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# Risk-averse expected utility maximiser, Jenny has initial wealth

Risk-averse expected utility maximiser, Jenny has initial wealth .

A risk-averse expected utility maximiser, Jenny has initial wealth w and utility function u. She faces a risk of a financial loss of L dollars, which occurs with probability π.

## Risk-averse expected utility maximiser, Jenny has initial wealth

1. A risk-averse expected utility maximiser, Jenny has initial wealth w and utility function u.
She faces a risk of a financial loss of L dollars, which occurs with probability π. An
insurance company offers to sell a policy that costs P pounds per pound of coverage. (per
pound paid back in the event of a loss). Denote by x the number of pounds of coverage. [30
marks]
a) [5 marks] Given x, Jenny faces the lottery below. Find payoffs w1 and w2.

b) [5 marks] Find Jenny’s expected utility V(x) as a function of x.

c) [10 marks] Suppose that ( ) = −

, = 1
4
⁄ , = 100, = 1/3. Find the
optimal value of x, as a function of λ and w by solving the first-order condition. Does
the optimal amount of coverage increase or decrease in λ?
d) [10 marks] Repeat b), but with = 1/6. Does the optimal amount of coverage increase
or decrease in λ? Compare it with the result in b)

2. Jimin is an expected utility maximiser, with the VNM utility function ( ) =
ln (x), > 0. [20 marks]

### a) [5 marks] What is Jimin’s certainty equivalent of the following lottery:

Probability Money
0.4 30
0.5 100
0.1 500
b) [5 marks] What is the risk premium Jimin is willing to pay to insure against this
uncertain prospect?

c) [10 marks] Suppose Jimin has £1,200,000 in wealth, and decides to become a
backsliding oil prospector. He finds a tract of land for sale for £1,000,000, which will
produce no return at all if no oil is found, or will yield £10,000,000 of income if oil is
found. Let p be the probability that oil is found. Specify the two lotteries that result
from the actions, “buy that land” and “not buy the land”. What probability p of finding
oil would make Jimin exactly indifferent between buying the land and not buying the
land?

#### 3. Tom produces widgets at a cost of 10 per widget.

He can only sell widgets to Jerry, and Jerry an only buy them from tom. Jerry values the widgets he buys according to the utility
function = 110 − 10
2
, where q is the number of widgets that Jerry buys from Tom
and UJ is Jerry’s utility. Tom’s utility is = −10q if q widgets are produced. The number
of widgets, q is determined by bargaining between Tom and Jerry. [25 marks]

3
a) [5 marks] What is the outcome of disagreement?

b) [5 marks] Find q which maximises the joint value.

c) [5 marks] Given q from part b), will Tom and Jerry reach an agreement? Explain.

d) [10 marks] If there has to be a side payment to agree with the quantity q from part b),
who will pay to whom and how much? (We assume that Tom and Jerry have equal
bargaining powers)

##### 4. A firm wants to hire you as a behavioural economist to consult their payment schemes.

To incentivise their employees to work hard in 40 projects over the year, the firm is considering
3 payment schemes:

(i) pay a fixed salary of £32,000; (ii) pay no fixed salary, but pay
£1000 for completion of each successful project, and nothing for unsuccessful projects;

(iii)
pay £40,000 as initial fixed salary, but then deduct £1000 for each unsuccessful project.
It is known that the average probability of doing the project successfully is 0.8, and the
probability of success increases with an increase in effort by the employees. While (ceteris
paribus) the employees prefer to exert less effort than more, the objective of the firm is to
obtain as many successful projects as possible.

Which scheme would you like to suggest to the firm? Explain your reasons based on the
behavioural theories covered in this class. Feel free to use any diagrams. (No more than
500 words) [25 marks]

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