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OLIGOPOLY Cournot Game In a Cournot game, each firm chooses its output level taking as given all other firms’ output levels. Model with 2 firms - 2 firms, index by i = 1, 2, producing homogenous good with output y1 and y2 and cost c1(y1) and c2(y2) - Aggregate output Y = y1 + y2 - Inverse demand p(Y) = p(y1 + y2) Firm i's maximization problem is Max Пi(yi, yj) = p(yi + yj) yi – ci(yi) i = 1, 2, i ≠ j yi Firm i's profit depends on the amount of output chosen by j. Hence, firm i must forecast firm j's output decision. A (pure-strategy) Cournot Nash equilibrium is a set of output (y1*, y2*) in which each firm is choosing its profit maximizing output level given its beliefs about the other firm’s choice and the beliefs are correct. Assuming interior optimum, y1 > 0 and y2 > 0 Solution to firm i's maximization F.O.C.: S.O.C.: ( ∂ Π i yi , y j ∂ yi ( ∂ Π i2 yi , y j 2 ∂ yi ) ) = 0 i = 1, 2, i ≠ j ≤0 i = 1, 2, i ≠ j From F.O.C.s, we get each firm reaction curve (best response function). Reaction curve gives firm i's optimal output as a function of its beliefs firm j's output choice. Denote the reaction curve of firm i by fi (yj), from F.O.C., we write ( ∂ Πi f i ( y j ), y j ∂ yi ) ≡0 To see how i changes its optimal output level as its belief about firm j's output changes, differentiate the above w.r.t. to yj ( ∂ Π 2i yi , y j ∂ y 2i ) f i′ (yj) + 2 ( ( ∂ Π 2i yi , y j ∂ yi ∂ yj Π i yi , y j f i′ (yj) = 2 ( ) =0 ) Π i yi , y j yi ∂ yj ) y 2i Denominator is negative due to S.O.C. Numerator is negative if inverse demand curve is downward (Y ) + p′ (Y ) yi < 0 sloping and not too convex such that p′ (yj) < 0 → fi′ If the reaction curves are downward sloping, players’ strategies are said to be strategic substitutes. If the reaction curves are upward sloping, players’ strategies are said to be strategic complements. System of differential equations dy1 = g1 ( y1 , y 2 ) dt dy2 = g 2 ( y1 , y 2 ) dt has a stable equilibrium if ∂ g1 ∂ y1 ∂ g2 ∂ y1 ∂ g1 ∂ y2 > 0 ∂ g2 ∂ y2 and ∂ g1 ∂ g2 + < 0 ∂ y1 ∂ y2 Suppose the firms adjust their outputs in the direction of increasing profits dy1 ∂ Π1 ( y1 , y 2 ) = α1[ ] dt ∂ y1 dy2 ∂ Π 2 ( y1 , y 2 ) = α2[ ] dt ∂ y2 Then the equilibrium is stable if Comparative Statics We want to see how optimal choice (say of firm 1) is affected if some parameter changes. Suppose that a is a parameter that shifts П1 ∂ Π1 ( y1 ( a ), y 2 ( a ), a ) ≡0 ∂ y1 ∂ Π 2 ( y1 ( a ), y 2 ( a ) ) ≡0 ∂ y2 by F.O.C.s Differentiating with respect to a and using the Cramer’s rule ∂ y1 to solve for ∂ , we get a Several Firms Suppose there are N firms p(Y ) + p′ (Y ) yi = ci′ ( yi ) i = 1, 2,…., N N where Y = ∑yi i =1 We can write p(Y )[1 + p′ (Y ) Y si ] = ci′ ( yi ) p i = 1, 2,…., N si p(Y )[1 + ] = ci′ ( yi ) i = 1, 2,…., N ε where si = yi/Y and ε = elasticity of market demand. Cournot model is in between monopoly and competitive models. if si = 1 → monopoly if si → 0 → competitive firms with infinitesimal market share. Adding up all N equations, we have N Np(Y ) + p′ (Y )Y = Σ ci′ ( yi ) i =1 Aggregate industry output Y only depends on the sum of the marginal costs, not on their distribution across firms. If all firms have the same constant MC = c then the equilibrium is symmetric si = 1/N and 1 p(Y )[1 + ] = c as N → ∞, price approaches marginal cost Nε Welfare Assuming a symmetric equilibrium, Cournot induxtry maximizes the weighted sum of consumers’ surplus and industry profit as given by (N-1)[U(Y) – cY] + [p(Y) – c]Y To see this, differentiate the above with respect to Y and set equal to 0 (Y ) – c] + p′ (Y ) Y + p(Y) – c = 0 (N-1)[ U ′ Y Note that U(Y) = ∫ 0p(x)dx U′ (Y ) = p(Y) Using this fact and rearrange the equation yeilds 1 p(Y )[1 + ]=c Nε This is the profit maximizing condition in a symmetric Cournot equilibrium. As N → ∞, more weight is put on the social objective of utility minus costs, as compared to private objective of profits. Alternatively, to maximize social objective Wsocial(Y) = U(Y) – cY (Y ) = p(Y) = c F.O.C.: U ′ 1 p ( Y )[ 1 + ] = c, However, Cournot optimization Nε as N → ∞, social objective is maximized. Bertrand Game In a Bertrand game, each firm chooses its price level taking as given all other firms’ price levels. Model with 2 firms - 2 firms, index by i = 1, 2, with a constant unit cost of c1 and c2 - Products are homogeneous and the demand for firm i is di(pi, pj) = { D( pi ) if pi < p j D( pi ) / 2 if pi = p j if pi > p j 0 where D(p) is the market demand curve Case 1): If c1 = c2 = c, a Bertrand Nash equilibrium is p1 = p2 = c and each firm gets half of the markets Proof.: If p1 > c, firm 2 can set p2 = p1 – ε and get the entire market, earning (almost) entire profit But if p2 > c, this cannot be an equilibrium because firm 1 will have incentive to undercut price further Case 2): If c2 > c1, a Bertrand Nash equilibrium has firm 1 sets p1 = c2 and gets the entire market and firm 2 sets p2 ≥ c2 and produces zero. Proof.: With only 2 firms, we get the result of firms setting price = MC, or the firm that is