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# ch5ans - Foundations of International Macroeconomics1...

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Foundations of International Macroeconomics 1 Workbook 2 Maurice Obstfeld, Kenneth Rogo ff , and Gita Gopinath Chapter 5 Solutions 1. Since consumption on date 2 is the sum of endowment and payments of contingent assets for the realized state, we have B 2 ( s ) = C 2 ( s ) Y 2 ( s ) , s = 1 , 2 . For s = 1, premultiply by p (1) / (1 + r ) and use eq. (16) in Chapter 5 to obtain the following: p (1) (1 + r ) B 2 (1) = π (1) β 1 + β " Y 1 + p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # p (1) 1 + r Y 2 (1) = π (1) β 1 + β Y 1 π (1) 1 1 + β " p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # + π (1) " p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # p (1) 1 + r Y 2 (1) = π (1) β 1 + β Y 1 π (1) 1 1 + β " p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # + π (1) p (2) Y 2 (2) 1 + r π (2) p (1) Y 2 (1) 1 + r = π (1) CA 1 + p (2) π (2) Y 2 (1) 1 + r " π (1) /Y 2 (1) π (2) /Y 2 (2) p (1) p (2) # . 1 By Maurice Obstfeld (University of California, Berkeley) and Kenneth Rogo ff (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 ff Book.html 48

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Here, the last equality comes from eq. (17), Chapter 5. Using eq. (80) from the chapter, we see that the autarky price of the Arrow-Debreu security for state 1 relative to that of the state 2 security is p (1) a p (2) a = π (1) /Y 2 (1) π (2) /Y 2 (2) . Substitution of the preceding into the expression for p (1) B 2 (1) / (1 + r ), gives the required result. The result for B 2 (2) follows from the identity CA 1 = p (1) 1 + r B 2 (1) + p (2) 1 + r B 2 (2) . The statement of the exercise provides the intuition. 2. The necessary ° rst-order conditions are p ( s ) 1 + r u 0 ( C 1 ) = π ( s ) β u 0 [ C 2 ( s )] , s = 1 , 2. For our utility function, u 0 ( C 1 ) = 1 /C 1 and u 0 [ C 2 ( s )] = 1, so that the above conditions imply p ( s ) 1 + r = π ( s ) β C 1 , s = 1 , 2. (1) (A similar relation holds for C 1 , the initial consumption of Foreign residents.) If we divide eq. (1) for s = 1 by its analog for s = 2 , we see that π (1) π (2) = p (1) p (2) , implying that equilibrium prices must be actuarially fair. Since p (1)+ p (2) = 1, it follows that the Arrow-Debreu prices equal the respective probabilities of the state occurring: p ( s ) = π ( s ) , s = 1 , 2 . Assuming that Home and Foreign share the same discount factor β , we may add eq. (1) for Home and for Foreign to obtain C 1 + C 1 = Y w 1 = 2 p (1) (1 + r ) π (1) β . 49
Because p (1) = π (1), we obtain an expression for the world interest rate 1 + r = 2 β Y w 1 . Using Euler eq. (1) again, but substituting in this expression for 1 + r , we see that equilibrium date 1 consumptions are: C 1 = C 1 = Y w 1 2 . The equilibrium intertemporal budget constraint of a Home resident is C 1 + π (1) 1 + r C 2 (1) + π (2) 1 + r C 2 (2) = Y 1 + π (1) 1 + r Y 2 (1) + π (2) 1 + r Y 2 (2) , so that date 2 consumptions must obey π (1) 1 + r C 2 (1) + π (2) 1 + r C 2 (2) = Y 1 Y w 1 2 + π (1) 1 + r Y 2 (1) + π (2) 1 + r Y 2 (2); (2) there is a similar equation for Foreign. Since utility is linear in date 2 con- sumption with weights π (1) and π (2), a Home resident is indi ff erent between any pair [ C 2 (1) , C 2 (2)] satisfying eq. (2). On date 2, however, goods-market equilibrium requires that C 2 ( s ) + C 2 ( s ) = Y w 2 ( s ) , s = 1 , 2 .

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ch5ans - Foundations of International Macroeconomics1...

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