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 periodst= 1 andt= 2 and is givenThours for each period. The hourly wage for

each period is given asw1andw2respectively. 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(TL1)+w2(TL2) We derived the solution to this problem{C1,L1,C2,L2}in a class.

a. Supposew2> w1. Show thatC1> w1(TL1). In other words, she has to borrowC1w1(TL1)>0 from a bank att= 1.

1. Consider a situation where there is no bank, and hence Worker A cannot borrow att= 1. In other words, she is now facing one more constraint:
2. C1=w1(TL1)
3. (1) Set up the Lagrangian with the above "additional" constraint other than the lifetime budget constraint.
4. (2) Derive the first order conditions.
5. (3) Find (C1, L1, C2, L2) satisfying the first order conditions.
6. Nowassume=0.5,w1=1,w2=2,T=10.
7. What is the value for the lifetime utility with a bank?
8. What is the value for the lifetime utility without a bank?