A framework for monetary-policy analysis Overview Basic concepts: Goals, targets, intermediate targets, indicators, operating targets, instruments

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1 Eco 504, part 1, Spring _L6_S06.tex Lars Svensson 3/6/06 A framework for monetary-policy analysis Overview Basic concepts: Goals, targets, intermediate targets, indicators, operating targets, instruments Commitment equilibrium, discretion equilibrium, reaction functions Monetary policy rules: Instrument rules, targeting rules Linear model, simple example c 2006 Lars E.O. Svensson. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personal use or distributed free. 1 Basic concepts Goals (price stability, high sustainable growth, full employment) Targets (operational goals: inflation target, zero output gap,...) Intermediate targets (fixed exchange rate, k% money growth) Not inherent goal Highly correlated with goals (targets) Easier to observe and fulfill than goals (targets) Indicators (information variables: state of the economy, stance of monetary policy, goal fulfillment) Operating target (almost perfect control, short-term target) Federal funds rate Repo rate Instrument (perfect control, used to implement policy) Monetary base, nonborrowed monetary base Simplification: Short interest rate 2

2 Monetary policy rules Targeting rule: A given condition for the target variables (or forecasts thereof) to be fulfilled (target criterion). Instrument rule: The instrument is a given function of observed variables Explicit instrument rule: The instrument is a given function of predetermined variables only Implicit instrument rule: The instrument is a given function also of forward-looking variables Simple instrument rule: The instrument is a function of only a few variables 3 Linear model (Lecture notes: Optimization under commitment... ) Xt+1 Xt C = A + Bi H+1 t + ε t 0 t+1 (1) X t predetermined variables (constant: X t =(1,...) 0 ), forward-looking variables, i t instruments (control variables), ε t iid zero-mean shocks with covariance matrix I Exogenous variables: predetermined variables who depend only on current shocks and lagged variables of themselves (X 1t below) X 1,t+1 X 2,t+1 A X X 1t 0 C 1 x1,t+1 t [H 11 H 12 ] x 2,t+1 t = A X21 A X22 A X23 A X24 X 2t A x11 A x12 A x13 A x14 x 1t + B X2 B x1 i t+ C 2 0 ε t+1 A 0 x21 A x22 A x23 A x24 x 2t B x2 0 Ax13 A H need not be invertible. x14 invertible; 3rd and 4th block determine x A x23 A t x24 4

3 Target variables: Endogenous variables that enter the loss function. For a target variable Y jt with target level Ŷjt, the loss function is increasing in Y jt Ŷjt. For simplicity, measure target variables as deviations from (constant) target levels Y t = D X t. (2) i t Period loss function L t = 1 2 Y t 0 ΛY t 1 X 0 t W X t, (3) 2 Λ and W D 0 ΛD are given symmetric positive semidefinite matrices. The elements of Λ are the weights on the target variables in the period loss function. Intertemporal loss function in period t be where 0 <δ<1 is a discount factor. X E t τ=0 i t i t (1 δ)δ τ L t+τ, (4) 5 Optimization under commitment: The commitment equilibrium Lagrange method: Lagrangian, FOCs, difference equation, solution Solving a system of linear difference equations with forward-looking variables Diagonalization Generalized Schur decomposition (Klein, Söderlind, Sims Gensys) Recursive saddlepoint method: Formulate recursive dual problem, Bellman equation, solve Lineare-quadratic regulator Optimization under discretion: The discretion equilibrium Recursive formulation, Bellman equation, iteration 6

4 Commitment equilibrium: Optimal policy under commitment (Söderlind 99, Svensson- Woodford 05) CB commits once and for all to a reaction function that minimizes (4) in period t 0, subject to X t0, (1) and (3). For t t 0 : Xt = F x (5) Ξ t 1 Xt i t = F i, (6) Ξ t 1 Xt+1 Xt C = M + ε Ξ t Ξ t 1 0 t+1. (7) Ξ t is a vector of Lagrange multipliers of lower block of (1), Ξ 1 =0. The matrices F x, F i,andm depend on A, B, H, D, Λ, andδ, but are independent of C. This demonstrates the certainty equivalence of the commitment solution: it is independent of the covariance matrix of the shocks to X t, CC 0, and the same as when that covariance matrix is zero. 7 History-dependence (Woodford) Xt t 0 Ξ t = M ΞX X t + M ΞΞ Ξ t 1 = M τ ΞΞ M ΞX X t τ τ=0 8

