T HE Z ERO L OWER B OUND : F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE Alexander W. Richter Auburn University Nathaniel A. Throckmorton DePauw University
I NTRODUCTION • Popular monetary policy rule due to Taylor (1993) r t = φ ˆ ˆ π t + ε t , ε t bounded support • Taylor principle requires φ > 1 (active monetary policy) ◮ Necessary and sufficient for unique bounded equilibrium • Three key assumptions 1. Fiscal policy is passive 2. Policy parameters are fixed 3. Zero lower bound (ZLB) never binds • Leeper (1991) relaxes the first assumption and Davig and Leeper (2007) relaxes the second assumption • This paper relaxes the third assumption R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
M AIN F INDINGS • We adopt a textbook New Keynesian model with two alternative stochastic processes: 1. 2-state Markov process governing monetary policy 2. Persistent discount factor or technology shocks • Convergence is not guaranteed even if the Taylor principle is satisfied when the ZLB does not bind. • The boundary of the convergence region imposes a clear tradeoff between the expected frequency and average duration of ZLB events ◮ Household can expect frequent—but brief—ZLB events or infrequent—but prolonged—ZLB events ◮ Parameters of the stochastic process affect convergence R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
A L ITTLE B ACKGROUND • Davig and Leeper (2007): Fisherian Economy φ ( s t ) π t = E t π t +1 + ν t , ν ∼ AR (1) p ij = Pr[ s t = j | s t − 1 = i ] and φ ( s t = j ) = φ j , s t ∈ { 1 , 2 } • Integration over s t E [ π t +1 | s t = i, Ω − s t ] = p i 1 E [ π 1 t +1 | Ω − s t ] + p i 2 E [ π 2 t +1 | Ω − s t ] , where Ω − s = { ν t , ν t − 1 , . . . , s t − 1 , s t − 2 , . . . } t • Define π jt = π t ( s t = j, ν t ) . The system is � φ 1 � � π 1 t � � p 11 � � E t π 1 t +1 � � ν t � 0 p 12 = + 0 φ 2 π 2 t p 21 p 22 E t π 2 t +1 ν t R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
D ETERMINACY : F ISHERIAN E CONOMY • The existence of a unique bounded MSV solution requires p 11 (1 − φ 2 ) + p 22 (1 − φ 1 ) + φ 1 φ 2 > 1 ( LRTP ) • Example determinacy/convergence regions p 11 = 0 . 8; p 22 = 0 . 95 p 11 = 0 . 5; p 22 = 0 . 95 1.4 1.4 φ 2 1.2 φ 2 1.2 1 1 0.8 1 1.2 1.4 0.5 1 1.5 φ 1 φ 1 R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
D ETERMINACY : NK E CONOMY • Example determinacy regions p 11 = 0 . 8; p 22 = 0 . 95 p 11 = 0 . 5; p 22 = 0 . 95 1.4 1.4 φ 2 φ 2 1.2 1.2 1 1 0.5 1 1.5 −1 0 1 φ 1 φ 1 • ZLB is similar to DL with φ 1 = 0 and φ 2 > 1 , but with a truncated distribution on the nominal interest rate R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
N ONLINEAR F ISHERIAN E CONOMY A second-order approximation of the Euler equation around the deterministic steady state implies π t +1 ] + E t [ˆ β t +1 ]) 2 ˆ r t + (ˆ r t − E t [ˆ = � �� � =0 ( First Order ) π t +1 ] − E t [ˆ π t +1 − ˆ β t +1 ) 2 ] − ( E t [ˆ π t +1 − ˆ β t +1 ]) 2 ) E t [ˆ β t +1 ] − ( E t [(ˆ � �� � =0 ( First Order, Jensen’s Inequality ) R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
N ONLINEAR F ISHERIAN E CONOMY 1.5 Nonlinear 1.4 ( ρ = 0 . 95 ) 1.3 Nonlinear Fixed Regime φ 2 1.2 ( ρ = 0 . 85 ) Convergence Region 1.1 Linear 1 0.8 0.9 1 1.1 φ 1 R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
N UMERICAL P ROCEDURE • We compute global nonlinear solutions to each setup using policy function iteration on a dense grid ◮ Linear interpolation and Gauss-Hermite quadrature ◮ Duration of ZLB events is stochastic ◮ Expectational effects of hitting and leaving ZLB • Algorithm is non-convergent whenever ◮ a policy function continually drifts from steady state ◮ the iteration step (max distance between policy function values on successive iterations) diverges for 100+ iterations • Algorithm is convergent whenever ◮ the iteration step is less than 10 − 13 for 10+ iterations R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
