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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) Thirddegree 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? Thirddegree 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. Thirddegree 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 firstorder 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 / Thirddegree 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 Thirddegree 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 firstorder 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 . Thirddegree 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 firstorder 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 bestreply 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 profitmaximizing 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 profitmaximizing choice of firm 1 given the conjecture that firm 2 chooses x2, and x2 is a profitmaximizing choice for 2 given the conjecture that 1 chooses x1. Cournot equilibrium is a combination of profitmaximization with rational expectations / selffulfilling 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 firstorder 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 selfenforcing? 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 firstorder condition of (1)). ...
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 '09
 SAMUELSON
 Microeconomics

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