Bill is a utility maximizer, and his utility function is = , where y = quantity of "goods" consumed per day and T = non-working hours per (24-hour) day. The price of "goods" is $1 per unit.

(a) Let the wage rate be $4 per hour. How many hours will Bill work? How much will he earn?

(b) Suppose a welfare system is introduced. It pays a daily benefit of $30, but for each $1 of wage income you earn, you lose 50 cents of the benefit. So if your wage income is $60, you get zero benefit. Draw a diagram showing Bill's original constraint, and his constraint under the welfare system. Then use your graph and the concepts of income and substitution effects to analyze what you might predict will be the effect of the welfare system on the number of hours Bill will choose to work. (Do not refer to Bill's specific utility function.)

(c) Calculate for Bill's utility function how many hours he will choose to work under the welfare system. Also find what his total income is, including any welfare benefit.