ps2soln - Problem Set 2 Solutions ECON105 Industrial...

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Problem Set 2 Solutions ECON105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego 1. Imagine there are ten consumers, each with individual demand curve: D i (q) = i – q, where i=1. .10. Note that consumers now have multiunit demands, but their demand curves differ. Assume a first degree price discriminating monopolist’s cost function is C(Q) = Q/2, where Q = q 1 +…+q 10 . If a monopolist can perfectly price discriminate, find its profits and the consumer surplus to each consumer. profit= [1-0.5] 2 /4+[2-0.5] 2 /4+…+[10-0.5] 2 /4=332.5, CS=0. 2. A monopolist faces the inverse market demand curve D(Q) = 1 - Q, and cost function C(Q) = cQ where c < 1. Assume the demand curve is made up of many consumers each with unit demand for this good (ie. each consumer will either buy one or nothing and the market demand curve is the aggregation of all these unit-demand consumers.) a. If the monopolist cannot price discriminate, find the equilibrium price, total quantity, profits, consumer surplus and welfare. (This is similar to questions 1a and 1j). 4 ) 1 ( 4 2 2 1 2 1 * 2 1 * 2 1 ) ( ) ( 2 1 1 ) ( 2 1 0 2 1 0 ) ( : ) 1 ( max ) ( ) ( ) ( 2 2 2 c c c c c c c c Q C Q P Q c Q Q P c Q c Q dQ Q d F OC cQ Q Q Q C Q Q P Q M M M M M M M M M M M Q Q Q 8 ) 1 ( 3 4 ) 1 ( 8 ) 1 ( 8 ) 1 ( 2 2 ) 1 1 ( * 2 ) 1 ( * 2 2 2 2 2 M M M M c c c CS W c Q Q Q P Q M M M M M b. What assumptions are necessary for the monopoly to be able to perfectly (1 st degree) price discriminate?
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A discriminating monopolist should be able to avoid arbitrage among customers and should be able to tell each customer’s own valuation perfectly . c. If the monopolist can perfectly (1 st degree) price discriminate, solve for profits, consumer surplus, and welfare. (Hint: since consumers have unit demand, the “package” contains exactly one good and every consumer receives a price equal to her valuation. Don’t be confused, the “package price” and the “per unit price” is the same thing, because of unit demand.) Compare these outcomes to part a. Intuitively, here the monopolist faces as many separate markets as consumers. In each market, the firm chooses the price of the unit good so as to maximize profits subject to the valuation (reservation price) of the consumer. Each maximization problem looks like this: 1 : _ ] * * [ max Qi c Pi Vi to subject c Pi where the subscript “i” stands for a given consumer with a certain valuation Vi. Obviously the solution to each problem is that the highest price Pi is just equal to the valuation Vi, so the first constraint should bind. Any higher price and the consumer won’t buy, any lower and it is possible to slightly increase price which will improve revenue without affecting cost. Obviously, for consumers for whom Vi<c it is best not to sell them the unit at all because it is too costly to produce given what we can charge them. So we only sell to those consumers for whom Vi≥c and always charge them Pi=Vi. The total
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This note was uploaded on 10/20/2010 for the course ECON ECON 105 taught by Professor Noel during the Fall '10 term at UCSD.

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ps2soln - Problem Set 2 Solutions ECON105 Industrial...

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