General Examination in Microeconomic Theory. Fall You have FOUR hours. 2. Answer all questions

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1 HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS General Examnaton n Mcroeconomc Theory Fall You have FOUR hours. 2. Answer all questons PLEASE USE A SEPARATE BLUE BOOK FOR EACH QUESTION AND WRITE THE QUESTION NUMBER ON THE FRONT OF THE BLUE BOOK. PLEASE PUT YOUR EXAM NUMBER ON EACH BOOK. PLEASE DO NOT WRITE YOUR NAME ON YOUR BLUE BOOKS.

2 Fall Mcro General Exam Glaeser Verbal Problem Remnder: For credt you must gve an algebrac answer or ustfcaton. Indvduals ether commt one or zero crmes and there s heterogenety n the populaton wth regards to propensty to commt a crme. (1) Derve comparatve statcs on the determnants of the amount of crme n socety wth respect to arrest probablty, punshment level, and opportunty cost of tme. (2) Make assumptons about the socal cost of crme and punshment and derve condtons for the optmal punshment holdng the arrest probablty fxed. Explan your result ntutvely. (3) Now assume that there are two perods, and that crmnals caught n the frst perod can be ncarcerated durng the second perod (put n al nstead of beng punshed wth a fne or other penalty). Assume that durng the perod of ncarceraton, the crmnal can nether commt a crme nor be economcally productve. Under what condtons s t optmal to ncarcerate f an ndvdual's beneft/cost of commttng a crme s constant over the two perods? How does your answer change f the ndvdual's propensty to commt a crme changes over tme? Explan your result ntutvely. (4) Can your answer to part three help explan the move from corporal punshment (physcal punshment and publc humlaton such as puttng people n stocks, beatng them, etc.) to ncapactaton durng the 19th century that s dscussed by Foucault and others? Provde an nterpretaton n the context of your model.

3 (1 hour) Consder the followng sgnallng game. The sender has two types, = H; L: The sender sees her type and choose an acton a 1 2 [0; 1) The buyer sees a 1 but not the seller s type, and chooses ether "accept" or" reect." If the buyer accepts, the seller s payo s a 1 and the buyer s payo s v a 1 : If the buyer reects, the seller s payo s w and the buyer s payo s 0. There s a common pror probablty p that the sender s type s H: Suppose that v H > v L > w H > w L > 0: a) Is there a poolng PBE (perfect Bayesan equlbrum)? b) Is there a seperatng PBE? c) What s the set of PBE outcomes? d) Is there a Bayesan Nash equlbrum whose outcome s not the outcome of a PBE? Now suppose that v H > w H > v L > w L > 0: e) Is there a poolng PBE? f) Is there a seperatng PBE? 1

4 1. Consder a closed economy/country A wth m consumers, n frms, and l goods. Preferences and producton sets are well-behaved. Denote a Walrasan equlbrum for ths economy by ( p *, x*, y*). The economy now opens tself up to free trade wth a second economy/country B, consstng of M consumers and N frms (and the same l goods). Ths second economy s also well-behaved. There s a new Walrasan equlbrum ( p ', x', y' ) after trade. Show that at least one consumer n country A s better off n the post-trade equlbrum. You are now asked to show that there exst lump sum transfers wthn the country A so that every consumer n A s better off n the post-trade equlbrum. To do ths you mght want to proceed as follows. For each prce vector p, let T = px * ( pw n + θ π ) 1 where π = Maxpy y εy shareholdngs n A. Show that, and the w and θ denote ndvdual endowments and m T Σ= 1 0. Now consder a regme n whch consumer n A receves the lump sum transfer 1 T ( T Tm ). m Argue nformally that a Walrasan equlbrum for the world economy n the presence of these lump sum transfers wll exst. Confrm that the lump sum * transfers sum to zero and that each consumer n A can afford to purchase x at any prces p ( where consumer receves x * n the pre-trade Walrasan

5 equlbrum ). Conclude that every consumer n A s better off n the post-trade Walrasan equlbrum. 2. Consder a moral hazard stuaton wth a rsk neutral prncpal and a rsk averse 1 2 agent. The agent s utlty functon s exp[ r( w ca )], where w s hs wage 2 and a hs effort (r and c are postve constants). Proft s gven by q = a + ε ε N o σ ε 2 where ~ (, ). The prncpal does not see a, but does observe q and 2 another sgnal z. Here ( ε, z ) are ont normal; the varance of z s σ Z and the covarance of ( ε, z) s σ εz. The prncpal offers the agent a lnear ncentve scheme w = α + βq + γz Show how to compute the optmal values of condtons under whch γ 0. β, γ. Dscuss and nterpret the

6 Throughout ths problem assume that utltes are all quas-lnear n money and all valuatons and wllngness to pay are expressed n the same monetary unts. Three travelers each have to make a trp from Boston to San Francsco by plane. Ther wllngnesses to pay for the trp are, $3500, $5000, $6000, respectvely. There are two possble ways to travel. They can rent a plane that seats up to two people for $3000. In that case the thrd person wll not travel. Alternatvely they can rent plane wth capacty three for $5000. a) What s the core of ths game? Is t empty or non-empty? b) What s the Shapley value of ths game? Now suppose that they each have to make two trps from Boston to San Francsco wthn a one-year perod, but that the dates of ther trps are exble. Per-trp valuatons and costs of rentng planes are exactly as above. The three players can plan ther travel n advance, negotatng wth each other about payments for planes, monetary transfers among themselves, and travel arrangements (.e. who travels wth whom). c) What s the core of ths game? Is t empty or non-empty? d) What s the Shapley value of ths game? e) Comment on the characterstc functon of the cooperatve game you are usng to analyze ths problem. How does the value of the coalton of all players relate to the payo these players can acheve f they all travel together? 1