Lecture 7 Applications_Perfect Bayesian equilibrium

3 Corporate Investment and Capital Structure (Myers and Majluf (1984))
The pro…t of the existing company can be either high or low: = L or H, where H > L > 0. A new project is attractive requiring an investment I, the payo¤ will be R. The potential investor’s alternative rate of return is r, and R > I (1 + r). (I (1 + r) is opportunity cost). The timing and game tree for this game:
The …rms cannot distinguish between workers and they have to pay everyone the same wage. To satisfy te zero-pro…t condition, the wage scheme must be
w = E ( ) = q H + (1 q) L
4
Then the investor (who is the receiver in this game) will accept the o¤er s if and only if the expected cash ‡ow based on the equity share s exceeds the opportunity cost, namely,
1
receive a wage w such that
u (w; e; ) = w C (e; )
and the …rms’pro…t function is
(w; ) = w
2.1 The perfect information solution
When …rms can observe each worker’s productivity, a Bertrand equilibrium imples
w (e) = (e) H + [1 (e)] L
belief system thus becomes
(e) = w (e) L 2 [0; 1]; w (e) 2 [ L; H]
H
L
2.3.1 Separating equilibrium
The low type worker chooses eL and high type worker chooses eH. The belief consistency condition requires that
e
e
since education does not enhance productivity. The optimal education levels are
eH = eL = 0
2.2 The no signaling solution
Suppose that education is banned so that eH = eL = 0
@C (e; ) @2C (e; ) @C (e; )
C (0; ) = 0; @e > 0; @e2 > 0; @
< 0 for all e 0
@2C (e; )
and
<0
@e@
Let u (w; e; ) denote the utility of a type worker who chooses education level e and
One possible pooling PBE: any education level e 2 [0; e^] can be supported by a wage schedule leading to a pooling equilibrium.
2.3.3 Equilibrium re…nement WPBE=PBE since there is only one subgame. No pooling equilibrium can be survived under ED criterion; But there will be a unique Pareto-e¢ cient separating PBE under CD/ED criterion.
if
she
does
not
like
him,
she
talks
to
him
with
probability
1 4
.
Suppose
she
talks to him. What is the probability that she likes him?
13
Pr(likejtalk) = 1
3
34
3 4
+
2 3
1:
s [qL + (1 q) H + R]
I (1 + r) , s
I (1 + r) qL + (1 q) H + R
While the otherside as for the entrepreneur (to is sender), has to trade o¤ between receiving the …nancing at the cost of an equity stake of s and foregoing the project but get a pro…t of existing company . That is,
4
2 Job market signaling
For simplicity, we restrict attention to the cases of two types of workers with productivities y ( ; e) = such that the productivity for the H-type worker is H and the low type worker has a productivity H, where H > L > 0 and q = Pr ( = H) 2 [0; 1]. Here we assume that education does nothing for a worker’s productivity. The cost of obtaining education level e for a type worker is given by the twice continuously di¤erentiable function C (e; ), with
One possible separating PBE, eL = 0; eH = e~;
(e) = w (e) L such that w (0) = HL
L; w (e~) = H
Another possible separating PBE: eL = 0; eH = e~; w (e) = H if e e~; w (e) = L if e < e~.
(1) For eah e, the forms’wage o¤er constitute a NE; (2) Thቤተ መጻሕፍቲ ባይዱ workers’strategy e ( ) is optimal given the …rms’strategy; (3) There is consistency between the belief and the actual probability such that
Pooling equilibrium: Both types of entrepreneurs will o¤er the same capital share, i.e., s = sL = sH. Thus the posterior probability of observing s is exactly equal to the prior probability distribution given by nature, namely, q Pr ( = Ljs) = p.
(1 s) ( + R)
,s
R +R
Combining (??) and (??) implies that a pooling equilibrium exists only if
I (1 + r)
R
qL + (1 q) H + R + R
case 1: p ! 0, (??) holds because R > I (1 + r); case 2: p ! 1, (??) holds only if
(e) = actual probability that a worker with e is of high productivity
Firms’wage o¤er is determined by a Bertrand equilibrium-wage o¤er must equal the expected productivity of worker:
(eL) = 0 and (eH) = 1
then according to (??)
w (eL) = L; w (eH ) = H
In any separating PBE, eL = 0, that is a low-ability worker chooses to get no education;