Oligopoly Theory (2) Quantity-Setting Competition Aim of this lecture (1) To understand the concept of quantity-setting competition (2) To understand the ideas of strategic substitutes and complements (3) To understand the relationship between the stability of Cournot equilibrium and comparative statistics Ologopoly Theory

Outline of the Second Lecture 2-1 Monopoly 2-2 Price-Setting or Quantity-Setting 2-3 Cournot Model 2-4 Strategic Complement and Strategic Substitute 2-5 Stability Condition 2-6 Stability Condition and Comparative Statistics 2-7 Stability Condition and Uniqueness of the Equilibrium 2-8 Cournot Limit Theorem and Perfect Competition Ologopoly Theory

Monopoly Producer P D MC MR Y Ologopoly Theory

Equilibrium of Monopoly Producer superscript denotes equilibrium values of monopoly P D PM MC MR superscript Mはmonopolyでの均衡を表す Y YM Ologopoly Theory

Marshallian View of the Market Monopolist chooses its output and the price is determined by the market clearing condition P D PM MC MR 独占企業が生産量を決めそれに対応する価格が市場で決まる Y YM Ologopoly Theory

Walrasian View of the Market P Monopolist chooses its output and the price is determined by the market clearing condition D PM MC MR 独占企業が生産量を決めそれに対応する価格が市場で決まる Y YM Ologopoly Theory

Why does not the monopolist choose both quantity and price ？ MR P D PM MC Y YM1 Ologopoly Theory

Why does not the monopolist choose both quantity and price ？ MR P D MC PM1 Y YM Ologopoly Theory

Duopoly Suppose that there are two or more firms in the market ～The price depends on both its own output and the rivals' outputs. ～The output depends on both its own price and the rivals' prices. ⇒The competition structure depends on whether firms chooses their outputs or prices. Quantity Competition Model (The second lecture) Price Competition Model (The third lecture) Which model should we use ？(The third lecture) Ologopoly Theory

Cournot Duopoly Firm 1 and firm 2 compete in a homogeneous product market. Each firm i independently chooses its price Yi. Each firm maximizes its own profit Πi. Πi＝P(Y)YiーCi(Yi), P: Inverse demand function, Y: Total output, Yi: Firm i's output, Ci: Firm i's cost function P' <0, C' >0, C'' ≧0 (Henceforth, I assume these conditions unless I explicitly make contradicting assumptions.) Ologopoly Theory

reaction function Reaction function of firm 1~R1(Y2)：Given the output of firm 2,Y2, Y1=R1(Y2) implies that Y1 is the optimal output for firm 1. The first order condition P+P‘Y1 =C1' ⇒ R1(Y2) is derived form this The second order condition 2P'+P''Y1 - C1''＜０ Henceforth, I assume the second order condition conditions unless I explicitly make contradicting assumptions. Ologopoly Theory

Cournot Equilibrium Nash Equilibrium of the Cournot Model Derivation of the Cournot Equilibrium Solving P+P'Ｙ1 =C1' , P+P'Ｙ２ =C２' Ologopoly Theory

Residual Demand residual demand P D MR MC Y2 Y Y1 Ologopoly Theory

Derivation of reaction function P residual demand D MR MC Y2’ Ｙ Y1’ Ologopoly Theory

reaction curve of firm 1 Y1 reaction curve of firm 1 Y2 Y2 Ologopoly Theory

slope of the reaction curve P+P'Y1 -C1'=0→dY1/dY2= - (P'+P''Ｙ1)/(2P'+P''Y1 - C1'') P'+P''Y1>0⇒upwprd sloping of the reaction curve (strategic complements) ～an increase in the rival's output increases the marginal revenue of the firm: unnatural in the context of Cournot competition, but it is possible. P'+P''Y1<0⇒downward sloping of the reaction curve (strategic substitutes) ～ an increase in the rival's output reduces the marginal revenue of the firm In this course, I assume P'+P''Y1<0 unless I make explicit contradicting assumptions. Ologopoly Theory

