Economic Theory Microeconomics Sequence
Economic Theory Microeconomics Sequence ECON 713
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Econ 713 Discussion Week 13 04 27 07 Danqing Hu XL Michael Rapp 1 Auctions with Reserve Price and with Risk Aversion Proposition 1 Symmetric equilibrium strategies in a rsteprice auction are given by E 90 EY1Y1 lt x where Y1 is the highest ofN 7 1 independently drawn values The equilibrium bid can be rewritten as where is the cdf onl and F is the cdf of bidder s ualuations Theorem 2 Revenue Equivalence Principle Suppose that values are in dependently and identically disributed and all bidders are risk neutral Then any symmetric and increasing equilibrium of any standard auction such that the ex pected payment of a bidder with value zero is zero yields the same expected revenue to the seller Proposition 3 Revenue maximizing seller should always set a reserve price that exceeds his or her ualue Proposition 4 Suppose that bidders are riskeauerse with the same utility func7 tions With symmetric independent private values the expected revenue in a rsteprice auction is greater than that in a secondeprice auction Econ 713 Discussion Week 8 03 16 06 Danqing Hu XL Michael Rapp 1 Bertrand Model with Product Differentiation Example 1 MWG 1202 Consider a city that can be represented as lying on a line segment of length 1 There is a continuum of consumers whose total number is M and who are assumed to be located uniformly along this line segment A consumer s location is indexed by z E 01 the distance from the left end of the city At each end of the city is located one supplier of widgets Firm 1 is at the left end rm 2 at the right Widgets are produced at a constant unit cost of c gt 0 Every consumer wants at most 1 widget and derives a gross bene t ofv from its consumption The total cost of buying from rm j for a consumer located a distance d from rmj is pj td7 where g gt 0 can be thought of as the cost or disutility per unit of distance traveled by the consumer in going to and from rm j s location The presence of travel costs introduces di erentiation between the two rms products because various consumers may now strictly prefer purchasing from one of the two rms even when the goods sell at the same price 8112 De nltlon 1 We say that s2 is a strategic complement of s1 if 8751 gt 0 as in the example on MWG 8b p399 s2 is a strategic substitute of s1 if lt 0 as in the example on MWG p392 51 Econ 713 Discussion Week 11 04 13 06 Danqing Hu XL Michael Rapp 1 Setting1 There is an agent that performs an unobservable action a E A C R The action results on pro ts at at a 9 that are accrued by the principal Note pro ts depend on some random variable 9 The principal s utility function is G w7 where G gt 0 and G S 0 The agent s utility function is H 107a Uw 7 V a where U gt 07 U lt 0 and V a gt 0 Pro ts are distributed with PDF f La where f and fga are well de ned 2 Second Best Since actions are unobservable7 the rst best solution can not be obtained We de ne the second best by max E G at 7 s saa 51 E H s 70L 2 H a E arg maxE H s 7 a a EA Note the rst constrain is a participation constrain or individual rationality IR constrain whereas the second constrain is an incentive compatibility IC constrain The interior solution for the problem is given by G39 x 7 we f m U39 was A quot f m 1 3 Example 1 The story is as follows The agent is a repairman The higher the effort 7117 he puts7 the less likely a failure in the machine That s why pro ts depend on at on a stochastic way Assume G w Uw 2V V a a2 1 as N exp 7 a In this setting condition 1 can be written We say at N exp if a2 own son Ap 7 1 1 2 7 1 1 2 1 Sl l1 59 For given values of the parameters 1 1 2 1 s at i ix 1 16 4 4 1The setting will follow Holmstrom Bengt quotMoral Hazard and Observability The Bell Journal of Economzcs Vol 10 No 1 Spring 1979pp 7491 Figure 1 Wage schedules with addtional information 4 Example 2 Let the utility functions be as in Example 1 Now however assume that the machine can also fail due to malfunction of components that are beyond the repairman control or if you prefer say that when it rains the machine works worse The important issue here is whether this information is observable for the principal To keep