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Unformatted text preview: Econ 145b. Practice Exercises with Solutions Recall the framework of Problem Set 2. A monopolistic seller with zero cost of producing a good maximizes expected profits. Given a pricing schedule T that determines the transfer to the seller as the function of quantity bought, a buyer buys quantity q ≥ that maximizes the net utility equal to gross utility U ( q ) minus transfer T ( q ) . The buyer’s gross utility from buying quantity q is U ( q ) = θ (1 exp ( q )) where θ equals θ L with probability λ and θ H with probability 1 λ . Assume that θ H > θ L > and that the seller needs to o ff er the same pricing options to both buyer types. The buyers are not coerced to buy; they will buy only if their net utility from buying is nonnegative. We will allow the seller to produce and sell an infinite quantity of good to any buyer θ ; in line with the above formulas, the buyer’s gross utility from buying an infinite quantity is θ . 1. Assume that the seller is o ff ering a menu of two (not necessarily di ff erent) twopart tari ff s T L ( q ) = p L + p L 1 q, T H ( q ) = p H + p H 1 q . (a) Solve for the optimal menu of this form. (b) How much the high type will buy? (c) What will be the low type’s net utility? 2. Assume that the seller is o ff ering a menu of two (not necessarily di ff erent) quantity price pairs ( q L , t L ) , ( q H , t H ) . The buyer can accept only one of the two pairs (or none); by accepting the pair ( q θ , t θ ) the buyer obtains quantity q θ and pays transfer t H . (a) Solve for the optimal menu of this form. (b) How much the high type will buy? (c) What will be the low type’s net utility? 1 Solutions 1(a) We may assume that the low type chooses to pay according to the tari ff T L and the high type chooses to pay according to T H . That means that the tari ff schedules are such that max q θ L ( 1 e q ) T L ( q ) ≥ (1) max q θ L ( 1 e q ) T L ( q ) ≥ max q θ L ( 1 e q ) T H ( q ) (2) max q θ H ( 1 e q ) T H ( q ) ≥ (3) max q θ H ( 1 e q ) T H ( q ) ≥ max q θ H ( 1 e q ) T L ( q ) (4) where the conditions formalize the following components of the above verbal statement: the low type prefers tari ff T L over not buying (1), low type prefers T L over T H (2), high...
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 Winter '10
 Obara
 Economics, pH, Trigraph, high type

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