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addendum3 - G s = T s M s − M s − 1 p s(6 and iterate...

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Addendum to Slide 116 Below you fi nd the explicit derivation of the right hand side of the last equation on slide 116 (the explicit solution for the constant consumption C ). This derivation can also be obtained by combining the intertemporal b.c. as is written on the slide or on page 534 of your book (see equation 38), X s = t 1 R s t [ C s + i s +1 1 + i s +1 M s p s ] = M s 1 p s 1 + X s = t 1 R s t ( Y s T s ) (1) together with footnote 26 page 537 of your book. The crucial question is: Why do not real money holdings appear in the solution for C ? Consider the modi fi ed budget constraint of the government: G s = T s + M s M s 1 p s (2) Following footnote 26 derive the intertemporal budget constraint of the gov- ernment. Forward the b.c. above by one period and divide by R. You obtain G s +1 T s +1 R = M s +1 /p s +1 R M s p s ( p s p s +1 R ) (3) Now recall from slide 114 that 1 p s p s +1 R = i s +1 1 + i s +1 (4) Hence, M s p s = G s +1 + T s +1 + M s +1 /p s +1 R + M s p s ( i s +1 1 + i s +1 ) (5) Substitute this expression of M s p s into

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Unformatted text preview: G s = T s + M s − M s − 1 p s (6) and iterate the procedure to obtain the intertemporal budget constraint of the government ∞ X s = t 1 R s − t ( G s − T s ) = − M s − 1 p s − 1 + ∞ X s = t 1 R s − t i s +1 1 + i s +1 M s p s (7) The simplest case is when G s = 0 for any s (this is the case we consider in the notes later on). In this case ∞ X s = t 1 R s − t ( − T s ) = − M s − 1 p s − 1 + ∞ X s = t 1 R s − t i s +1 1 + i s +1 M s p s (8) 1 Substituting into the intertemporal b.c. of the representative agent and considering that C is constant you get C = r 1 + r ∞ X s = t 1 R s − t Y s (9) The more general case is the one you have in the notes (in this case we are not assuming that G s = 0 yet!). 2...
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addendum3 - G s = T s M s − M s − 1 p s(6 and iterate...

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