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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 23 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) Third-degree Price Discrimination Suppose there are two types of consumers in the population, A and B , with proportions A and B . The monopolist knows the aggregate demand function for each population. Suppose the monopolist can charge different unit prices to different populations, but, given a population, it cannot charge different prices for different units. What is the optimal pricing strategy of the monopolist? Third-degree Price Discrimination What does it mean to assume that the monopolist can charge different prices to different populations? That is an informational assumption. It means whether a consumer belongs to one population or another is verifiable information. Example: student discounts, senior discounts, discounts for professional associations, etc. Third-degree Price Discrimination Suppose A D B D 1=2. Monopolist solves x A ;x B max p A.x A/x A C p B .x B /x B C.x A C x B / That's easy, no? Take first-order conditions: .p A/0.x A/x A C p A.x A/ D C 0.x A C x B / .p B /0.x B /x B C p B .x B / D C 0.x A C x B / Third-degree Price Discrimination Same as p .x / 1 B B p .x / 1 A A 1=j" .x /j D C 0.x A C x B / B B 1=j" .x /j D C 0.x A C x B / A A or, equivalently, C 0.x A C x B / p .x / D 1 1=j"A.x A/j A A C 0.x A C x B / p .x / D 1 1=j"B .x B /j B B Third-degree Price Discrimination Consider the case of constant marginal cost C 0 D M C . Linear demand case pA D a Then, bx A and p B D c dx B ; where a > c > M C: j"Aj D .a Therefore, Da 2bx A bx A/=.bx A/ and j"B j D .c dx B /=.dx B / .a .c ,, , bx bx /.1 / D MC A a bx dx B B dx /.1 / D MC B , c dx ... A Dc 2dx B ...,, A ) ) MC a C MC A x D ; p D > MC 2b 2 c C MC c MC ; pB D > MC xB D 2d 2 A a Linear Demand Case Thus, p A > p B . How about the comparison between x A and x B ? Notice that, by the first-order conditions, p A > p B implies 1 1=j"A.x A/j < 1 1=j"A.x B /j H) j"A.x A/j < j"B .x B /j: But whether x A is greater than or smaller than x B depends on b and d . Constant Elasticity Case Now suppose p A D a.x A/ where 1="A and p B D b.x B / 1="B ; 1 < "A < "B : Then, MC 1 1="A MC B p D 1 1="B pA D and hence pA > pB : Again, comparison between x A and x B will depend on parameters a and b . Third-degree Price Discrimination Populations with higher price elasticity are charged lower prices. Example: students, senior citizens. Cournot Oligopoly Consider two firms, 1 and 2, facing a downward sloping aggregate demand p.x/. Suppose each firm choose its output xi to maximize profits. This is called Cournot competition. Cournot Oligopoly Let us model the firm's reasoning. What is the profit of firm 1? p.x1 C x2/x1 What is firm 1's first-order condition? C1 .x1/: p 0.x1 C x2/x1 C p.x1 C x2/ D M C1 Notice: this depends on x2! Cournot Oligopoly Consider the linear demand case: p D a b.x1 C x2/. x1 .x2/ a bx2 2bx1 D M C1 H) D a bx2 M C1 2b Here, x1 is firm 1's best-reply to x2, i.e. the profit maximizing output of firm 1 under the conjecture that 2 chooses x2. It depends on x2! Cournot Oligopoly Thus, in order for firm 1 to find its profit-maximizing optimal x1 it ought to reason about the choice x2 of firm 2! Moreover, firm 1 also understands that firm 2 will be reasoning about firm 1. Thus, firm 1 should reason about firm 2's reasoning about firm 1. Ans also about firm 2's reasoning about firm 1's reasoning about firm 2, and so on. We are in the realm of Game Theory, which is the study of strategic thinking. Cournot Equilibrium The solution .x1; x2/ of x1 .x2/ D x1 x2 .x1/ D x2 In words, .x1; x2/ is a Cournot equilibrium if x1 is a profit-maximizing choice of firm 1 given the conjecture that firm 2 chooses x2, and x2 is a profit-maximizing choice for 2 given the conjecture that 1 chooses x1. Cournot equilibrium is a combination of profit-maximization with rational expectations / self-fulfilling prophecies. Cournot Equilibrium Linear demand / constant marginal cost case: x1 D x2 D Solving for it: a a bx2 M C1 2b bx1 M C2 2b x2 D a 3 x2 D 4 Likewise a bx1 M C2 a M C2 D 2b 2b 2M C2 C M C1 4b x1 D a H) x2 D a bx2 M C1 4b 2M C2 C M C1 3b a 2M C1 C M C2 3b Cournot Equilibrium Consider the homogeneous case: M C1 D M C2 D M C . Then, x1 D x2 D Total output a MC : 3b MC/ 3b x1 C x2 D This is greater than 2.a a MC D monopoly output; 2b MC D competitive output: b and less than a Collusion Suppose the firms could form a cartel. That is, they get together and act as a monopolist who maximizes the joint profits of the two firms. Then, the cartel solves max p.x1 C x2/.x1 C x2/ x1 ;x2 C1 .x1/ C2.x2/ The first-order conditions yield 0 c 0 c c c c c c p 0.x1 C x2/.x1 C x2 / C p.x1 C x2 / D C1.x1 / D C2.x2 / Collusion Is the collusion agreement self-enforcing? That is, given that firm 2 will follow the agreement, does firm 1 have an incentive to do so? Firm 1 would like to solve max p.x1 C x2/x1 x1 ;x2 c c but, denoting by x c D x1 C x2 , C1 .x1/ (1) 0 c c 0 c c c c c p 0.x c /x1 Cp.x c / D p 0.x c /.x1 Cx2 /Cp.x c / x2 p 0.x c / D C1.x1 / x2 p 0.x c / > C1.x1 / Thus, c 0 c p 0.x c /x1 C p.x c / > C1.x1 / and hence, firm 1 has is not maximizing profits! It has an incentive to c produce more than x1 . (Check the first-order condition of (1)). ...
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