Calculus : Utility theory

[Utility theory is a powerful tool, and has been used to study a wide range of

issues. Questions 3 and 4 will expose you to two of the many applications of

this theory; you will see more applications in other economics courses.]



  1. 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 benefits 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? Specifically, as the wage rises, does she have more or less children?



  1. [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 inflation in the economy. His income at period


 = 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




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 inflation 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 results from part(a) to avoid redoing many of the

calculations for part (b).]


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