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Econ 4460, Sept 2016,

Assignment 1

Problem 1: A Grossman-type model

We represent the life-time utility of an individual as the

sum of utilities in three equally long periods indexed by t=1,2,3 (representing youth, working age, and old age). In each

period, utility depends on consumption c

and health status h, and is given by

the expression .gif”>. The individual assigns

equal value to present and future utility, so her life-time utility U is given by

.gif”>

The individualâs health status h in period t depends on

her âhealth capitalâ which is denoted by k.

Specifically, her health status in period t is given by .gif”>.

She is born with an amount .gif”>, but health capital

depreciates at the rate of 50% per period. However, health capital can also be

rebuilt by the use of medical services m.

Consequently, the individualâs health capital in working age and old age are

given by

.gif”>

Each individual in the community is given a one-time amount

of 2 as income support at the beginning of her life, out of which she has to

pay for her consumption and medical care. She can lend or borrow at the rate r=100% per period, and can also augment

her income by working during period 2 (her working age). When working she earns

a wage of 5 per unit of labour she supplies. The amount of labour she is able

to supply in period 2depends on her health capital during this period, and is

given by .gif”>. The individualâs life-time

budget constraint, using the summation notation, therefore is

.gif”>

The individualâs problem can now be stated as follows:

Maximize life-time utility U subject

to the life-time budget constraint, by choice of .gif”> ,

and given .gif”>. Answer the following

questions.

1. Discuss whether in this model there is a consumption

motive, or an investment motive, or both, for investing in health.

2. Write down the Lagrangean expression you would use to find

the solution to this problem, using the symbol .gif”> for the Lagrange multiplier. To simplify the

notation, substitute the numbers corresponding to .gif”> , t=1,2,3,

when r=100% (they are 1, 0.5, 0.25).

3. Find the derivatives of the Lagrangean with respect to .gif”> and write down the three equations that are

the first-order conditions for the optimal choices of the .gif”>.

4. Assuming that the individual chooses .gif”> such that her health capital k remains at 2 during all three period,

find the values of the amounts .gif”> she would need to spend in order to do so.

5. Using a calculator or computer, verify that the optimal

choices of .gif”> when her health capital k remains at 2 during all three period are roughly .8, 1.12, and

1.59.

6. Explain why in this example, the optimal value of .gif”> is .gif”>.

7. According to my calculations, the optimal choices of .gif”> are roughly 1.18 and 0.52, leading to optimal

values .gif”> and consumption in all three periods that are

higher than in question 5 above. Comment on the logic of this result.

Problem 2. A PPC and Expected Utility

Consider

an economy with 1 thousand people. Each person has 2 units of labour and 4

units of capital, so the total amounts of capital and labour in the economy,

denoted by K and L, are given by .gif”>. The capital and labour in the economy can be used to

produce either health care or consumption goods, according to

.gif”>

where

.gif”> are parameters. In

any given year, an individual in this population may be either well (subscript W) or ill (subscript I). It is not known in advance whether a

person will be well or ill, but statistics have shown that on average, 15% of

the population will be ill at any given time. For a typical person who is well,

utility depends only on consumption, so it is given by .gif”>. (The symbol âlnâ denotes the natural logarithm.) For a

typical person who is ill, utility depends both on consumption and on the

quantity of health services received, according to the formula

.gif”>

where .gif”> and .gif”> are parameters, .gif”> is the amount of the

consumption good that the well people receive, and .gif”> is the amount of

health services that is used to treat each ill person. The utility function for

sick people implies that a sick person derives less utility from a given amount

of consumption than a person who is well, and that her utility rises, but at a

diminishing rate, when the amount of health services .gif”> increases.

Answer the following.

1. Suppose first that 1000 units of labour and 1000 units of

capital are allocated to producing healthcare, and the rest to consumption.

Find the number of units of each that will be produced.

2. Show that the quantities that you computed in your answer

to 1 is not on this economyâs production possibilities curve.

[Hint: You can do this by taking the amount of health care

that is produced as given, and then maximizing C, subject to the constraint that the amount of H is constant.]

3. After dividing your answers to 1 by 1000 to get the

quantities per person in the economy, discuss the question how these quantities

should best be allocated between the sick and well members of the population.

[Hint: In your answer, use the concept of expected utility, and

consider how much healthcare is available to each ill person. Also consider how

total consumption should best be divided between sick and well people.]