roles (hence, asymmetries) in oligopoly as the result of a trade off between the value of ex ante commitment and the value of ex post flexibility in a stochastic environment 3) The main difference between these papers and the present analysis is the timing of decisions as compared to the publication of information on the realization of the stochastic variable In this paper, the players are fully informed about the innovation in stochastic demand when they decide whether or not to adjust, whereas the two papers mentioned above assume that players have to decide on their timing of moves before knowing the realization of the stochastic variable Consequently, in both types of models, decisions can be related to the variance (in our model size) of the shock, but our modelling is distinct in the sense that it allows decisions to be contingent on the sign of the shock, which is, of course, impossible if the formation of roles is decided ex ante This latter feature of our model plays an important role in the formal analysis below Before plunging into the model, let us be a little more specific about the exact game the duopolists are playing As noted above, we shall consider simple two-period duopoly games However, the first-period merely serves to fix the status quo, and all the interesting strategic interaction takes place in the second period We may therefore consider the first period as the past, at the end of which play has settled into a symmetric Nash Equilibrium either of the Cournot or the Bertrand variety The second period may be interpreted as the present, and before any play takes place, the firms observe a common shock to demand Then, in a standard duopoly we would expect to see immediate adjustment by the two firms to the new symmetric Nash equilibrium, and it seems hard to endogenize a sequential decision structure Assume, however, that adjustments have to be consummated by a market-maker at an organized exchange Transactions, ie adjustments, at this exchange are costless, but agents (firms) have to show up to actively play Then, we may interpret the first-period choices (of prices or quantities) as standing offers, that can only be changed by showing up at the exchange at the beginning of period two It is costless to show up, and everybody present observe who else is present before any trading takes place Hence, if all players are present, second period choices immediately jump to the new simultaneous move Nash equilibrium If one player remains absent, her first period offer stands and the other player responds optimally (ie chooses the unique best response) If both are absent, no choices are changed, and the market-maker consummates trades given the standing offers 4) For this game we show that, depending on the realization of demand in period two, an asymmetric (subgame perfect) equilibrium is possible where only one firm changes its choice in period two Such an asymmetric equilibrium can arise when the demand shock is non-negligible and negative, whereas there will be full adjustment to a new symmetric outcome when the shock is either positive or sufficiently large negative Hence, we find that adjustment asymmetries are possible, but there will always be some adjustment, ie both firms staying with the first period choices can never be a subgame perfect equilibrium Moreover, the result that only one firm may choose to adjust in case of a small negative shock turns out to be remarkably robust In particular, the results 3 The seminal paper on the trade off between commitment and flexibility is by Vives (1986) In a duopoly setting, Daughety and Reinganum (1994) have studied the scope for endogenous role formation by augmenting the market interaction by an information collection stage Prior to the market interaction firms may decide to collect information about demand, and it is shown how (in equilibrium) only one firm may decide to collect information and then subsequently take on the role as leader This is referred to as an endogenously generated signaling game 4 A real world institution with a somewhat similar flavour could be price catalogues If the firms indicate that the prices in the catalogues are valid until a new catalogue is issued, we have a two stage decision game In stage 1 it is decided whether to issue new catalogues, and in stage prices are determined

4 4 hold irrespective of whether the actions of the firms are strategic substitutes or strategic complements In other words, the results hold in a Cournot game as well as in a Bertrand game, and they hold when goods are substitutes as well as when goods are complements in consumption The paper is organized as follows In the following section, we discuss the basic idea of the paper and it is argued that our qualitative results are fairly robust to the specification of the second period interaction In section 3, we set up a simple, fully specified, model, and the equilibria of the model are derived in section 4 In section 5, we discuss our results, and, finally, we have a few concluding comments in section 6 The proof of our central result is in the appendix The Basic Plot One of the main points of this paper is that strategic behaviour implies an asymmetry between adjustment to positive and negative shocks in the sense that, in a duopoly, both firms always adjust to a positive demand shock, but, if there is a negative shock which is not "too large" in size, only one firm adjusts In other words, the market responds less flexibly to negative shocks than to positive shocks and, in the case of a small negative shock, the symmetric duopoly ends up in an asymmetric equilibrium This result will be derived formally in a simple setting in the following sections, but the point is far more general than our formal model may suggest, and this can be shown graphically Let us consider a game with two symmetric players Moreover, let us start by assuming that the actions of the two players are strategic complements, that is, reaction functions are positively sloped (see eg Bulow, Geanakoplos and Klemperer, 1985), and that there is a unique and stable Nash equilibrium 5) These assumptions are sufficient to generate our qualitative results In Figure 1, a i is the action of player i (i = 1,) and R i is her reaction function (D,D) represents the initial Nash equilibrium Now, assume that the players are hit by a symmetric shock which is negative in the sense that the reaction functions shift inwards (downwards) The new reaction functions become R' i (i = 1,) Figure 1 A negative shock 6) 5 The reaction functions only cross once, and the reaction function of player 1 is steeper than the reaction function of player 6 In the figure we have drawn linear reaction functions However, the arguments should make clear that this is not crucial to the results

