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Suppose an individual has preferences over two goods that can be represented by the utility function u(x,y) = x 1/2 y1/2, where x and y represent the quantities consumed of each good. Calculate the marginal utility of each good for generic quantities x and y. Calculate the marginal rate of substitution. Do these preferences exhibit diminishing marginal returns? How do you know? Suppose the goods have prices p x = 3 and p y = 5 and the individual has an income of 120. Graph the individual’s budget line for these two goods. Given your above results, find the individual’s optimal consumption bundle (x∗, y∗). What two conditions hold at this point? Illustrate on a diagram the income and substitution effects to show what would happen to the individual’s optimal choices if the price of good x fell to 2. Calculate the new optimal consumption bundle. (You do not need to calculate the decomposition of the change into substitution and income effects.) Assume a consumer’s utility is a function of units of consumption C and hours of lesisure R, given by the function U(C , R) = 48R +RC −R2. Let w be the person’s wage rate and suppose the time period has 24 hours, which must be divided between leisure and work. Hence, hours of work, L, is given by L= 24 − R. Formulate the consumer’s budget constraint by setting expenditures equal to income. (Assume there is no non-labour income.) What is the slope of the budget line, dC/dR? Determine the marginal rate of subsitution between leisure and consumption, dC/dR. Show that this consumer’s optimal number of hours of work increases with the wage rate. Show that, irrespective of the wage rate, the individual will never work more than 12 hours per day Consider a simple intertemporal choice problem in which there are only two periods: today (t = 0) and tomorrow (t = 1). In this problem, we are interested with the effects of imperfect capital markets. An individual has three options. The first is to pursue a job immediately (option W ) and earn a salary of $25,000 each period. Secondly, she could obtain a bachelor’s degree (option B). This would requiring forgoing some current income and earning $15,000 today but would yield a salary of $40,000 tomorrow. Finally, she could pursue a doctoral degree (option D) and earn $5,000 today and $60,000 tomorrow. Plot the three options facing the individual. That is, graph her possible endowments. Suppose that capital markets are perfect and that the individual can borrow and save at an interest rate of 25%. If the individual chooses based on maximizing the net present value of her earnings, which option would she choose? Suppose the individual’s preferences are given by U (c0, c1) = ln c0 +
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