逆向选择模型-

1、 Lecture on Contract Theory We can show C T as figure 4-1: wF C G* UB C B* UB UG VG = 0 VB =0 wS p B xS + (1 - p B xF p G xS + (1 - p G xF Fig. 4-1 Note 1 Gs indifference curve is more sloping than that of B, which implies that G will succeed with larger probability. Because p G u¢( wS p B u

2、62;( wS ¶U G / ¶wS dwF dw | EU G = - = - < - = F | EU B G G B dwS (1 - p u ¢( wF (1 - p u¢( wF dwS ¶U / wF Note 2 The optimal equilibrium contracts are on the 45 line. Note 3 with asymmetric information, all the B will have incentive to pretend to be G, as Fig. 4-1. (2 Po

3、oling equilibrium If a pooling equilibrium exist, it must satisfy expression (4-1, and on the line o V T = pT xS + (1 - pT xF - pT wS - (1 - pT wF = 0 Where p = qp + (1 - q p , and q is the probability of type G . T G B If the pooling equilibrium C T exists in which both G and B get the same payment

4、, it must on the cross of indifference curves of G and B , and principals zero profit budget line, as figure 4-2. 6 Lecture on Contract Theory VG = 0 wF UG VT = 0 CT UB wS pT xS + (1 - pT xF Notice that if P offer a new contract C T¢ Fig. 4-2 in the shade area of Fig. 4-2, G will accept for his

5、 T G utilities rise, and P will get positive profit ( V < V , but B will get less benefit. By doing so, C T ¢ is Pareto efficient for G and P, which leads to B will be screened. Hence, C T is impossibly pooling equilibrium. In a word, pooling equilibrium doesnt exist. (3 Separating equilibri

6、um Remember that the problem is that B want to pretend to be G , which is contrary to previous subsection. At first, we can find out a contract C efficient. Secondly, we find out a contract C G B* which is on the line V B = 0 and is Pareto = 0 and on B s which is on the line V B* G indifference curv

7、e. Equilibrium contracts pair (C , C is shown as figure 4-3. G wF UB VG = 0 C B* UG CG VB =0 wS 7 Lecture on Contract Theory Fig. 4-3 Technically, C G is defined by following expressions: (4-5 (4-6 G G u ( wB* = p B u ( wS + (1 - p B u ( wF p G xS + (1 - p G xF - p G wS + (1 - p G wF = 0 Expression

8、(4-5 indicates that there is no arbitrage advantage for B type, and (4-6 means that C G is also on zero profit line. Actually, it must above the line V = 0 , otherwise T principals will get positive profit, as the shade area of figure 4-4. wF VT = 0 UB VG = 0 C B* UG CG VB =0 Fig. 4-4 wS In fact, th

9、e little q , the more down line V = 0 . So, in order to assure that separating T equilibrium exists, we hope that q is sufficiently small. (4 Conclusions If q £ q , (C , C is a separating equilibrium; G B* n n n If q > q , equilibrium does not exist; In equilibrium state, the lowest efficien

10、t types get his utility as that under symmetric information, but the highest efficient types get less utility than that under symmetric information (C < C G G* . The loss of G type is the shade area of the Fig. 4-3. Maybe it is contrary to simple adverse selection problem. However, the property that “top component is not distorted” also holds. Differently, in this subsection the top component is B type that has incentive to pretend to be other types as the highest e