5 5 Before the shock, the outcome of the game is given by the initial Nash equilibrium where the two reaction functions, R 1 and R, intersect Now, it is easily seen that, after the shock, it can never be a Nash equilibrium that none of the players adjust Given that one of the players chooses not to adjust, it will always be optimal for the other player to adjust, and to act in accordance with her new reaction function Therefore, let us say that player adjusts Then we have to check whether it is also optimal for player 1 to adjust 7) If player 1 chooses not to adjust, the outcome will be (D,A), whereas if she chooses to adjust, the outcome in stage will be (A,A) In the figure, we have drawn an iso-payoff locus for player 1 which passes through (D,A) as well as (A,A) Therefore the circumstances given in Figure 1, player 1 is indifferent between adjusting and keeping her action unchanged However, it is easily seen that, if the shock were smaller in size, R 1() and R' 1() would be closer, and the iso-payoff locus through (A,A) would be below (D,A), and it would be optimal not to adjust (the payoff of player 1 is increasing in a ) In contrast, if the shock were larger, R 1() and R' 1() would be farther apart, and the iso-payoff locus through (A,A) would be above (D,A) Hence, the payoffs would be larger for player 1 if she were to adjust It is obvious that, when the shock is "sufficiently large", both players adjust However, it will also be the case that, if the shock is "sufficiently small", only one player adjusts To see this, note that (D,A) is always to the right of (A,A), and, since (A,A) is on the reaction function of player 1, the iso-payoff locus passing through (A,A) has a horizontal tangent at (A,A) Therefore, if (D,A) is "sufficiently close" to (A,A), the iso-payoff locus through (A,A) will be below (D,A) That it is impossible to obtain asymmetric outcomes, where only one firm adjusts, when the shock is positive (ie reaction functions shift upwards (outwards)) is illustrated in Figure Figure A positive shock 7 When we get an equilibrium where only one of the players adjusts, it is of course arbitrary to choose player to be the one who adjusts This is just for expositional reasons, and we actually have two asymmetric equilibria One where player 1 adjusts and one where player adjusts

