Second compulsory assignment:
Emphasis on economics of information
ECON4240 Game Theory and Economics of Information—Spring semester 2007
Proposed solutions to Problems 3 and 4
Problem 3
All potential contracts can be represented in a plane, where along the axes. No insurance is represented by the origin,

and
1

are measured
2
 
1
 
.
2
0 way: If
A consumer's marginal rate of substitution between

and
1

is changed to
1

1
  
1
and

is changed to
2

2
  
2

can be found this
2
, the change in the expected utility is (to first-order approximation):

EU
  
1

1
 
G
   
1
   G
   
2

Here G is equal to either H or L and refers to the two groups of consumers. Setting

EU

0 , this gives
(1) MRS
G 




2
1


1


G

G

  


1
2


Note that increased

is a disadvantage for the consumer while increased
1

is an
2 advantage; hence indifference curves have a positive slope. If insurance is not complete, we have
(2)
This is seen as follows: Incomplete insurance means that w T

2 w

1
MRS
G 
1



G
. Since u is concave, this gives
G

 
2
 
T , or equivalently

2
    
1

, and (2) follows from (1) . Intuitively, (2) says that a risk-averse consumer wants full insurance if it can be bought at actuarially fair terms.
Comparing the two groups of consumers, we get
MRS
L 
MRS
H
That is, in every point the indifference curve for the L -group is steeper than that for the
H -group.
(a)

Assume that the equilibrium consists in there being offered two different contracts,
 
1 2
L
 and

 
1
H
2

, each being chosen by the appropriate group of consumers. That is, the L -group prefers (at least weakly)

 
2
L

to

 
2
H

, while the H -group has the opposite (weak) preference. Together with the statement made about the steepness of indifference curves, this implies
1


1
L  
H

2
L  
(Consider the two indifference curves through
1

H
2
  the one for group L being the steeper. The contract
H
2

. They both have positive slope;

 
2
L

must lie above the one for the L -group and below the one for the H -group. The inequalities follow.)
Hence the deductible, T
   
2

is higher for the L -group.
If even the H -group pays a deductible, that is, if increase

and
1
H

1
H  
2
H 
T , it is possible to

, moving along the H -groups indifference curve through
2
H

 
1
H
2
 until

1
H  
2
H 
T . This makes the contract worse for type L consumers, so they do not choose it. The expected profits of the insurance company increases due to the change; this follows from (2) . Hence a company will prefer offering the modified contract, and the original situation was not an equilibrium.
The conclusion follows.
Could there exist a different type of equilibrium, in which only one contract is offered and accepted by every consumer? In that case the deductible, if any, would be the same for each group.
Such an equilibrium is impossible. Proof by contradiction:
Let
  
2

be such a contract. In order for the companies to break even or make a profit, we must have
 
1
 
2
. An intruder into the market can offer a contract

1
,
2

2

, with
 
1 2

0 so that MRS
L 


2
1

MRS
H
. This will be chosen only by consumers of type L . Getting rid of the type H consumers increases the company's profit. Hence the original situation was not an equilibrium.
(b)
If you have full insurance, that is, a contract without a deductible, your type does not matter for your expected utility, since your final wealth with be w
 
1 w T

2
in any case. It is (at least weakly) better to be of type L and consuming contract

 
2
L
 than to be of type L and consuming contract

 
2
H

; otherwise, type L consumers would have chosen

 
1 2
H

.
Faced with the hypothetical choice described, therefore, one would prefer "being a consumer with the low accident probability and consuming its contract with a deductible".
(c)
In the situation described in (a), consumers in the L -group do not get full insurance, although they are risk averse. The company can offer them a little more insurance at a rate which is not actuarially fair, But nevertheless have that offer accepted. That is,
2

instead of

 
2
L

, the company offers offer


1
L
 
1
L
,
 
L
2

0 and
1



L
L





1
2
L
L
   
2
L

MRS
L
  
2
L

, where
This can be advantageous to the type L consumers and increase the profit the company gets from them. The problem is that the change may lead to the type H consumers choosing this contract as well. The company will loose money on them, but if they are few enough ("there is a high proportion of consumers with the low accident probability"), it is nevertheless possible that its profit increases, and the original situation was not an equilibrium.
Problem 4
Watson Exercise 28.7 is based on Exercise 25.5. It is necessary first to solve parts (a) and (b) of the latter problem; part (c) being irrelevant. (Watson Chapter 25 has not been specifically covered in the lectures but occurs on the reading list. In any case, Exercise
25.5 can be solved based on what has been covered in the course.)
25.5(a)
The worker accepts the offer if firm's payoff is at most 180
  w y 100 , that is, w

100
 w 180
 
100
 y
 
80
 y . y . Hence the
25.5(b)
The worker accepts the offer if w

E
   
100 q

100 , that is,
Hence the firm's payoff is at most 200 w
 w
  q
    q

.
  q

.
28.7(a)
The parameters for the L -type are denoted y
L
and q , and similarly for the H -type.
L
For the L -type, the safe job gives the firm a profit of 80 gives the firm a profit of
 y
L

145 , while the risky job
  q
L
 
140 . Hence the firm offers the safe job, z

0 , and wage 100
 y
L

35 .
For the H -type, the safe job gives the firm a profit of 80 gives the firm a profit of
 y
H

155 , while the risky job
  q
H
 
160 . Hence the firm offers the risky job, z

1 , and wage
  q
H
 
40 .
28.7(b)
By taking the safe job, the H -type get a payoff of w
0  y
H

35 75 110 . If the firm wants the H -type to choose the risky job, w
1 
E
   w
1  w 1
must be chosen so that
100 q
H

110 , that is w
1 
50 .
3

Suppose the firm offers w
1 
50 and type and gives the firm a profit of 200 w
0 
35
 w
1 
. The former contract is chosen by the
150
H -
, while the latter contract is chosen by the L -type and gives the firm a profit of 180
145 p
   p
 
150 5 p . If w
1 
50
 w
0 
145 . Hence the expected profit is
, everybody chooses the safe job, and the profit is 145 for sure. Choosing w
1 
50 just reduces the firm's profit compared to w
1 
50 .
Hence the optimal value of w
1
is 50.
(To induce even the L -type to choose the risky job, w
1 
E
   w
1 
100 q
L
 w
0  y
L
, which requires w
1 w
1
must be chosen so that

60 , giving the firm a profit of
140. Hence this is not optimal.)
28.7(c)
The strategy of (b) gives expected profit 150 5 p .
Any strategy which induces both types to take the same job, gives a lower profit than this.
This follows from the arguments of (b).
There is one more possibility. The firm can try to attract the H -type to the risky job, making the safe job so unattractive that nobody accepts it, for example, by choosing w
0 
0 . Attracting the H -type to the risky job requires w
1 
40 , cf. (a). With the profit on the H -type worker is 160. Hence the expected profit is
  w
1 
40
160 1
 p . This
, strategy is better than the one of (b) if
  p
 
150 5 p or p

10

2
155 31
.
4