Question

Consider an economy with two goods, consumption c and leisure l, and a representative

consumer. The consumer is endowed with 24 hours of time in a day. A consumer’s daily

leisure hours are equal to l = 24 − h where h is the number of hours a day the consumer

chooses to work. The price of consumption p is equal to 1 and the consumer’s hourly

wage is w. The consumer faces an ad valorem tax on their earnings of τ percent. The

consumer also receives some exogenous income Y that does not depend on how many

hours she works (e.g. an inheritance). The consumer’s preferences over consumption and

1+1 hours of work can be represented by the utility function U (c, h) = c − h  .

(a) What is this consumer’s budget constraint? [5 marks]
(b) Solve for the consumer’s utility maximizing hours of work h(w,1−τ) and consump-

tion c(w, 1 − τ, Y ). [10 marks]
(c) Repeat part (b) for a consumer with the utility function U (c, h) = αlog(c) − βlog(h).

[10 marks]