Advanced Microeconomics Leonardo Felli EC441: Room D.106, Z.332, D.109 Lecture 8 bis: 24 November 2004
Monopoly Consider now the pricing behavior of a profit maximizing monopolist: a firm that is the only producer of a commodity. Let the aggregate demand of this commodity at price p be x(p) assumed to be continuous, strictly decreasing and such that x(p) > 0. Assume that there exists a price p < + such that x(p) = 0 for every p p. Assume that the monopolist knows x(p) and is endowed with a technology characterized by the cost function c(q). Slide 1
The monopolist s problem is then: max p p x(p) c(x(p)) Equivalent formulation in terms of quantity choice q is derived using the inverse demand function P ( ) = x 1 ( ): max q 0 P (q) q c(q) We focus on this (equivalent) formulation and assume that: P ( ) and c( ) are twice continuously differentiable, P (0) > c (0) and there exists a unique output q c such that P (q c ) = c (q c ). Slide 2
The solution to the monopolist s problem q m satisfies the following necessary first order conditions: P (q m ) q m + P (q m ) c (q m ) with equality if q m > 0 The left-hand-side is known as the marginal revenue and it is equal to the derivative of the revenue function R(q) = P (q) q. The right-hand-side is the familiar marginal cost. Slide 3
Since P (0) > c (0) the necessary first order conditions can only be satisfied at q m > 0. Therefore the monopolist s optimal quantity choice is the one that sets marginal revenue equal to marginal cost: P (q m ) q m + P (q m ) = c (q m ) In the typical case P (q) < 0 we obtain that: P (q m ) > c (q m ) The price under monopoly exceeds marginal cost. Slide 4
Correspondingly: q m < q c A reduction in the quantity sold by the monopolist allows him to increase the price charged on the remaining sales. The effect on profits is captured by the term P (q m ) q m. The welfare loss, known as the deadweight loss of monopoly is measured by the change in surplus: q c q m [P (s) c (s)] ds > 0 Slide 5
Notice that the deadweight loss is absent in the special case of a perfectly elastic demand: P (q) = 0 for all q. In this case the monopolist sells the same quantity as a perfectly competitive firm: q m = q c. In other words, competitive firms perceive demand as perfectly elastic: the price is given in the market. Slide 6
Consider now the special case of a monopolist whose technology is fully described by the cost function that exhibits constant returns to scale with no fixed costs: c(q i ) = c q i i {1, 2}. The consumer behavior is summarized by the following linear inverse demand function: { a q if q a P (q) = 0 if q a We assume for viability of the economy c < a. Slide 7
The monopolist s problem is now: max {q} Π(q) = q [p(q) c] = q [a q c] The solution to this problem characterizes the monopolist s quantity: q m = (a c) 2 This is strictly lower than the perfectly competitive quantity: q c = (a c) Slide 8
Duopoly We distinguish between different types of duopoly (oligopoly) depending on: whether the firms involved in the market compete in quantity or in prices, and whether they decide their strategy simultaneously or sequentially. Slide 9
Consider a Cournot Duopoly (Cournot 1838) model in which two firms compete in their choice of the quantity produced and take these decisions simultaneously. Assume that both firms produce a perfectly homogeneous good. The consumers are then indifferent on whether they consume the good produced by one firm or the other. Therefore N = {1, 2}, A i = R + the set of positive quantities denoted q 1 and q 2. Slide 10
To be able to define the payoff function for each firm we need to specify the technology of the firm and the demand function in the market. Assume that the firms technologies are identical and fully described by the cost function that exhibits constant returns to scale with no fixed costs: c(q i ) = c q i i {1, 2}. The consumer behavior is summarized by the inverse demand function faced by both firms: P (q 1 + q 2 ) = { a (q1 + q 2 ) if q 1 + q 2 a 0 if q 1 + q 2 a Slide 11
We assume for viability of the economy c < a. Firm i s profit (payoff) function is then: Π i (q 1, q 2 ) = q i [P (q 1 + q 2 ) c] = = q i [a (q 1 + q 2 ) c] i {1, 2}. A pair of quantities (q1, q 2 ) is a Nash equilibrium of the Cournot duopoly game if and only if: q i = argmax Π i (q i, q j ) i j, i, j {1, 2}. q i R + Slide 12