6 6 After a shock, reaction functions are given as R' 1 and R' Again, it cannot be a Nash equilibrium that no player adjusts So, assume that player adjusts Then, it is easily checked that player 1 always wants to adjust, too (D,A) will always be to the left of (A,A), and, therefore, the iso-payoff locus through (A,A) will always be above (D,A) To provide some intuition, we can picture it as if each player has two interests that may or may not run in the same direction First, player i would like to be on her reaction function, ie play a best response Second, player i would like that player j chooses a high value of a j (i g j) Now, if the payoffs of the players are hit by a negative shock, player i has to compare the gain from adjusting to the loss stemming from player j choosing a lower value of a j if player i adjusts This is easily seen in Figure 1, since a is lower in (A,A) than in (D,A) Therefore, when the shock is negative, it will only be optimal for both players to adjust if the gain from adjusting is very large, ie if the shock is large In contrast, if the shock is positive, player i still gains directly from adjusting, but, in addition, an adjustment also implies that a j becomes higher than if player i does not adjust This is seen in Figure where a is higher in (A,A) than in (D,A) Another way to argue is that both players want to keep a 1 as well as a high In the case of a negative shock, one way to keep a 1 and a relatively high is that one player chooses not to adjust This is not possible in the case of a positive shock since, in this case, the highest values of a 1 and a are achieved when both players adjust Until now we have assumed that the actions of the players are strategic complements, but by using similar figures as above, we can illustrate exactly the same results when actions are strategic substitutes In this case reaction functions are negatively sloped, and player i would prefer that player j chooses a low value of a j Therefore, if there is a negative shock, one way for player i to keep the value of a j low is to choose not to adjust In this way a i is kept high in a form of imperfect Stackelberg leadership When there is a positive shock, player j chooses a lower value of a j if player i chooses to adjust than if she chooses not to adjust Hence, in this case there is no incentive to commit to an unchanged value of a i Above we have been arguing in terms of strategic substitutes and strategic complements, and, qualitatively, we obtain similar results in the two cases In a duopoly, the actions of the firms are strategic complements in a Bertrand game when goods are substitutes, and in a Cournot game when goods are complements The actions become strategic substitutes in a Cournot game when goods are substitutes, and in a Bertrand game when goods are complements Hence, qualitatively our results are very robust with respect to the mode of competition and with respect to how goods are related in demand This is remarkable compared to what is found in the literature on oligopoly where there is typically a qualitative difference between cases where actions are strategic complements and where they are strategic substitutes (see eg Bulow, Geanakoplos and Klemperer, 1985, and Fudenberg and Tirole, 1984) In the following sections, we set up a relatively standard duopoly model to formally demonstrate the results outlined above In section 5, we discuss the results and how they generalize to an oligopoly (ie an n-players game), and we also comment on the robustness of the results with respect to other assumptions 3 The Model We set up a model of a symmetric differentiated duopoly The notation follows Vives (1984) and Albæk ( 1990) closely The demand side of the market is derived from a continuum of identical consumers with utility of the form u U(q 1,q ) m (1) where q i 0, i = 1, are the quantities consumed of the differentiated goods and m the amount of a numeraire good The quasi-linearity of the utility function implies that one may concentrate on solving

7 7 Max U(q 1,q ) (p 1 q 1 p q ) () q 1,q where p i 0, i = 1, are the prices of the differentiated goods For simplicity, U is specified as a quadratic, strictly concave, symmetric function of q 1 and q : U(q 1,q ) (q 1 q ) ½(q 1 q 1 q q ) (3) with > 0, > 0 The goods are substitutes when > 0, independent when = 0, and complements when < 0 The goods become perfect substitutes as and "perfect complements" as - From consumer optimization we derive inverse demand functions of the linear form q i p i q i q j, i,j1,, igj (4) and, likewise, linear direct demand functions p i p, j ij 1, i gj (5) We assume that prices are non-negative throughout Nash equilibria, when either prices or quantities are decision variables, may now be computed on the assumption that marginal costs of production are constant and normalized at zero (or already subtracted from ) The resulting firstperiod prices, quantities, and profits may be found in Table 1 Choice variables Table 1: Prices, quantities, and profits in the first period Prices Quantities Profits Bertrand ( ) Cournot Now assume that a completely unanticipated change of preferences alters by an amount û Once the change has occurred it becomes public information and is fully known Firms now have to decide whether to adjust their choice variable If they do not adjust, they are stuck with the old value of the variable 4 Equilibria We study two types of games corresponding to the underlying structure in Table 1: 1) A game with simultaneous adjustment decisions and prices as choice variables; and ) a game with

8 8 simultaneous adjustment decisions and quantities as choice variables 8) In both games, the actual market interaction is a simultaneous move game, if both firms have decided to adjust Thus, the general structure of the games is as follows: In the first period either a Bertrand or a Cournot game is played, given the initial state of demand Then a new state of demand is drawn by Nature and is made publicly observable Play in period two evolves in two stages: In stage one players decide (irrevocably) whether to adjust their choices or not These decisions are simultaneous and are made publicly observable before stage two In stage two three kinds of situations may arise: If no firm decided to adjust in stage one, the choice variables stay at the first-period levels; if only one has decided to adjust, the adjusting firm optimizes against the first-period choice of the nonadjusting firm given the new state of demand; and if both firms have decided to adjust, they play a simultaneous move game with the same choice variables as in period one and given the new state of demand We treat the two games in turn 41 Prices as Choice Variables There may be four outcomes of the game corresponding to the situations in which both firms do not adjust (D,D), one firm adjusts while the other does not (A,D) or (D,A), and both firms adjust (A,A) This gives rise to the simultaneous move game in Table The label B captures that we are in the Bertrand case The first subscript denotes the action of Firm 1, whereas the second denotes the action of Firm Table : Simultaneous price-adjustment game D Firm A Firm 1 D B 1 DD B DD B 1 DA B DA A B 1 AD B AD B 1 AA B AA The payoffs are defined as follows: B 1 DD B DD û ûb B 1 AD B DA û û 4 ûb û 4 8 Two further variants, where the adjustment decisions are made sequentially, are discussed in Hansen et al (1996a) The main distinguishiing feature of these cases is that uniqueness is ensured, cf below

