Microeconomic Theory - Assignment 4
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Microeconomic Theory - Assignment 4
1. (25) Abigail is a consumer whose utility is a function of her total wealth W . u(W) = log W.
Suppose that Abigail begins with initial wealth of A = 100 but will sufer a serious illness with probability 
= 0.15 which will require extensive treatment costing L = 80. To hedge against this risk, Abigail considers buying a health insurance policy. She may buy as much insurance I as she wishes at a cost of p per dollar of coverage, so her payofs in each state are
Healthy Ill
Probability 0.85 0.15
No Insurance 
100 20
Claim 0 I
Premium —pI —pI
(a) (5) Show that Abigail is risk averse.
(b) (5) Suppose that the insurance premiums are actuarially fair so that p = 0.15. Find
Abigail’s expected wealth E[W] and expected utility E[u(W)] as functions of how much insurance she buys I.
(c) (5) How much insurance should Abigail buy?
(d) (5) Now suppose that the insurance company raises premiums to p = 0.2 so that they are no longer actuarially fair. Find Abigail’s expected wealth E[W] and expected utility E[u(W)].
(e) (5) How much insurance should Abigail buy now?
2. (25) The Arrow-Pratt measures of absolute and relative risk aversion respectively describe the willingness of a consumers to risk a ixed amount of wealth or a ixed fraction of their wealth. This problem will demonstrate this by setting up a simple investment problem. Suppose that consumers begin with initial wealth W0 and may buy shares of a risky asset whose payof per share is given by
= 
Therefore buying 
shares of the risky asset yields inal wealth
W˜ = W0 + 
.
Suppose that each consumer may buy an unlimited number of shares, and seeks to maximize expected utility of inal wealth
max E[u(W˜ )].
(a) (5) Expand the consumer’s expected utility maximization problem, and ind the irst
order condition.
(b) (5) Let Cara be a consumer whose utility function exhibits constant absolute risk aver-
sion
uA (W) = 1 — e −αW .
Find Cara’s optimal number of shares 
A(∗) and show that it does not depend on her starting wealth W0 .
(c) (5) What condition does Cara require to buy a positive number of shares? How does her investment vary with her coe
cient of absolute risk aversion α?
(d) (5) Let Cirra be a consumer whose utility function exhibits constant relative risk aversion
uR (W) = W1−ρ — 1
Find Cirra’s optimal number of shares 
R(∗) and show that it is proportional to her starting wealth W0 .
(e) (55) What condition does Cirra require to buy a positive number of shares? How does
her investment vary with her coe
cient of relative risk aversion ρ?
3. (25) Elsa and Anna are two students each considering whether to pursue a college education before entering the job market. For each worker, an education level of E = 1 indicates that she has completed college and earned a degree, while E = 0 indicates she has not. Elsa can generate VE = 36 worth of output per hour while Anna can only produce VA = 12 per hour.
Obtaining a degree is costly, and the students must pay for their educations by taking out student loans that will be repaid during their working years. Elsa is a naturally capable student so the degree costs her the equivalent of CE = 18 per hour in future wages, whereas Anna struggles with her studies and must pay CA = 27 per hour. Note that their education levels do not impact their productivity.
Hans is an employer who cannot observe the workers’innate abilities, but can observe their education levels. The labor market is competitive, so that Hans must always ofer a wage equal to his reservation price, and thus makes zero proit.
(a) (5) Suppose that Hans ofers the same wage W to all workers, regardless of whether
they attended college or not. What education levels do Elsa and Anna choose? What wage does Hans ofer?
(b) (5) Now suppose that Hans decides to ofer higher wages to workers with college degrees
W1 than to those without W0 . What education levels must he expect Elsa and Anna to choose in order to justify this? What wages does Hans ofer?
(c) (5) Show that both Elsa and Anna have an incentive to choose the education levels that match Hans’expectations.
(d) (5) Now suppose that the costs of a college education fall, so that now CE = 14 for Elsa and CA = 21 for Anna. What education levels do they choose now, and what wages does Hans ofer?
(e) (5) In this model, how does reducing the cost of education afect how well educated
people are overall?
4. (25) One of the key provisions of the Afordable Care Act was the ensuring coverage for“pre- existing conditions,”which had formerly been excluded from insurance plans as people with known medical conditions that will require treatment with certainty, the costs of which are virtually guaranteed to be more than the premiums that insurance companies could charge them. Suppose we have a population of patients with diferent health proiles, so that the
cost of future treatment X is uniformly distributed on the interval [0, 1000], 
~ U[0, 1000].
Suppose that we have an insurance company that is bound to ofer the same plan to all customers who wish to buy it, and cannot deny coverage based on pre-existing conditions. The market for insurance is competitive, so that the insurance company must set its premium equal to the expected future payout.
P = E[
]
(a) (5) Another key provision of the ACA was the“individual mandate,”which required
everyone to buy insurance so that young and healthy individuals would reduce the overall risk in the pool of insurees. With a mandate in place, what is the premium P charged by the insurance company?
(b) (5) The mandate proved to be an unpopular policy, and was eventually repealed at the
federal level. Suppose that all customers are now free to choose whether to buy health insurance. Let IX indicate whether a patient with health costs X buys insurance, so that
IX = 
If the premium P remains at the level you found in the last part, what is the payof 
X to buying insurance for a patient with costs X? Who will buy insurance (IX = 1) and who will not (IX = 0)?
(c) (5) The insurance company now realizes that, with the mandate no longer in force, their plan is no longer correctly priced. As P had been chosen under the assumption that everyone would be buying insurance, patients opting out will alter the overall risk of the remaining pool. The insurance company must still price its plan competitively, but now must set a new premium based on the expected cost of treating the insured patients.
P/ = E[X|IX = 1].
Suppose that all patients in the last part continue buy insurance, what premium P/ should the company charge now? Is it higher or lower than before?
(d) (5) Now that the premium has changed, people must reconsider whether to buy insurance yet again. Of course, this means that the insurance company must adjust its premium once again. What is the Nash equilibrium outcome? Who buys insurance in this world?
(e) (5) Why do the resolutions typically proposed for other instances of the lemons problem
fail to address issues of adverse selection in the health insurance market?
2022-07-30