exercise 4

Exercise 4. Consider a consumer who is deciding how many hours of leisure L she should enjoy on a typical day, and how much she should consume. The consumer's utility from consumption and leisure time is of the form u(c, l) = ac + v(L), where v(.) is a strictly increasing and strictly concave function with v (0) = O and a is a scalar. The price of the consumption good is p = 1. a) Compute the marginal utility of consumption au(c. L)/ ac and the marginal utility of leisure time au(c. L)/aL. What are special features of this utility function? b) The consumer maximizes u(c. L) subject to a budget constraint that combines a time constraint and an income constraint. Every hour of werk is subtracted from leisure. so L = 24H. where H is the number of hours worked. The consumer starts the period with nonlabor income M. and earns a hourly wage rate w > O for each hour worked. Write down the budget constraint as a function of c and L. Justify the steps. c) Write down the consumer's utility maximization problem (UMP). Include the nonnegativity constraints for c and L, but then neglect them from now on. cl) Assuming that the budget constraint holds with equality. write down the Lagrangian function and derive the first order conditions with respect to c, L, and A. e) Solve for 1* as a function of the parameters on, M, w? f) In the UMP. the parameter 7:\" measures the increase in maximized utility from receiving an extra dollar of non-labor income (marginal utility from non-labor income). Does the consumer have a decreasing marginal utility from non-labor income? How do you interpret this finding