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ch3ans - Foundations of International Macroeconomics1...

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Foundations of International Macroeconomics 1 Workbook 2 Maurice Obstfeld, Kenneth Rogo ff , and Gita Gopinath Chapter 3 Solutions 1. (a) Because r = 0 , an individual°s desired consumption when young would be c y = 1 3 [ y y + (1 + e ) y y ] = 1 3 (2 + e ) y y if unrestricted borrowing were possible. In general c y = min y y , 1 3 (2 + e ) y y ° . Obviously the borrowing constraint will bind only when e > 1. Since β = 1, c o = c m = c y when the borrowing constraint of the young doesn°t bind. If it does (that is, if e > 1), then c o = c m = 1 2 (1 + e ) y y . The saving of a young person is s y = max 0 , y y 1 3 (2 + e ) y y ° = max 0 , 1 e 3 y y ° . That of a middle-aged person is s m = (1 + e ) y y (1 + e ) y y + s y 2 = (1 + e ) y y s y 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 23

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and that of an old person is s o = ( s y + s m ) = " (1 + e ) y y + s y 2 # . (b) Now 1+ g is the gross growth rate of a young-person°s output. Aggregate saving out of total output is s y t + s m t + s o t y y t + y o t , or Aggregate saving rate = max 0 , 1 e 3 y y t ° + (1+ e ) y y t 1 s y t 1 2 " (1+ e ) y y t 2 + s y t 2 2 # [(1 + g ) + (1 + e )] y y t 1 . (1) Let us ° rst assume that e 1, so that the young do not wish to borrow a positive amount. In that case, eq. (1) becomes Aggregate saving rate = 1 e 3 (1 + g ) y y t 1 + 1 + 2 e 3 y y t 1 2 + e 3(1 + g ) y y t 1 [(1 + g ) + (1 + e )] y y t 1 = (1 e )(1 + g ) 2 + (1 + 2 e )(1 + g ) (2 + e ) 3 [(1 + g ) 2 + (1 + g )(1 + e )] . (2) Borrowing by the young becomes an issue only if e > 1. If the young cannot borrow s y = 0 and aggregate saving is computed accordingly, taking account of the fact s m and s o are not the same as when the young can borrow freely. In the borrowing-constrained case, eq. (2) is replaced by Aggregate saving rate = s m t + s o t y y t + y o t = (1 + e ) y y t 1 2 (1 + e ) y y t 2 2 [(1 + g ) + (1 + e )] y y t 1 = (1 + e ) 2 (1 + e ) 2(1 + g ) (1 + g ) + (1 + e ) = g (1 + e ) 2 [(1 + g ) 2 + (1 + g )(1 + e )] . (3) 24
(c) Take the derivative with respect to e of the numerator of eq. (2). It is (1 + g ) 2 + 2(1 + g ) 1 = g 2 < 0 . Because, in addition, the denominator of (2) rises when e rises, steeper in- come growth between youth and middle age depresses saving. Intuitively, the young and old save less and the middle-aged save more, but with positive overall economic growth ( g > 0), it is the e ff ects on the young and old that dominate. When the young can°t borrow and e > 1 , the derivative is com- puted from eq. (3) and is proportional to 2(1+ g ) g > 0. The preceding e ff ect is reversed: the positive e ff ect of a higher e on middle-age saving dominates. (d) Observe that s y t = 2 3 y y t 1 3 y m t +1 , s m t = 2 3 y m t 1 3 y y t 1 , s o t = 1 3 y y t 2 + y m t 1 · .

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ch3ans - Foundations of International Macroeconomics1...

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