The Basic New Keynesian Model by Jordi Gal November 2010 - PowerPoint PPT Presentation

The Basic New Keynesian Model by Jordi Galí November 2010

Motivation and Outline Evidence on Money, Output, and Prices: � Short Run E¤ects of Monetary Policy Shocks (i) persistent e¤ects on real variables (ii) slow adjustment of aggregate price level (iii) liquidity e¤ect � Micro Evidence on Price-setting Behavior: signi…cant price and wage rigidities Failure of Classical Monetary Models A Baseline Model with Nominal Rigidities � monopolistic competition � sticky prices (staggered price setting) � competitive labor markets, closed economy, no capital accumulation

Households Representative household solves 1 X � t U ( C t ; N t ) max E 0 t =0 where �Z 1 � � � � 1 C t ( i ) 1 � 1 C t � � di 0 subject to Z 1 P t ( i ) C t ( i ) di + Q t B t � B t � 1 + W t N t � T t 0 for t = 0 ; 1 ; 2 ; ::: plus solvency constraint.

Optimality conditions 1. Optimal allocation of expenditures � P t ( i ) � � � C t ( i ) = C t P t implying Z 1 P t ( i ) C t ( i ) di = P t C t 0 where �Z 1 � 1 1 � � P t ( i ) 1 � � di P t � 0 2. Other optimality conditions � U n;t = W t U c;t P t � U c;t +1 � P t Q t = �E t U c;t P t +1

Speci…cation of utility : 1 � � � N 1+ ' U ( C t ; N t ) = C 1 � � t t 1 + ' implied log-linear optimality conditions (aggregate variables) w t � p t = �c t + 'n t c t = E t f c t +1 g � 1 � ( i t � E t f � t +1 g � � ) where i t � � log Q t is the nominal interest rate and � � � log � is the discount rate . Ad-hoc money demand m t � p t = y t � �i t

Firms � Continuum of …rms, indexed by i 2 [0 ; 1] � Each …rm produces a di¤erentiated good � Identical technology Y t ( i ) = A t N t ( i ) 1 � � � Probability of resetting price in any given period: 1 � � , independent across …rms (Calvo (1983)). � � 2 [0 ; 1] : index of price stickiness 1 � Implied average price duration 1 � �

Aggregate Price Dynamics � t ) 1 � � � 1 � ( P t � 1 ) 1 � � + (1 � � )( P � P t = 1 � � Dividing by P t � 1 : � P � � 1 � � t � 1 � � = � + (1 � � ) t P t � 1 Log-linearization around zero in‡ation steady state � t = (1 � � )( p � t � p t � 1 ) (1) or, equivalently p t = �p t � 1 + (1 � � ) p � t

Optimal Price Setting 1 X � � �� � k E t P � max Q t;t + k t Y t + k j t � � t + k ( Y t + k j t ) P � t k =0 subject to Y t + k j t = ( P � t =P t + k ) � � C t + k for k = 0 ; 1 ; 2 ; ::: where � C t + k � � � � P t � Q t;t + k � � k C t P t + k Optimality condition: X 1 � � �� � k E t P � Q t;t + k Y t + k j t t � M t + k j t = 0 k =0 where t + k j t � � 0 t + k ( Y t + k j t ) and M � � � � 1

Equivalently, � � P � �� 1 X � k E t t Q t;t + k Y t + k j t � M MC t + k j t � t � 1 ;t + k = 0 P t � 1 k =0 where MC t + k j t � t + k j t =P t + k and � t � 1 ;t + k � P t + k =P t � 1 Perfect Foresight, Zero In‡ation Steady State: P � MC = 1 t Q t;t + k = � k = 1 ; � t � 1 ;t + k = 1 ; Y t + k j t = Y ; ; P t � 1 M

Log-linearization around zero in‡ation steady state: X 1 p � ( �� ) k E t f c t � p t � 1 = (1 � �� ) mc t + k j t + p t + k � p t � 1 g k =0 where c mc t + k j t � mc t + k j t � mc . Equivalently, 1 X p � ( �� ) k E t f mc t + k j t + p t + k g t = � + (1 � �� ) k =0 � where � � log � � 1 . Flexible prices ( � = 0 ): p � t = � + mc t + p t = ) mc t = � � (symmetric equilibrium)

Particular Case: � = 0 (constant returns) = ) MC t + k j t = MC t + k Rewriting the optimal price setting rule in recursive form: p � t = ��E t f p � t +1 g + (1 � �� ) c mc t + (1 � �� ) p t (2) Combining (1) and (2): � t = �E t f � t +1 g + � c mc t where � � (1 � � )(1 � �� ) �

Generalization to � 2 (0 ; 1) (decreasing returns) De…ne mc t � ( w t � p t ) � mpn t 1 � ( w t � p t ) � 1 � � ( a t � �y t ) � log(1 � � ) 1 Using mc t + k j t = ( w t + k � p t + k ) � 1 � � ( a t + k � �y t + k j t ) � log(1 � � ) , � mc t + k j t = mc t + k + 1 � � ( y t + k j t � y t + k ) �� 1 � � ( p � = mc t + k � t � p t + k ) (3) Implied in‡ation dynamics � t = �E t f � t +1 g + � c mc t (4) where � � (1 � � )(1 � �� ) 1 � � � 1 � � + ��

