Analysing shocks and policy in the NKM Alex Karalis Isaac University of Warwick March 2022 Karalis (University of Warwick) EC924 March 2022 1 / 34 Preview of the lecture 1 IRFs and shocks in the BNKM I Solving backward and forward looking difference equations I Substituting the short rate through the NKM I Gali’s matrix format I Solution and responses to shocks (IRFs) 2 From IRFs to policy analysis I Optimal Policy I Determinacy and (optimal) policy in the NKM I Welfare calculations I An implementable Taylor Rule F The implied policy-technology shock I Welfare with implementable Taylor rules Karalis (University of Warwick) EC924 March 2022 2 / 34 Solving difference equations Before studying Gali’s NKM and policy it is useful to recall difference eqns Consider the backward looking equation yt = γyt 1 + σet (1) where et ～iid (0, 1) and |γ| < 1 Then yt+s Etyt+s = s 1 ∑ k=0 γkσet+s k yt+s et = γsσ γkσ are the dynamic multipliers of the shocks Plotting γkσ as a function of k is referred to as the IRF Karalis (University of Warwick) EC924 March 2022 3 / 34 Backward looking equation: lag operator solution We can re-write (1) (I γL)yt = σet where Lyt = yt 1. Provided |γ| < 1 yt = (I γL) 1σet = (I + γL+ γ2L2 + γ3L3 + . . . )σet = ∞ ∑ k=0 γkσet k To see this define (I γL) 1 = (I + ψ1L+ ψ2L2 + . . . ) and solve (I γL) 1(I γL) = I , by equating coefficients on Li to 0 for i ≥ 1 Leading the last line forward s periods we again see yt+s et = γ sσ Karalis (University of Warwick) EC924 March 2022 4 / 34 Solving forward looking difference equations Macro models often have forward looking components The scalar forward-looking difference equation yt = φEtyt+1 + σut (2) can be written in terms of F where Fyt = Etyt+1: (1 φF )yt = σut yt = (1 φF ) 1σut = σ ∞ ∑ k=0 φkEtut+k provided |φ| < 1, by a similar argument to the backward-looking equation. Karalis (University of Warwick) EC924 March 2022 5 / 34 Solving forward looking difference equations In forward looking equations it is typical to consider autoregressive shocks ut = ρut 1 + et where et is iid as before Solution of the forward looking equation is completed by noting that Etut+k = ρ kut So that yt = σ ∞ ∑ k=0 (φρ)kut = σ 1 φρut a simple case of the result that the purely forward looking model implies no endogeneous dynamics Karalis (University of Warwick) EC924 March 2022 6 / 34 Policy and shocks in the NKM Technology shocks drive movements in unobservable natural rates, I Monetary policy based on observables implies policy errors. While it is possible to add other shocks (demand, costs push) to the NKM much policy analysis therefore assumes I A technology shock (NKM followed from RBC literature) I A monetary policy shock Recall the NKM y t = Et y t+1 1σ (rt r n t ) pit = βEtpit+1 + κy t (3) it = ρ+ φy y t + φpipit + vt Karalis (University of Warwick) EC924 March 2022 7 / 34 Adding shocks to the NKM There is a rule it = ρ+ φy y t + φpipit written for the nominal rate I Is it a Taylor rule To which we have appended a shock vt = ρvvt 1 + evt which we later relate to policy errors arising from unobservability of y t ynt = ya + ψ n yaat is the natural rate; y t = yt ynt Driven by at = ρaat 1 + ea,t Welfare consequences turn on how nearly a monetery policy sets y t = 0 Karalis (University of Warwick) EC924 March 2022 8 / 34 Dynamic responses to monetary and technology shocks Thus model responses to shocks ea,t , ev ,t drive results. Reprise: To examine responses to shocks we can put (3) in a form similar to (2): Substitute it into the IS eqn y t = Et y t+1 1σ (φy y t + φpipit + vt Etpit+1 r n t ) pit = βEtpit+1 + κy t[ σ+φy σ φpi σ κ 1 ] [ y t pit ] = [ 1 1σ 0 β ] Et [ y t+1 pit+1 ] + [ 1 σ 0 ] (r nt vt) (4) where r nt = r n t ρ are deviations in the natural rate due to ea,t Karalis (University of Warwick) EC924 March 2022 9 / 34 Gali solution Standard matrix algebra renders (4) in the format of section 3.4 Gali (2008):[ y t pit ] = ATEt [ y t+1 pit+1 ] + BT (r n t vt) (5) Provided eigenvalues of AT are < 1 in absolute value then, by analogy to (2), (5) solves forward: [ y t pit ] = [I2 ρxAT ] 1 BT xt (6) where ρx is the persistence of the composite shock xt = (r nt vt) IRFs follow by leading (6) forward and replacing future shocks with expected values: xt+i = ρ i xxt Note: in the book Gali derives (6) equation by equation, using undertermined coefficients. Karalis (University of Warwick) EC924 March 2022 10 / 34 Some features of the solution The pure forward looking model displays clearly the property that Estrella and Furher (2002) document: I The two variables are not tied to past values, but are free to jump down instantly in response to the shock I Solution format (6) makes it clear the whole expected future impact of the shock is accounted for in the initial response Initially we examine effects of vt for at fixed I Both the output gap and inflation jump down in response to the contractionary shock I The effect is muted for inflation compared to the gap, κ = 0.128, in the std calibration Karalis (University of Warwick) EC924 March 2022 11 / 34 Gali solution Consider first the response to monetary shock vt Specialise xt = vt , putting ea,t = 0 From (6) you can see, as Ch. 3 y t = ψvyvt pit = ψypivt Exercise: derive as in Gali and check against (6) Karalis (University of Warwick) EC924 March 2022 12 / 34 Gali IRFs for monetary shock (interest rate) 0 2 4 6 8 10 12 -0.2 -0.1 0 output gap to vt 0 2 4 6 8 10 12 -0.05 0 to vt 0 2 4 6 8 10 12 -0.2 -0.1 0 Aunnual to vt Karalis (University of Warwick) EC924 March 2022 13 / 34 Features of the response Contractionary shock unambiguously raises real rate above its natural counterpart I The more persistent the policy shock the longer the deviations last This induces an output gap which reduces employment and consumption Inflation also falls on impact, exacerbating effect of any increase in nominal rates on the real rate Karalis (University of Warwick) EC924 March 2022 14 / 34 Response to a technology shock Consider next the response to a technology shock. Putting vt = 0 xt = r n t = r n t r = ρ+ σψnyaEt [ at+1] ρ = σψnya(ρa 1)at (7) The technology shock boosts the natural rate of output I But the effect of monopoloy and price dispersion keeps output response smaller by about 10% in this calibration I Employment also falls in this case; a substitution effect Gali suggests is in keeping with recent evidence The negative output gap induces a negative inflation rate and (see the book) the central bank drops nominal and real rates to accomodate this Although we set the monetary shock, vt = 0, policy rule above may not close the output gap instantly Karalis (University of Warwick) EC924 March 2022 15 / 34 Response to a technology shock 0 2 4 6 8 10 12 -0.2 -0.1 0 Output gap to at 0 2 4 6 8 10 12 -0.1 -0.05 t to at 0 2 4 6 8 10 12 -1 -0.5 0 Annualized t to at Karalis (University of Warwick) EC924 March 2022 16 / 34 From IRFs to policy analysis We want to consider how to design policy rules/actions I What goes in the rule, what parameters to choose Start from the ideal case and examine welfare losses Which may arise from the presence of I Monopoly I Sticky prices In the absence of policy And in the presence of technology shocks which are key to RBC agenda Karalis (University of Warwick) EC924 March 2022 17 / 34 Efficient Allocation What is that The best possible outcome: Solve the planning problem maxU(Ct ,Nt) s.t. Ct = ( ∫ 1 0 Ct(i) e 1 e ) e e 1 Ct(i) = AtNt(i) 1 α Nt = ∫ 1 0 Nt(i)di Why is this not a dynamic problem Karalis (University of Warwick) EC924 March 2022 18 / 34 Efficient Allocation From the text book, or as an exercise, Efficiency requires us to : I Equate the Marginal Rate of Substitution (between...) to the Marginal Product of Labour I Equalise consumption, production and employment across varieties (because...) Very much as in the RBC (think UG - tangency of PPF and IC) There are two sources of distortions Karalis (University of Warwick) EC924 March 2022 19 / 34 Distortions Distortions from monopoly power I A constant distortion I Tackeld by a subsi
