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Unformatted text preview: Economics 350a Truman Bewley Faii 2009 Homework No. 8
(Due Thursday, November 5) 1) Do probiem 1 of chapter 7 of the text on page 274. 2) A worker-consumer provides labor and consumes one good. This person suffers from
migraines, and the occurrence of migraines is independently and identically distributed each
week, with the probability of occurrence in any one week being one half. The person suffers
terribiy from the migraines, but works harder when they happen as a way to distract from the
pain. The person has one unit of iabor per week. His or her utility function when weil is u(x, L)=x—X_2—.L_2.
20 2 and the utility function when sick is u(x, L) = x—i — 100,
20 where x is consumption in one week and t. is labor supplied during one week. Notice that the
person has no disutility from labor when sick. The wage of labor and the price of the
consumption good are both 1. The worker, not the employer, chooses L. The worker-
consumer's weekly consumption equals her or his entire weekly wage income plus any other
receipts or minus any other payments. A risk neutrai insurer is willing to pay the person a benefit when sick in exchange for a
premium when the person is wait. The insurance is actuariain fair and there are no . administrative costs, so that if y is paid to the person when sick, y is collected from the person when well. The consumer worker chooses the value of y so as to maximize her or his own
expected utility. (if y is negative, the benefit is, in fact, a premium and the premium is a
benefit.) a) Set up and solve the person’s utility maximization problem when weli and given a fixed value
of y. b) Set up and soive the person’s utility maximization problem when sick and given a fixed value
of y. c) Find the vaiue of y that maximizes the person’s expected utility. d) Can you explain intuitively why y turns out to have the sign that it does? 3) A worker is unemployed in one week with probability 1/10 and employed with probability
9ND. The worker has ten units of time available for iabor or ieisure during the week (never
mind what the units of time are). If the worker is unemployed, aft of this time is enjoyed as
leisure. If the worker is employed, oniy half of it is enioyed as leisure. The worker‘s weekly
pay it employed is 1 and is nothing if unemployed. The worker's consumption each week equals
her or his pay plus any other receipts or minus any other payments. The worker has a von Neumann-Morgenstern utility function. The worker’s utility in one week is 2M, where is x the amount of money spent on consumption during the week and E is the amount of leisure enjoyed during the week. The worker's empioyer offers actuarially fair insurance against
unemployment. Under this insurance, the worker pays the employer a premium of p when
employed and the employer pays the worker a benefit of b when she or he is unemployed. The
worker chooses the amount of the premium or benefit so as to maximize her or his expected
utility. a) Describe formatly the worker’s optimization problem. Be sure to include the constraint that
the insurance be actuariain fair. b) Solve the worker's optimization probiem. c) Compare the worker's total consumption when unemployed with that when employed under
the optimal insurance. Can you explain intuitively the resutt of this comparison? 4) Robinson Crusoe and Friday farm an island together. With probability 1/2, Crusoe's part of
the island is flooded. Friday‘s crop is to, whether there is a flood or not. Crusoe’s crop is 20
if there is no flood and 10 if there is a flood. Crusoe and Friday each consume their own crop,
unless they arrange insurance. Each ot'them has a von Neumann-Morgenstern utility function
for the crop with Bernoulli utility function u(x) = ln(x), where x is the amount of
consumption. a) Set up a formal model describing this situation. That is, develop notation for the endowments
and aliocations of Crusoe and Friday and so on. b) Solve for an Arrow-Debreu equiiibrium, where the prices of one unit of the crop in each
state sum to i. A tribe on a nearby island offers to give either Friday or Crusoe or both of them one unit of crop
in the event of a flood in exchange for one unit if there is no flood and a fee of 3 units of crop to
be paid no matter what happens. 0) Suppose there is no insurance, so that Friday and Crusoe consume their endowments. Would
either Friday or Crusoe accept the neighboring tribes offer for some non-negative levei of fee
s? It so, what would be the maximum acceptable level of s for each of them? (You wiit need a
catculator to find the maximum level of s.) ,.
i ?? d) Suppose there is insurance. That is, suppose Friday and Crusoe arrange an Arrow-Debreu
equilibrium. Suppose also that after the insurance contracts have been agreed on, the
neighboring tribe makes its otter. is there a non-negative levet for the fee such that either
Friday or Crusoe would accept the neighboring tribe's offer? It so, what would be the maximum
acceptable levei of s for each of them? e) Compare the total of the maximum tees coltected from Friday and Crusoe in part c with the
total collected in part d. That is, does insurance decrease or increase the totai fees coltected? 5) Each of two consumers A and B tosses a fair coin and the outcomes are mutually independent.
It a consumer‘s coin comes up heads, she or he receives $200 from an outside source. It the
coin comes up tails, she or he receives $100 from the same source. Each consumer has a
differentiable and strictly increasing von Neumann-Morgenstern utility function. The two
consumers use Arrow-Debreu markets for contingent ctaims to insure each other against the
outcomes of the two tosses. (Remember that in general equilibrium, agents take prices as given.
They do not bargain.) a) Suppose both consumers have the same strictiy concave Bernoulli utitity function.
What will each consumer’s consumption be in all possible outcomes as a result of the
insurance? Please observe that the two consumers cannot exchange money with anybody
other than each other in arranging insurance. b) Now suppose consumer A is risk neutral and that consumer B is strictly risk averse.
What witl each consumer receive in ali possible outcomes as a result of the insurance? ...
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