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]
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Assume that a consumer’s satisfaction depends upon leisure (L) and income (Y), so that the utility function is: U = 48L + LY – L2 Let N denote the hours of work by the consumer and W the hourly wage rate. Consequently, Y = WN and L + N = 24. If the hourly wage rate is $15 per hour: a) What are the utility maximizing hours of leisure, L*? b) What are the utility maximizing hours of work, N*?
Consider a representative consumer who has preferences over an aggregate consumption good c and leisure l. Her preferences are described by the utility function: U(c,l) = ln(c) + ln(l) The consumer has a time endowment of h hours which can be used to work at the market or enjoyed as leisure. The real wage rate is w per hour. The worker pays a proportional wage tax of rate t, so the worker’s after-tax wage is (1−t)w. The consumer also has...
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