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{"ops":[{"insert":"Exercise 1 ([Indifference Curves). Consider the utility function U(C1, C2) = ln(C1) + ln(C2).\n1. Using a program of your choice (say excel, or matlab) plot indifference curves in the space (C1, C2) for\nU \u0304 = -0.5, -1, and -1.5. Consider values of C1 in the interval (0, 1]. Set the range of the vertical axis to\n[0, 2].\n2. Find an analytical expression for the slope of the indifference curve and show that it is equal to the\n(negative) of the marginal rate of substitution.\n3. Show analytically that the indifference curves are convex.\n4. For the 3 indifference curves plotted above, find the slope of the indifference curve at the point C1 = 1\nand the corresponding value of C2. Explain why the indifference curves at C1 = 1 become flatter as\nthe level of welfare declines.\nExercise 2 (The Saving Schedule). Consider a two-period economy populated by identical households with\npreferences defined over consumption in period 1, C1, and consumption in period 2, C2, and described by\nthe utility function\n\np\nC1 +\np\nC2.\n\nAssume that households are endowed with Y1 kilos of apples in period 1 and with Y2 kilos of apples in\nperiod 2. Let P1 and P2 denote the price of apples in periods 1 and 2. Households can save (or borrow)\nat the nominal interest rate i. Let r denote the real interest rate, so that the gross real interest rate is\n1 + r =\nP1\nP2\n(1 + i). Let St denote saving in kilos of apples.\n1. State the household\u2019s budget constraints in periods 1 and 2.\n2. Derive the household\u2019s intertemporal budget constraint in terms of C1, C2, Y1, Y2, and r.\n3. State the household\u2019s utility maximization problem.\n4. Find the optimal level of consumption in period 1, C1, in period 2, C2, and the associated level of\nsaving, S1. Express your answer in terms of Y1, Y2, and r.\n5. Now assume that output is 10 kilos of apples in both periods (Y1 = Y2 = 10) and that the real interest\nrate is 0 percent (r = 0). Find C1, C2, and S1. (Your answer should be 3 numbers.) Finally, compute\nthe same 3 numbers but under the assumption that the real interest rate is 10 percent (r = 0.1). Is\nsaving increasing in r? Provide intuition.\nExercise 3 (An Economy Driven by Natural-Rate Shocks). Consider a two-period sticky-price economy\npopulated by identical households with preferences defined over consumption in period 1, C1, and consumption\nin period 2, C2, and described by the utility function\nlnC1 + \u03b2 lnC2,\n1\n\nwhere \u03b2 = 1\/1.1 is the subjective discount factor. In both periods, potential output (Y ) is equal to 10 kilos\nof apples. Let P1 and P2 denote the price levels in periods 1 and 2, respectively. Assume prices are fixed\nat P1 = P2 = 1 and that the economy is always in full employment in period 2 (the long run). The central\nbank uses the nominal interest rate, denoted i, as its monetary instrument. The nominal interest rate is\nsubject to the zero lower bound (ZLB) constraint.\n1. Assume further that the central bank sets the nominal interest rate so as to maximize employment\nand minimize excess aggregate demand for goods. Denote this interest rate by i\n\u2217\n. Find i\n\u2217\n.\n2. Now suppose that a financial panic causes households to become more patient. Specifically, suppose\nthat the subjective discount factor increases to 1\/0.9. Suppose that the central bank is slow to react\nand keeps the interest rate at i\n\u2217\n(the level of i obtained in question 1. Find the output gap, defined as\n\n(Y \/Y\u00a0\u0304\n1 \u2212 1)100, where Y1 denotes output in period 1.\n3. Now suppose that contrary to the assumption in question 2, the central bank acts quickly and changes\nthe interest rate to minimize unemployment. Denote this interest rate by i\n\u2217\u2217. Find i\n\u2217\u2217 and the output\n\ngap.\n4. Consider the scenario of question 3, that is, i = i\n\n\u2217\u2217. Suppose that the fiscal authority decides to\nalso intervene. Let G\u2217 denote the lowest level of government spending that eliminates involuntary\nunemployment. Find G\u2217\n\n. Calculate private consumption in period 1.\n\n5. Suppose that the government miscalculates G\u2217 and instead sets government spending equal to G \u0303, where\nG \u0303 is 10 percent higher than G\u2217\n\n. Assume further that realizing this situation, the central bank changes\nthe interest rate to avoid excess aggregate demand, while still maintaining full employment. Find the\nnew interest rate and private consumption. Comment.\n"}]}

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