To identify the best reply of each firm we need to solve the following problem: R i (q j ) = argmax q i R + q i [a (q 1 + q 2 ) c]. If q j < a c this is characterized by the following set of necessary and sufficient conditions: a 2q i q j c = 0 Which gives us the following pair of best reply functions: R i (q j ) = 1 2 (a q j c) i j, i, j {1, 2} Slide 13
These two (linear) best reply functions are represented in the following graph: q 2 a c R 1 (q 2 ) q m 2 = (a c) 2 R 2 (q 1 ) (0, 0) q m 1 = (a c) 2 a c q 1 Slide 14
The Nash equilibrium of the Cournot game is then the solution to the following problem: q 1 = 1 2 ( a q 2 c) q 2 = 1 2 ( a q 1 c). This solution is unique and is: q 1 = q 2 (a c) =. 3 Slide 15
The Nash equilibrium is the intersection of the best reply functions: q 2 a c R 1 (q 2 ) q m 2 = (a c) 2 (q 1, q 2 ) R 2 (q 1 ) (0, 0) q m 1 = (a c) 2 a c q 1 Slide 16
An alternative way: iterated elimination of strictly dominated strategies. We proceed in the following way: The monopolist quantity q m = strictly dominates any higher quantity for each player. (a c) 2 Given that quantities in excess of q m are never chosen by a firm then quantity R i (q m ) strictly dominates any quantity below it.. The only quantity left to choose for each player is then qi. Notice that this process does not converge to a unique point if instead of a Cournot duopoly we consider a Cournot oligopoly in which three firms compete. Slide 17
The iterated elimination of strictly dominated strategies: q 2 a c R 1 (q 2 ) q m 2 = (a c) 2 R 2 (q m 1 ) (0, 0) (q1, q 2 )......... R 2 (q 1 )... q m 1 R 1 (q m 2 ). = (a c) 2 a c q 1 Slide 18
A set of sufficient conditions that guarantee existence of a Nash equilibrium of a Cournot duopoly game are that the profit function of each firm is quasi-concave and twice differentiable. Quasi-concavity is checked by establishing that 2 Π 1 q 2 1 0 whenever Π 1 q 1 = 0 and 2 Π 2 q 2 2 0 whenever Π 2 q 2 = 0 These conditions imply: 2 P (q 1 + q 2 ) + q i P (q 1 + q 2 ) c (q i ) 0 (P 0 then P 0 sufficient condition) Slide 19
Consider a Bertrand Duopoly (Bertrand 1883) model in which two firms compete in their choice of prices and take these decisions simultaneously. Assume for simplicity that the consumer demands only one unit of the homogeneous good produced by either firm. Let v be the utility that the consumer derives from this unit of a good. A consumer will buy the good from the firm that charges the lowest price p i if v p i. When the price is the same he will buy the good from firm 2. Slide 20
Then N = {1, 2} and A i = R + the set of all positive prices p i. Assume that firm i produces this unit of the good at a cost c i and where v > c 1. c 1 > c 2. The payoffs to each firm are then: Π 1 (p 1, p 2 ) = { p1 c 1 if p 1 < p 2 0 if p 1 p 2 Slide 21
and { p2 c 2 if p 2 p 1 Π 2 (p 1, p 2 ) = 0 if p 2 > p 1 The best reply for firm 1 is then: p 1 p 2 if p 2 c 1 p 1 < p 2 if p 2 > c 1 while for firm 2 is instead: p 2 > p 1 if p 1 < c 2 p 2 p 1 if p 1 = c 2 p 2 = p 1 if p 1 > c 2 Slide 22
The Nash equilibria of the Bertrand duopoly game are then: a choice of prices: p 2 = p 1 = p c 2 p c 1. in equilibrium the consumer buys the unit of the good from firm 2. Notice the multiplicity of equilibria in terms of price choices: a whole interval of prices can be equilibria of this game. Slide 23
The best replies are represented in the following graph. p 1 c 2......... c 1 p 2 45 Slide 24
Odds and ends on normal form games: We have restricted attention to iterated elimination of strictly dominated strategies, why not weakly dominated? Consider the following game: T 1 T 2 S1 10, 0 5, 2 S2 10, 1 2, 0 Notice S2 is weakly dominated by S1. Slide 25
However (S2, T 1) is acceptable outcome provided that player 1 believes that player 2 will play T 1 with probability 1. T 1 T 2 S1 10, 0 5, 2 S2 10, 1 2, 0 Notice that (S2, T 1) is a Nash equilibrium. Moreover iterating the elimination of weakly dominated strategies creates problems: the outcome obtained is path dependent. This is not the case for iterated elimination of strictly dominated strategies. Slide 26
Consider the following game: T 1 T 2 T 3 S1 10, 0 5, 0 4, 200 S2 10, 100 5, 0 0, 100 If we start with player 1 we obtain that S2 is weakly dominated by S1. Then for player 2 T 3 is strictly dominated. Outcomes: (S1, T 1) and (S1, T 2). Slide 27
Now T 1 T 2 T 3 S1 10, 0 5, 0 4, 200 S2 10, 100 5, 0 0, 100 If we start with player 2 we obtain that both T 2 and T 3 are, respectively, weakly and strictly dominated by T 1. Then for player 1 neither strategy is dominated. Outcomes: (S1, T 1) and (S2, T 1). A different outcome depending on which player we start from. Slide 28