Fall 2023: Practice Problem set 3
Micro 1
November 29, 2023
Total: 220 points
1. Consider the following vNM utility function u : [0, 200] → R where
√
u(z) = z.
Consider the lottery L that gives z = 64 with probability 0.25 and z = 36 with
probability 0.75.
(a) (5 points) Calculate the EU of the lottery L.
(b) (5 points) Find the certainty equivalent for the lottery L.
(c) (5 points) Show that the utility function represents a risk-averse preference.
2. For some increasing utility function u show the following lotteries over monetary
prizes in a two-dimensional plane, where the x axis shows the monetary prize and
the y axis shows the utility of lotteries.
(a) (10 points) L1 = [($25, 0.8), ($100, 0.2)]
(b) (20 points) L2 = [($30, 0.2), ($60, 0.2), ($90, 0.6)]
3. Suppose in an economy there are two types of consumers, young and old. The
young people have a probability 0.25 of being sick and the old people has a prob-
ability of 0.45 of being sick. Both types have an income 100 in good state and 80
in bad state, because they have to pay for the hospitalization cost when they fall
ill. Suppose, an insurance company is offering a premium of 0.5 per dollar.
(a) (15 points) Show that the young people will not buy non-zero amount of
insurance (we assume that the young and the old agents cannot sell insurance).
(b) (15 points) Solve for the optimal insurance decision of the old people.
(c) (10 points) Suppose the government forces all individuals to buy full insur-
ance (Y = 20) at a concession price of γ = 0.4. Compare the EU of both
types of agents before and after the policy change.
(d) (10 points) In light of your answer in part (c), what is your opinion about a
government policy that ensures that every agent in an economy have to buy
an insurance but offers it at a price lower than the market price.
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4. (15 points) Consider the following consumer who is deciding whether to buy fire
insurance in the face of fire risk. The probability of a fire incident is .4. In case of
a fire, her total wealth is 1000 whereas if there is no fire incident her wealth would
be 5000. Suppose each unit of insurance is sold at q = 0.5. Suppose the utility
function is given by
u(x) = ln x.
Find the optimal level of insurance bought by the consumer.
5. Consider a consumer who is deciding how to allocate his wealth 1000 between two
types of assets. If he buys a risk-free asset, e.g., a govt bond, it would give him a
10% return ($1 becomes $1.1). The risky asset gives 20% return with probability
.8 and −20% return with probability .2. The utility function of the risk-averse
consumer is as follows:
u(x) = ln x.
(a) (5 points) Write down the EU of the consumer when he invests $x in the risky
asset and the rest in the risk-free asset.
(b) (15 points) Find the optimal value of x.
6. Solve the life-cycle optimal choice for a consumer who has the period utility func-
tion is v(x) = ln x and β = 0.5 in the following contexts:
(a) (20 points) When r = 10% and his endowment is (500, 500).
(b) (20 points) When r = 100% and his endowment is (1000, 0).
(c) (20 points) When r = 20% and his endowment is (0, 1000).
7. (30 points) Consider a consumer who earns Y = 1 in two periods but faces a
possibility of loss of the magnitude L = 0.4 with probability p = 0.6. If his period
utility function is given by u(x) = ln x with β = 0.9 and the rate of interest is
r = 0. Then find the optimal consumption path (life cycle consumption choice) for
this consumer.
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