Imperfect Competition


 Erik Craig
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1 Imperfect Competition Lecture 5, ECON 4240 Spring 2018 Snyder et al. (2015), chapter 15 University of Oslo iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
2 Outline Our general equilibrium analysis assumed firms are pricetakers (many firms) Now we abandon this assumption: limited number of firms that interact strategically Remainder of class Next: give up assumption of private goods Second half of the course: give up assumption of complete information iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
3 Oligopoly  Short Run Decisions Definition 1 An oligopoly is a market with relatively few firms, but more than one. Firms act strategically. Our focus: Firms shortterm decisions in oligopolistic markets: choice of price and/or quantity ( longrun decisions = entry, investment, research and development, advertising are the topic of ECON 4820 on Strategic Competition ) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
4 Oligopoly  Short Run Decisions Definition 1 An oligopoly is a market with relatively few firms, but more than one. Firms act strategically. Our focus: Firms shortterm decisions in oligopolistic markets: choice of price and/or quantity ( longrun decisions = entry, investment, research and development, advertising are the topic of ECON 4820 on Strategic Competition ) Bertrand model: Firm s set prices Cournot model: Firm s set quantities Cartels: Firms collude (somewhat different) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
5 The Bertrand Model Setting 2 identical firms, called 1 and 2: identical costs identical products (Same results with n firms, but more striking with just 2 firms) Marginal costs or production c are constant equal to the average cost Many identical consumers Demand strictly decreasing iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
6 The Bertrand Model Setting 2 identical firms, called 1 and 2: identical costs identical products (Same results with n firms, but more striking with just 2 firms) Marginal costs or production c are constant equal to the average cost Many identical consumers Demand strictly decreasing... and what makes it a Bertrand competition: Firms choose their prices p 1 and p 2 simultaneously Sales go to firm with the lowest price, and are split evenly if p 1 =p 2 iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
7 Solution Concept The firms act strategically so we face a game. A game is characterized by iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
8 Solution Concept The firms act strategically so we face a game. A game is characterized by Players  here: 1 and 2 Strategies  here: player i sets price p i Note: Continuous control Payoffs  here: profits (outcomes: strategies decide outcomes which map to payoffs) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
9 Solution Concept The firms act strategically so we face a game. A game is characterized by Players  here: 1 and 2 Strategies  here: player i sets price p i Note: Continuous control Payoffs  here: profits (outcomes: strategies decide outcomes which map to payoffs) Our solution concept for the game will be the Nash equilibrium. So no longer the Walrasian equilibrium. Say n players: iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
10 Solution Concept The firms act strategically so we face a game. A game is characterized by Players  here: 1 and 2 Strategies  here: player i sets price p i Note: Continuous control Payoffs  here: profits (outcomes: strategies decide outcomes which map to payoffs) Our solution concept for the game will be the Nash equilibrium. So no longer the Walrasian equilibrium. Say n players: Definition 2 A strategy profile S =(s 1,s 2,...,s n) constitutes a Nash equilibrium if for all i {1,...,n} the strategy s i is player i s best response to the rival s equilibrium strategies S i. iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
11 Nash Equilibrium of the Bertrand Game Claim: The only (purestrategy) Nash equilibrium of the Bertrand game is p 1 =p 2 =c Nash equilibrium in this context: p 1 =p 1 is the best response of firm 1 to p 2 =p 2. At the same time, p 2 =p 2 is the best response of firm 2 to p 1 =p 1 iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
12 Nash Equilibrium of the Bertrand Game Claim: The only (purestrategy) Nash equilibrium of the Bertrand game is p 1 =p 2 =c Nash equilibrium in this context: p 1 =p1 is the best response of firm 1 to p 2 =p2. At the same time, p 2 =p2 is the best response of firm 2 to p 1 =p1 1) p1 =p 2 =c is a Nash equilibrium: Firm 1 has no profitable deviation: At p2 =c, setting p 1 =c implies profits of 0 For p1 >c firm 1 does not sell anything (profit also 0) For p1 <c firm 1 sells to all consumers, but it incurs a loss for every unit sold a symmetric reasoning holds for firm 2 iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
13 Nash Equilibrium of the Bertrand Game 2) p1 =p 2 =c is THE ONLY Nash equilibrium: Without loss of generality (WLOG) assume p 2 p 1 (why is that WLOG?) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
14 Nash Equilibrium of the Bertrand Game 2) p1 =p 2 =c is THE ONLY Nash equilibrium: Without loss of generality (WLOG) assume p 2 p 1 (why is that WLOG?) If p 1 >c, If p 1 <c, If p 1 =c and p 2 >p 1, So only p 1 =p 2 =c can be a Nash equilibrium (in pure strategies, but also holds in mixed strategies if p such that demand is zero above p) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
15 Nash Equilibrium of the Bertrand Game 2) p1 =p 2 =c is THE ONLY Nash equilibrium: Without loss of generality (WLOG) assume p 2 p 1 (why is that WLOG?) If p 1 >c, then firm 2 would be better off choosing p 1 ε for some small enough ε rather than p 2 p 1 (and get to satisfy the full market demand) If p 1 <c, If p 1 =c and p 2 >p 1, So only p 1 =p 2 =c can be a Nash equilibrium (in pure strategies, but also holds in mixed strategies if p such that demand is zero above p) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
