Cournot Oligopoly and the Theory of Supermodular Games

More documents

Recommendations

Info

112 RABAH AMIRThis strand of the literature may be divided into two groups, according to oneimportant aspect of the information structure of the dynamic game under consideration,namely, perfect information vs simultaneous moves.In the former category, at least three separate areas figure prominently. A notionof consistent planning/altruistic growth, introduced by Strotz and Pollack, hasbeen thoroughly investigated by Leininger (1986) and Bernheim and Ray (1983).See also Lane and Leininger (1986) and Bernheim and Ray (1987). A modelof dynamic duopoly has been the object of several studies: Cyert and DeGroot(1970), Maskin and Tirole (1988a,b). Finally various racing models have beenanalyzed; see, e.g., Harris and Vickers (1987) and Budd et al. (1992).Most notably, very general pure-strategy existence results have been derivedfor dynamic games of perfect information: Harris (1985) and Hellwigand Leininger (1987) consider history-dependent equilibria while Hellwig andLeininger (1988) deal with Markovian equilibria. A common feature of allthe studies in this class of dynamic games is the deterministic nature of thedynamics.For games with simultaneous moves, the dominant setting has been that ofstrategic capital accumulation/resource extraction, initiated by the pioneeringwork of Levhari and Mirman (1980). Existence of a stationary equilibrium forthe deterministic version of this class of games is established in Sundaram (1989)and Amir (1990). See also Amir (1989). An extension to the stochastic case isgiven in Majumdar and Sundaram (1991) and Dutta and Sundaram (1990).It should come as no surprise that no general pure-strategy existence resultsare available for such games. Indeed, the simultaneity of moves in a dynamiccontext often implies that quasi-concavity, which is a basic requirement for theapplication of most fixed-point theorems, is not preserved, even under mostfavorable assumptions on the data of the game. Note that such difficulties arealso encountered when dealing with stationary equilibria of games with perfectinformation.Somewhat more surprisingly, it turns out that the available abstract subgameperfectequilibrium existence results in mixed (or behavioral) strategies forstochastic games (see Raghavan et al., 1991, and references therein) are notapplicable to most economic models. The primary reason is that these results involvethe finiteness of the action spaces or the continuity in the variation norm ofthe transition law with respect to the actions. In this context, the basic difficulty isthe absence of upper hemi-continuity of the best response correspondence in theweak-star topology on the strategy spaces (consisting of transition probabilitiesfrom the history or state space to the action spaces).Another related strand of the literature on dynamic games builds on the workof Boldrin and Montrucchio (1986) on the indeterminancy of (optimal) capitalaccumulation paths, and extends it to various dynamic games with sequential aswell as simultaneous moves, including Cournot adjustment processes. See Danaand Montrucchio (1991) for a survey.
STOCHASTIC CAPITAL ACCUMULATION GAMES 113The present paper reconsiders strategic capital accumulation/resource extractionwith bounded one-period capacities and convex transitions as a discountedstochastic game. While previous work on this problem extended the existence resultof the deterministic counterpart under a symmetry assumption on the players(Dutta and Sundaram, 1990), our analysis goes further in a variety of directions:(i) the assumption of symmetric players is removed, (ii) the stationary equilibriumstrategies are continuous and nondecreasing, in addition to having, as inthe deterministic case, all slopes bounded above by one, and (iii) uniqueness ofMarkovian equilibrium for every finite-horizon truncation is established. Whilepresented in the context of two players only, our results do extend to the n-playercase, n > 2.These surprising conclusions are reached at the expense of a new naturalconvexity assumption on the stochastic technology, represented as a transitionprobability mapping today’s joint investment into tomorrow’s random output.This key assumption may be simply stated as follows: The probability that thenext output is at or below any given level is a convex function of current investment.A class of examples given in the next section suggests that this convexitycondition is rather general; in particular it allows for atoms in the productionprocess.Nevertheless, this notion of convexity is intrinsically stochastic and has nomeaningful deterministic analog. Its convexifying effect may be expressed by thefollowing elementary but potent observation: The integral of any nondecreasingfunction with respect to a transition probability which is convex in its parameteris a concave function of that parameter. In the context at hand, think of the typicalvalue function as the integrand and of the joint investment as the parameter, inthe above statement (cf. Lemma 1.2).It is worthwhile to observe that many of the qualitative divergences betweenthe one-player (optimal growth) and the multiplayer cases prevailing in the deterministicframework are no longer present in the stochastic convex setting.The equilibrium strategies have marginal propensity of consumption between 0and 1, as does the (one-player) optimal policy, cf. Brock and Mirman (1972).Thus the classical properties of consumption functions are restored. Under theassumption of bounded extractions capacities, Markovian equilibria are evenunique for any finite horizon. Elsewhere, similar results are shown to hold forthe consistent planning problem under the same technology.It is instructive to point out that our key assumption on the stochastic technologymay also be interpreted along lattice-theoretical lines, cf. the proof ofLemma 1.3. Furthermore, Topkis’ theorem is conveniently invoked a numberof times in our analysis. Finally, it is shown that an N-period horizon problemmay be analyzed, via the dynamic programming recursion, as a sequence of Nparametrized one-shot supermodular games, cf. Topkis (1979), Vives (1990),Milgrom and Roberts (1990), and Sobel (1988). Finally, it is our belief that thestochastic technology introduced here will turn out to be of crucial importance in
Page 1: GAMES AND ECONOMIC BEHAVIOR 15, 111
Page 5 and 6: STOCHASTIC CAPITAL ACCUMULATION GAM
Page 21: STOCHASTIC CAPITAL ACCUMULATION GAM

Cournot Oligopoly and the Theory of Supermodular Games

Create successful ePaper yourself

Delete template?

Save as template?