902: Issues in Economic Systems and Institutions Department of Economics, Delhi School of Economics
Solutions to nal exam. Winter Semester, 2016-17.
PART B 1. (a) Informed leader's optimal e ort:
e1( ) arg max
e1 1 (e1+e2) 1
2e21 = 1 Uninformed follower's optimal e ort:
e2 = arg max
e2 2
1
2(4 + 6) (e1+e2) 1
2e21 = 5 1 (b) The answer is the solution to
max
1; 2
1
2:4(4 1+5 2)+1
2:6(6 1+5 2) 1 2
1
24 21+ 1
26 21 1
25 22 subject to 1+ 2 = 1 The solution is
1 = 5
9; 2 = 4 9
(c) In a separating equilibrium, let eh and el be the leader's e ort choice when productivity is high and low respectively (i.e., = 6 and 4). Since follower learns the value of , his e ort choice is given by
e(eh) = 6 2 = 3 e(el) = 4 2 = 2
since 2 = 12. Also, el = 12:4 = 2, since the leader has no reason to signal extra hard when signaling low productivity. eh must satisfy two incentive constraints that make sure the leader does not want to switch to the high (low) productivity e ort level when actual productivity is low (high):
IC ( = 6): 1
2:6(eh+ 3) 1
2e2h 1
2:6(2 + 2) 1
2:22 )eh 3 +p 7
1
(we choose the higher root).
IC ( = 4): 1
2:4(eh+ 3) 1
2e2h 1
2:4(2 + 2) 1
2:22 )eh 4 Combining:
4 eh 3 +p 7
So the lowest e ort level ebh which will \separate" the high productivity type from the low productivity type is 4. In the separating equilibrium with lowest e ort, the value of eh is
maxfebh; e1(6)g= minf4;3g
[Note that if lowest e ort level needed to separate from the low productiv- ity type is lower than the high productivity type's optimal e ort under full information, she will choose the latter].
(d) In this example, due to signaling motives, the leader supplies higher e ort under incomplete information than under full information (but still less than
rst best). Hence, team surplus is higher.
2. (a) (i) Babbling. (ii) 2-interval partition: [0;187), [187;1]:(iii) 3-interval partition:
[0;19), [19;49), [49;1].
(b) Optimal ceiling
arg min
a
Za
0
b2d +
Z1
a
( a)2d = 1 b= 17 18
Calculate welfare under the optimal ceiling, using the formula above, and under cheap talk using the derivation in the slides.
(c) The equilibrium strategies are as follows: (i) the unbiased type (b= 0) sends the message m = for < and m = m for (ii) the biased type (b = 1), who always prefers a higher action in the [0;1] interval, sends the message m=m for all . The receiver's response is
a(m) = m if m <
= if m=m 2
By Bayes' rule
Pr(b = 1jm=m) = = :1
:1 + (1 )(1 )
Note that the biased type sends the message m with higher probability than the unbiased type, hence beliefs about types have to be updated after m. If the biased type has sent m, expected value of is 12. If the unbiased type has sent it, that expectation is 1+2 . The message m must lead to an expected value of equal to .
E( jm=m) = :1
2+ (1 )(1 + )
2 =
Use the last two equations to solve for . It is the solution to the quadratic (1 ) 2+ 2(2 1) + 1 2 = 0
3. (a) (i) Majority rule: if the voting threshold isx, a yes vote implies the conditional expectation of the corresponding random variable is 1+x2 , while a no vote implies the conditional expectation is x21. Being pivotal means there is 1 yes vote and 1 no vote. If the voter's own random variable is x, she should be indi erent between a yes and a no vote, i.e.,
x0 +x+ 1 +x
2 +x 1
2 = 0)x= x0 2
(ii) Unanimity rule: being pivotal means other 2 votes are yes. Using the indi erence condition:
x0+x+ 2 1 +x
2 = 0 ) 1 +x0 2
(b) (i) Majority rule: investment happens if there are 2 or 3 yes votes, i.e., with probability
3 1 x
2
2 1 +x
2 + 1 x
2
3
= 1
4 1 + x0 2
2
2 + x0 2 3
(ii) Unanimity rule: investment happens if all 3 votes are yes, i.e., with prob- ability
1 x
2
3
= 3 +x0 4
3
(c) (i) Under majority rule, probability of a yes vote (usingx0 = 0) is 12. Expected pro t:
3: 1 2
2 1
2 2:1 2
1
2 + 1 2
3
3:1 2 = 3
8
(ii) Under unanimity rule, probability of a yes vote (using x0 = 0) is 12. Expected pro t:
3 4
3 1
2 1 1
2 = 81 256 Higher expected pro t under majority rule.
4