dy not monetary policy Distortions from price sticky-ness I Aggregate distortion I Price dispersion I Target of monetary policy to eliminate these suboptimalities Karalis (University of Warwick) EC924 March 2022 20 / 34 Distortions Monopoloy Inefficiently low production, employment, consumption Set price as mark up over Marginal Cost Pt =M Wt MPNt So that Uct Unt = Wt Pt = MPNt M First equality is delivered by markets; second violates efficiency Subsise employment at rate τ to restore desired equaliy Karalis (University of Warwick) EC924 March 2022 21 / 34 Distortions Pricy stickiness Aggregate inefficiency: output will differ from the flexible price equilibrium due to variations marginal costs across firms I A time-varying wedge between the MRS and the MPN Mt = Pt (1 τ)(Wt/MPNt) = PtM Wt/MPNt so that MRS=MPN can only be restored by stabilising mark-up at the frictionless level M Price Dispersion: I Products enter preferences symmetrically, but price stickyness induces relative prices 6= 1 so that demand is asymetric. I Although technology is symmetric, production is not, and some firms are too large (high MC) or small (low MC). From symmetric equilibrium MP to remove incentive to change prices Karalis (University of Warwick) EC924 March 2022 22 / 34 From IRFs to policy analysis To compare the performance of monetary rules a Welfare criterion is useful Rotemburg and Woodford (1999) show that an appropriate criterion is W = 1 2 E0 ∞ ∑ t=0 [( σ+ + α 1 α ) y 2t + e λ pi2t ] following from a second order approximation to lifetime utility U Output gap and inflation deviations reduce welfare with I Output gap deviations worse for higher risk aversion and Frisch labour elasticity, which amplify wedge between MRS and MPN I Counterintuitively, inflation deviations are more harmful for increases in substitutability (which reduces monopoly power) due to an increased utility loss from price dispersion in this case I Inflation losses are also increasing in average contract duration, i.e. in θ, as this increases the price dispersion resulting from any given inflation deviation Karalis (University of Warwick) EC924 March 2022 23 / 34 Optimal policy In an ideal world welfare losses would be 0 What is the policy that attains such an outcome Consider it = r n t This will always instantly close the output gap With y t = 0 marginal costs do not vary, prices are fixed Price stickiness is non binding While simple there are two problems I Determinacy I Implementability Karalis (University of Warwick) EC924 March 2022 24 / 34 Optimal policy: indeterminacy Bullard and Mitra (2002) show that stability of AT under the rule in (3) requires κ(φpi 1) + (1 β)φy > 0 (8) φpi > 1 if φy = 0 Our optimal policy is consistent with sunspot equilibria! Such equililbria are very bad for welfare To stabilise optimal policy introduce feedback which ‘leans against the wind’ it = r n t + φy y t + φpipit The threat of policy action deters coordination on alternative equilibria Karalis (University of Warwick) EC924 March 2022 25 / 34 A simple rule Unfortunately this is not an implementable rule as rnt is unobservable I For obvious reasons, estimating the natural rate is an active field Attention therefore shifts to implementable rules, based on observables The classic Taylor rule is one example it = ρ+ φy y t + φpipit (9) where y t = yt y , assuming y is known, or estmated from time-series data Karalis (University of Warwick) EC924 March 2022 26 / 34 Simple rules and monetary shocks In the case that at variation drives natural rate variation The observable rule induces a monetary policy shock φy (yt y) = φy (yt ynt + ynt y) = φy y t + φy y n t = φy y t + vt y nt is the deviation in the natural rate induced by the tech shock and missed by the observable rule The standard deviation of the policy shock is now proportional to the weight on the output gap in the Taylor rule Karalis (University of Warwick) EC924 March 2022 27 / 34 Simple rules and monetary shocks We have a model where there are tehcnology shocks, at = ρaat 1 + ea,t and