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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 24 Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Monopoly (Ch. 24, 25) Oligopoly (Ch. 27) Cournot Oligopoly N firms, facing downward sloping demand p./. Firm i has cost function Ci . A Cournot equilibrium is an output vector .x1 ; x2 ; : : : ; xN / such that for each firm i D 1; : : : ; N , output xi solves xi max p 0 X j i xj C xi xi Ci .xi / First-order conditions: p0 N X j D1 xi xi C p N X j D1 xi D Ci0.xi/ Cournot Oligopoly Suppose all firms have the same cost function C . Then, the first-order conditions are p0 N X j D1 xi xi C p N X j D1 xi D C 0.xi/ In this case, the unique equilibrium is symmetric: x1 D : : : D xN D x Plugging into the first-order conditions p 0.N x /x C p.N x / D C 0.x / Cournot Oligopoly Linear demand case: p.Q/ D a bQ; C.x/ D F C cx First-order condition: bx C a bN x D M C ) a MC x D b.N C 1/ Total output and price: .a M C /N Q D Nx D ; b.N C 1/ pDa .a M C /N a N D CM C N C1 N C1 N C1 Cournot Oligopoly .a M C /N QD b.N C 1/ As N ! 1, a N and p D C MC N C1 N C1 MC b Q! a and p ! M C: That is, outcome approximates perfect competition when number of firms is very large. Optimal Collusion col col Cartel chooses .x1 ; : : : ; xN / to maximize joint profit: N X i D1 N X i D1 N X i D1 x1 ;:::;xn max p xi xi Ci .xi / F.O.C.: 0 N X i D1 N X col i D1 p xi xi col Cp N X i D1 xicol D Ci0.xicol/; all i: Instability of Cartels Suppose all firms produce according to the optimal collusive agreement. Does firm i want to follow that agreement? Or does it want to cheat on the other firms? col col Consider the marginal revenue of firm i when all firms follow .x1 ; : : : ; xN /: X X col xicol/ p. xi /xi C p. 0 i i We will show that this is greater than Ci .xicol /. Instability of Cartels Let us show that, at the collusive outcome, the marginal revenue exceeds marginal cost. We have: N X N X p 0 xj j D1 col , xicol Cp col xj j D1 ...,, N N N N X X X X col X col 0 col col 0 col xj p xj xj xj C p xj Dp j D1 j D1 j D1 j D1 j i <0 ...,, N X X col 0 col xj > Ci0.xicol/; p xj DCi0 .xi /; by the FOC of the cartel D Ci0.xicol/ as was to be shown. , j D1 j i Instability of Cartels This implies that firm i can increase profits by increasing xi , because MRi > M Ci . In particular, firm i does not want to stick to the collusive agreement of the cartel. Bottom line: absent an external enforcement mechanism, the cartel breaks down. Effect of Long-run Relationships We have assumed that the firms compete only in one period, and then they "die." But, in reality, firms compete with each other over and over again, over a long period of time. Flow of profits of a firm: 1 ; 2 ; : : : ; t ; : : : where t D profit in period t Firm's discounted profit / present value: 1 C i2 C i 23 C C i t 1 t C where iD and r is the interest rate. 1 ; 1Cr Effect of Long-run Relationships The long-term interaction between the firms makes collusion possible, even without any external enforcement mechanism. Collusion becomes self-enforcing. Here are (dynamic) strategies for the members of the cartel which will give rise to collusion: In period t D 1, producing collusive output. In any period t > 1, consider the history of output up to time t 1: if so far all firms have produced the collusive output, then produce the collusive output; otherwise, produce the Cournot output. Thus, if any firm ever cheats, the Cartel switches to the Cournot output forever after. Effect of Long-run Relationship Let us check that these strategies are self-enforcing. Notation: ...Cournot D static Cournot profit ...col D static collusive profit ...cheat D static profit of a firm that unilaterally cheats on the cartel Thus, ...cheat > ...col > ...Cournot: Effect of Long-run Relationship Suppose all firms but firm i follow the specified dynamic strategies. If firm i follows that strategy, then it gets a profit of ...col in every period. This yields a present value of ...col D ...col .1 C i C i 2 C / 1 i If instead, the firm cheats on the cartel, then for one period it enjoys the profit ...cheat, but then, in all subsequent periods, the static Cournot output obtains. Thus, the deviating firm gets a present value of: cheat ... ...Cournot Ci D ...cheat C ...Cournot.i C i 2 C / 1 i Effect of Long-run Relationship Thus, the dynamic strategies are self-enforcing if the firm i prefers to follow the strategy than to deviate from it: ...col 1 i that is, ... cheat ...Cournot Ci ; 1 i i or equivalently, ...cheat ...col ; cheat Cournot ... ... ...cheat ...Cournot ...cheat ...col 1 r Effect of Long-run Relationship Punch line: If firms are sufficiently patient, then they can use dynamic strategies to create intertemporal incentives that sustain the collusive behavior. The strategies involve an implicit threat of perpetual reversion to the static Cournot equilibrium. This threat is never carried out, but its possibility is precisely what makes collusion possible. Bertrand Competition 2 identical firms, price setting game, constant marginal cost c 2 .0; 1/. Demand: 8 ^ 1 if pi < pj < Di D 0 if pi > pj ^ : 1=2 if p D p i j .Di .pi ; pj / c/pi Profit of firm i : Bertrand equilibrium: .p1 ; p2 / such that pi solves max .Di .pi ; pj / pi c/pi Bertrand Competition "Best"-reply of firm i : 8 ^ pj " < pi D c ^ : anything strictly above p where " is positive but arbitrarily small. j if if if pj > c pj D c pj < c Thus, p1 D p2 D c is the unique Bertrand equilibrium. New Topic: Externalities Consider an economy with 2 consumption goods (fish and manufactured good), 2 firms (fishery and manufacturer) and the representative consumer. The representative consumer has utility function u.xf ; xm/: The production function of the firm that produces the manufactured good is: ym D F m.`m; x/ where `m is the labor input and x is the amount of pollution generated. Externalities The production function of the fishery is: yf D F f .`f ; x/ where `f is the labor input and x is the amount of pollution generated by the other firm. We assume that @F f < 0: @x so that pollution is bad for the fishery. We also assume @F m m @F m m .` ; 0/ > 0 > .` ; x/ N @x @x for some x sufficiently high. N Externalities Pareto efficiency requires: max u F .`; x/; F .L `;x f m `; x/ F.O.C.: M Uf MP` f M Um MP`m D 0 ) ) MRS D MRT @F m=@x MRS D @F f =@x @F f @F m M Uf C M Um D0 @x @x This implies that at a Pareto efficient allocation @F m=@x > 0: Externalities In a competitive equilibrium, we have MRS D f pf pm w pf w MP`m D pm @F m=@x D 0 MP` D Thus, MRS D MRT but @F m=@x MRS @F f =@x Externalities Pigouvean tax: must pay t per unit of pollution x Competitive equilibrium: pf MRS D pm w f MP` D pf w m MP` D pm t m @F =@x D m p If set t D p f @F f =@x then we get Pareto efficiency. Caveat: To set t , policy maker needs to have precise information about magnitude of the externality. ...
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This note was uploaded on 03/10/2012 for the course ECON 121 at Yale.

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