5 Commitment in a timeless perspective (Woodford, Svensson-Woodford 05). Recommitment in period t, taking into account Ξ 1 6=0. Modified intertemporal loss function in period t X E t τ=0 (1 δ)δ τ L t+τ + 1 δ δ Ξ 0 t 1H (8) Discretion equilibrium (Söderlind 99, Svensson-Woodford 05): CB minimizes (3) in each period t, subject to X t,(1), +1 = G t+1 X t+1, and reoptimization in period t +1. Equilibrium reaction function under discretion i t = ˆF ix X t (9) = ˆF xx X t No history-dependence Stabilization bias, ˆF ix 6= F ix, ˆF xx 6= F xx. Discretion equilibrium for modified intertemporal loss function (8) results in commitment in timeless perspective (cf. recursive saddlepoint method) 9 Given (explicit) reaction function (policy function) i t = FX t (10) F matrix of response coefficients Respond to predetermined variables only, else equilibrium condition For given F, add (10) to (1), solve system of difference equations ( Solving a system... : Xt+1 A11 + B = 1 F A 12 Xt C + ε H+1 t A 21 + B 2 F A 22 0 t+1 where A11 A A = 12 B1, B =. A 21 A 22 B 2 This results in in = GX t (11) X t+1 =(A 11 + A 12 G + B 1 F )X t + Cε t+1 (12) Implicit reaction function, equilibrium relation i t = FX t + F (13) Circularity, equilibrium condition. Implementation? Iteration? Solve system of difference equations Equilibrium reaction function ( = GX t in equilibrium) i t =(F + FG)X t (14) 10

6 Simple example (Svensson EER 97, 99, 03 appendix) Lags, imperfect control 2-year control lag for inflation, 1-year control lag for output (VAR studies, Leeper- Sims-Zha) Similarity to models used by CBs Simplest possible (compromise on expectations, Lucas critique) Accelerationist Phillips curve π t+1 = π t + α + ε t+1 (15) Aggregate demand (output gap) +1 = β x β r (i t π t+1 t r)+η t+1 (16) (π t+τ t E t π t+τ, r average real interest rate). Years, π t = p t p t 1 inflation, (log) output (gap), i t instrument (short interest rate), and ε t and exogenous η t zero-mean iid shocks, variances σ 2 ε, σ2 η. All parameters 0. Note that 1-period inflation expectations are predetermined in period t π t+1 t = π t + α (17) 11 Objectives: Loss function Stabilize inflation around constant inflation target π Stabilize output gap Period loss function L(π t, )= 1 h i (π t π ) 2 + λx 2 t, (18) 2 λ 0 (relative) weight on output-gap stabilization λ>0 flexible inflation targeting, λ =0 strict inflation targeting State-space form πt+1 1 α πt 0 εt+1 X t+1 = + (i +1 β r β x + αβ r β t r)+ r η t+1 πt π Y t = 1 0, W = 0 λ No forward-looking variables, no difference discretion-commitment 12

7 Optimal policy Rewrite AS π t+2 = π t+2 t+1 + ε t+2 π t+2 t+1 = π t+2 t + ε t+1 + αη t+1 π t+2 t = π t+1 t + α+1 t (19) +1 = +1 t + η t+1 Note that 1 E t 2 [(π t+1 π ) 2 + λx 2 t+1] = 1 2 [(π t+1 t π ) 2 + λx 2 t+1 t ]+1 2 (σ2 ε + λσ 2 η) Consider problem subject to min E 0 X t=0 (1 δ)δ t [(π t+1 t π ) 2 + λx 2 t+1 t ] π t+2 t+1 = π t+1 t + α+1 t + ε t+1 + αη t+1 π t+1 t + α+1 t + θ t+1 where θ t ε t + αη t, and consider +1 t the control variable in period t. Then, given π t+1 t, +1 t and, use (16) to choose i t according to i t = r + π t+1 t 1 β r (+1 t β x ). 13 Solve by dynamic programming (problem set) V (π t+1 t )=min ½(1 δ) 1 h πt+1 t π i ¾ 2 + λx 2 +1 t 2 t+1 t + δe t V (π t+2 t+1 ) (20) subject to π t+2 t+1 = π t+1 t + α+1 t + θ t+1 (21) Alternative solution: Lagrange problem 14

8 Introduce the variables Constraint Period loss function π t π t+1 t +1 t π t+1 = π t + α + θ t+1 (22) L t = 1 2 [( π t π ) 2 + λ x 2 t] Lagrangian L 0 =E 0 X t=0 (1 δ)δ t { 1 2 [( π t π ) 2 + λ x 2 t]+δϕ t+1 ( π t+1 π t α θ t+1 )}, (23) where ϕ t+1 is the Lagrange multiplier of the constraint (22). Note that π t is predetermined in period t, and consider the first-order conditions for an optimum, with respect to π t+1 and. 15 They are with respect to π t+1,and E t π t+1 π +E t ϕ t+1 δe t ϕ t+2 =0 (24) λ δαe t ϕ t+1 =0 (25) with respect to. From (25), we have E t ϕ t+1 = δα x λ t. 16

