Monetary Policy Tradeos by Jordi Gal November 2010 Policy Tradeos - PowerPoint PPT Presentation

Monetary Policy Tradeo¤s by Jordi Galí November 2010

Policy Tradeo¤s and the New Keynesian Phillips Curve � t = �E t f � t +1 g + � ( y t � y n t ) Criticism: no policy tradeo¤s, optimality of strict in‡ation targeting y e t � y n Implicit assumption: t = � Alternative: time-varying y e t � y n t gap � t = �E t f � t +1 g + �x t + u t where x t � y t � y e t and u t � � ( y e t � y n t )

The Monetary Policy Problem 1 X � t � � � x x 2 t + � 2 min E 0 f g (1) t t =0 subject to: � t = �E t f � t +1 g + �x t + u t where f u t g evolves exogenously according to u t = � u u t � 1 + " t In addition: x t = � 1 � ( i t � E t f � t +1 g � r e t ) + E t f x t +1 g (2) Note: utility based criterion requires � x = � �

Optimal Discretionary Policy Each period CB chooses ( x t ; � t ) to minimize � x x 2 t + � 2 t subject to � t = �x t + v t where v t � �E t f � t +1 g + u t is taken as given. Optimality condition : x t = � � � t (3) � x Equilibrium � t = � x � u t (4) x t = � � � u t (5) i t = r e t + �[ �� (1 � � u ) + � x � u ] u t (6) 1 where � � � 2 + � x (1 � �� u )

Implementation: t + [(1 � � u ) �� i t = r e + � u ] � t � x uniqueness condition: �� � x > 1 (likely if utility-based: �� > 1 ) Alternatively, i t = r e t + �[ �� (1 � � u ) + � x � u ] u t + � � ( � t � � x � u t ) uniqueness condition: � � > 1 :

Optimal Policy with Commitment State-contingent policy f x t ; � t g 1 t =0 that minimizes X 1 � t � � � x x 2 t + � 2 E 0 t t =0 subject to the sequence of constraints: � t = �E t f � t +1 g + �x t + u t Lagrangean: X 1 L = � 1 � t [ � x x 2 t + � 2 2 E 0 t + 2 � t ( � t � �x t � �� t +1 )] t =0 First order conditions: � x x t � �� t = 0 � t + � t � � t � 1 = 0 for t = 0 ; 1 ; 2 ; ::: and where � � 1 = 0 .

Eliminating multipliers: x 0 = � � � 0 (7) � x x t = x t � 1 � � � t (8) � x for t = 1 ; 2 ; 3 ; ::: .. Alternative representation: x t = � � b p t (9) � x for t = 0 ; 1 ; 2 ; ::: where b p t � p t � p � 1

Equilibrium p t = a b b p t � 1 + a�E t f b p t +1 g + au t � x for t = 0 ; 1 ; 2 ; ::: where a � � x (1+ � )+ � 2 Stationary solution: � p t = � b b p t � 1 + (1 � ��� u ) u t (10) 1 � p 1 � 4 �a 2 for t = 0 ; 1 ; 2 ; ::: where � � 2 (0 ; 1) : 2 a� ! price level targeting ! �� x t = �x t � 1 � � x (1 � ��� u ) u t (11) for t = 1 ; 2 ; 3 ; ::: as well as �� x 0 = � � x (1 � ��� u ) u 0