Monetary Policy Design in the Basic New Keynesian Model by Jordi Galí October 2015
The E cient Allocation where C t R 1 0 C t(i) 1 1 1 di Optimality conditions: max U (C t ; N t ; Z t ) subject to: C t (i) = A t N t (i) 1 ; all i 2 [0; 1] N t = Z 1 0 N t (i)di C t (i) = C t, all i 2 [0; 1] N t (i) = N t, all i 2 [0; 1] U n;t U c;t = MP N t where MP N t (1 )A t N t
Sources of Suboptimality of Equilibrium 1. Distortions unrelated to nominal rigidities: Market power Optimal price setting: P t = M W t MP N t, where M " " 1 > 1 =) U n;t = W t = MP N t U c;t P t M < MP N t Assume employment subsidy : Under exible prices, P t = M (1 )W t MP N t. =) U n;t U c;t = W t P t = MP N t M(1 ) Optimal subsidy: M(1 ) = 1 or, equivalently, = 1 ".
2. Distortions associated with the presence of nominal rigidities: Markup variations resulting from sticky prices (assuming optimal subsidy): M t = P t (1 )(W t =MP N t ) = P t M W t =MP N t U n;t =) = W t M = MP N t 6= MP N t U c;t P t M t E ciency requirement: average markup = desired markup, all t 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] Optimal policy: replicate exible price equilibrium allocation. Implementation: commit to stabilizing 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 natural counterparts.
Equilibrium under the Optimal Policy y t = y n t ) ey t = 0 t = 0 for all t. i t = r n t Implementation: Some Candidate Interest Rate Rules Non-Policy Block: ey t = 1 (i t E t f t+1 g rt n ) + E t fey t+1 g t = E t f t+1 g + ey t where rt n = (1 a ) ya a t + (1 z )z t
An Exogenous Interest Rate Rule Equilibrium dynamics: where eyt t i t = r n t = A O Et fey t+1 g E t f t+1 g A O 1 1 + Shortcoming: the solution ey 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 ey t Equilibrium dynamics: eyt t = A T Et fey t+1 g E t f t+1 g where 1 A T + y + 1 + ( + y )
Existence and uniqueness condition: (Bullard and Mitra (2002)): ( 1) + (1 ) y > 0 Taylor-principle interpretation (Woodford (2000)): lim k!1 di t+k d t = + y lim k!1 dey t+k d t = + y(1 )
Figure 4.1 Determinacy and Indeterminacy Regions: Standard Taylor Rule 2 1.5 Determinacy? : 1 0.5 Indeterminacy 0 0 0.5 1 1.5 2? y
A Forward-Looking Interest Rate Rule Equilibrium dynamics: where A F i t = r n t + E t f t+1 g + y E t fey t+1 g eyt t = A F Et fey t+1 g E t f t+1 g 1 1 y 1 ( 1) (1 1 y ) 1 ( 1) Existence and uniqueness conditions (Bullard and Mitra (2002): ( 1) + (1 ) y > 0 ( 1) + (1 + ) y < 2(1 + )
Figure 4.2 Determinacy and Indeterminacy Regions: Forward Looking Taylor Rule 25 20 15 Indeterminacy?: 10 5 Determinacy 0 0 0.5 1 1.5 2? y
Shortcomings of Optimal Rules assumed 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 Simple rules : 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: X 1 W E 0 t Ut U c C t=0 U n t = 1 2 E 0 1X t=0 t + ' + ey t 2 + 1 2 t =) expected average welfare loss per period: L = 1 + ' + var(ey t ) + var( t ) 2 1
A Taylor Rule Equivalently: i t = + t + y by t i t = + t + y ey t + v t where v t y by n t Equilibrium dynamics: eyt t where A T = A T Et fey t+1 g E t f t+1 g 1 + ( + y ) + B T (br n t v t ) 1 ; B T and 1 + y + : Note that br n t v t = ya ((1 a ) + y )a t + (1 z )z t Exercise: a t AR(1) + modi ed Taylor rule i t = + t + y y t
Table 4.1 Evaluation of Simple Rules: Taylor Rule Technology Demand 1:5 1:5 5 1:5 1:5 1:5 5 1:5 y 0:125 0 0 1 0:125 0 0 1 (y) 1:85 2:07 2:25 1:06 0:59 0:68 0:28 0:31 (ey) 0:44 0:21 0:03 1:23 0:59 0:68 0:28 0:31 () 0:69 0:34 0:05 1:94 0:20 0:23 0:09 0:10 L 1:02 0:25 0:006 7:98 0:10 0:13 0:02 0:02
Money Growth Peg Money demand: where l t m t p t. where 2 [0; 1). m t = 0 l t = y t i t t t = t 1 + " t b lt = ey t + by n t bi t t Letting l t + l t + t bi t = 1 (ey t + by t n b l + t ) Imposig the assumed rule m t = 0, and clearing of the money market: b l + t 1 = b l t + + t t
Equilibrium dynamics: 2 A M;0 4 ey t t b l + t 1 3 2 5 = A M;1 4 E t fey t+1 g E t f t+1 g b l + t 3 2 5 + B M 4 brn t by n t t 3 5 where A M;0 2 4 1 + 0 0 1 0 0 1 1 3 5 ; A M;1 2 4 1 0 0 0 0 1 3 5 ; B M 2 4 1 0 0 0 0 0 0 1 3 5
Table 4.2 Evaluation of Simple Rules: Constant Money Growth Technology Demand Money Demand (y) 1:72 0:59 1:07 (ey) 0:92 0:59 1:07 () 0:35 0:12 0:55 L 0:29 0:04 0:69
The Taylor Rule (Taylor 1993) i = 4 + 1.5( π 2) + 0.5 y t t t
Source: Taylor 1999
Clarida, Galí and Gertler (QJE 2000) i = ρi + (1 ρ)[ r+ π + βe{ π π } + γe{ y y }] * * * t t 1 t t+ 1 t t+ 1 t+ 1
Orphanides (JME 2003)