takehome_exam

Make-up Exam Economics  b — First Half Note: There is no point to your taking this make- up examination if you received a score of  or better on the first exam. A score of  or more on the first exam means you already have a grade of a or better for the first-half of Economics  b . Instructions. This is a take-home examination. You may consult whatever material you wish. You may not , however, discuss the exam with anybody or otherwise communicate with any person about it. The only two exceptions to this last rule are Ben Hermalin and Leonidas Enrique de la Rosa (Kike). You may not email or otherwise distribute this copy of the exam to anyone. You have  hours from the time that you are emailed this exam to turn in your answers to it. Acceptable means of turning in your answers are (i) as a Word attachment to an email to Ben Hermalin (hermalin@haas.berkeley.edu); (ii) as a standard L A T E X attachment to an email to Ben Hermalin (herma- lin@haas.berkeley.edu); (iii) as a pdf file attachment to an email to Ben Herma- lin (hermalin@haas.berkeley.edu); or (iv) in blue books turned into Ben Herma- lin ( f  Haas School). Whatever method you choose, please make sure your name is on any materials turned in (or in any file submitted electronically). Answer all five (  ) questions. Each question counts equally. Neither the gsi nor I are trained in reading cuneiform, linear b , or hieroglyphics, so please write neatly. Questions 1. A theater faces demand D ( p ) = 1000 − 20 p for seats to a performance, where p is the price of a ticket. The marginal cost of providing a seat is so negligible that you may consider it to be zero. (a) What is the profit-maximizing price for the theater to charge? (b) What is the deadweight loss if the theater charges the profit-maxi- mizing price? 2. A theater has 100 seats in the orchestra section (near the stage) and 100 seats in balcony section (far from the stage). It faces two types of customers for a particular concert. Assume there are N 1 consumers of type 1 and N 2 consumers of type 2. The theater cannot distinguish between the two types by observation or any other attribute that would permit third- degree price discrimination. Assume a type-1 consumer values a seat in Copyright c 2003 Benjamin E. Hermalin. All rights reserved.

Economics  b Make-up Exam Page 2 the orchestra section at r 1 (“r” for o r chestra) and a seat in the balcony at b 1 . Assume a type-2 values both kinds of seats equally at ν 2 . Both types of consumer have a quasi-linear utility of the form δv + y , where δ ∈ { 0 , 1 } denotes whether a seat is purchased ( δ = 1) or not ( δ = 0), v is the value for the kind of seat purchased, and y is consumption of the numeraire good. Assume that r 1 > b 1 > ν 2 > 0. Finally, presume that the marginal cost of providing a seat is so negligible that you may consider it to be zero. (a) Assume that N 1 = N 2 = 100 and 2 ν 2 > b 1 . What are the profit- maximizing prices for the theater to charge for orchestra seats and for balcony seats? (b) Assume that N 1 = N 2 = 100 but 2 ν 2 < b 1 . Now what are the profit-maximizing prices? (c) Assume that N 1 > N 2 = 100. Return to the assumption that 2 ν 2 > b 1 . As a function of N 1 , what are the profit-maximizing prices to charge? (d) A “fair-seating” law is passed such that all seats in a theater must be offered at the same price ( i.e., the theater can no longer charge different prices for orchestra seats as opposed to balcony seats). As- sume, as in (a), that N 1 = N 2 = 100 and 2 ν 2 > b 1 . Derive conditions under which such a fair-seating law is welfare reducing . 3. A benevolent social planner is deciding how much park land to develop for the citizens of a given region. Let x denote the acres of such park land. There are N citizens in the relevant society. Each citizen, n , has a quasi-linear utility function of the form θ n log( x )+ y , where θ n ∈ Θ ⊂ R + , log( · ) denotes the natural logarithm, and y is the amount of the numeraire good. The cost of x acres of park land is 1000 x . This cost will be evenly divided among the citizens; that is, citizen n must pay 1000 x/N in taxes to fund the provision of x acres of park land. The social planner wishes to maximize net social welfare from park land; that is, she wishes to maximize N n =1 θ n log( x ) total benefit − 1000 x total cost . (1) Each citizen’s θ is his or her private information. Design a dominant- strategy mechanism that allows the social planner to maximize (1) for all realizations of ( θ 1 , . . . , θ N ) ∈ Θ N . 4. Consider a hidden-information principal-agent model of the sort examined in “Hidden-Information Agency.” Assume the agent’s utility is s − (2 − θ ) x 2 / 2 ,