Question

Let Tom's utility function be U(C, L) =C2+X×L2. Suppose he has 100 hours to split between work and leisure and he has no non-labor income. Derive Tom's optimal choice of consumption and leisure as a function of the wage and X. What is Tom's reservation wage?(Hint:Graphing an indierence curve before solving the problem might be useful.)

*This is the correct utility function, copied directly from the homework.

Answer 1

We have the following utility function and the time constraint,

where L= hours of leisure and W= hours of work.

Let the wage per hour be m, then Tom's total income is m(100-L)

Since there are no savings in the given case, Tom consumes whatever he earns. Thus, C=m(100-L)

As a rational consumer, Tom would maximize his utility from L and C

Using first order conditions,

Reservation wage is the wage at which Tom is indifferent in working and resting.

If Tom does not work at all, L=100. C=0, U(C=0, L=100)=10000X

If reservation wage is r, then L=W-100, C=r(100-L),

If Tom is indifferent between working and resting,