9 Using this in (24), we can write a consolidated (without Lagrange multipliers) first-order condition as E t π t+1 π + λ δα ( δe t +1 )=0 (26) for t 0. In order to find the equilibrium, rewrite (22) as = 1 α (E t π t+1 π t ) (27) and use this to eliminate in (26). This results in a difference equation for π t, E t π t+1 π + λ δα 2[(E t π t+1 π t ) δ(e t π t+2 E t π t+1 )] = 0. For the case of flexible inflation targeting, λ>0, rewrite the difference equation as where E t [( π t+2 π ) 2a( π t+1 π )+ 1 δ ( π t π )] = 0, 2a 1+ 1 δ + α2 (28) λ (since π t is given, it is natural to express the difference equation in terms of the inflation forecasts). 17 By standard methods, the solution to this difference equation can be shown to fulfill E t π t+1 π = c( π t π ) (29) for t 0 (recall that π t isgiveninperiodt). Here, the coefficient c fulfills 0 <c<1and is the smaller root of the characteristic equation, hence given by μ 2 2aμ + 1 =0; δ (30) r c a a 2 1 δ. (31) Furthermore, c is an increasing function of λ, c(λ), whichfulfills c(0) = lim λ 0 c(λ) =0, c( ) lim λ c(λ) =1(show!). Forthecaseofstrictinflation targeting, λ =0,wehavec(0) = 0, so (29) is replaced by for t 0. It follows from (22) fulfills E t π t+1 π =0 = 1 c α ( π t π ). 18

10 Optimal reaction function By (16), the optimal interest setting in period t then follows i t = r + π t+1 t 1 β r +1 t + β x β r = r + π t 1 β r + β x β r = r + π t + 1 c ( π t π )+ β x µ αβ r β r = r + π c(λ) (π t+1 t π )+ β x (32) αβ r β r ī + f π (λ)(π t+1 t π )+ f x (33) Express in terms of π t and : µ i t =ī c(λ) (π t π ) µ αβ r + α 1+ 1 c(λ) + β x (34) αβ r β r ī + f π (λ)(π t π )+f x (λ) (35) Definition: A(n explicit) reaction function expresses the instrument as a function of predetermined variables. 19 Properties of the optimal reaction function. Dependence on λ. Properties of coefficients Coefficients are functions of parameters of AD, AS and LF. Coefficients of π t+1 t, f π (λ) 1, f π λ < 0, Coefficients of π t,, fx > 0, f x λ =0, f π (λ) 1, f π λ < 0, f x(λ) > 0, f x λ < 0 Result: Optimal to respond to all of π t,, (or π t+1 t, ). Result: Optimal response implies i t π t increasing in π t (or i t π t+1 t increasing in π t+1 t ). (Taylor principle, Woodford) Result: Coefficient response to λ intricate. Not necessarily f x λ > 0. 20

11 Strict inflation targeting, λ =0 c(0) = 0 First-order condition π t+2 t = π (36) 2-year-ahead inflation forecast on target (targeting rule) Optimal reaction function µ i t = ī (π t+1 t π )+ β x µ αβ r β r =ī µ (π t π )+ α β x αβ r αβ r β r Flexible inflation targeting (λ >0), λ>0 0 <c<1 π t+2 t π = λ δα (δ+2 t +1 t ) 2-year-ahead inflation-forecast gap proportional to forecast of output-gap change (targeting rule) 21 Strict output stabilization, λ c( ) =1 +1 t =0 Optimal reaction function i t = r + π t+1 t + β x µ β r = r + π t + α + β x β r 22

12 Relative variability of π t and π t+2 t+1 π = c(λ) π t+1 t π + ε t+1 + αη t+1 π t+2 = π t+2 t+1 + ε t+2 +1 t = 1 πt+2 t π t+1 t α = 1 c(λ) (π t+1 t π ) α +1 = +1 t + η t+1 π t is an AR(1) plus serially correlated (s.c.) noise. is proportional to an AR(1) (plus s.c. noise), and hence itself an AR(1) plus s.c. noise. Var[π t ] increasing, Var[ ] decreasing in λ (show!) Under strict inflation targeting, π t is constant plus s.c. noise. Under strict output stabilization, is constant plus noise. π t has unit root. 23 Sum up Basic concepts: Goals, targets, intermediate targets, indicators, instruments Predetermined/forward-looking variables Reaction function, implicit reaction function Monetary policy rules: Targeting rules, instrument rules Discretion/commitment equilibria Simple example Endogenous predetermined variables No forward-looking variables: No difference discretion-commitment AS, AD, LF Reasonable control lags Dynamic programming (alternative, Lagrangian) First-order condition Optimal reaction function Respond to all (relevant) predetermined variables (all determinants of forecasts of target variables) Response coefficient properties, model-dependent Targeting rule 24