Assignment:
Suppose that individuals potentially live for two periods. The utility function in each period is given by:
u(ci)= c^(1/2)
where ci is period i's consumption. Every individual receives income m, in the first period. This income can be used to finance consumption in that period, or it can be saved at zero interest to finance consumption in period two. The individual receives no income in the second period. The individual's budget constraint is thus
c1+c2=m
An individual must determine his consumption level before knowing whether he will survive into the second period. He will survive into period two with probability p. Suppose the utility of being dead is zero. The individual maximizes expected utility:
(c1)^(1/2)+p(c2)^(1/2)
subject to the budget constraint above.
a) Show the optimal consumption levels are given by:
c1(p,m)= m/1+p^2, c2(p,m)= p^2m/1+p^2
Why is c1(p,m)> c2(p,m)? What is the expected bequest (this is the value of the assets individuals leave when they die) Remember, some die at the end of the period one, the rest at the end of period two] Show that the indirect utility function is given v(p,m)= ((1+p^2)m))^(1/2)
b) Let pi(p,m) be the amount an individual is willing to pay in the first period so that he is guaranteed to survive into the second period. Show that:
pi(p,m)= (1-p^2)m/2
c) Using your result from part (a), fine the expenditure function e(p,u). Use the expenditure function to compute the equivalent and compensating variation for the change in p described in part (b). How are they related to your answer in part (b)?
(d) Suppose we change the problem so that instead of possibly dying, the individual can get sick before the start of the second period with probability 1-p. If the individual does become ill, he must incur an expenditure h, to make himself well. Assume that actuarially fair health insurance is available before the start of period two and before the individual knows whether or not he will become ill. Let x be the amount of insurance coverage the individual purchases. If he gets sick the insurance pays the individual x. Show that the individual will fully insure, i.e x=h. Show the the optimal consumption in each period is
c1(p,m)=c2(p,m)= m-(1-p)h/2
Why is the total second period expenditure c2+(1-p)h, greater than the first period expenditure, c1?