Cournot Oligopoly and the Theory of Supermodular Games

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128 RABAH AMIRFIG. 1. Reaction functions for G x v .This is because the existence of slopes of c ∗ 1 strictly larger than −1 within [b 1, b 2 ]would imply the presence of slopes less than −1 also, for a 1 − a 2 =−(b 1 −b 2 )to hold. Furthermore, c ∗ 1 is clearly interior on (b 1, b 2 ) as a result of (3.17), or inother words (see Fig. 1)0 < c ∗ 1 (c 2)
STOCHASTIC CAPITAL ACCUMULATION GAMES 129[b 1 , b 2 ], (3.17) implies that c1 ∗(c 2)+c 2 is a constant, say C. We now claim that theintegral term in (3.19) is differentiable at x − c1 ∗(c 2) − c 2 = x − C. Suppose not.Then, from (3.19), we conclude that Vv x(c2) − u[c1 ∗(c 2)] is not differentiable atany c 2 ∈ [b 1 , b 2 ], a contradiction, since both Vv x(·)and u[c∗ 1(·)] are differentiablefor almost all c 2 , the latter being so in view of (3.17). Therefore, invoking also(3.18), the following first order condition must hold, for almost all c 2 ∈ [b 1 , b 2 ],for the maximization in (3.13a),{∫}∣u ′ 1 [c∗ 1 (c d∣∣∣z=x−c2)] = δ 1 v 1 (x ′ )dF(x ′ | z). (3.20)dz∗1 (c 2)−c 2Let ¯c 2 and ĉ 2 be two distinct points in [b 1 , b 2 ] for which (3.20) holds. Sincec1 ∗(¯c 2) +¯c 2 =c1 ∗(ĉ 2)+ĉ 2 =C, the RHS of (3.20) is the same when c 2 =ĉ 2 aswhen c 2 =¯c 2 . Hence, u ′ 1 [c∗ 1 (¯c 2)] = u ′ 1 [c∗ 1 (ĉ 2)], or, in view of the monotonicityof u ′ 1 , c∗ 1 (¯c 2) = c1 ∗(ĉ 2), a contradiction to (3.17). Hence there cannot be morethan one equilibrium for Gv x , and this completes the proof of Lemma 2.3.Now we are ready for theProof of Proposition 2. We present an argument by induction on n. Forn=0, i.e., for the one-period game G o , Player i solves the following optimizationproblem max{u i (c i ): c i ∈ [0, K i (x)]}. Therefore, the unique equilibriumconsumption pair is (K 1 (x), K 2 (x)), with corresponding value functions(V1 1,V 2 1)= (u 1[K 1 (x)], u 2 [K 2 (x)]).Then, the two-period problem (or n = 1) has payoffs given by (3.12) withv i replaced by Vi 1 , i = 1, 2. The action spaces are, of course, independent ofn. Since Vi1 ∈ CM i , Lemma 2.1 guarantees the existence of an equilibriumin LCM 1 × LCM 2 for the game G 1 = G V 1. But, by Lemma 2.3 we haveuniqueness of the equilibrium actions of the game G x , for every x ∈ [0, +∞).V 1Therefore G V 1 has a unique equilibrium strategy pair, and furthermore, this pairis in LCM 1 × LCM 2 .Let the equilibrium payoffs of G 1 be (V1 2,V 2 2 ) and repeat the above argumentfor n = 2, 3,.... This process clearly yields the desired conclusion.REFERENCESAmir, R. (1989). “On the Continuity of the Best-Response Map in Some Dynamic Games,” unpublishednote.Amir, R. (1990). “Strategic Common Property Capital Accumulation: Existence of Nash Equilibria,”preprint.Bartle, R. (1976). The Elements of Real Analysis. New York: Wiley.Basar, T., and Olsder, G. J. (1982). Dynamic Non-cooperative Game Theory. New York: AcademicPress.
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