Question

Sarah has the following utility function and has a market wage of $10 per hour, and can work upto 2000 hours per year.

U = 100* lnC + 175* lnL ; Where C is the consumption and L is the leisure

A) Determine the utility maximizing levels of C and L for Sarah. Also, determine the corresponding maximum level of utility of Sarah.

Now assume that she is subject to a TANF program that features a benefit guarantee of $5000 and a benefit reduction rate of 50%. As a result, Sarah’s budget constraint will have two segments, one without the TANF and other with TANF benefits.

B) Determine the two segments of her budget constraints

Answer 1

Maximize U(c,l)= [\sqrt{cl}] s.t. [5(168-l) +100 = pc] , where p is price of consumption.

At point of tangency of utility function and constraint, the slopes will be equal.

[-l/c = -p/5] , which is also First Order condition.

[l = cp/5] ..Putting this in the constraint, we get [5(168-cp/5) +100 = cp]

[840-cp +100 = cp]

[940 = 2cp]

[c = 470/p] and [l = (470/p)(p/5) = 94]

Hence total working hours = 168 - 94 = 74 hours

B) Let t denote labor hours, then

[l = cp/5] , l + t = 168 hours

[168-t = cp/5]

[t = 168-cp/5]