eco204_summer_2009_HW_3

eco204_summer_2009_HW_3 - University of Toronto, Department...

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain HW 3 Please help improve the course by sending me an email about typos or suggestions for improvements Question 1 In this question, you will practice the intertemporal consumption model with complements preferences and inflation. Consider a 2 period model in which a consumer has real income Y1 and Y2 in T = 1, 2 respectively (the real income is measured in terms of goods review your ECO 100 macro notes; see for example RSM 332 HW 1, question 1, in which the agent receives "corn" income). The consumer can borrow or lend at nominal interest rate i. Assume T = 1 is the "base" year so that P1 = 1. There may be inflation or deflation so that (maybe) P2 1. (a) Suppose the consumer has the utility function U = min (C1, C2). Solve for the optimal consumption in T = 1, 2. (b) Interpret your answer for the case when r = 0. (c) Interpret your answer for the case when r = 1. (d) Use the results for C1 = C2 from part (a). When will this consumer be a borrower (lender) in T = 1? Question 2 In this question, you will practice the intertemporal consumption model with imperfect substitutes preferences and inflation. Consider a 2 period model in which a consumer has real income Y1 and Y2 in T = 1, 2 respectively (the real income is measured in terms of goods review your ECO 100 macro notes; see for example RSM 332 HW 1, question 1, in which the agent receives "corn" income). The consumer can borrow or lend at nominal interest rate i. Assume T = 1 is the "base" year so that P1 = 1. There may be inflation or deflation so that (maybe) P2 1. You will investigate the effect of higher real interest rates on whether someone who is a lender (borrower) continues to be a lender (borrower), and, the impact on utility. 1 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain You will show: If the consumer is a lender at T = 1, then as the real interest rates rise, she will continue to be a lender. This should be intuitive: at the current real interest rates, if I am lending money, then as real interest rates rise, lending becomes even more attractive and so I will continue to be a lender. The impact of rising real interest rates on the lenders' utility is obvious: they will be "happier" If the consumer is a borrower at T = 1, then as real interest rates rise, she will may switch to being a lender. This should be intuitive: at the current real interest rates, if I am borrowing money, then as real interest rates rise slightly, I may continue borrowing. However, there may come a threshold real interest rate, where borrowing becomes too expensive or put it another way, lending becomes more attractive so that I will switch from being a borrower to a lender. The impact of rising real interest rates on the borrower's utility, if they continue to be a borrower, is obvious: they will be "sadder". For this question, assume the consumer has CobbDouglas utility function between C1 and C2: U = C1 C2 Intuitively, the consumer is willing to tradeoff consumption today against consumption tomorrow. She's not like the consumer with U = min (C1, C2) who will do consumption smoothing. In contrast the consumer U = C1 C2 won't necessarily smooth consumption she may or may not. (a) Solve the consumer's intertemporal UMP. Hint: it's easier to work with the log transformation of the utility function. (b) When is the consumer a borrower in period 1? What about a lender in period 1? (c) Suppose this consumer is a lender in period 1. Show that if the real interest rate rises, she will continue to be a lender. If it helps you can assume that = = , Y1 = 10, Y2 = 5, r = 10%. (d) Suppose this consumer is a borrower in period 1. Show that if the real interest rate rises, she may become a lender. If it helps you can assume that = = , Y1 = 5, Y2 = 10, r = 10%. (e) We have seen that a consumer with complements preferences U = min (C1, C2) will by definition do consumption smoothing, i.e. C1 = C2. Hence, if you want to model consumption smoothing you could resort to complementary preferences (see RSM 332 HW 1, question 1, for example). But this does not mean other preferences cannot generate consumption smoothing. Use your answer in part (a) for when a consumer with preferences U = C1 C2 will "smooth consumption". 2 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Question 3 You've seen that one convenient way to model consumption smoothing is to assume the agent has the utility function U = min(C1 , C2 ). We have also investigated intertemporal consumption with utility function U = C1 C2. Now, from consumer choice, recall that we can use a variety of utility functions to model various situations this begs the question, do perfect substitute preferences have interesting implications for intertemporal consumption? Investigate this issue by assuming the consumer has the utility function U = C1 + C2 and must consume some positive amount in each period; i.e. consumption cannot be zero. Consider a 2 period model in which a consumer has real income Y1 and Y2 in T = 1, 2 respectively (the real income is measured in terms of goods review your ECO 100 macro notes; see for example RSM 332 HW 1 question 1, in which the agent receives "corn" income). The consumer can borrow or lend at nominal interest rate i. Assume T = 1 is the "base" year so that P1 = 1. There may be inflation or deflation so that (maybe) P2 1. What is the pattern of intertemporal consumption? Question 4 In this question, you will practice the intertemporal consumption model with quasilinear utility function and inflation. Consider a 2 period model in which a consumer has real income Y1 and Y2 in T = 1, 2 respectively (the real income is measured in terms of goods review your ECO 100 macro notes; see for example RSM 332 HW 1 question 1, in which the agent receives "corn" income). The consumer can borrow or lend at nominal interest rate i. Assume T = 1 is the "base" year so that P1 = 1. There may be inflation or deflation so that (maybe) P2 1. The consumer has the quasilinear utility function: U = f(C1) + C2 Where f(x) is some function. Suppose f(C) = log C. Derive the optimal consumption in T = 1 and 2. By the way, it will be helpful to compare this question from last week's HW problem on quasilinear utility. Question 5 In this question, you will practice the intertemporal consumption model with complements preferences and inflation. Consider a 3 period model in which a consumer has real income Y1 , Y2 and Y3 in T = 1, 2, 3 respectively (the real income is measured in terms of goods review your ECO 100 macro notes; see for example RSM 332 HW 1 question 1, in which the agent receives "corn" income). The consumer can borrow or lend at nominal interest rate i. Assume T = 1 is the "base" year so that P1 = 1. There may be inflation or deflation so that (maybe) P2, P3 1. Suppose the consumer has the utility function U = min (C1, C2, C3). Solve for the optimal consumption in T = 1, 2, 3. Interpret your result for the case r = 0. Hint: Starting from the 3 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain intertemporal budget constraint: C2 + C1(1 + r) = Y1 (1 + r) + Y2 write down the PV budget constraint for the 2 period problem; from this, try to guess the form of the 3 period intertemporal budget constraint. Or you can do it from 1st principles: (consumption at T = 3) = (Income at T = 3) + (Savings from T = 2) + (Savings from T = 1). 4 ...
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