325_Lecture20_April13

325_Lecture20_April13 - MONETARY POLICY IN THE...

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1 M ONETARY P OLICY IN THE I NFINITE- P ERIOD A NALYSIS: THE N EUTRALITY D EBATE AND THE S TEADY S TATE A PRIL 13, 2011 April 13, 2011 2 B ASICS Introduction ± Timeline of events ± Notation ± c t :c o n s u m p t i o n i n p e r i o d t ± P t :n o m i n a l p r i c e o f c o n s u m p t i o n i n p e r i o d t ± Y t o m i n a l i n c o m e i n p e r i o d t (“falls from the sky”) ± a t- 1 :r e a l s t o c k h o l d i n g s a t b e g i n n i n g o f p e r i o d t /end of period t -1 ± M t- 1 o m i n a l m o n e y h o l d i n g s a t b e g i n n i n g o f p e r i o d t /end of period t -1 ± B t- 1 o m i n a l b o n d h o l d i n g s a t b e g i n n i n g o f p e r i o d t /end of period t -1 ± S t o m i n a l p r i c e o f a u n i t o f s t o c k i n p e r i o d t ± D t o m i n a l d i v i d e n d p a i d i n p e r i o d t by each unit of stock held at the start of t ± P b t o m i n a l p r i c e o f a b o n d i n p e r i o d t ± i t o m i n a l i n t e r e s t r a t e o n a b o n d p u r c h a s e d i n t and which pays off in t +1 ± Ⱥ t +1 e t i n f l a t i o n r a t e b e t w e e n p e r i o d t and period t +1 ± y t e a l i n c o m e i n p e r i o d t ( = Y t / P t ) Now three types of assets consumers can use for savings purposes
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2 April 13, 2011 3 2 111 2 22 11 1 1 1 1 1 1 1 1 2 22 2 21 (, / ) ( , / ) / ) () ( ) . ) ( .. tt t t t t t b t t t t t t tt b t tt tt t t t t uc M P uc M P P YSD a M B P cS aMP B YS D a M B P c S a MP B D a M EE O EO ±±± ± ±± ²² ² ±± ± ± ± ± ± ± ± ± ± ªº ± ² ² ² ² ¬¼ ± ± ± ² ² ² ² ± ± 112 2 2 3 33 3 32 2 2 3 3 3 3 3 ... b ttt t t b t t t t t t BP aM P B D a MB P c S a B ± ± ± ± ± ± ±² ² ² ² ± ± ± ² ² ² ² ± L AGRANGE A NALYSIS: S EQUENTIAL A PPROACH Infinite-Period Model: Sequential Formulation ± Step 1: Construct Lagrange function (starting from t ) ± Step 2: Compute FOCs with respect to c t , a t , B t , M t , with respect to c t : with respect to a t : with respect to B t : with respect to M t : …then the period t constraint… …then the period t +1 constraint… Infinite number of terms First the lifetime utility function….(no different than Chapter 8) …then the period t +3 constraint… …then the period t +2 constraint… 1 0 t ucM P P ² 1 0 b t P OE ± ² ± 1 0 t t t SS D ± ² Equation 1 Equation 2 Equation 3 2 1 0 t t P OEO ± ² Equation 4 (need chain rule of calculus to derive this) April 13, 2011 4 A SSET P RICING R EVISITED Finance Fundamentals ± Equation 2 Æ ± Equation 3 Æ ± Recall ± Æ can express pricing kernel as 1 t t t S SD ± ± ± §· ± ¨¸ ©¹ STOCK-PRICING EQUATION 1 0 t P ² 1 0 b t P ± ² 1 0 t t t D ± ² Equation 1 Equation 2 Equation 3 2 1 0 t t P ± ² Equation 4 Future return Pricing kernel x Period- t stock price = 1 b t t t P ± BOND-PRICING EQUATION 1 1 b t t P i ± 1 1 1 t i ± ±
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3 April 13, 2011 5 C ONSUMPTION- M ONEY O PTIMALITY C ONDITION Money Demand Begin with equation 4: Use ǃNJ t +1 = NJ t P b t from equation 3 2 1 (, / ) tt t t ucM P P O EO ± ² ² 2 b t t t P P ² Divide through by NJ t 2 1 b t t P P ² ² 2 1 1 b t t t P ²
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325_Lecture20_April13 - MONETARY POLICY IN THE...

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