N UMERICAL P ROCEDURE • Our algorithm yields the same determinacy regions Davig and Leeper analytically derive in their Fisherian economy and New Keynesian economy ◮ When the LRTP is satisfied (not satisfied), our algorithm converges (diverges) • Within the class of MSV solutions, there is a link between the convergent solution and determinate equilibrium ◮ Non-MSV solutions with fundamental or non-fundamental components may still exist ◮ Finding locally unique MSV solutions with a ZLB is helpful since most research is based on MSV solutions. • Not a proof, but it provides confidence that our algorithm accurately captures MSV solutions R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
L ITERATURE • Linearized models with a singular ZLB event ◮ Eggertsson and Woodford (2003), Christiano (2004), Braun and Waki (2006), Eggertsson (2010, 2011), Erceg and Linde (2010), Christiano et al. (2011), Gertler and Karadi (2011), and many others • Nonlinear models with recurring ZLB events ◮ Judd et al. (2011), Fern´ andez Villaverde et al. (2012), Gust et al. (2012), Basu and Bundick (2012), Mertens and Ravn (2013), Aruoba and Schorfheide (2013), Gavin et al. (2014) • Determinacy in Markov-switching models ◮ Davig and Leeper (2007), Farmer et al. (2009,2010), Cho (2013), Barth´ elemy and Marx (2013) • Determinacy in models with a ZLB constraint ◮ Benhabib et al (2001a), Alstadheim and Henderson (2006) R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
K EY M ODEL F EATURES • Representative Household ◮ Values consumption and leisure with preferences ∞ � β t { log( c t ) − χn 1+ η � E 0 / (1 + η ) } t t =0 ◮ Cashless economy and bonds are in zero net supply ◮ No capital accumulation • Intermediate and final goods firms ◮ Monopolistically competitive intermediate firms produce differentiated inputs ◮ Rotemberg (1982) quadratic costs to adjusting prices ◮ A competitive final goods firm combines the intermediate inputs to produce the consumption good R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
E XOGENOUS ZLB E VENTS • The monetary authority follows � r ( π t /π ∗ ) φ ¯ s t = 1 for r t = 1 s t = 2 for r = 1 . 015 , π ∗ = 1 . 005 , and φ ∈ { 1 . 3 , 1 . 5 , 1 . 7 } Baseline: ¯ • s t follows a 2 -state Markov chain with transition matrix � Pr[ s t = 1 | s t − 1 = 1] � � p 11 � Pr[ s t = 2 | s t − 1 = 1] p 12 = Pr[ s t = 1 | s t − 1 = 2] Pr[ s t = 2 | s t − 1 = 2] p 21 p 22 R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
C ONVERGENCE (S HADED ) R EGIONS 0.5 Non-Convergence 0.4 0.3 p 22 0.2 0.1 φ = 1 . 7 φ = 1 . 5 φ = 1 . 3 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p 11 R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
C ONVERGENCE (S HADED ) R EGIONS 0.4 Non-Convergence 0.35 Prob. of Going to ZLB ( p 12 ) 0.3 φ = 1 . 7 0.25 0.2 φ = 1 . 5 0.15 0.1 0.05 φ = 1 . 3 0 1 1.2 1.4 1.6 1.8 2 2.2 Avg. Duration of ZLB Event (1 /p 21 ) R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
E NDOGENOUS ZLB E VENTS • The monetary authority follows r ( π t /π ∗ ) φ } r t = max { 1 , ¯ • Discount Factor ( β ) or Technology ( a ) follows z ) ρ z exp( ε t ) , ε t ∼ N (0 , σ 2 z t = ¯ z ( z t − 1 / ¯ ε ) • Let z t − 1 ∈ { z 1 , . . . , z N } . Probability of going to or staying at the ZLB given z t − 1 is � Pr { s t = 2 | z t − 1 = z i } = π − 1 / 2 φ ( ε j | 0 , σ ε ) j ∈J 2 ,t ( i ) J 2 ,t ( i ) is the set of indices where the ZLB binds given the state z t − 1 = z i . φ are the Gauss-Hermite weights. R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
Z ERO L OWER B OUND P ROBABILITIES 1 Prob. of Going to/Staying at ZLB ( ρ a , σ ε ) = (0 . 70 , 0 . 0180) ( ρ a , σ ε ) = (0 . 75 , 0 . 0155) ( ρ a , σ ε ) = (0 . 80 , 0 . 0133) 0.8 0.6 0.4 0.2 0 −2 0 2 4 6 8 Technology R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
C ONVERGENCE (S HADED ) R EGIONS 0.03 Non-convergence 0.025 φ = 1 . 7 0.02 φ = 1 . 5 σ ε 0.015 0.01 φ = 1 . 3 0.005 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 ρ a R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
E XPECTATIONAL E FFECT π = 0 . 5 ¯ 1 Inflation Rate 0 −1 ( ρ a , σ ε , σ z ) = (0 . 70 , 0 . 0187 , 0 . 0262) ( ρ a , σ ε , σ z ) = (0 . 90 , 0 . 0091 , 0 . 0209) −2 −4 −2 0 2 4 6 8 Technology R ICHTER AND T HROCKMORTON : T HE ZLB: F REQUENCY , D URATION , AND N UMERICAL C ONVERGENCE
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