ch2ans - Foundations of International Macroeconomics1...

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Foundations of International Macroeconomics 1 Workbook 2 Maurice Obstfeld, Kenneth Rogo f , and Gita Gopinath Chapter 2 Solutions 1. (a) The current account identity can be written as B s +1 =(1+ r ) B s + TB s . Now just plug in the assumed trade balance rule. (b) Using the answer to part a, for any ξ > 0 , X s = t 1 1+ r s t ξ rB s = X s = t 1 1+ r s t ξ r [1 + (1 ξ ) r ] s t B t = ξ rB t 1 1+(1 ξ ) r 1+ r = (1 + r ) B t . (c) Under the rule above, debt grows without bound if ξ < 1. But once the debt is as big as Y/r , the country can honor its foreign commitments only if debt stops growing and consumption is zero forever. Thus, the suggested rule must entail negative consumption levels at some point, which are inad- missible. To see directly why, consider the constant-output case, in which TB s = Y C s = ξ rB s so that the payback rule implies C s = Y + ξ rB s . Notice that since B s →−∞ ,C s must at some point become negative. The rule therefore is consistent with intertemporal solvency only if we counterfac- tually allow for negative consumption levels: the price of high consumption today would be infeasibly high trade surpluses later on. In general, suppose output grows at the gross rate 1 + g ,sothat Y s =(1+ g ) s t Y t .Un le s s1+ g 1 By Maurice Obstfeld (University of California, Berkeley) and Kenneth Rogo f (Prince- ton University). c MIT Press, 1996. 2 c MIT Press, 1998. Version 1.1, February 27, 1998. For online updates and correc- tions, see http://www.princeton.edu/ObstfeldRogo
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is at least as great as the gross growth rate of debt, which was shown to be 1+ r (1 ξ ) in part a, the external debt-output ratio is unbounded. Thus the minimal payback fraction ξ consistent with intertemporal solvency and positive consumption is ξ =1 ( g/r ) (which is positive if we assume that g<r ). 2. (a) The expected utility E t U t is a weighted average over di f erent life spans, with weights equal to the survival probabilities: E t U t =( 1 ϕ )[ u ( C t )] + ϕ (1 ϕ )[ u ( C t )+ β u ( C t +1 )] + + ϕ 2 (1 ϕ ) h u ( C t )+ β u ( C t +1 )+ β 2 u ( C t +2 ) i + .... (b) The result follows simply by expanding the expression in part a and grouping terms together. 3. Recall that with isoelastic utility, u ( C )= C 1 1 σ 1 1 σ , σ > 0 , log C, σ =1 . Using the intertemporal Euler equation, we thus obtain, (1 + r ) β =1= 1 E t ( C t +1 C t 1 / σ ) . (1) Since consumption has a conditional lognormal distribution, the natural log of the gross consumption growth rate is conditionally normally distributed: log C t +1 C t N E t log C t +1 C t ± , Var t
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This note was uploaded on 10/21/2009 for the course ECON ECONOMICS taught by Professor Yu-chinche during the Winter '08 term at University of Washington.

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ch2ans - Foundations of International Macroeconomics1...

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