Problem Set V - Investments, Risk, and Uncertainty
ECON 511
Spring 2022
Exercise 1: Optimal Safe Investment
Anna has access to a perfect capital market with interest rate r = 0.15 (i.e. 15%) per period.
She also has the opportunity to purchase a private asset for price P . If she purchases the asset, she will
e able to invest any amount z ≥ 0 in this private asset. If Anna invests z ≥ 0 now, the asset will return
R(z) = 80.5 ·
√
z
next period.
What is the largest amount P that Anna would be willing to pay for the private asset?
[Hint: What is the net present value, NPV (z), of the private investment opportunity? What is the optimal
investment z∗ that maximizes NPV (z).]
1
Exercise 2: Investment and Optimal Consumption Choice
Kate has endowment E = (2775, 3000), i.e. she receives E1 = 2, 775 in the first period and E2 = 3, 000 in
the second period. Kate has access to a perfect capital market with interest rate r = 0.2 (i.e. 20%) pe
period.
She also has access to a private investment opportunity. If she invests z in the private investment
opportunity in the first period, it will return
R(z) =
180 ·
√
z − 100 if z > 100
0 if z ≤ 100
in the second period.
Kate’s preferences can be represented by the utility function
u(x, y) = x · y,
where x is the number of dollars available to purchase goods in the first period and y is the number of dollars
available to purchase goods in the second period.
1. What is the net present value, NPV (z), of the private investment opportunity?
2. What is Kate’s budget constraint (including the investment opportunity)?
3. What is Kate’s optimal consumption bundle (x∗, y∗)?
4. How will Kate finance this consumption plan (i.e. how much does she bo
ow/save/invest in the first
period and payback
eceive in the second period)?
5. What is the effect of the private investment opportunity on Kate’s consumption choice? That is,
compare Kate’s optimal choice with and without the private investment opportunity.
ECON 511, Spring XXXXXXXXXXB.Klose 2 Problem Set V
Exercise 3: Risky Investment
Charlie has von Neumann-Morgenstern utility function u(x) = lnx and has wealth W = 250, 000.
She is offered the opportunity to purchase a risky project for price P = 160, 000.
With probability p =
1
2
the project will be a success and return V > 160, 000. With probability 1−p = 1
2
the project will fail and be worthless (i.e. it returns 0).
For simplicity assume there is no interest between the time of the investment and the time of its return,
that is r = 0 .
How large must V be in order for Charlie to want to purchase the risky project?
[Hint: What is Charlie’s expected utility is she does not purchase the project? What is Charlie’s expected
utility is she purchases the project?]
Exercise 4: Insurance
Fiona has von Neumann-Morgenstern utility function u(x) =
√
x and initial wealth 640, 000. She faces a
25% chance of losing L = 280, 000.
1. Is Fiona risk averse?
2. What is Fiona’s utility if no loss occurs, what is her utility if the loss occurs? What is Fiona’s expected
utility?
3. What is the cost of fair insurance against the possible loss?
Suppose Fiona is able to choose insurance with any coverage z ∈ [0, 1] (i.e. 0 ≤ z ≤ 1). If she buys insurance
coverage at level z, she will get reimbursed z · 280, 000 if the loss occurs. Insurance coverage at level z costs
C(z) =
c0 + z · c1 if z > 00 if z = 0 .
4. Suppose c0 = 0 and c1 = 70, 000. Is insurance at coverage level z > 0 fair insurance? What coverage
level z∗ would Fiona choose? Explain.
5. Suppose c0 = 100 and c1 = 70, 000. Is insurance at coverage level z > 0 fair insurance? What coverage
level z∗∗ would Fiona choose? Explain. (Note that c0 = 100 is an ”avoidable fixed cost” which is only
paid if she chooses strictly positive insurance coverage. However, the ”marginal cost” of additional
insurance, c1 = 70, 000, is the same as in the previous part.)
6. Suppose c0 = 100 and c1 = 72, 000. Is full insurance, that is, coverage level z = 1 optimal? Explain.
ECON 511, Spring XXXXXXXXXXB.Klose 3 Problem Set V
Exercise 5: Insurance
Consider two individuals, Dave and Eva. Both Dave and Eva have initial wealth 810, 000 and face a 40%
chance of losing L = 450, 000. Dave has von Neumann-Morgenstern utility function uD(x) = x and Eva has
von Neumann-Morgenstern utility function uE(x) =
√
x.
1. What do you know about Dave’s and Eva’s risk preferences?
2. What is the most Dave would be willing to pay for complete insurance against the loss?
3. What is the most Eva would be willing to pay for complete insurance against the loss?
Suppose they are each able to choose insurance with any coverage level z ∈ [0, 1] (i.e. 0 ≤ z ≤ 1). If an
individual buys insurance coverage at level z, they will get reimbursed 450, 000·z if the loss occurs. Insurance
coverage at level z costs c(z) = z · 200, 000.
4. What coverage level z∗D would Dave choose? Explain.
5. Based on your previous results, try to explain that Eva chooses a strictly positive coverage z∗E > 0.
6. Is Eva’s optimal choice full insurance, i.e. z∗E = 1?
ECON 511, Spring XXXXXXXXXXB.Klose 4 Problem Set V