9 9 B 1 DA B AD û ûb 4 B 1 AA B AA û (û) ûb û where B 1 >0 Note that the first-period profits have been subtracted from the second-period profits, and it follows that the payoffs in the table represent the change in profits compared to period 1 Table allows us to examine under which conditions (D,D), (A,D), (D,A) and (A,A) are Nash equilibria in the price-setting game 4 Quantities as Choice Variables As before, there may be four outcomes: (D,D), (A,D), (D,A) and (A,A) where the decision to adjust (A) or not (D) now refers to quantity decisions This gives rise to the simultaneous move game in Table 3 Again the first-period Cournot profits have been subtracted from the second period profits so the payoffs in the table represent the change in profit resulting from the shock, û Table 3: Simultaneous quantity-adjustment game D Firm A Firm 1 D C 1 DD C DD C 1 DA C DA A C 1 AD C AD C 1 AA C AA In this case the payoffs are as follows: C 1 DD C DD û C 1 AD C DA û û 4 ûc ûc û 4

10 10 C 1 DA C AD û ûc 4 C 1 AA C AA 1 where C >0 û û ûc û 43 How Nash Equilibria depend on the Parameters Theorem 1: Suppose û g 0 Then, in both the price-adjustment game and the quantityadjustment game: 1) (D,D) is never an equilibrium ) If û > 0, then (A,A) is the only equilibrium û 3) If, then (A,D), (D,A) and a mixed strategy profile 1,0 where firms randomize between A and D are the only equilibria û 4) If, then (A,D), (D,A) and (A,A) are the only equilibria 1 û 5) If, then (A,A) is the only equilibrium < 1 Proof: See Appendix b Corollary: (A,A) is the only NE if = 0, ie the goods are independent In other words: the asymmetric equilibria, (A,D) and (D,A), "vanish" as /0 5 Discussion In this section we will give an interpretation of the results and state their implications What does Theorem 1 tell us? First of all it should be noticed that the results hold independently of whether the price-adjustment or the quantity-adjustment game is played This follows immediately from the fact that results are independent of the sign of (ie whether the goods are substitutes or complements in consumption), and that a Cournot game with substitutes (complements) is always qualitative similar to a Bertrand game with complements (substitutes), when the optimality of the different strategies is considered This can easily be established by comparing the entries in Table and Table 3 Except for the constant term (-)/(+) which nets out when the different entries are compared for finding the optimal strategy, the only difference between Cournot and Bertrand competition is the terms involving (+) and (-) respectively Hence, it should be quite clear

