# Utility function.

.Consider a person’s decision problem in trying to decide how many
children to have. Although she cares about children and would like to have
as many as possible, she knows that children are “costly” in the sense that
there are costs to their upbringing as well as the time that she will have to
take off from work in order to have children. Her utility function over her
own consumption () her own leisure () and the number of children () is
given by the following utility function:
() = √ + √
For tractability (and to be able to use calculus), we will assume that the
number of children,  is a continuous variable (i.e. it can take any nonnegative value, including decimal values like 215 etc.).
This individual is endowed with a total of  units of time in her life,
which she can divide between working, leisure and having children. For
having each child, she will have to take time  off from work, during which
she will not earn anything. Besides this, there is a per child cost of  for
upbringing expenses.
Her wage rate is ; she uses her total income to purchase good  for her
own consumption, as well as to provide for the upbringing expenses of her
children. Assume that good  is priced at  per unit.
(a) Write the consumer’s optimization problem with the appropriate resource constraint, and derive her Marshallian demand for children 
[ Hint: Instead of redoing the whole calculations, can you make use of your
results from Problem 1?]
(b) Suppose the government introduces child bene fits i.e. for every child
she has, the government provides her an amount . How will this affect her
decision on how many children to have?
(c) How does the wage  effect the individual’s decision of how many children
to have? Speci fically, as the wage rises, does she have more or less children?
4. [Savings problem] Suppose an individual lives for two periods,  = 1 2
He consumes only one good,  whose price is  in period  = 1 and is (1+)
at  = 2 Thus  is the rate of in flation in the economy. His income at period
2
 = 1 is  but he has no income in period  = 2 and must depend on his
savings from the first period. Savings earn an interest rate of  from the
bank.
The individual’s life-time utility is given by:
( 1 2) = √ 1 + √ 2
where  is his consumption of the single good in period 
(a) What is the individual’s optimal savings decision? How is his savings
affected by the interest rate  and by the in flation rate  (i.e. do they increase
or decrease with changes in these variables)?
(b) Suppose the government introduces a pension plan in which the individual’s income in period  = 1 is taxed at the rate  but he will be given an
amount  in period  = 2 What is the individual’s savings decision now?
Can it happen now that he may not save at all?
[ Hint: Try to use your res ul ts from part(a) to avoid redoing many of t he
calculations for part (b).]
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