Consider a consumer that chooses a bundle of leisure time (r) and a composite good (c) to maximize the utility function U(c, r) = c + 16(r1/2), where c is the number of units of the composite good consumed, and r is the number of hours of leisure consumed each month. The consumer's monthly income equals hours of labor supplied (ls) times the hourly wage rate (w). Every hour spent not sleeping (480 hours per month) is either consumed as leisure or supplied as labor, so the quantity of labor supplied each month is ls = 480 - r. Every dollar earned from working is spent on the composite good. The wage rate (w) earned by the consumer for each hour supplied as labor is $10. The price of a unit of the composite good (p) is $20.

a. At these prices, what is the maximum amount of the composite good that the consumer could purchase each month? What is the maximum amount of leisure the consumer could consume?

b. Write down the equation for the consumer's monthly budget line, and draw it. Be sure to indicate the magnitude of both the horizontal-axis and vertical-axis intercepts.

c. What is the opportunity cost of an hour spent at leisure, measured in units of the composite good?

d. Under these conditions, what is the optimal bundle of leisure and the consumption good? Hint: the marginal utility functions are MUc = 1 and MUr = 8/(r1/2). What is the optimal quantity of labor supplied?

e. Suppose the consumer's wage rate increases from $10/hour to $20/hour. What is the new equation for the budget line? What is the new (optimal) quantity of labor supplied?

f. What is the substitution effect (its magnitude and sign) of the wage increase on the quantity of labor supplied? What is the income effect (its magnitude and sign) of the wage increase on the quantity of labor supplied?