Cournot Oligopoly and the Theory of Supermodular Games
Cournot Oligopoly and the Theory of Supermodular Games
Cournot Oligopoly and the Theory of Supermodular Games
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118 RABAH AMIRprominent setting <strong>of</strong> noncooperative common-property resource extraction, AssumptionA.3 is undoubtedly appropriate <strong>and</strong> is easily interpreted in terms <strong>of</strong>capital <strong>and</strong> labor (bounded) capacities.In <strong>the</strong> following, it is convenient to refer to <strong>the</strong> (n + 1)-period horizon game(including <strong>the</strong> 0th period) as G n , with G ∞ being <strong>the</strong> infinite horizon game.Ano<strong>the</strong>r related (auxiliary game) is defined in Section 3.We are now in a position to state <strong>the</strong>MAIN THEOREM. The infinite-horizon game G ∞ has a stationary equilibrium.Fur<strong>the</strong>rmore, every finite-horizon game G n has a unique Markovian equilibrium.In both cases, equilibrium strategies consist <strong>of</strong> (consumption) decisionfunctions which are continuous <strong>and</strong> nondecreasing <strong>and</strong> have all slopes below 1,<strong>and</strong> corresponding value functions are continuous <strong>and</strong> nondecreasing.3. PROOFSThe pro<strong>of</strong> <strong>of</strong> <strong>the</strong> Main Theorem proceeds via two intermediate propositions,each <strong>of</strong> which requires, in turn, a few lemmas. Many <strong>of</strong> <strong>the</strong>se results are <strong>of</strong>independent interest. Note that we assume <strong>the</strong> reader is familiar with Topkis’<strong>the</strong>orem (Topkis, 1978).We start with two lemmas which establish useful facts invoked repeatedlythroughout our analysis.LEMMA 0.1. Let F 1 <strong>and</strong> F 2 be probability distributions over [0, +∞), withits Borel subsets. Then F 2 first-order stochastically dominates F 1 , or∫ ∫F 1 (x) ≥ F 2 (x), ∀x if <strong>and</strong> only if v dF 1 ≤ vdF 2 ,for all real-valued nondecreasing functions v on [0, +∞).Pro<strong>of</strong>. This result is well known. See, e.g., Stoyan (1983).LEMMA 0.2. Let U: [0,+∞) → [0, +∞) be a bounded concave function.Let h: [0,+∞) → [0, +∞) satisfy h(0) = 0 <strong>and</strong> [h(x 1 ) − h(x 2 )]/(x 1 −x 2 ) ≤ 1, for all distinct x 1 , x 2 ∈ [0, +∞). Then (i) <strong>the</strong> set ϕ ={(x,y):x∈[0, +∞) <strong>and</strong> y ∈ [0, x − h(x)]} is a lattice, <strong>and</strong> (ii) U[x − y − h(x)] is supermodularon ϕ.Pro<strong>of</strong>. (i) This follows directly from <strong>the</strong> fact that x − h(x) is nondecreasingin x. (ii) We show <strong>the</strong> equivalent fact that U[x − y − h(x)] has nondecreasingdifferences in (x, y) ∈ ϕ. Let x 2 ≥ x 1 , <strong>and</strong> y 2 ≥ y 1 be fixed with 0 ≤ y i ≤x i − h(x i ), i = 1, 2. Then, clearly for i = 1, 2, we havex 2 − y 1 − h(x 2 ) ≥ x i − y i − h(x i ) ≥ x 1 − y 2 − h(x 1 ),