# Carleton Econ 4460 Assignment 1 – A Grossman-type model

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## Carleton Econ 4460 Assignment 1 – A Grossman-type model

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 individualâs health status h in period t depends on
her âhealth 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 individualâs 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 individualâs life-time
budget constraint, using the summation notation, therefore is

.gif”>

The individualâs 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 âlnâ 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 economyâs 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.]

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