This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Foundations of International Macroeconomics 1 Workbook 2 Maurice Obstfeld, Kenneth Rogo f , 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) = (1) 1 + " Y 1 + p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # p (1) 1 + r Y 2 (1) = (1) 1 + Y 1 (1) 1 1 + " p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # + (1) " p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # p (1) 1 + r Y 2 (1) = (1) 1 + Y 1 (1) 1 1 + " p (1) Y 2 (1) + p (2) Y 2 (2) 1 + r # + (1) p (2) Y 2 (2) 1 + r (2) p (1) Y 2 (1) 1 + r = (1) CA 1 + p (2) (2) Y 2 (1) 1 + r " (1) /Y 2 (1) (2) /Y 2 (2) p (1) p (2) # . 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 f Book.html 48 Here, the last equality comes from eq. (17), Chapter 5. Using eq. (80) from the chapter, we see that the autarky price of the ArrowDebreu security for state 1 relative to that of the state 2 security is p (1) a p (2) a = (1) /Y 2 (1) (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 & rstorder conditions are p ( s ) 1 + r u ( C 1 ) = ( s ) u [ C 2 ( s )] , s = 1 , 2. For our utility function, u ( C 1 ) = 1 /C 1 and u [ C 2 ( s )] = 1, so that the above conditions imply p ( s ) 1 + r = ( s ) 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 (1) (2) = p (1) p (2) , implying that equilibrium prices must be actuarially fair. Since p (1)+ p (2) = 1, it follows that the ArrowDebreu prices equal the respective probabilities of the state occurring: p ( s ) = ( s ) , s = 1 , 2 . Assuming that Home and Foreign share the same discount factor , we may add eq. (1) for Home and for Foreign to obtain C 1 + C 1 = Y w 1 = 2 p (1) (1 + r ) (1) . 49 Because p (1) = (1), we obtain an expression for the world interest rate 1 + r = 2 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 ....
View
Full
Document
This note was uploaded on 10/21/2009 for the course ECON ECONOMICS taught by Professor Yuchinche during the Winter '08 term at University of Washington.
 Winter '08
 YuChinChe

Click to edit the document details