the effort and sales should be higher.

Transcription

1 10.3 An Example of Postcontractal Hidden Knowledge: The Salesman Game ð If the cstomer type is a Pshoer, the efficient sales effort is low and sales shold be moderate. ð If the cstomer type is a Bonanza, the effort and sales shold be higher. The Salesman Game ð Players r a manager and a salesman

2 ð The order of play 1 The manager offers the salesman a contract of the form [ wm ( ), qm ( )], where w is the wage, q is sales, and m is a message. 2 The salesman decides whether or not to accept the contract. 3 Natre chooses whether the cstomer type t is a Bonanza or a Pshoer with probabilities 0.2 and 0.8. The salesman obseres the type, bt the manager does not.

3 4 If the salesman has accepted the contract, he chooses his effort e. His sales leel is q œ e, so his sales perfectly reeal his effort. 5 The salesman's wage is wm ( ) if he chooses eœ qm ( ) and zero otherwise.

4 ð Payoffs r The manager is risk-netral and the salesman is risk-aerse. r If the salesman rejects the contract, his payoff is U _ œ 8 and the manager's is zero. r If he accepts the contract, then 1 œ q w, and 1 œ U( e, w, t), manager 2 2 salesman where `UÎ`e 0, ` UÎ`e 0, `UÎ`w 0, 2 2 and ` UÎ`w 0. ð The manager can perfectly dedce effort, een ot of eqilibrim.

5 The optimal contract ð The manager's indifference cres are straight lines with slope 1. ð The salesman's indifference cres slope pwards, and are conex. r The salesman has two sets of indifference cres, solid for Pshoers and dashed for Bonanzas.

6 ð Figre 10.1 r The optimal trth-telling contract is the pooling contract that pays the intermediate wage of w 3 for the intermediate qantity of, and q 3 zero for any other qantity, regardless of the message. r The pooling contract is a second-best contract, a compromise between the optimm for Pshoers and the optimm for Bonanzas. r The contract mst satisfy the participation constraint, 0.8 U( q, w, Pshoer) 0.2 U( q, w, Bonanza)

7 ð The natre of the eqilibrim depends on the shapes of the indifference cres. ð Figre 10.2 r The eqilibrim is separating, not pooling, and there does exist a first-best, flly reealing contract. r The contract indces the salesman to be trthfl, and the incentie compatibility constraints are satisfied. ð The idea is to reward salesmen not jst for high effort, bt for appropriate effort.

8 Another way to look at a separating eqilibrim is to think of it as a choice of contracts rather than as one contract with different wages for different otpts. ð In this interpretation, the manager offers a men of contracts and the salesman selects one of them after learning his type.

9 The Salesman Game illstrates a nmber of ideas. ð It can hae either a pooling or a separating eqilibrim. ð The reelation principle can be applied to aoid haing to consider contracts in which the manager mst interpret the salesman's lies. ð It shows how to se diagrams when the algebraic fnctions are intractable or nspecified.

10 10.4 The Groes Mechanism ð The principal is an altristic goernment that cares directly abot the tility of the agents. r a beneolent goernment ð The mayor is considering installing a streetlight costing $100. r He will only install it if he decides that the sm of the residents' alations for it is greater than or eqal to the cost. r The mayor's problem is to discoer their alations.

11 The Streetlight Game ð Players r the mayor and fie hoseholders ð The order of play 0 Natre chooses the ale i that hoseholder i places on haing a streetlight installed, sing distribtion f( ). Only hoseholder i obseres. i i i

12 1 The mayor annonces a mechanism, M, which reqires a hoseholder who reports m to pay w( m) if the streetlight is installed, and installs the streetlight if gm, ( m, m, m, m) m jœ1 j 2 Hoseholder ireports ale with all other hoseholders. m i simltaneosly 3 If gm, ( 1 m, 2 m, 3 m, 4 m5 ) 0, the streetlight is bilt and hoseholder ipays w( m i ).

13 ð Payoffs r The mayor tries to maximize social welfare, inclding the welfare of taxpayers besides the 5 hoseholders. r His payoff is zero if the streetlight is not bilt. r Otherwise, it is 5 1 mayor œ j jœ1 100, 5 sbject to the constraint that wm ( ) 100, jœ1 so he can raise the taxes to pay for the light. j

14 r The payoff of hoseholder i is zero if the streetlight is not bilt. r Otherwise, it is 1 i m, 1 m, 2 m, 3 m, 4 m5 œ i wm i ( ) ( ).

15 Mechanisms ð Mechanism M 1 r w( m ) œ 20, Bild iff m 100 i 5 jœ1 j r Talk is cheap, and the dominant strategy wold be to oerreport or nderreport. r a flawed mechanism

16 ð Mechanism M 2 r w( m ) œ Max { m, 0}, Bild iff m 100 i i j jœ1 5 r If all the hoseholders knew each other's ales perfectly, then there wold be a continm of Nash eqilibria that attained the efficient reslt. r Each hoseholder wold annonce p to his alation if necessary.

17 ð Mechanism M 3 r w( m ) œ 100 m, Bild iff m 100 i j j jái jœ1 5 r a Nash eqilibrim in which all the players are trthfl r a dominant-strategy mechanism ñ Trthflness is weakly dominant. ñ The players are strictly better off telling the trth wheneer lying wold alter the mayor's decision. r It is not bdget-balancing. r The total tax reene cold easily be negatie.

