Monetary Policy Design in the Basic New Keynesian Model by Jordi - PowerPoint PPT Presentation

Monetary Policy Design in the Basic New Keynesian Model by Jordi Galí November 2010

The E¢cient Allocation max U ( C t ; N t ) hR 1 i � � � 1 subject to: 0 C t ( i ) 1 � 1 where C t � � di C t ( i ) = A t N t ( i ) 1 � � ; all i 2 [0 ; 1] Z 1 N t = N t ( i ) di 0 Optimality conditions : C t ( i ) = C t , all i 2 [0 ; 1] N t ( i ) = N t , all i 2 [0 ; 1] � U n;t = MPN t U c;t where MPN t � (1 � � ) A t N � � t

Sources of Suboptimality of Equilibrium 1. Distortions unrelated to nominal rigidities: MPN t , where M � " W t � Monopolistic competition : P t = M " � 1 > 1 � U n;t = W t = MPN t = ) < MPN t U c;t P t M Solution: employment subsidy �: Under ‡exible prices, P t = M (1 � � ) W t MPN t . � U n;t = W t MPN t = ) = U c;t P t M (1 � � ) Optimal subsidy: M (1 � � ) = 1 or, equivalently, � = 1 " . � Transactions friction (economy with valued money): assumed to be negligible

2. Distortions associated with the presence of nominal rigidities: P t � Markup variations resulting from sticky prices: M t = (1 � � )( W t =MPN t ) = P t M W t =MPN t (assuming optimal subsidy) � U n;t = W t M = ) = MPN t 6 = MPN t U c;t P t M t Optimality requires that the average markup be stabilized at its frictionless level. � Relative price distortions resulting from staggered price setting: C t ( i ) 6 = C t ( j ) if P t ( i ) 6 = P t ( j ) . Optimal policy requires that prices and quantities (and hence marginal costs) are equalized across goods.

Optimal Monetary Policy in the Basic NK Model Assumptions: � optimal employment subsidy = ) ‡exible price equilibrium allocation is e¢cient � no inherited relative price distortions, i.e. P � 1 ( i ) = P � 1 for all i 2 [0 ; 1] = ) the e¢cient allocation can be attained by a policy that stabilizes marginal costs at a level consistent with …rms’ desired markup, given existing prices : � no …rm has an incentive to adjust its price, i.e. P � t = P t � 1 and, hence, P t = P t � 1 for t = 0 ; 1 ; 2 ; ::: (aggregate price stability) � equilibrium output and employment match their counterparts in the (undistorted) ‡exible price equilibrium allocation.

Equilibrium under the Optimal Policy e y t = 0 � t = 0 i t = r n t for all t . Implementation: Some Candidate Interest Rate Rules Non-Policy Block: y t = � 1 � ( i t � E t f � t +1 g � r n e t ) + E t f e y t +1 g � t = �E t f � t +1 g + � e y t

An Exogenous Interest Rate Rule i t = r n t Equilibrium dynamics: � � � � e E t f e y t y t +1 g = A O � t E t f � t +1 g where � � 1 1 A O � � � � + � � Shortcoming : the solution e y t = � t = 0 for all t is not unique: one eigenvalue of A O is strictly greater than one. ! indeterminacy. (real and nominal).

An Interest Rate Rule with Feedback from Target Variables i t = r n t + � � � t + � y e y t Equilibrium dynamics: � � � � e E t f e y t y t +1 g = A T � t E t f � t +1 g where � � 1 � 1 � �� � A T � �� � + � ( � + � y ) � + � y + �� �

Existence and Uniqueness condition: (Bullard and Mitra (2002)): � ( � � � 1) + (1 � � ) � y > 0 Taylor-principle interpretation (Woodford (2000)): di = � � d� + � y d e y � � � � + � y (1 � � ) = d� �

A Forward-Looking Interest Rate Rule i t = r n t + � � E t f � t +1 g + � y E t f e y t +1 g Equilibrium dynamics: � � � � e E t f e y t y t +1 g = A F � t E t f � t +1 g where � � 1 � � � 1 � y � � � 1 ( � � � 1) A F � � (1 � � � 1 � y ) � � �� � 1 ( � � � 1) Existence and Uniqueness conditions (Bullard and Mitra (2002): � ( � � � 1) + (1 � � ) � y > 0 � ( � � � 1) + (1 + � ) � y < 2 � (1 + � )

Shortcomings of Optimal Rules � they assume observability of the natural rate of interest (in real time). � this requires, in turn, knowledge of: (i) the true model (ii) true parameter values (iii) realized shocks Alternative: “simple rules” , i.e. rules that meet the following criteria: � the policy instrument depends on observable variables only, � do not require knowledge of the true parameter values � ideally, they approximate optimal rule across di¤erent models

Simple Monetary Policy Rules Welfare-based evaluation: � U t � U n � X 1 X 1 � t � � = 1 � t t y 2 t + �� 2 � e W � � E 0 2 �E 0 t U c C t =0 t =0 = ) expected average welfare loss per period: L = 1 2 � [ � var ( e y t ) + � var ( � t )] See Appendix for Derivation.

