(The lifetime value of finance)

Consider the two period life-cycle labor supply problem that we solved in a class.

Worker A's utility per period is given:

U(Ct,Lt)=lnCt +(1)lnLt

She lives for two periods t = 1 and t = 2 and is given T hours for each period.

The hourly wage for each period is given as w1 and w2 respectively.

Worker A maximizes the lifetime utility:

U (C1 , L1 , C2 , L2 ) = U (C1 , L1 ) + U (C2 , L2 )

subject to the lifetime budget constraint:

C1 +C2 =w1(T L1)+w2(T L2)

We derived the solution to this problem {C1,L1,C2,L2} in a class.

a. Suppose w2 > w1. Show that C1 > w1(T L1). In other words, she has to borrow C1 w1(T L1) > 0 from a bank at t = 1.

b. Consider a situation where there is no bank, and hence Worker A cannot borrow at t = 1. In other words, she is now facing one more constraint:

C1 = w1(T L1)

(1) Set up the Lagrangian with the above "additional" constraint other than the lifetime budget constraint.

(2) Derive the first order conditions.

(3) Find (C1, L1, C2, L2) satisfying the first order conditions.

Now assume =0.5,w1 =1,w2 =2,T =10.

1. What is the value for the lifetime utility with a bank?
2. What is the value for the lifetime utility without a bank?