Identifying Reaction Functions in a Differential Oligopoly Game with Sticky Prices

Lambertini, Luca; Mantovani, Andrea Working Paper Identifying Reaction Functions in a Differential Oligopoly Game with Sticky Prices Quaderni - Working Paper DSE, No. 530 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca; Mantovani, Andrea (2004) : Identifying Reaction Functions in a Differential Oligopoly Game with Sticky Prices, Quaderni - Working Paper DSE, No. 530, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4760 This Version is available at: https://hdl.handle.net/10419/159371 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Identifying Reaction Functions in a Differential Oligopoly Game with Sticky Prices1 Luca Lambertini2and Andrea Mantovani3 October 2004 1We thank Roberto Cellini, Ulrich Doraszelski and the audience at the ASSET Annual Meeting (Barcelona, November 4-6, 2004) for useful comments and suggestions. The usual disclaimer applies. Financial support by the University of Bologna and the Italian Ministry of University and Research is gratefully acknowledged. 2Department of Economics, University of Bologna, Strada Maggiore 45, I-40125 Bologna, Italy, [email protected]; and ENCORE, Faculty of Economics & Econometrics, University of Amsterdam, Roetersstraat 11, WB1018 Amsterdam, The Netherlands. 3CORE, Universit´e Catholique de Louvain, 34 Voie du Roman Pays, B-1348, Louvain-la-Neuve, Belgium, mantov[email protected]e; and Department of Economics, University of Bologna, Strada Maggiore 45, I-40125 Bologna, Italy, [email protected]. Abstract We investigate the issue of strategic substitutability/complementarity in a Cournot differential game with sticky prices. We show that first order conditions do not produce instantaneous best reply functions. However, we identify negatively sloped reaction functions in steady state, with the open-loop best reply being flatter than its closed-loop counterpart. JEL classification: C73, D43, D92, L13. Keywords: complementarity/substitutability, differential games, reaction functions, price stickiness 1Introduction The issue of super-/submodularity has been investigated mostly in static games, and refers to the slope of reaction functions in the (stage) game, as initially pointed out by Bulow, Geanakoplos and Klemperer (1985).1A potential development of this discussion consists in investigating whether the same properties can be reconstructed in a differential game, and to what extent.2 To the best of our knowledge, only Jun and Vives (2004) have considered intertemporal strategic complementarity/substitutability. They compare steady states of open-loop and stable closed-loop equilibria in a general symmetric differential duopoly model with adjustment costs, as in Reynolds (1987) and Driskill and McCafferty (1989). One of the most interesting result appears in the “mixed” case of price competition and production adjustment costs: the strategic complementarity of the static game turns into an intertemporal strategic substitutability, given that, from the standpoint of any given firm, a price cut today makes the rival smaller in the future by raising its short-run marginal cost. 1The concept of economic complementarity has recently emerged as a leading theme of economic research and has benefitted from the development of the theory of supermodular games, introduced by Topkis (1978) and based on lattice-theoretic arguments. The analysis has been focused on games with strategic complementarities and their use in industrial economics (Vives, 1990; Milgrom and Roberts, 1990 and Amir, 1996) and in comparative statics analysis (Milgrom and Shannon, 1994). In the presence of complementarity relationships between different units of a global system, a separate study of any unit alone, ceteris paribus, could lead to a wrong interpretation of the phenomenon under consideration. 2See Dockner et al. (2000) and Mehlmann (1988) for the theory of differential games and its appications to economics. 1 A related issue is that of conjectural variations. Dockner (1992) shows that any closed-loop (i.e., subgame perfect) equilibrium coincides with a conjectural variations equilibrium. Using a Cournot model with linear demand and quadratic production costs, he proves that the dynamic conjectural variations consistent with the closed-loop equilibrium are negative constants which, depending upon the level of the discount rate, vary between zero and the consistent conjectures characterising the static version of the same game. The aim of the present paper is to identify best reply functions in a Cournot differential game with sticky prices `alaSimaan and Takayama (1978) and Fershtman and Kamien (1987). In general, a dynamic game of the type proposed here may either generate instantaneous best replies directly from the first order conditions on controls, or yield best replies at the steady state only. The emergence of the first or the second case ultimately depends upon whether the first order condition taken w.r.t. the output level of any given firm contains the outputs of her rivals or not. In the model we investigate, the second case