A Taylor Rule i t = � + � � � t + � y b y t Equivalently: i t = � + � � � t + � y e y t + v t y n where v t � � y b t Equilibrium dynamics: � � � � e E t f e y t y t +1 g r n y n + B T ( b t � � y b = A T t ) � t E t f � t +1 g where � � � � � 1 � �� � 1 A T � � ; B T � � �� � + � ( � + � y ) � 1 t = � n r n y n � + � y + �� � : Note that b t � � y b and � � ya [ � (1 � � a ) + � y ] a t Exercise: � a t � AR (1) + modi…ed Taylor rule i t = � + � � � t + � y � y t

Money Growth Peg � m t = 0 money market clearing condition b t � � b y n l t = e y t + b i t � � t where l t � m t � p t and � t is a money demand shock following the process � � t = � � � � t � 1 + " � t De…ne l + t � l t � � t . = ) i t = 1 b t � b y n l + � ( e y t + b t ) b t � 1 = b l + l + t + � t � � � t

Equilibrium dynamics: 2 3 2 3 2 3 r n e E t f e b y t y t +1 g t 4 5 = A M ; 1 4 5 + B M 4 5 y n b � t E t f � t +1 g A M ; 0 t l + l + � � t t � 1 t 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 Simulations and Evaluation of Simple Rules

Table 4.1: Evaluation of Simple Monetary Policy Rules Taylor Rule Constant Money Growth � � 1 : 5 1 : 5 5 1 : 5 - - � y 0 : 125 0 0 1 - - ( � � ; � � ) - - - - (0 ; 0) (0 : 0063 ; 0 : 6) � ( e y ) 0 : 55 0 : 28 0 : 04 1 : 40 1 : 02 1 : 62 � ( � ) 2 : 60 1 : 33 0 : 21 6 : 55 1 : 25 2 : 77 welfare loss 0 : 30 0 : 08 0 : 002 1 : 92 0 : 08 0 : 38

Technical Appendix: Derivation of Second-Order Approximation of Welfare around the Undistorted Flexible Price Equilibrium Allocation We derive a second order approximation of utility around the e¢cient equilibrium allocation. Under our assumptions the latter corresponds to the ‡exible price equilibrium allocation. All along we assume that utility is separable in consumption and hours (i.e., U cn = 0 ). In order to lighten the notation we de…ne U t � U ( C t ; N t ) , U n t � U ( C n t ; N n t ) , and U � U ( C; N ) . A second order Taylor of expansion of U t yields: � C t � C n � � N t � N n � U t � U n U n c;t C n t + U n n;t N n t = t t t C n N n t t � C t � C n � 2 � N t � N n � 2 +1 + 1 2 U n cc;t ( C n t ) 2 t 2 U n nn;t ( N n t ) 2 t C n N n t t � � X t Letting e x t � log denote log-deviations from ‡exible price equilibrium values, we can write: X n t � � � � c t + 1 � � n t + 1 + ' U t � U n t = U n c;t C n c 2 + U n n;t N n n 2 e e e e t t t t 2 2 where we use the approximation X t � X n x t + 1 t x 2 ' e 2 e t X n t � � � � � 1 � � R 1 1 � 1 � � di , P t ( i ) Y t The next step consists in rewriting e n t in terms of the output gap. Using the fact that N t = A t 0 P t we have (1 � � ) e n t = e y t + d t

�R 1 � � � � � 1 � � di P t ( i ) where d t � (1 � � ) log : The following lemma shows that d t is proportional to the cross-sectional 0 P t variance of relative prices and, hence, of second order. � 1 � � Lemma 1: up to a second order approximation , d t = 2� var i f p t ( i ) g , where � � 1 � � + �� . See proof at the end of the present appendix. Accordingly we have: � � � � + U n n;t N n y t + 1 � � 1 + ' t + � t U t � U n t = U n c;t C n y 2 y 2 e e e 2(1 � � ) e y t + 2� var i f p t ( i ) g t t 2 1 � � where we have made use of the market clearing condition e c t = e y t for all t . Finally, we recall that when the optimal subsidy is in place, the ‡exible price allocation is e¢cient, thus implying � U n n;t N n t = U n c;t C n t (1 � � ) Hence, up to second order, we have � � + ' + � (1 � � ) � t = � 1 t + � (1 + � ( � � 1)) U t � U n 2 U n c;t C n y 2 e var i f p t ( i ) g t 1 � � 1 � � Next we derive a …rst order approximation to U n c;t C n t about the steady state: � C n � t � C U n c;t C n = U c C + ( U cc C + U c ) t C c n = U c C + U c C (1 � � ) b t It follows that, up to second order,