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# Class 5 monetary policy u c c t m t β v b b t m t u

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Class 5 Monetary Policy

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u 0 c ( c t , m t ) = β V 0 b ( b t , m t ) u 0 c ( c t , m t ) ± u 0 m ( c t , m t ) = β V 0 m ( b t , m t ) Envelop conditions V 0 b ( b t ± 1 , m t ± 1 ) = ( 1 + r t ± 1 ) u 0 c ( c t , m t ) V 0 m ( b t ± 1 , m t ± 1 ) = 1 1 + π t u 0 c ( c t , m t ) Class 5 Monetary Policy
u 0 c ( c t , m t ) = β V 0 b ( b t , m t ) u 0 c ( c t , m t ) ± u 0 m ( c t , m t ) = β V 0 m ( b t , m t ) Envelop conditions V 0 b ( b t ± 1 , m t ± 1 ) = ( 1 + r t ± 1 ) u 0 c ( c t , m t ) V 0 m ( b t ± 1 , m t ± 1 ) = 1 1 + π t u 0 c ( c t , m t ) Using the envelop conditions at period t + 1 into the °rst order condition yields The Euler conditions (easy to interpret) u 0 c ( c t , m t ) = β ( 1 + r t ± 1 ) u 0 c ( c t + 1 , m t + 1 ) u 0 c ( c t , m t ) ± u 0 m ( c t , m t ) = β 1 1 + π t + 1 u 0 c ( c t + 1 , m t + 1 ) Class 5 Monetary Policy

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Now divide the two °rst order equations, one by the other. One °nds u 0 m ( c t , m t ) u 0 c ( c t , m t ) = 1 ± 1 ( 1 + π t + 1 ) ( 1 + r t ± 1 ) To conclude, assume that the CES function u ( c t , m t ) = ² ac 1 ± 1 ε t + bm 1 ± 1 ε t ³ 1 1 ± 1 ε ε > 0 One °nds b a ° c t m t ± 1 ε = 1 ± 1 ( 1 + π t + 1 ) ( 1 + r t ± 1 ) Class 5 Monetary Policy
Or, at the °rst order log m t = ω + log c t ± ε log r n t where r n t = r t + π t + 1 . Simple LM curve "microfounded". Can be uses with data to measure ε . One °nds a value between . 19 and . 5 . Class 5 Monetary Policy
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