Reaction Curve of Firm 2 Y2 Y1 Ologopoly Theory

Reaction Curve of Firm 1 Y1 Y2 Ologopoly Theory

Cournot Equilibrium superscript C denotes Cournot Equilibrium reaction curve of firm 1 Y2 reaction curve of firm 2 Y2C Y1 Y1C superscript C denotes Cournot Equilibrium Ologopoly Theory

Existence of the Equilibrium From the definitions of the reaction function and the Cournot equilibrium, we have R1(R2(Y1C)) =Y1C , 　R2(R1(Y2C)) =Y2C We can use the fixed point theorem to show the existence of the Cournot equilibrium. Substituting Y1≠Y1C into the above system →The discussions on the stability and the uniqueness of the equilibrium. Ologopoly Theory

Stability |R1(R2(Y1))－Y1C| ＜ |Y1 －Y1C | |R2(R1(Y2))－Y2C| ＜ |Y2 －Y2C | ~Starting form the non-equilibrium point and consider the best reply dynamics →The distance form the equilibrium point is decreasing ⇒Cournot equilibrium is stable. Ologopoly Theory

Stable Cournot Equilibrium Reaction Curve of Firm 1 Ｙ２ Reaction Curve of Firm 2 Ｙ２C1 Ｙ２C Y１C2 Y1 Y１C1 Y１C Ologopoly Theory

Unstable Cournot Equilibrium Reaction Curve of Firm 2 Ｙ２ Reaction Curve of Firm 2 Ｙ２C1 Ｙ２C Y１C１ Y１ Y１C2 Y１C Ologopoly Theory

A Sufficient Condition for the Stability of the Cournot equilibrium |Ri'| ＜ 1　The absolute value of the reaction curve is smaller than one. ~On unit increase of the rival's output changes the optimal output of the firm less than one unit. Ologopoly Theory

Why do we often assume the stability condition? Cournot Model is a One-Shot Game. It seems nonsense to discuss the dynamic adjustment. However, most IO papers assume this condition. Why? (1) This condition is plausible since it is satisfied under standard settings of cost and demand conditions. (2) evolution, learning (3) for comparative statistics (4) uniqueness Ologopoly Theory

Stabilityの条件の意味 (1) 通常の設定ではこの条件は満たされるから 反応曲線の傾き dY1/dY2= - (P'+P''Ｙ1)/(2P'+P''Ｙ1 - C1'') 戦略的代替（P'+P''Ｙ1<0) , C1''≧0 →-1 <R1'<0 同質財でなかったり、限界費用が逓減していたりすれば戦略的代替のケースでも安定性の条件を満たさないことはある。いずれにせよ、この分野でよく使われる標準的な設定では安定性の条件は満たされる事が多い。 Ologopoly Theory

Stabilityの条件の意味 (3)比較静学をするため ・通常満たされないような変な条件下で通常と逆のcounter intuitiveな結果を出してもしょうがない Ologopoly Theory

Question: Suppose that firm 2's marginal cost is constant Question: Suppose that firm 2's marginal cost is constant. Consider the effect of a reduction of firm 2's marginal cost on firm 2's reaction curve. Y2 Y1 Ologopoly Theory

企業２の費用と反応曲線 反応曲線は上方にシフト Y2 反応曲線は上方にシフト Y1 Ologopoly Theory

企業2の費用とCournot均衡 Y1 ポイント：企業2の費用が低下すると企業２の生産量が増え 企業１の生産量が減る。 Y2 企業１の 反応曲線 Y1 Ologopoly Theory

Unstable Cournot Equilibrium 均衡が(局所的にでも) unstableな状況だとcounter intuitiveな結果が出てくる。→普通の需要関数、費用関数では出てこない状況を考えてcounter intuitiveな結果を出してもしょうがない。 Y2 企業１の 反応曲線 Y1 Ologopoly Theory

Stabilityの条件の意味 (4) 均衡のuniqueness 安定性の条件が大域的に満たされている →前述の対応を考えることによって、Contraction Mapping Theoremがそのまま使える →均衡の一意性が保証される。 (Remark) 安定性の条件はあくまで十分条件であって必要条件ではない Ologopoly Theory