thing simple there s an observable variable y E 0 1 that alters the distribution 1 a k 0 7 Haw 7a aexplt ak 96 1 a k 1035721 170 EeXP W35 This new information can be useful In fact the same calculations that led to 1 now lead to G 7 I SWJDAhw m U 51711 Navy 117a Note that in 2 the wage schedule 5 3531 may depend on y In our example that leads to s at 1 s 10 Those wage schedules are presented on Figure 1 for some speci c values of the parameters 5 Value of information Note that equation 2 characterizes the interior optimal contract Note that if the right hand side was constant in y then the left hand side would need to be so Then the wage schedule would not depend on y In other words y will not be valuable if the following condition holds gate x faw ihltgt a We can rewrite that equation as2 f 731711 906711 Mama 4 The above condition can be interpreted in the following way Note that at holds all the information available about a In other words7 take a to be a random variable and suppose we observe both Ly lf 4 holds7 we say at is a suf cient statistic for at y with respect to a Problem 1 14B8 Amend the twore ortrchoice hidden action model as follows Suppose the principal can for a cost of c7 observe an extra signal 17 of the agent s e ort Pro ts 7T and the signal y have a joint distribution f myle conditional on e The decision to investigate the value ofy can be made after observing 7T A contract now speci es a wage schedule w 7139 in the event of no investigation a wage schedule w 71397 y if an investigation occurs and a probabilityp 7139 of investigation conditional on 7T Characterize the optimal contract for implemening e ort level eH 2Note that we are solving the di erential equation 3 whioh should hold almost everywhere For the cases of interest 3 and 4 will be equivalent Econ 713 Discussion Week 4 02 16 06 Danqing Hu XL Michael Rapp 1 Subgame Perfect Equilibrium SPE De nition 1 Subgame A subgame Iquot of an extensive form game I is a subset ofl which 1 Begins at a node x6 and contains this node all of its successors and no other nodes 2 Is closed under information sets ie ifx E lquotx E h and y E h then y E Iquot De nition 2 SPE A Po le 039 is a SPE off if it induces a NE in every subgame off 2 Perfect Bayesian Equilibrium PBE De nition 3 Belief Beliefs are a map u D gt gt 0 l for all decision nodes which satis es l 191 for all information sets h De nition 4 Sequentially Rational A strategy pro le 039 is sequentially rational given beliefs u if for each player i and information set h 6 Hi player is behavior conditional on h being reached maximizes his expected utility given 0 and u De nition 5 Bayesian beliefs The beliefs u are Bayesian given pro le 039 if If x E h then u 0 whenever R7 h gt 0 De nition 6 PBE The strategyebeliefs pair 0i is a weak Perfect Bayesian Equilibrium if 1 039 is sequentially rational given u 2 u is Bayesian given 039 Exercise 1 Consider the following interaction between two entrepreneurs players 1 and 2 who are working on a joint project and a venture capitalist player 5 who is a potential investor in the project First the entrepreneurs simultaneously decide whether to devote high or low e ort to preliminary work on the project They then make a presentation to the venture capitalist If both entrepreneurs chose high e ort the presentation goes well otherwise it goes poorly The venture capitalist only observes whether the presentation goes well or poorly he does not directly observe the entrepreneurs e ort levels The payo s are as follows Each entrepreneur obtains 5 if the venture capitalist invests and 0 otherwise In addition choosing high e ort costs an entrepreneur 1 while choosing low e ort is free Investing costs the venture capitalist 2 but if he invests he gains 3 for each entrepreneur who chose high e ort If the venture capitalist does not invest his payo is 0 All players are risk neutral 1 Draw an extensive form representation of this game 2 Find all perfect Bayesian equilibria in which all players choose pure strategies Figure Exitmpn39e39rnml1 3F game Exercise 2 1 Give an example of a game that has a weak perfect Bayesian equilibrium but is not sabgame perfect 2 Give an example of a game that has a sabgame perfect equilibrium but is not a weak perfect Bayesian