16 Nash Equilibrium of the Bertrand Game 2) p1 =p 2 =c is THE ONLY Nash equilibrium: Without loss of generality (WLOG) assume p 2 p 1 (why is that WLOG?) If p 1 >c, then firm 2 would be better off choosing p 1 ε for some small enough ε rather than p 2 p 1 (and get to satisfy the full market demand) If p 1 <c, then firm 1 incurs losses on each unit sold (p 1 c < 0), and firm 1 would be better off charging a price strictly larger than p 2 (selling nothing) If p 1 =c and p 2 >p 1, So only p 1 =p 2 =c can be a Nash equilibrium (in pure strategies, but also holds in mixed strategies if p such that demand is zero above p) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
17 Nash Equilibrium of the Bertrand Game 2) p1 =p 2 =c is THE ONLY Nash equilibrium: Without loss of generality (WLOG) assume p 2 p 1 (why is that WLOG?) If p 1 >c, then firm 2 would be better off choosing p 1 ε for some small enough ε rather than p 2 p 1 (and get to satisfy the full market demand) If p 1 <c, then firm 1 incurs losses on each unit sold (p 1 c < 0), and firm 1 would be better off charging a price strictly larger than p 2 (selling nothing) If p 1 =c and p 2 >p 1, then firm 1 would be better off increasing the price a little So only p 1 =p 2 =c can be a Nash equilibrium (in pure strategies, but also holds in mixed strategies if p such that demand is zero above p) iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
18 Bertrand Paradox The paradox Bertrand model suggests that in a market with only 2 firms we should expect same allocations as under perfect competition (setting prices equal to marginal costs) Why is that an issue? iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
19 Bertrand Paradox The paradox Bertrand model suggests that in a market with only 2 firms we should expect same allocations as under perfect competition (setting prices equal to marginal costs) Why is that an issue? Not because of social welfare outcome (nice!), but Because it seems descriptively inaccurate iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
20 The Cournot Model  2 firms Setting Similar to Bertrand, except and what makes it a Cournot competition: Firms choose their output quantities q 1 and q 2 simultaneously Price is given by demand at Q :=q 1 +q 2 We solved Bertrand competition by analyzing different cases because the strategic response of each firm was discontinuous. Cournot is better behaved and we calculate the optimal response. iacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
21 The Cournot Model  2 firms Notation (and an assumption): Let P(Q) be the inverse of the demand function (demand fct strictly decreasing invertible, and P (Q)<0) Profits: π i =(P(Q) c)q i for i = 1,2 Nash equilibrium in this context: acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
22 The Cournot Model  2 firms Notation (and an assumption): Let P(Q) be the inverse of the demand function (demand fct strictly decreasing invertible, and P (Q)<0) Profits: π i =(P(Q) c)q i for i = 1,2 Nash equilibrium in this context: each q i maximizes π i for given q i (where i is the other firm ) firstordercondition (FOC): π i q i =P(Q)+P (Q)q i =c secondordercondition (SOC): 2 π i 2 q i = 2P (Q)+P (Q)q i < 0 assume here that the (SOC) holds acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
23 The Cournot Model  2 firms Interpretation: P(Q)+P (Q)q i = marginal revenue c= marginal cost As usual marginal revenue=marginal cost BUT: Marginal revenue accounts for price and resulting revenue decrease when increasing output by one more unit acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
24 The Cournot Model  2 firms Interpretation: P(Q)+P (Q)q i = marginal revenue c= marginal cost As usual marginal revenue=marginal cost BUT: Marginal revenue accounts for price and resulting revenue decrease when increasing output by one more unit Note: As P (Q)q i < 0, we have P(Q)>c, unlike Bertrand or Walrasian equilibrium So price too high and quantity too low as compared to welfare maximizing allocation acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
25 Excursion: The Cartel Solution  2 firms As before: Let P(Q) be the inverse of the demand function (demand fct strictly decreasing invertible, and P (Q)<0) Profits: π i =(P(Q) c)q i for i = 1,2 In a cartel the two firms maximize joint profits: (so act essentially as a monopoly) firm s maximize together Π=π 1 + π 2 firstordercondition (FOC): π q i =P(Q)+P (Q)q i +P (Q)q i =c π q i =P(Q)+P (Q)Q =c secondordercondition assumed to hold acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
26 Excursion: The Cartel Solution  2 firms As before: Let P(Q) be the inverse of the demand function (demand fct strictly decreasing invertible, and P (Q)<0) Profits: π i =(P(Q) c)q i for i = 1,2 In a cartel the two firms maximize joint profits: (so act essentially as a monopoly) firm s maximize together Π=π 1 + π 2 firstordercondition (FOC): π q i =P(Q)+P (Q)q i +P (Q)q i =c π q i =P(Q)+P (Q)Q =c secondordercondition assumed to hold Wedge between price and marginal costs P (Q)Q takes into account effect of changing q i on revenue loss for all firms acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
27 The Cournot Model  n firms Let s extend the Cournot model to n firms: We focus on symmetric equilibrium: q i = Q n for all i = 1,..,n Derivation almost the same Equilibrium: firstordercondition: P(Q)+P (Q) Q n =c The wedge between price and marginal cost is P (Q) Q n, which is decreasing in n: for very large n: Cournot gives same prediction as perfect competition for n=1 Cournot is equal to the monopoly Unlike Bertrand, Cournot competition results in a continuous interpolation from monopoly to perfect competition. acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23
28 Prices or Quantities? Difference between Bertrand and Cournot: In Bertrand, starting from equal prices, a small reduction in price allows a firm to steal the entire market: competition is very intense In Cournot, starting from equal quantities, a small reduction or increase in quantity of one firm has only a marginal effect on price and market shares: competition is softer Realism: Cournot quantity mechanism seems less realistic: In real world firms set prices. Yet: In real world Bertrand outcome seems unrealistic: Markets with just two firms do not usually lead to Walrasian equilibria acquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring /23