These three equations can be derived from an explicit

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These three equations can be derived from an explicit model. Class 5 Monetary Policy
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Advanced Monetary Economics How to model money demand. ? Two main hypothesis : Cash-in-advance constraint : Simply assume that households need money to consume. Introduce a constraint M t ³ P t c t Class 5 Monetary Policy
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Advanced Monetary Economics How to model money demand. ? Two main hypothesis : Cash-in-advance constraint : Simply assume that households need money to consume. Introduce a constraint M t ³ P t c t Money-in-the Utility Function : If households hold money it must be useful for something : Introduce the real value of money in the utility function u ° c t , M t P t ± Class 5 Monetary Policy
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Advanced Monetary Economics How to model money demand. ? Two main hypothesis : Cash-in-advance constraint : Simply assume that households need money to consume. Introduce a constraint M t ³ P t c t Money-in-the Utility Function : If households hold money it must be useful for something : Introduce the real value of money in the utility function u ° c t , M t P t ± If you need money to consume a simple for is the following (CES function) u ° c t , M t P t ± = ac 1 ± 1 ε t + b ° M t P t ± 1 ± 1 ε ! 1 1 ± 1 ε ε > 0 The two approaches are in fact very close (Walsh, chap 3). Class 5 Monetary Policy
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A simple monetary model (New keynesian economics) The goal : Find the simple expression for money Demand Assume that households maximize max f c t , m t g t = 0 β t u ° c t , M t P t ± P t c t + B t + M t = W t + ( 1 + r n t ± 1 ) B t ± 1 + M t ± 1 where r n t ± 1 is the nominal interest rate between t ± 1 and t and B is the nominal quantity of bonds (+ transversality conditions) . Class 5 Monetary Policy
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A simple monetary model (New keynesian economics) The goal : Find the simple expression for money Demand Assume that households maximize max f c t , m t g t = 0 β t u ° c t , M t P t ± P t c t + B t + M t = W t + ( 1 + r n t ± 1 ) B t ± 1 + M t ± 1 where r n t ± 1 is the nominal interest rate between t ± 1 and t and B is the nominal quantity of bonds (+ transversality conditions) . The budget constraint can be written as c t + b t + m t = w t + ( 1 + r t ± 1 ) b t ± 1 + m t ± 1 1 + π t where m t = M t / P t , b t = B t / P t r t is the real interest rate and π t = ( P t ± P t ± 1 ) / P t ± 1 Class 5 Monetary Policy
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Bellman Equations V ( b t ± 1 , m t ± 1 ) = max m t , b t u ( c t , m t ) + β V ( b t , m t ) c t + b t + m t = w t + ( 1 + r t ± 1 ) b t ± 1 + m t ± 1 1 + π t Class 5 Monetary Policy
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Bellman Equations V ( b t ± 1 , m t ± 1 ) = max m t , b t u ( c t , m t ) + β V ( b t , m t ) c t + b t + m t = w t + ( 1 + r t ± 1 ) b t ± 1 + m t ± 1 1 + π t Substitute c t using the budget constraint. One obtains the °rst order conditions V ( b t ± 1 , m t ± 1 ) = max m t , b t u ° w t + ( 1 + r t ± 1 ) b t ± 1 + m t ± 1 1 + π t ± b t ± + β V ( b t , m t ) First order conditions are ...
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