ECON30001 Advanced Microeconomics Final Assessment, Semester 1 2021/22.
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ECON30001 Advanced Microeconomics
Final Assessment, Semester 1 2021/22.
Question 1 (15 marks).
a)Considerasetofmonetaryoutcomes
= {
1 , … , 
}wheretheoutcomesarearranged in increasing order 
1 < ⋯ < 
. Let 
and 
be simple lotteries. We say that lottery 
first-order stochastically dominates lottery 
, if the following holds:
∑ 
(
) ⩾ ∑ 
(
) for all = 1, … 
,
= 
=
with at least one of the inequalities being strict. Explain the intuition of first-order stochas- tic dominance. Show that, for expected utility preferences with vNM utility 
that is strictly increasing in money, if 
first-order stochastically dominates 
then 
≻ 
.
b) Consider the following decision problems:
|
Decision 1: Choose between
A. A sure gain of £240.
B. A 25% chance to gain £1000, and a 75% chance to gain nothing. |
|
Decision 2: Choose between
C. A sure loss of £750.
D. A 75% chance to lose £1000, and a 25% chance to lose nothing. |
The lotteries in each decision are independent. When people are offered both decisions,
and it is explained they will receive both of their choices, it has been found that most people
choose 
and 
, so receive 
+ 
. Discuss why such preferences might occur. Compare
this pair of choices with the other possible pairs and discuss.
Question 2 (15 marks).
a) For each bargaining game 
a solution concept 
selects an alternative 
(
) ∈ 
. If the solution concept selects alternative 
(
) = (
1 (
), 
2 (
)), then player 1 receives utility 
1 (
) and player 2 receives utility 
2 (
). Suppose that a solution concept 
satisfies “Property A”:
|
Property A: For bargaining games, |
Discuss Property A and prove / disprove if the Nash solution satisfies this property.
b) Two players, both with zero wealth, bargain over how to divide £
> 0 between them. Failure to reach agreement means both get nothing. Both players are expected utility max- imisers. Player 1 has utility 
(
) = 
, where 0 < 
< 1. Player 2 has utility 
(
) = 
, where 0 < 
< 1. Determine the Nash solution for this problem and discuss.
Question 3 (20 marks). A firm’s revenue 
is stochastically related to the effort exerted by its employee. Effort is a continuous variable. The employee can choose any level of
effort 
∈ [0, ∞). The choice of effort affects revenue so that:
(
"
) = 
and 
(
"
) = 1
where 
(
"
) and 
(
"
) denote the expected value and variance, respectively, of rev- enue when the employee exerts effort level 
. The employer cannot observe the level of
effort exerted by the employee. The employer wants to design a wage contract 
based on the revenue and considers only contracts of the form:
= 
+ 
,
and so the employee is guaranteed a payment 
and then a bonus payment 
which de- pends on revenue. The employee is a risk-averse expected utility maximiser. A contract 
gives expected utility:
(
"
) = 
(
"
) − 
(
"
) − 
(
)
where 
(
"
) and 
(
"
) denote the expected value and variance of the contract, re- spectively, conditional on effort 
, 
is a parameter of risk aversion, and 
(
) denotes the disutility of effort. For this employee, 
(
) = 
2 . If the employee rejects the contract, they
receive reservation utility of zero.
Explain how the employer can implement a level of effort 
.
Show that the optimal contract has the property that the bonus payment decreases with the level of risk aversion.
Note that for a random variable 
and constants 
and 
we have , 
(
) = 
2 
(
), 
(
+ 
) = 
(
), 
(
) = 
(
) and 
(
+ 
) = 
(
) + 
.
2022-01-28