11 11 that Bertrand competition with complements (<0) is identical to Cournot competition with substitutes (>0) So there is no difference between using price or quantity as the choice variable for observing the different types of equilibria But the implications for equilibrium prices and quantities naturally differ in the two cases as explained below Furthermore in the comparison of A and D, when the opponent chooses A, enters squared Hence, the ranking of strategies is independent of the sign of Now let us turn to the different possible outcomes of the game We have shown that (D,D) is never an NE for any shock, positive as well as negative, if > 9) For any positive and for a sufficiently large negative shock, (A,A) is the unique NE, where sufficiently large is given by the condition in the theorem The intuition for this result is clear from inspecting the figures in Section The most striking result is that (A,D) and (D,A) can be the only pure-strategy NE if the shock is negative and relatively small 10) That is, in a model with completely symmetric firms, we obtain an asymmetric equilibrium, in which only one of the firms adjust its strategy variable to a negative shock This has the important implication that, in the aggregate, the response to negative shocks is more rigid than to positive shocks Whether the symmetric or asymmetric NE is the outcome of a negative shock depends on the degree to which goods are substitutes or complements, but not whether they are one or the other Figure 3 illustrates when the different strategy combinations are Nash equilibria depending on the size of the shock and on the degree of substitutability or complementarity of the goods FIGURE 3 ABOUT HERE From the figure it can be seen that the larger the substitutability or the complementarity between the goods, the more likely it is that we obtain an asymmetric Nash Equilibrium; and the more negative the shock, the less likely it is that (A,D) or (D,A) is an NE The intuition behind this it quite clear Let us look at the arguments in the case of Bertrand behaviour and start by considering a negative shock of a given size We know that (D,D) is never an NE, so at least one of the firms adjusts its price The optimal outcome for both firms is to maintain as high a price as possible, and this can be achieved by having only one of the firms adjusting instead of both The larger the degree of substitutability or complementarity between the goods, the more likely it is that only one of the firms adjusts, because if the goods are strong substitutes or complements the effect on an individual firm s price from an adjustment of the competing firm s price is high, making it more desirable to obtain an asymmetric equilibrium Hence, (A,D) is an NE This also illustrates the corollary: if the goods are independent (=0), the only NE is (A,A) If the goods are independent each firm acts as a monopolist Then, clearly, we cannot observe an asymmetric NE, since firms are identical and each has monopoly power in its own market This has the important implication that it is the strategic interaction per se that creates the partial rigidity Consequently, when studying the aggregates, the results can be altered dramatically by having imperfect competition represented by a monopoly rather than by a model involving real strategic interaction 11) 9 In the limit where = everything collapses into (D,D) This is because we have normalized marginal costs to zero, and with perfect substitutes and Bertrand competition we know p=mc, which is zero So it is impossible to set another price than the initial 10 Whether (A,D) or (D,A) is the Nash equilibrium is, of course, indeterminate in the simultaneous move game Moreover, as stated in Theorem 1, there is also an equilibrium in mixed strategies, but in this discussion we restrict attention to pure strategy equilibria 11 In fact, the Corollary to Theorem 1 indicates why strictly positive adjustment costs are needed in the monopoly setting of Mankiw (1985) to generate price-rigidity This point is developed further in a related note, see Hansen et al (1996b)

12 1 For any given degree of substitutability or complementarity (A,A) is an NE for sufficiently large shocks The intuition is that each firm s incentive to change its price is increasing in the shock, since the individual gain from adjusting is increasing in the size of the shock It is noteworthy that, as the goods become close substitutes or strong complements, we require negative shocks of substantial magnitude to ensure two-sided adjustment As an example, suppose / = 4/5, then (A,D) and (D,A) are the only pure strategy equilibria for û/ (-8/5,0) So, for negative shocks to the intercept of up to more than 30%, only one firm will adjust The line of argument here has been based on Bertrand competition, but it goes along the same lines for Cournot competition However, as pointed out above, the implications for prices and quantities differ This can come as no surprise since in the Bertrand case prices are the choice variables, whereas in the Cournot case quantities are If a positive shock hits the economy, the Nash equilibrium is for both firms to adjust, and whether prices and quantities are higher or lower in Bertrand than Cournot is not that interesting It is merely a difference between the two modes of competition What is interesting, however, is to compare the asymmetric equilibrium, where some rigidity exists, to the fully flexible symmetric outcome In that way the distinction between having prices or quantities as choice variables becomes clear Suppose there is a negative shock to demand Then, in the aggregate, quantities are higher in Cournot and lower in Bertrand competition in the asymmetric equilibrium compared to the symmetric outcome In contrast, the price level is lower in Cournot and higher in Bertrand in the asymmetric equilibrium than at the fully flexible symmetric outcome Let us take the example with substitute goods and quantities If the demand shock is negative and we play the quantity-adjustment game, both firms will set a lower quantity at a symmetric outcome But in the asymmetric case, one of the firms keeps the initial quantity Hence, the aggregate quantities are higher at the asymmetric than at the symmetric outcome Since quantities are higher, prices are of course lower In other words, in Cournot competition prices are less flexible upwards than downwards, whereas, in Bertrand competition prices are less flexible downwards than upwards This should be compared to a similar analysis in the model of the kinked demand curve (see Tirole, 1988, p 44) where it is found that, if the initial price is the monopoly price (which is often assumed in this literature), prices will always be less flexible upwards than downwards Finally, let us take a look at the robustness of our results For ease of exposition, our analytical results have been derived for a specific model, but, as argued in Section, the qualitative results are far more general than the formal model indicates First, the discussion in Section should make it clear that our functional forms are not crucial to the main qualitative results Secondly, whereas we have confined ourselves to consider a duopoly in this paper, it can be shown that our main results generalize to an n-firm oligopoly in the following way: If there is a positive demand shock, the only Nash equilibrium is that all firms adjust, and this will also be the only equilibrium if a sufficiently large negative shock hits the market Moreover, as in the duopoly case, there will be asymmetric equilibria, where only some of the firms choose to adjust, if the market is hit by a small negative shock Third, we have confined ourselves to a completely unanticipated shock This simplifies matters considerably since we can use the static Nash equilibrium as the initial outcome It is of course important to know whether our results generalize to a setting of Rational Expectations where the firms, in period 1, know that there may be a change in demand in period One simple version of such a model would be to asssume perfect foresight where a change in demand would be fully anticipated It turns out that our results are quite robust to such a change in the model 1) If we restrict attention to pure strategy equilibria, (A,A) will be the only equilibrium when the change in demand in period is relatively large When the change in demand is relatively small, (A,D) and (D,A) become equilibria One difference in the results compared to the results in Theorem 1 is that (A,D) and (D,A) are also 1 The solution of this model is available from the authors upon request