in operation sets its price = marginal cost of firm with higher cost. Differentiated Products Inverse demand functions of two goods that are not perfect substitutes p1 = α1 – β1y1 – γy2 p2 = α2 – β2y2 – γy1 Measure of product differentiation γ2/ β1β2 Cournot Competition for firm i Max [αi – βiyi – γyj] yi yi Direct Demand functions y1 = a1 – b1p1 + cp2 y2 = a2 – b2p2 + cp1 Bertrand Competition for firm i Max [ai – bipi + cpj] pi pi Outputs of firms are strategic substitutes → increasing yj makes it less profitable for firm i to increase its output Prices are strategic complements → an increase in pj makes it more profitable for firm i to increase its price. Quantity Leaership (Stackelberg Model) - A sequential game with 2 stages - First stage: Leader moves by choosing its output level - Second stage: Follower chooses its output level after observing the leader’s output. Stackelberg with homogeneous products (perfect substitutes) Let leader = firm 1, follower = firm 2 Second stage Firm 2 chooses y2 given y1 This is essentially reaction curve of firm 2, f2(y1) in Cournot game First stage Firm 1 maximizes Revealed Preference: The profit of the leader in Stackelberg equilibrium will usually be higher than his profit in Cournot equilibrium. Leadership Preferred: A firm always weakly prefers to be a leader. Proof. This can be shown using the fact that with homogeneous products i) П1(y1, y2) is a strictly decreasing function of y2 and vice versa ii) The reaction curves f1(y2) and f2(y1) are strictly decreasing functions Price Leadership - Heterogeneous products - Demand for output of firm i = xi(p1, p2) - 2 firms, leader = firm 1, follower = firm 2 Second stage Firm 2 max First stage Firm 1 max If goods are substitutes with high degree of substitution, we may expect reaction curves to be upward sloping. Following Preferred If both firms have identical cost and demand functions and if reaction curves are upward sloping. → each firm prefers to be the follower to being the leader Intuition: Leader has to support high price by producing low output whereas follower can set price equal to leader’s price (or undercut leader’s price a little) and produce as much as it wants. Capacity Constraint Game - 1st period, each firm chooses its production capacity yi - 2nd period, they play Bertrand game The outcome of this game is typically a Cournot equilibrium - The price charged at 2nd period is p(y1 + y2) which is just the inverse demand at capacity - Choosing capacity at 1st period is then just a Cournot game. Conjectural Variations If firm 1 makes a conjecture about firm 2 responds to its choice of output, denote it by y 2v ( y1 ) and let v12 ∂ y 2v ( y1 ) = ∂ y1 v Firm 1 max p(y1 + y 2 ( y1 ) ) y1 – c1(y1) F.O.C.: i) If v12 = 0 → Cournot model, each firm believes that the other firm’s choice is independent from its own ii) If v12 = -1 → Competitive model ( y1 ) = slope of reaction function → Stackelberg iii) If v12 = f 2′ iv) If v12 = y2/y1 → Cartel Criticism: It is a static model but is should be a dynamic model as it is modelling how other firms should respond. Collusion If firms act as cartel, they maximize the sum of profits by choosing y1, y2,….,yN simultaneously N max y1, y2,..,yN p(Y)Y – Σ ci ( yi ) i =1 F.O.C.: However, the solution is not stable, there is incentive to cheat If firm 1 maximizes its own profit at cartel’s solution Y* If firm 1 believes that other will employ cartel output, then it would benefit to increase its own output. If it does not believe that other firms will set cartel output, generally, not optimal for it to maintain cartel output either. To make cartel possible, often use effective punishment to deter cheating. Repeated Oligopoly Games Example: Punishment Strategy - chooses cartel output as long as no one cheats - if someone cheats, chooses Cournot output in all future periods Let * ∏ i be one-period profit of firm i from cartel outcome C ∏ i be one-period profit of firm i from Cournot outcome D ∏ i be one-period profit of firm i from deviation (cheating) Infinitely Repeated Games Note: i) cooperation is not part of SPE in finitely repeated game. ii) Abreu (1986) shows that one-period punishment will typically be sufficient to support cartel output. Limit Pricing Limit pricing = pricing to prevent entry Example: Perfect knowledge - 2 firms, incumbent and a potential entrant - 2 periods: 1st period, incumbent sets price and quantity (observed by entrant) 2nd period, if entry occurs, they play duopoly game, if there is no entry, the incumbent will charge monopolist price. Solution If duopoly profit for entrant > 0, will enter regardless of what incumbent does previously. Hence, incumbent should choose monopolist price in 1st the period. This is because the first period price conveys no information. Example: Imperfect Information - uncertainty about incumbent’s cost (entrant does not know if the incumbent has low or high cost) Incumbent can reveal that it is a low cost by set low enough price at first period such that a high-cost incumbent will have no incentive to do so.