18 10.5 Price Discrimination ð A problem of mechanism design nder aderse selection Varian's Nonlinear Pricing Game ð Players r one seller and one byer ð The order of play

19 0 Natre assigns the byer a type, s. The byer is "nenthsiastic" with tility fnction or "aling" with tility fnction, with eqal probability. The seller does not obsere Natre's moe, bt the byer does. 1 The seller offers mechanism { w, q } nder which the byer can annonce his type as m and by amont q for lmp sm w. m m m m 2 The byer chooses a message m or rejects the mechanism entirely and does not by at all.

20 ð Payoffs r The seller has a zero marginal cost, so his payoff is w w. r The byers' payoffs are 1 œ q ( ) w and 1 œ q ( ) w if q is positie, and 0 if q œ 0, w w ww ww with, 0 and, 0. r The marginal willingness to pay is greater for the aling byer: for any q, w w ( q) ( q). (10.27)

21 ð Condition (10.27) is an example of the single-crossing property. r Combined with the assmption that (0) œ (0) œ 0, it also implies that q ( ) q ( ) for any ale of q.

22 Perfect Price Discrimination ð The game wold allow perfect price discrimination if the seller did know which byer had which tility fnction. ð The seller's maximization problem r Maximize w w q, q, w, w sbject to the participation constraints ñ q ( ) w 0 ñ q ( ) w 0

23 ð The constraints will be satisfied as eqalities. r w œ q ( ) r w œ q ( ) ð The seller's maximization problem rewritten r Maximize ( q) ( q) q, q w * w * ð ( q ) œ 0 ( q ) œ 0 * * * * w œ q ( ) w œ q ( ) r The entire consmer srplses are eaten p.

24 Interbyer Price Discrimination ð The interbyer price discrimination problem arises when the seller knows which tility fnctions Smith and Jones hae and can sell to them separately. ð Assme that the seller mst charge each byer a single price per nit and let the byer choose the qantity.

25 ð The seller's maximization problem r Maximize p q p q q, q, p, p sbject to the participation constraints ñ q ( ) pq 0 ñ q ( ) pq 0 and the incentie compatibility constraints ñ q œ argmax [ ( q ) p q ] ñ q œ argmax [( q ) p q ]

26 ð The byers' qantity choice problems r w ( q ) p œ 0 r w ( q ) p œ 0 ð The seller's maximization problem rewritten r Maximize ( q ) q ( q ) q q, q w w sbject to the participation constraints w ñ q ( ) ( q) q 0 w ñ q ( ) ( q) q 0

27 ð The participation constraints will not be binding. r q ( ) ( q) q is increasing in q. w r q ( ) ( q) q is increasing in q. w ð The first-order conditions ww w r ( q ) q ( q ) œ 0 ww w r ( q ) q ( q ) œ 0 r two independent problems ð If the cost fnction were a more general conex fnction cq ( q), the two first-order conditions wold hae to be soled together, becase each condition wold depend on both q and q.

28 Back to Nonlinear Pricing ð Interqantity price discrimination r The seller charges different nit prices for different qantities. ð Neither the perfect price discrimination nor the interbyer problems are mechanism design problems. r The seller is perfectly informed abot the types of the byers. ð The original game is a problem of mechanism design nder aderse selection. r Separation is the seller's main concern. r The seller designs incenties to separate the types of the byers.

29 The eqilibrim mechanism ð The seller's maximization problem r Maximize w w q, q, w, w sbject to the participation constraints ñ q ( ) w 0 ñ q ( ) w 0 and the self-selection constraints ñ q ( ) w q ( ) w ñ q ( ) w q ( ) w

30 ð Not all of these constraints will be binding. r In a mechanism design problem like this, what always happens is that the contracts are designed so that one type of agent is pshed down to his reseration tility. Sppose that the optimal contract is in fact separating, and also that both types accept a contract. The nenthsiastic consmer's participation constraint is binding. r w œ q ( )

31 The aling consmer's self-selection constraint is binding. r w œ w q ( ) q ( ) ð The seller's maximization problem reformlated Maximize q ( ) q ( ) q ( ) q ( ) q, q ð The first-order conditions w w w r ( q ) { ( q ) ( q )} œ 0 r w ( q ) œ 0

32 The aling type bys a qantity sch that his last nit's marginal tility exactly eqals the marginal cost of prodction. r w ** ( q ) œ 0 r His consmption is at the efficient leel. The nenthsiastic type bys less than his first-best amont. r the single-crossing property that ( q) ( q) w w w r ( q ) { ( q ) ( q )} œ 0 w w r w ** ( q ) 0

33 The seller mst sell less than first-best optimal to the nenthsiastic type so as not to make that contract too attractie to the aling type. On the other hand, making the aling type's contract more alable to him actally helps separation, so q is chosen to maximize social srpls. ** ** q q r the single-crossing property that ( q) ( q) r ww ( q) 0 w ** w ** r ( q ) 0 and ( q ) œ 0 w w

34 The eqilibrim is separating, not pooling. A corner soltion ð Despite facing a monopolist, the aling type can end p retaining consmer srpls an informational rent. r a retrn to his priate information abot his own type

35 The Single-Crossing Property Condition (10.27) is an example of the single-crossing property, since it implies that the indifference cres of the two agents cross at most one time. The aling byer has stronger demand than the nenthsiastic byer. w r ( q) ( q) for all q w

36 Two cres satisfying the single-crossing property r q ( ) œ q r q ( ) œ 2 q It is often natral to assme that the single-crossing property holds, and it is a sefl sfficient condition for separation to be possible, bt it is not a necessary condition.