13 13 equilibria when there is a small increase in demand However, there is still an asymmetry in the sense that the interval for negative changes in demand, where (A,D) and (D,A) are equilibria is larger than the interval for positive changes where (A,D) and (D,A) are equilibria Another difference compared to the results in Theorem 1 is that for some sizes of the change in demand (A,A), (D,A) and (A,D) are all equilibria In spite of these differences, our main results seem to be surprisingly robust to changes in the model 6 Concluding Remarks In this paper we analysed a model for a duopoly market which is hit by an unanticipated demand shock One main result is that, starting out in a symmetric equilibrium, a small negative demand shock implies that the firms end up in an asymmetric equilibrium where only one firm adjusts to the shock The other firm gets a leadership role by choosing inactivity If there is a positive shock, or a large negative shock, both firms always choose to adjust, and the new equilibrium becomes symmetric It is remarkable that these results hold in the case of Bertrand as well as in the case of Cournot competition, and they hold in the case where goods are substitutes as well as in the case where goods are complements A second result is that we may explain price or quantity rigidity as a result of strategic behaviour If firms compete à la Bertrand, prices will be downward rigid, whereas, Cournot competition implies that quantities will be downward rigid These results can be related to a large literature in macroeconomics which combine adjustment costs (menu costs) and imperfect competition to explain aggregate price rigidity (see eg Mankiw, 1985, and Ball and Romer, 1991) In our model, we could easily introduce adjustment costs, and it would strengthen the results on price rigidity However, what is particularly interesting in our model is that we may explain price rigidity as a consequence of strategic behaviour without referring to adjustment costs 13) With respect to rigidity or flexibility of prices, we also find that in Bertrand competition, prices are less flexible downwards than upwards, whereas, in Cournot we get the opposite result, ie prices are less flexible upwards than downwards References Albæk, S, 1990, Stackelberg Leadership as a Natural Solution Under Cost Uncertainty, Journal of Industrial Economics 38, Albæk, S, 199, Endogenous Stackelberg Leadership when Costs are Private Information, Recherches Économiques de Louvain 58, Ball, L and D Romer, 1991, Sticky Prices as Coordination Failure, American Economic Review 81, Boyer, M and M Moreaux, 1987, Being a Leader or a Follower, International Journal of Industrial Organization 5, Bulow, J, J Geanakoplos and P Klemperer, 1985, Multimarket Oligopoly: Strategic Substitutes and Complements, Journal of Political Economy 93, Daughety, AF and JF Reinganum, 1994, Asymmetric Information Acquisition and Behavior in Role Choice Models: an Endogenously Generated Signaling Game, International Economic Review 35, Dowrick, S, 1986, von Stackelberg and Cournot Duopoly: Choosing Roles, Rand Journal of Economics 15, Fudenberg, D and J Tirole, 1984, The Fat Cat Effect, the Puppy Dog Ploy and the Lean and Hungry Look, American Economic Review, Papers and Proceedings 74, In the menu cost literature the work horses have either been monopoly models (Mankiw, 1985) or models of monopolistic competition (Ball and Romer, 1991), ie models that have no real strategic interaction between firms