Stable Cournot Equilibrium Y2 reaction curve of firm 1 reaction curve of firm 2 Y2C Y1 Y1C Ologopoly Theory

Unstable Cournot Equilibrium Reaction Curve of Firm 2 Y2 Reaction Curve of Firm 1 Y2C Cournot均衡３つ、内点は不安定 Y1 Y1C Ologopoly Theory

Unstable Cournot Equilibrium 前ページの例 Cournot均衡３つ、内点は不安定。この内点に注目して比較静学する意味あるか？ このような無意味な議論を避けるために安定性の条件を課すことが多い。 Ologopoly Theory

問題：企業２の利潤はどの点が一番大きいか？ Y2 Y1 Ologopoly Theory

企業２の生産量と利潤 Y2 より利潤が大きい Y1 Ologopoly Theory

問題:企業2の利潤はどちらの方が大きいか？ Y2 Y1 Ologopoly Theory

企業２の利潤 こちらがより利潤が大きい Y2 Y1 Ologopoly Theory

企業２の利潤 等利得曲線 Y2 Y1 Ologopoly Theory

Cournot均衡と効率性 企業1の 反応曲線 Y2 企業1の 等利潤曲線 企業２の 等利潤曲線 Y2* 企業2の 反応曲線 Y1* Y1 Y1* Y1 両企業の利潤が 増加する領域 Ologopoly Theory

Welfare Implications ・各企業はライバルの利潤に与える影響を考えないで生産量を決定 →Cournot均衡での生産量は企業部門全体の利潤最大化の観点からは過大生産 ・各企業の限界費用は価格を下回る。 →Cournot均衡での生産量は総余剰最大化の観点からは過小生産 Ologopoly Theory

Cournot Oligopoly 企業１、企業２、．．．企業ｎが同質財市場で競争 企業２は同時に独立に各自の生産量を決定 各企業の利得は自社の利潤 Ologopoly Theory

Cournot Equilibrium Cournot均衡の導出 P+P'Ｙ1 =C1' 、 P+P'Ｙ２ =C２'、．．． P+P'Ｙn =Cn'の連立方程式を解くだけ 全ての企業がsymmetricなら P+P'Ｙ1 =C1'、Y=nＹ1 の連立方程式から対象均衡を導出できる Ologopoly Theory

Symmetric Equilibrium 企業1の反応曲線 n-1 n-1 Y1 Y1* Ologopoly Theory

Cournot Limit Theorem 企業1の一階条件 P+P'Ｙ1 =C1' P(1+P' Y/P ・Ｙ1/Y)=C1' Ｙ1/Y→０ 　P → C1'　(Cournotの極限定理の世界) Cournotの極限定理　企業数が十分大きくなれば価格は限界費用に近づく Ologopoly Theory

Marginal Revenue for Small Firms P MR →P if Y is sufficiently small residual demand MR 生産量がゼロに近づくと限界収入が価格と等しくなる Y Ologopoly Theory

perfect competition 価格受容者：価格を与えられた者として行動する者 自分が生産量を増やしても価格が変わらないと思い込んでいる者 実際には需要の価格弾力性が無限大でない限り価格は変化する。変化の程度はその企業が大きかろうと小さかろうと同じ。 「価格受容者＝価格に影響を与えられないほど小さな事業者」という説明は変。大きさにかかわりなく価格は変化する⇒完全競争というのはフィクション Ologopoly Theory

Cournotの極限定理 Cournotの極限定理 企業数が十分大きくなれば価格は限界費用に近づく 完全競争は現実の近似。 「企業が十分小さい⇒価格受容者として近似できる」 Ologopoly Theory

strategic complements 企業1の 反応曲線 Y2 Cournot 均衡 企業2の 反応曲線 Y2* 数量競争の文脈ではstrategic complementsになることはあまり普通ではないがあり得なくはない strategic complementsになることはあまり普通ではないがあり得なくはない Y1* Y1 Ologopoly Theory

strategic complements 企業1の 等利潤曲線 企業1の 反応曲線 Y2 企業2の 反応曲線 企業2の 等利潤曲線 Y2* Y1* Y1 両企業の利潤が 増加する領域 Ologopoly Theory