Nancy is deciding on an optimal schooling strategy. She currently has zero years

Question

Nancy is deciding on an optimal schooling strategy. She currently has zero years of schooling and knows she will live for 60 more years. In any given year she can either choose to add to her current level of schooling or she can work. If she decides to work, she will have an annual salary of ?S, where S is her current level of schooling (measured in years of schooling). If she decides to go to school, she will not have any income during that year, but her schooling

level will increase by 1 year. Schooling is otherwise free. Nancy's discount rate is r =0.

Nancy's objective is to maximize lifetime income.

1. What is Nancy's lifetime income as a function of her level of schooling, S?

2. What is Nancy's lifetime income if she gets no schooling? What is it if she goes to school for all 60 remaining years of her life? In words, describe the "cost" to Nancy of choosing to attend school for 1 additional year.

3. Determine Nancy's optimal schooling level (hint: you can do this with calculus or by graphing her lifetime income and ?nding the level of schooling that maximizes that income).

4. During which part of Nancy's remaining life will she get her schooling?

Explanation / Answer

a) at time t = 0, she'll live for 60 more years.

first year, she either goes to school , in which case her schooling will become S = 1 and income = 0 or she'll do a job in which case she'll earn income = 0 and S = 0

total income at the end of 1 year = 0

If your final amount of schooling = S, it is better to do schooling continuously for the first S years and then do the job for the next 60 - S years, because in this way the total income will be (60-S)*sqrt(S)

In any other way in which we achieve S amount of schooling and doing job in between we earn < sqrt(S) for atleast one year of the rest 60 - S years

So in the optimal case, we have, Lifetime income = (60 - S)*sqrt(S)

b) if she gets no schooling lifetime income = 0

and if she attends school for all 60 years lifetime income = 0 by putting 0 and 60 in the above equation

cost to nancy of choosing to attend school for 1 additional year = (60 - (S+1))*sqrt(S+1) - (60-S)*sqrt(S)

c) to find the optimal schoolinf level, we need to maximizze her lifetime income,

maximize (60-S)*sqrt(S) = max(f(S))

f ' (S) = (60-S)*(1/2)*(1/sqrt(S)) + sqrt(S)*(-1) = 0

==> 2S = 60 -S

==> 3S = 60

==> s = 20 is the optimal schooling level

4) She'll always get her schooling towards the first part of her life to maximize her lifetime income

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