ch1ans - Foundations of International Macroeconomics1...

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Foundations of International Macroeconomics 1 Workbook 2 Maurice Obstfeld, Kenneth Rogo f , and Gita Gopinath Chapter 1 Solutions 1. (a) The intertemporal budget constraint can be expressed as C 2 =(1+ r )( Y 1 C 1 )+ Y 2 . Substitute this expression for C 2 into lifetime utility U ( C 1 ,C 2 )toobta in U = U [ C 1 , (1 + r )( Y 1 C 1 )+ Y 2 ] . (1) Taking the total derivative of U with respect to C 1 and equating it to zero, one gets the following Euler equation for optimal consumption: U ( C 1 ,C 2 ) C 1 =(1+ r ) U ( C 1 ,C 2 ) C 2 . (b) Total di f erentiation of expression (1) with respect to r gives d U d r = U ( C 1 ,C 2 ) C 1 d C 1 d r + U ( C 1 ,C 2 ) C 2 " ( Y 1 C 1 ) (1 + r ) d C 1 d r # = " U ( C 1 ,C 2 ) C 1 (1 + r ) U ( C 1 ,C 2 ) C 2 # d C 1 d r + U ( C 1 ,C 2 ) C 2 ( Y 1 C 1 ) = U ( C 1 ,C 2 ) C 2 ( Y 1 C 1 ) , (2) where the Euler equation from part a gives the last equality. (Notice another instance of the envelope theorem.) 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
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(c) Recall the relationship derived in part b, d U ( C 1 ,C 2 ) d r = U ( C 1 ,C 2 ) C 2 ( Y 1 C 1 ) . Because the marginal utility of consumption is positive, the country will bene t from an increase in r if and only if Y 1 C 1 > 0, that is, if it is a n e tl end e rinpe r iod1 . Intha tca s e ,th ein c r ea s ein r corresponds to an (d) Write expression (1) as U = U [ C 1 , (1 + r )( W 1 C 1 )] . Di f erentiation with respect to W 1 gives d U d W 1 =(1+ r ) U ( C 1 ,C 2 ) C 2 . (By the envelope theorem, the two terms in d C 1 / d W 1 cancel each other.) If d W 1 = b r ( Y 1 C 1 ), where b r d r/ (1 + r ), then by the Euler equation, d U =(1+ r ) U ( C 1 ,C 2 ) C 2 b r ( Y 1 C 1 )= U ( C 1 ,C 2 ) C 1 b r ( Y 1 C 1 ) . Bu teq . (2 )inpa r tbimp l ie stha tthewe l fa ree f ect of a percentage gross interest rate change b r is the same, d U = U ( C 1 ,C 2 ) C 2 ( Y 1 C 1 )d r = U ( C 1 ,C 2 ) C 1 b r ( Y 1 C
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into the intertemporal budget constraint gives C 1 ( r )= 1 1+ β Y 1 + Y 2 1+ r . (b) Home saving is derived as
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ch1ans - Foundations of International Macroeconomics1...

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