monetary shocks vt We can analyse these using the dynamic responses captured in (5) as the at shock enters through the natural rate deviation r nt Both the deviation of the natural rate in the IS eqn,r nt , and the monetary shock, vt , are driven by the same underlying process Use the natural real rate and output equations rnt = ρ+ σ n yaEt at+1 ynt = n yaat + n y to define y nt and r n t , then combine xt = r n t + vt to see that xt = n ya(σ(ρa 1) φy )at (10) so the shock is really a re-scaled technology shock, c.f. (7) Karalis (University of Warwick) EC924 March 2022 28 / 34 Exploring policy rules The parameters φpi and φy are central bank choices What values should the central bank choose We can compare performance of alternative choices underW I W microfounds the old literature of LQR p policy assessment Social welfare losses,W are increasing in y 2t , pi 2 t To gain intuition about the performance of different rules we can look at their IRFs in response to (10) Karalis (University of Warwick) EC924 March 2022 29 / 34 Candidate Taylor Rules Gali (2008) considers four candidate Taylor rules on y t , pit : I (φpi, φy ) = (1.5, 0.125) ‘Taylor (1993)’ I (φpi, φy ) = (1.5, 0) ‘No output gap’ I (φpi, φy ) = (5, 0) ‘Hawk ’ I (φpi, φy ) = (1.5, 1) ‘Dove ’ Which regime entails the most aggressive action against inflation Which induces the smallest ouptut gap Which will be the best regime I The one which prodcues the smallest deviations in the weighted average of y t , pit Karalis (University of Warwick) EC924 March 2022 30 / 34 Candidate Taylor Rules Gali (2008) considers four candidate Taylor rules on y t , pit : I (φpi, φy ) = (1.5, 0.125) ‘Taylor (1993)’ I (φpi, φy ) = (1.5, 0) ‘No output gap’ I (φpi, φy ) = (5, 0) ‘Hawk ’ I (φpi, φy ) = (1.5, 1) ‘Dove ’ Which regime entails the most aggressive action against inflation Which induces the smallest ouptut gap Which will be the best regime I The one which prodcues the smallest deviations in the weighted average of y t , pit Karalis (University of Warwick) EC924 March 2022 30 / 34 IRFs to unit at shock, alternative Taylor rules 0 5 10 15 20 25 30 35 40 -0.6 -0.4 -0.2 0 output gap to obs policy Base No y =5 y=1 0 5 10 15 20 25 30 35 40 -0.8 -0.6 -0.4 -0.2 0 to obs policy Karalis (University of Warwick) EC924 March 2022 31 / 34 Welfare losses, alternative Taylor rules Gali runs a simulation exercise and compares losses underW : Taylor Rules φpi 1.5 1.5 5 1.5 φy 0.125 0 0 1 σ(y ) 0.55 0.28 0.04 1.40 σ(pi) 2.60 1.33 0.21 6.55 Loss 0.30 0.08 0.002 1.92 The results confirm the picture from the IRF analysis A strong response to inflation closes the inflation and output gaps I UnderW there are no trade offs between stablisations Karalis (University of Warwick) EC924 March 2022 32 / 34 Discussion of these results To stabilise inflation is to stabilise output I The Phillips Curve is a function of the output gap Output stabilisation is not improved by targeting the output gap High φy does not mean CB cares more about employment stabilisation I At least, not in the BNKM I In the BNKM underW there is no such thing as Hawks and Doves I There is no reason here to be ‘smooth’ with the interest rate I Things may be different in real life! But σ(v) = φy so raising φy increases variance of policy-error component This reduces the performance of the rule Output gap is unobserved, but pit is an excellent signal target the signal, don’t put weight on your noise! Karalis (University of Warwick) EC924 March 2022 33 / 34 Optimal policy A policy which could ensure y t = pit = 0 t would suffer no losss At the natural rate of output, the economy attains the efficient allocation I Recall the RBC roots and the consumer optimisation problem At least if a subsidy is in place to offset ss. costs of monopoly I Gali suggests a subsidy to equate the real wages and MPNt In the presence of sticky prices, price stability pi = 0 is the only way to avoid costs of price dispersion and attain the flexible price equilibrium Karalis (University of Warwick) EC924 March 2022 34 / 34