Question

Suppose Mandy’s earnings are given by w(S, A) = (SA+1)^(1/2) , where S denotes years of schooling and A denotes ability. Assume that Mandy lives forever, discounts future income at a rate of r, earns nothing while in school, and supplies labor inelastically thereafter. Assume that A is exogenous and time is continuous, Mandy’s objective is to choose S to maximize the present value of lifetime earnings.

(a) Calculate Mandy’s present value of lifetime earnings.

(b) Find Mandy’s optimal choice of years of schooling S* .

(c) Find the partial derivatives of S* with respect to A. Is the relationship between S* and A intuitively reasonable? Explain

Answer 1

W(S,A) = (SA+1)1/2 for every year, after the year of labour service is completed.

Let Mandy undergo S years of schooling and have A ability. Hence after S years of schooling, she starts to earn.

Hence the first income will come after S+1 years and second income will come after S+2 years.

a)

Hence Present Value of Future Earnings(PV) = W/(1+r)s+1 + W/(1+r)s+2 + ....

Using the formula for a sum of a geometric series, we get,

PV = (W/(1+r)S+1) / (1 - (1/(1+r)))

Hence PV = W/(r*(1+r)s)

Now we substitute the value of W and get

PV = (SA+1)1/2 / (r*(1+r)S)

b)

Mandy will choose to maximise the PV w.r.t S

Hence we have to differentiate PV wrt S and equate it to zero.

d(PV)/dS = 0

c)

This result is intuitive, as if your ability is high, then you have a lesser requirement of schooling and hence if A increases then optimal S should decrease, which is consistent with our result.

Note: Mandy's earnings are symmetric is S and A, meaning if we replace S by A and A by S, her earnings remain the same. Also to increase her earnings, increasing either S or A or both would increase her earnings. Hence to keep earnings constant, SA should be constant. Hence for earnings to be constant and we increase A, then we would have to reduce S. This is consistent with the result obtained in part c.