holds both under the open-loop solution and the closed-loop one. This implies that, at any time during the game, each firm has a dominant strategy independent of the rivals’ behaviour. A proper strategic interaction only emerges when one imposes stationarity on the dynamics of state and control variables. At the steady state, best replies are negatively sloped, with closed-loop best replies being always steeper than open-loop ones, to indicate that strategic interaction is stronger in the former case than in the latter. Moreover, we also show that, if price stickiness is infinitely high, the types of equilibria coincide with the perfectly competitive outcome which can be computed in the static model. The plan of the paper is as follows. Section 2 introduces the issue of 2 identifying reaction functions using a general differential game framework. The sticky price game, and its open-loop and closed-loop solutions are investigated in section 3. Section 4 concludes. 2 Preliminaries Consider a generic differential game, played over continuous time, with t∈ [0,∞).3The set of players is P≡{1,2, ...N}.Moreover, let xi(t)andui(t) define, as usual, the state variable and the control variable pertaining to player i. Assume there exists a prescribed set Uisuch that any admissible action ui(t)∈Ui.The dynamics of player i’s state variable is described by the following: dxi(t) dt ≡. xi(t)=fi(x(t),u(t)) (1) where x(t)=( x1(t),x 2(t), ...xN(t)) is the vector of state variables at time t, and u(t)=( u1(t),u 2(t), ...uN(t)) is the vector of players’ actions at the same date, i.e., it is the vector of control variables at time t. That is, in the most general case, the dynamics of the state variable associated with player idepends on all state and control variables associated with all players involved in the game. The value of the state variables at t= 0 is assumed to be known: x(0) = (x1(0) ,x 2(0) , ...xN(0)) . Each player has an objective function, defined as the discounted value of the flow of payoffs over time. The instantaneous payoffdepends upon the choices made by player ias well as its rival, that is: πi≡πi(x(t),u(t)) .(2) 3One could also consider a finite terminal time T. The specific choice of the time horizon is immaterial to the ensuing analysis, provided that terminal conditions are appropriately defined. 3 Player i’s objective is then, given uj(t),j6=i: max ui(·)Ji≡Z∞ 0 πi(x(t),u(t))e−ρtdt (3) subject to the dynamic constraint represented by the behaviour of the state variables, (1), ui(t)∈Uiand initial conditions x(0) = (x1(0) ,x 2(0) , ...xN(0)) . The Hamiltonian of player iis: Hi(x(t),u(t)) ≡e−ρt [πi(x(t),u(t)) + λii(t)·fi(x(t),u(t)) + +X j6=i λij(t)·fj(x(t),u(t))] ,(4) where λij(t)=µij(t)eρt istheco-statevariable(evaluatedattimet)that firm iassociates with the state variable xj(t). The interesting property, in the present perspective, is summarised by the second cross-derivative w.r.t. controls: ∂2Hi ∂ui∂uj (5) Two cases are possible: •If ∂2Hi ∂ui∂uj 6=0 (6) then the first order condition (FOC) ∂Hi ∂ui =0 (7) yields the instantaneous best reply function of player iagainst any admissiblechoiceofplayerjat any time t, and sgn µ∂2Hi ∂ui∂uj¶(8) is the slope of such best reply function. 4 •If instead ∂2Hi ∂ui∂uj =0∀j6=i, (9) then the FOC does not yield an instantaneous best reply function. The necessary and sufficient condition for (9) to hold is additive separability of the Hamiltonian of player iw.r.t. control variables. In this situation, it must be nonetheless true that the expression ∂Hi/∂uicontains the co-state variables. Hence, in order to solve for the equilibrium path of ui, one has to take the derivative of (7) w.r.t. t. This yields: · ui=ziµ· λii,· λij,· xi,· xj¶(10) where · xi,· xjare given by state equations (1), and the dynamics of the co-state variables λii and λij comes from the co-state equations: −∂Hi ∂xi −X j6=i ∂Hi ∂uj ∂u∗ j ∂xi =· λii −ρλii (11) If (11) contains (ui,u j),then, by substitution, we will observe · ui= wi(ui,u j). Imposing · ui=0 ,one obtains u∗ i=vi(uj)representingthe best reply against the choice of jin the steady state equilibrium. The difference between the two cases lies in the fact that while in the first case we observe an instantaneous reaction function characterising the optimal behaviour of player iat any time during the game, in the second case we only observe player i’s best reply at the steady state equilibrium, while i’s optimal behaviour during the transition to the steady state can be characterised in terms of states and co-states only, regardless of any player j’s control. This amounts indeed to saying that along the path to the steady state each player has a dominant strategy. This discussion is summarised by: 5 Remark 1 If player i’s Hamiltonian is additively separable w.r.t. controls, then ∂2Hi/∂ui∂uj=0and each player ihas a dominant strategy at every instant. To better illustrate this point, we resort to a model with sticky prices `a la Fershtman and Kamien (1987). 