Microeconomic Theory II

The First-Best OutcomeSuppose that r is perfectly observable. Then: e i D 0w i D r iJianfei Shen (Shandong University) 5/32

The Extensive-FormStage 1:The worker chooses e to maximize his expected utility.Let p i 2 .R C / denote the mixed strategy of the worker oftype i.Stage 2:The wage is determined through bargaining, where theworker has all the bargaining power:Let ˇ.r i j e/ denote the firm’s revised beliefs aboutproductivity upon observing e.Then the equilibrium wage isw.e/ D ˇ.r H j e/r H C ˇ.r L j e/r L :Jianfei Shen (Shandong University) 6/32

Perfect Bayesian Equilibrium (PBE)DEFINITION A PBE is a triple ˝p L ; p H ; ˇ.r i j e/˛ such that: p i .e / > 0 H) e 2 argmax e˚ˇ.rH j e/r H C ˇ.r L j e/r L i e ˇ.r i j e/ D†p i .e/ˇip L .e/ˇL C p H .e/ˇHif p L .e/ C p H .e/ > 0any value in Œ0; 1 if p L .e/ C p H .e/ D 0: w.e/ D ˇ.r H j e/r H C ˇ.r L j e/r L .Jianfei Shen (Shandong University) 7/32

Some PBEs in the Education ModelThere are many PBEs. We focus on:Separating PBEs:type exactly.The signal chosen by the worker identifies hisPooling PBEs:The observed signal reveals no additionalinformation about the worker’s type.Jianfei Shen (Shandong University) 8/32

Separating PBEDEFINITIONA separating equilibrium is a PBE where each typeof worker chooses a different education level in equilibrium:e L ¤ e H so that ˇ.r H j e H / D 1, ˇ.r L j e L / D 1, and w i D r i .Jianfei Shen (Shandong University) 9/32

Separating Equilibrium Levels of EducationPROPOSITIONeducation is given by(S s DThe set of separating equilibrium levels of.e L ; e H / W e L D 0 and e H 2Proof. I will prove everything in class.rHr L L; r H r L H ) :Jianfei Shen (Shandong University) 10/32

Separating PBEwu L D r Lu H D r H H e Hr Hw.e/r Lw.e/0 r Hr Le Hr H r L eL HJianfei Shen (Shandong University) 11/32

Pooling PBEwr Hu L D r p L e pu H D r p H e pr pr Le 0 p r p r Le LJianfei Shen (Shandong University) 14/32

1 Spence’s Model of Education as a Signal2 Refinement: Intuitive Criterion3 Cheap TalkJianfei Shen (Shandong University) 15/32

The Basic IdeaThere may be so many different PBEs.We introduce restrictions on the set of allowableout-of-equilibrium beliefs using the observation that sometypes of players are more likely to choose some deviationsthan others.Jianfei Shen (Shandong University) 16/32

Cho and Kreps’ Intuitive CriterionDEFINITION Let u iD wi .e i/ i e i denote the equilibriumpayoff of type i. Thenˇ r j j e D 0 for e ¤ e i ; e j ;whenever r H j e < uj and r H i e > u i .Jianfei Shen (Shandong University) 17/32

Cho-Kreps Unstable Pooling PBEwr Hu L D r p L e pu H D r p H e pr pr Le 0 p r p r Le d e LJianfei Shen (Shandong University) 18/32

Least-Cost Separating Equilibriumwu L D r Lu H D r H H e Hr Hw.e/r Lw.e/r Hr L0 r H r L eL HJianfei Shen (Shandong University) 19/32

1 Spence’s Model of Education as a Signal2 Refinement: Intuitive Criterion3 Cheap TalkJianfei Shen (Shandong University) 20/32

The ModelTwo player: a Sender (S) and a Receiver (R). The state of the world is distributed uniformly on a set .S observe his type, 2 , and sends a costless messagem 2 to R.R processes the information in S’s message and chooses anaction y, which determines players’ payoffs:U R .y; / D .y / 2 ;U S .y; ; b/ D Œy . C b/ 2 ;where b > 0 is a “bias” parameter that measures how closelyaligned the preferences of the two are.Jianfei Shen (Shandong University) 21/32

PBEDEFINITION A (pure strategy) PBE consists of a strategy for S,denoted q./, a strategy for R, denoted y.m/ and a belief system,denoted ˇ. j m/ 2 b./, such thatGiven R’s strategy y.m/, the sender of type send messageq./ so thatq./ 2 argmax U S y.m/; ; b :m2The receiver’s action y.m/ satisfiesZy.m/ 2 argmax U R .y; /ˇ. j m/ d:yˇ.m/ is derived via the Bayesian rule whenever possible.Jianfei Shen (Shandong University) 22/32

Two States of the WorldPROPOSITION Suppose that D f L ; H g with L < H . Ifb 6 . H L /=2, then the strategy profile q. i / D i , y. i / D i andthe beliefs system ˇ. i j i / D 1 consists of a PBE.Jianfei Shen (Shandong University) 23/32

Proofyy S D C by R D H L C b L0 L HJianfei Shen (Shandong University) 24/32

Three States of the WorldPROPOSITION L < M < H . IfSuppose that D f L ; M ; H g withb 6 minH2 M; M L;2then the strategy profile q. i / D i , y. i / D i and the beliefssystem ˇ. i j i / D 1 consists of a PBE.Jianfei Shen (Shandong University) 25/32

Proofy H C b H M C b M L C b L0 L M H Jianfei Shen (Shandong University) 26/32

Continuous Information Let D Œ0; 1.As the number of possible states grows, full revelationbecomes ever more difficult to achieve.For a sender to be able to distinguish among all possiblestates, b must be smaller than one-half of the distancebetween any two of them. But as the number of states tendsto infinity, this requirement becomes impossible to fulfil.Jianfei Shen (Shandong University) 27/32

No Full RevelationPROPOSITIONentail some information loss.If the sender is even slightly biased, all equilibriaJianfei Shen (Shandong University) 28/32

ExampleSuppose S uses the following strategy:„m 1 if 2 Œ0; a/q./ Dm 2if 2 .a; 1m 1 or m 2 if D a:If R observes m 1 , he believes that lies in Œ0; a/ uniformly,and if R observes m 2 , he believes that lies in .a; 1 uniformly.Then R uses the following strategy:y.m/ D˚a2 µ y 1 if m D m 11Ca2µ y 2 if m D m 2 :Jianfei Shen (Shandong University) 29/32

We first verify that there exists a 2 .0; 1/ for some b such thatthe profile .q; y/ and the belief system ˇ described aboveconsist of a PBE.Since the a-type is indifferent between m 1 and m 2 , whichimplies that he is indifferent between y 1 and y 2 , we have.a C b/ y 1 D y 2 .a C b/ Itherefore,a D 1 22b: (1)It follows from (1) that there exist a unique a wheneverb 2 .0; 1=4/.Jianfei Shen (Shandong University) 30/32

yy S D C by 2y 1a0 1 Jianfei Shen (Shandong University) 31/32

Partition EquilibriumPROPOSITION All equilibria partition the state space into afinite number of intervals. The information conveyed in the mostinformative equilibrium is decreasing in the bias of the sender.Jianfei Shen (Shandong University) 32/32