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An agent has utility function u(c; l) = c l, where c is greater or equal to 0 is a consumption good, and l is greater or equal to 0 is leisure. The agent initially has 24 hours of labor endowment per day and 0 units of c. The unit price of the consumption good is 1 and the unit price of leisure is r per hour.
(a) Formulate the agent's utility maximization problem.
(b) Obtain the first order condition and solve them. (You must also prove that the solution to the first order conditions solves the utility maximization problem.)
(c) Suppose that the agent now has another option. He can quit his job (in which case he has 24 hours of leisure) and obtain a welfare payment of W per day from the government. Prove that if W > 6r then the agent would prefer to be on welfare rather than work.
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