3Stickyprices This model dates back to Simaan and Takayama (1978) and Fershtman and Kamien (1987). Consider an oligopoly where, at any t∈[0,∞),Nsingleproduct firms produce quantities qi(t),i∈{1,2, ...N},of the same homogeneous good at a total cost Ci(t)=cqi(t)+[ qi(t)]2/2,c>0.In each period, market demand determines the notional price level: bp(t)=A− N X i=1 qi(t). In general, however, bp(t) will differ from the current price level p(t),due to price stickiness, and price moves according to the following equation: dp(t) dt ≡· p=s{bp(t)−p(t)}(12) Notice that the dynamics described by (12) establishes that price adjusts proportionately to the difference between the price level given by the inverse demand function and the current price level, the speed of adjustment being determined by the constant s∈[0,∞). This amounts to saying that the price mechanism is sticky, that is, firms face menu costs in adjusting their price to the demand conditions deriving from consumers’ preferences: they may not (and, in general, they will not) 6 Note that N/(2N+1)>1/3 for all N>1,which entails that Lemma 3 If price adjustment is instantaneous, the associated best reply at the closed-loop memoryless equilibrium is steeper than the static Cournot best reply for all N>1. The above Lemma entails that the limit of the optimal output and price levels at the closed-loop equilibrium cannot coincide with the equilibrium output and price generated by the static game. To ascertain this property, we may invoke symmetry, and rewrite (36) as follows: · q=ρ(c−p+q)+s[A−p+q−N(p−c)] (41) As in the open-loop case, · q= 0 is a linear relationship between pand q. This, together with · p=0 ,which is also a linear function, yields pCL =A[ρ+s(N+1)]+N(ρ+sN)c (N+1)ρ+( N2+N+1)s;(42) qCL =(A−c)( ρ+sN) (N+1)ρ+( N2+N+1)s. as the unique steady state of the system. The dynamic system can be immediately rewritten in matrix form to verify that the pair ©pCL ,q CLªis stable in the saddle sense. The proof of this is omitted for the sake of brevity. Using (42), we may compute: lim s→0qCL = lim ρ→∞ qCL =A−c N+1 ; lim s→0pCL =lim ρ→∞ pCL =A+Nc N+1 ;(43) lim s→∞ qCL =lim ρ→0qCL =N(A−c) N2+N+1 ; lim s→∞ pCL =lim ρ→0pCL =A(N+1)+cN2 N2+N+1 . (44) Clearly, the output and price levels in (44) are, respectively, larger and smaller than the corresponding equilibrium values for the static game. 13 From expression (30) and (43), we immediately draw the following implication: Proposition 4 If the price is infinitely sticky, both the open-loop equilibrium and the closed-loop memoryless one coincide with perfect competition for all N≥1. Note that this result is not true in general for any closed-loop equilibria. For instance, the feedback equilibrium investigated in Fershtman and Kamien (1987) and Cellini and Lambertini (2004) does collapse into perfect competition in the limit. Our last result concerns the comparative evaluation of the slope of the best reply across settings. Evaluating (26) against (39), we find: ¯¯¯¯ ∂q∗ i(t) ∂qj(t)¯¯¯¯CL >¯¯¯¯ ∂q∗ i(t) ∂qj(t)¯¯¯¯OL (45) for all N>1. Proposition 5 The best reply associated with the closed-loop equilibrium is steeper than the best reply associated with the open-loop equilibrium for all N>1. The reason is that, when taking into account feedback effects (35) for the closed-loop solution, by definition each firm becomes more sensitive to the rival’s behaviour, which makes her best reply steeper than in the open-loop game. For a given intercept of the best reply function, this would imply that firms produce more at the open-loop solution than at the closed-loop one. However, comparing equilibrium outputs in the two settings, we have that qCL >q OL,as we know from Fershtman and Kamien (1987) and Cellini and Lambertini (2004). This is due to the fact that, indeed, the intercept of 14 the closed-loop best reply is larger than the intercept of the open-loop best reply, to such an extent that the resulting equilibrium outputs are higher in the closed-loop case. 4Conclusion We have investigated the issue of intertemporal strategic interaction in differential games. We have considered a Cournot model with sticky prices where first order conditions do not identify best replies at any time during the game. They only emerge in steady state, where one can check that (i) reaction functions are negatively sloped, (ii) the feedback effects accounted for in the closed-loop solution entail that, in such a case, best replies are steeper than under the open-loop solution; (iii) this notwithstanding, steady state outputs at the closed-loop equilibrium are larger than the corresponding output levels at the open-loop equilibrium, due to the fact that closed-loop best replies are steeper but shifted outwards w.r.t. open-loop best replies. Finally, inthecaseofinfinitely sticky prices, then both types of equilibria replicate the perfectly competitive outcome generated by the static model. 15 References [1] Amir, R. (1996), “Cournot Oligopoly and the Theory of Supermodular Games”, Games and Economic Behavior,15,132-148. [2] Bulow, J. I., Geanakoplos, J. D. and P. D. Klemperer (1985), “Multimarket Oligopoly: Strategic Substitutes and Complements”, Journal of Political Economy,93,488-511. 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