6001: ECON Solution to Assignment 8
(1) An agents utility function is u = x1 x2 . The market prices are p1 =
p2 = 1. The consumers income however is uncertain: the income
could be m1 = 100 or m2 = 50, each event being equally likely. What
is the maximum expected utility of the consumer?
Given any income m, the utility maximizing bundle is x1 = m/2 and
x2 = m/2. Hence the maximum utility that the the agent can get
when his income is m is V (m) = m2 /4. Thus the expected utility of
the consumer from the lottery is 1/2(100)2 /4 + 1/2(50)2 /4.
(2) Consider two identical individuals with utility function u(x) = x.
Each has an initial wealth of 100. Moreover, each can invest in a
project that will pay either 400 or 100 with equal probability. It costs
200 to invest in this project. Borrowing is possible at zero rate of
interest.
(i) Show that no single individual will invest in the project.
If the agent invests, he will get 1/2u(400 100) + 1/2u(0) =
300/2. If he does not invest, his utility is u = 100 = 10.
Clearly 10 > 3002 and thus the agent is better o not investing.
(ii) Show that investment will be protable if these individuals form
a partnership.(Hint:use the equal sharing rule).
If they form a partnership with equal sharing rule, the expected
utility of an agent is given by
200 + 50
1/2u(200) + 1/2u(50) =
2
It is easy to check that this amount is greater than 10, the utility
from not investing.
(3) Consider a modied version of Ellisberg paradox discussed in the lecture notes: There is an urn that contains 300 colored balls. 100 of
these balls are red, the remaining are either all blue or all green. It
is also known that the probability that the remaining balls are blue
equals p. Mr. X is supposed to pick a ball from this urn. The lotteries
faced by X are as follows:
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Lottery A: get $1,000 if the ball is red and zero otherwise.
Lottery B: get $1,000 if the ball picked is blue and zero otherwise.
Lottery C: get $1,000 if the ball picked is not red.
Lottery D: get $1,000 if the ball is not blue.
Show that if X is an expected utility maximizer and if X prefers lottery
A over lottery B , then he prefers lottery D over lottery C .
Let u(.) be the utility function of X . Since X prefers lottery A over
B and he is an expected utility maximizer, we have
u(1000) 2pu(0)
(1 + 2(1 p))u(0) 2pu(1000)
+
>
+
3
3
3
3
(0.1)
From the rst equation , we get
u(1000)[1 2p] > u(0)[1 2p]
(0.2)
Since u(.) is increasing, it follows from equation (2) that
1 2p > 0
(0.3)
u
Now the expected utility from lottery C is EU (C ) = u(0)+23 (1000) while
))
the expected utility from lottery D is EU (D) = 1+2(1p3 u(1000) +
2pu(0)
3 . EU (D ) > EU (C ) if and only if u(1000)(1 2p) > u(0)(1 2p)
which holds because of equation (3), i.e, 1 > 2p.
(4) Consider the TV game show discussed in the class. Suppose the contestant has chosen Door 1. If the car is behind this door, then the host
will decide to open any other door with equal probability. However, if
the car is not behind this door, then the host can only open the door
that does not have the car. Suppose the host opens Door number 3.
Show that the probability that the car is behind door 2 is 2/3.
Let A be the event that door 2 has the car and let B be the unconditional probability that door 3 will be opened. Observe that door
3 will be opened with probability if 1 door 2 has the car and with
probability 1/2 if door 2 does not have the car.1 Thus, probability
1
Door 3 can not be opened if it has the car.
2
of B equals 1/3(1) + 1/3(1/2) = 1/2. Thus by Bayes rule, probability that door 2 has the car given that door 3 was opened is given by
(
P (A|B ) = P PABB ) . Note that if door 2 has 2 has the car, then door
()
3 must be opened with probability 1. Thus, P (A B ) = P (A) = 1/3
1/3
and P (A|B ) = 1/2 = 2/3.
(5) Suppose you have 1 dollar to invest in a project. The outcome of the
project depends on the state of the world. There are two states of
the world: 1 and 2 and they are equally likely. In state 1, the project
returns 4, 2 or nothing. Each event has an equal chance. In state 2,
the return is zero for sure.
(i) Will you invest in the project if you are i) risk-neutral, ii) risk
averse?
The net expected return from investing in this project is 1/2(2) +
1/2(0) 1 = 0. A risk neutral agent is indierent between investing or otherwise. A risk averse person will strictly prefer not to
invest.
(ii) Before making the investment, you can consult a nancial analyst
who can tell you the true state of the world. If you are riskneutral, what is the maximum amount that you will be willing to
pay for the analysts service?
If F is the amount that you have to pay,then your expected return
from consulting the analyst is
1/2(2 1) + 1/2(0) F
This payo must be non negative and thus F 1/2.
(6) Suppose a risk averse agent has an asset valued at W = 100. With
probability 1/2, there will be a loss and the value of the asset will
be zero while with probability 1/2, the asset continues to retain its
value of 100. An insurance company oers a contract that provides a
complete insurance. If the utility function of the agent is u(x) = x1/2 ,
(a) What is the maximum premium that this agent is willing to pay
to buy this insurance?
Let X be the total amount of the premium that agent is willing
to pay. Since the contract provides full insurance, the payo
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to the agent is
independent of the asset value and is given by
u(100 X ) = 100 X . If he does not buy the insurance, his
expected utility is 1/2u(100) = 1/2 100 = 5. Thus, 100 X =
5 and X = 75.
(b) If the insurance company is risk neutral, what is the minimum
value of the premium at which the company will be willing to
oer this contract?
Given X , the value of the total premium, the expected payo to
the risk-neutral company is X 1/2(100). This payo must be
non negative. Thus, the minimum value of X is 50.
Assume now that the agent can take an action (call it care) which
aects the probability of the loss being realized. In particular, assume
that agent can spend an amount C = 2 which will reduce this probability to 1/4. However, the loss probability remains at 1/2 if the agent
does not spend this amount.
(a) If the agent had no insurance, will he take care ?
If he does not take care, his payo is 5 (this is 1/2u(100)). If he
takes care, his expected utility is 3/4u(100) C = 3/4(10) C .
Since C = 2, this payo equals 11/2. Thus, the agent will take
care.
(b) If the company provides complete insurance, what is the maximum insurance premium that the agent will be willing to pay?
Under complete insurance, the agent will not take care. Thus
u(100 X ) must equal 11/2, thus the maximum value of X is
given by X = 100 121/4.
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