14 14 Gal-Or, E, 1985, First and Second Mover Advantages, International Economic Review 6, Hall, RL and CJ Hitch, 1939, Price Theory and Business Behaviour, Oxford Economic Papers, 1-45 Hamilton, JH and SM Slutsky, 1990, Endogenous Timing in Duopoly Games: Stackelberg or Cournot Equilibria, Games and Economic Behavior, 9-46 Hansen, PS, HP Mølgaard, PB Overgaard and JR Sørensen, 1996a, Strategic Adjustment Asymmetries, University of Aarhus, mimeo, October Hansen, PS, HP Mølgaard, PB Overgaard and JR Sørensen, 1996b, Asymmetric Adjustment in Symmetric Duopoly, Economics Letters 53, Mankiw, NG, 1985, Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly, Quarterly Journal of Economics 100, Martin, S, 1993, Advanced Industrial Economics, Blackwell Maskin, E And J Tirole, 1988, A Theory of Dynamic Oligopoly II: Price Competition, Kinked Demand, and Edgeworth Cycles, Econometrica 56, Robson, AJ, 1990, Duopoly with Endogenous Strategic Timing: Stackelberg Regained, International Economic Review 31, Rotemberg, JJ and G Saloner, 1986, A Supergame-Theoretic Model of Price Wars during Booms, American Economic Review 76, Sadanand, A and V Sadanand, 1996, Firm Scale and the Endogenous Timing of Entry: a Choice Between Commitment and Flexibility, Journal of Economic Theory 70, Spencer, BJ and JA Brander, 199, Pre-commitment and flexibility: applications to oligopoly theory, European Economic Review 36, Sweezy, PM, 1939, Demand under Conditions of Oligopoly, Journal of Political Economy 47, Tirole, J, 1988, The Theory of Industrial Organization, MIT Press Vives, X, 1984, Duopoly Information Equilibrium: Cournot and Bertrand, Journal of Economic Theory 34, Vives, X, 1986, Commitment, Flexibility, and Market Outcomes, International Journal of Industrial Organization 4, 17-9

15 15 Appendix Proof of Theorem 1: For the price-adjustment game, the results follow by comparing payoffs in Table : i) For (D,D) to form an NE, we require û û û (A1) 4 û since >0 But this reduces to 0 which is impossible 4 ii) (A,A) is an NE if and only if û 4 û û which can be written as û û 0 (A) This is always fulfilled with strict inequality if û > 0 To see that (A,A) is the unique NE in this case, it suffices to note that (D,D) is never an NE and that for (A,D), (D,A) or any mixture to be in equilibrium, we would have to reverse (A) iii) Given the impossibility of (A1) and reversing (A), we have that if û < 0, then û 0 (A3) is sufficient for (A,D) and (D,A) to be in equilibrium We can rewrite (A3) as û 1 b (A4) which ensures that D is a best response to A That A is a best response to D follows from the reversal of (A1)

16 16 iv) If (A4) holds with equality, then both A and D are best responses to A In fact, A weakly dominates D, and (A,D), (D,A) and (A,A) are all equilibria û v) If the inequality in (A4) is strict, ie, then (A,D) and (D,A) are 1,0 vi) strict NE, from which it follows immediately that a mixed strategy equilibrium also exists If û < 0, and the reverse of (A4) holds with strict inequality, then A is the unique best response to A, and (A,A) is the only equilibrium For the quantity-adjustment game, we compare payoffs in Table 3: I) For (D,D) to form an NE, we require (since C > 0) û û û (A5) 4 II) which reduces to 0 û 4 But this is impossible as in the price-adjustment game (A,D) and (D,A) are NE if A is a best response to D, and D is a best response to A This amounts, firstly, to a reversal of (A5), which is satisfied by construction, and, secondly, to the requirement û 4 û(û) which can be written as 0 û û (A6) This can never be fulfilled if û > 0, which establishes that (A,A) is the unique NE for positive shocks If û < 0, (A,D) and (D,A) are NE if û 1 (A7) while (A,A) is an NE if (A7) is reversed III) If (A7) holds with strict inequality and û < 0, then a mixed strategy equilibrium exists This completes the proof of Theorem 1 b

17 Figure 3 17