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# Optimal consumption choice with uncertainty about survival

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?

#### Solution Preview

The text below is copied for your convenience, but please refer to the attached .doc file for correct display of all the formulas and text.

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Problem:
Consumption Levels, two periods
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?

Solution:

First of all, what we are doing here is trying to finding the optimal combination of consumption in time periods 1 and 2. This optimal combination maximizes utility and so, this is a maximization problem subject to constraints.
What we are maximizing is utility, so let's write that down:
U (c1,c2) = (c1)1/2+ p(c2)1/2
What about the constraint? As given, it is simply that the present value of today's and tomorrow's consumptions must not exceed given income, m. So, ...

#### Solution Summary

This problem looks at the optimal behaviour of an individual that is facing a likelihood of dying. There are two main questions solved: 1. what is the optimal consumption choice and 2. how is it affected by availability of an actuarially fair insurance scheme that covers expenditures in case of sickness.

Please note that this problem is solved using particular assumptions, specifically the utility function of the consumer in each period is given as u(ci)= ci^(1/2), where ci is the consumption at time period i. Other assumptions are used (please see the long description of the problem.

The solution provided assumes knowledge of calculus at a generic 3rd year Economics major level (e.g. maximization of functions of several variables) and doesn't contain detailed mathematical solution.

However, a student with a good background in Economics and Mathematics will be able to follow the general steps and apply them to a problem with a different utility function and other modified assumptions.

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