Equilibrium Goods markets clearing Y t ( i ) = C t ( i ) for all i 2 [0 ; 1] and all t . �R 1 � � 0 Y t ( i ) 1 � 1 � � 1 , Letting Y t � � di Y t = C t for all t . Combined with the consumer’s Euler equation: y t = E t f y t +1 g � 1 � ( i t � E t f � t +1 g � � ) (5)

Labor market clearing Z 1 N t = N t ( i ) di 0 � Y t ( i ) � Z 1 1 1 � � = di A t 0 � Y t � � P t ( i ) � � 1 � � Z 1 1 � 1 � � = di A t P t 0 Taking logs, (1 � � ) n t = y t � a t + d t R 1 � 0 ( P t ( i ) =P t ) � 1 � � di (second order). where d t � (1 � � ) log Up to a …rst order approximation: y t = a t + (1 � � ) n t

Marginal Cost and Output mc t = ( w t � p t ) � mpn t = ( �y t + 'n t ) � ( y t � n t ) � log(1 � � ) � � � + ' + � y t � 1 + ' = 1 � �a t � log(1 � � ) (6) 1 � � Under ‡exible prices � � � + ' + � t � 1 + ' y n mc = 1 � �a t � log(1 � � ) (7) 1 � � ) y n = t = � � y + ya a t where � y � ( � � log(1 � � ))(1 � � ) 1+ ' > 0 and ya � � + ' + � (1 � � ) . � + ' + � (1 � � ) � � � + ' + � ( y t � y n ) c = mc t = t ) (8) 1 � � where y t � y n t � e y t is the output gap

New Keynesian Phillips Curve � t = �E t f � t +1 g + � e y t (9) � � � + ' + � where � � � . 1 � � Dynamic IS equation y t +1 g � 1 � ( i t � E t f � t +1 g � r n y t = E t f e e t ) (10) where r n t is the natural rate of interest , given by r n t � � + � E t f � y n t +1 g = � + � ya E t f � a t +1 g Missing block: description of monetary policy (determination of i t ).

Equilibrium under a Simple Interest Rate Rule i t = � + � � � t + � y e y t + v t (11) where v t is exogenous (possibly stochastic) with zero mean. Equilibrium Dynamics: combining (9), (10), and (11) � � � � e E t f e y t y t +1 g r n + B T ( b = A T t � v t ) (12) � t E t f � t +1 g where � � � � � 1 � �� � 1 A T � � ; B T � � �� � + � ( � + � y ) � 1 and � � � + � y + �� �

Uniqueness ( ) A T has both eigenvalues within the unit circle Given � � � 0 and � y � 0 , (Bullard and Mitra (2002)): � ( � � � 1) + (1 � � ) � y > 0 is necessary and su¢cient.

E¤ects of a Monetary Policy Shock r n Set b t = 0 (no real shocks). Let v t follow an AR(1) process v t = � v v t � 1 + " v t Calibration: � v = 0 : 5 , � � = 1 : 5 , � y = 0 : 5 = 4 , � = 0 : 99 , � = ' = 1 , � = 2 = 3 , � = 4 . Dynamic e¤ects of an exogenous increase in the nominal rate (Figure 1) : Exercise: analytical solution

E¤ects of a Technology Shock Set v t = 0 (no monetary shocks). Technology process: a t = � a a t � 1 + " a t : Implied natural rate: r n b t = � � ya (1 � � a ) a t Dynamic e¤ects of a technology shock ( � a = 0 : 9 ) (Figure 2) Exercise: AR(1) process for � a t

Equilibrium under an Exogenous Money Growth Process � m t = � m � m t � 1 + " m (13) t Money market clearing b y t � � b l t = b i t (14) t � � b y n = e y t + b i t (15) where l t � m t � p t denotes (log) real money balances. Substituting (14) into (10): y t +1 g + b r n y n (1 + �� ) e y t = ��E t f e l t + �E t f � t +1 g + � b t � b (16) t Furthermore, we have b l t � 1 = b l t + � t � � m t (17)

Equilibrium dynamics 2 3 2 3 2 3 e E t f e r n y t y t +1 g b t 4 5 = A M ; 1 4 5 + B M 4 5 y n � t E t f � t +1 g b A M ; 0 (18) t b b � m t l t � 1 l t � 1 where 2 3 2 3 2 3 1 + �� 0 0 �� � 1 � � 1 0 4 5 4 5 4 5 A M ; 0 � � � 1 0 ; A M ; 1 � 0 � 0 ; B M � 0 0 0 0 � 1 1 0 0 1 0 0 � 1 ) A M � A � 1 Uniqueness ( M ; 0 A M ; 1 has two eigenvalues inside and one outside the unit circle.

E¤ects of a Monetary Policy Shock r n t = y n Set b t = 0 (no real shocks). Money growth process � m t = � m � m t � 1 + " m t where � m 2 [0 ; 1) Figure 3 (based on � m = 0 : 5 ) E¤ects of a Technology Shock Set � m t = 0 (no monetary shocks). Technology process: a t = � a a t � 1 + " a t Figure 4 (based on � a = 0 : 9 ). Empirical Evidence