Changes in the Inflation Target and the Comovement between Inflation and the Nominal Interest Rate

Transcription

1 Economics Working Paper Series Changes in the Inflation Target and the Comovement between Inflation and the Nominal Interest Rate Yunjong Eo and Denny Lie July 2018

2 Changes in the Inflation Target and the Comovement between Inflation and the Nominal Interest Rate Yunjong Eo Denny Lie July 25, 2018 Abstract Does raising an inflation target require increasing the nominal interest rate in the short run? We answer this question using a New Keynesian model calibrated to the U.S. economy in which firms explicitly take into account changes in the inflation target in their price setting behavior. We find that the short-run comovement between the nominal rates and inflation conditional on the change in the inflation target is most likely positive. While this so-called Neo-Fisherism is less likely to hold the more backward-looking elements are incorporated into the model, two features play an important role in generating the positive comovement in spite of rich backward-looking elements in our model. First, a Taylor-type rule mitigates the impact of the backward-looking elements on forming inflation expectations compared to strict inflation targeting. Second, the additional forward-looking effect driven by firms explicit consideration of inflation target adjustment in setting their prices enlarges the region of the parameter space exhibiting Neo-Fisherism. Our results are robust to empirically plausible parameterizations of the model. JEL Classification: E12; E32; E58; E61; Keywords: Neo-Fisherism; inflation expectations; a Taylor-type rule; strict inflation targeting; hybrid NKPC; inflation target adjustment; School of Economics, The University of Sydney, NSW 2006, Australia; yunjong.eo@sydney.edu.au School of Economics, The University of Sydney, NSW 2006, Australia; denny.lie@sydney.edu.au

3 1 Introduction Since the global financial crisis (GFC) and the subsequent Great Recession, nominal interest rates in the U.S. and other developed economies have been persistently lower than before the GFC. Inflation rates, however, have also continued to be low in these economies and in some cases in inflation-targeting economies, have been below the targets. This low inflation, low interest rate environment presents a challenge for central banks. When the short-term nominal interest (policy) rate is low, the central bank s ability to use its conventional monetary policy tool and cut the rate during a recession or an economic downturn is more limited. As shown by Kiley and Roberts (2017), in such an environment the frequency and length of hitting the effective lower bound (ELB) on the nominal interest rate are higher, and it may lead to costly economic performance associated with inflation and economic activity being more volatile and systemically falling short of their desirable levels. To alleviate these concerns, several alternative policy frameworks have been proposed. One such framework is for an inflation-targeting central bank to simply raise its inflation target, as proposed by Blanchard, Dell Ariccia and Mauro (2010), Ball (2014), and Krugman (2014), among others. 1 Raising the inflation target, especially in a low interest rate environment, in turn poses a substantive policy question: Does raising the target entail an increase in the nominal interest rate? The question has an important policy implication because if a higher inflation target entails a reduction in the nominal rate, policy implemented to avoid the ELB may, in fact, result in hitting the ELB. The answer to this question is relatively clear in the long run. From the Fisher equation, i t = E t π t1 r t (1) where i t is the nominal interest rate, r t is the real interest rate, and E t π t1 is the oneperiod ahead expected inflation the nominal interest rate and expected inflation, and hence inflation, move together one-for-one in the long run insofar as the classical dichotomy holds, i.e., the long-run real interest rate is independent of nominal variables and is solely deter- 1 Two other notable alternative frameworks that have been proposed are price-level targeting (e.g., Gaspar, Smets and Vestin (2010), Bernanke (2017), and Williams (2017)) and nominal-income targeting (e.g., McCallum and Nelson (1999), Frankel (2013), and Williams (2016)). For other studies on changing the inflation target, see Williams (2016), Rosengren (2018), and Summers, Wessel and Murray (2018). 2

4 mined by macroeconomic fundamentals such as the discount rate and the long-run output growth. Built on this long-run relationship, Cochrane (2016) and Williamson (2016) argue that a central bank can raise inflation by setting a higher interest rate consistent with an inflation target. They dub this property Neo-Fisherism. In the short run, however, the answer is not so clear-cut. The presence of nominal frictions such as price and wage rigidities complicates the short-run relationship between the inflation target, inflation, and the nominal interest rate. The comovement between inflation and the nominal interest rate may break down, as nominal shocks (e.g., an increase in the inflation target) have short-run effects on the real interest rate. Garín, Lester and Sims (2018) argue that increasing the inflation target and inflation, in fact, necessitates a short-run decrease in the nominal interest rate in a prototypical, textbook New Keynesian model. In this paper, we investigate the Neo-Fisherian property, which we define, following Garín, Lester and Sims (2018), as a positive comovement between the nominal interest rate and inflation conditional on a change in the inflation target. We do this first within a prototypical New Keynesian model, where the closed-form analytical solution is readily available, and then in a more general model calibrated to the U.S. economy. Two features of the models are worth mentioning. First, in our New Keynesian models considered in this paper, firms explicitly take into account the current inflation target in their price-setting process through an indexation mechanism. We adopt this specification from Fève, Matheron and Sahuc (2010), who find gradual changes in the inflation target have been a major driving force of business cycle fluctuations in the euro area. 2 This specification leads to a generalized New Keynesian Phillips curve (NKPC) where inflation depends directly on the inflation target. Second, for the analytical study using the prototypical New Keynesian model, we consider a Taylor-type monetary policy rule in addition to strict inflation-targeting rule. As we will discuss later, these two features turn out to be important in generating a short-run comovement between inflation and the nominal interest rate. Our findings can be summarized as follows. Using the prototypical model, we find that forward-looking elements make it more likely for the model to exhibit Neo-Fisherism, while 2 We examine the role of the inflation target in explaining inflation dynamics for the U.S. economy when the model is calibrated in Section 4. 3

5 the backward-looking elements such as past price indexation and habit formation make it less likely. This finding is similar to the finding in Garín, Lester and Sims (2018), which also investigates the Neo-Fisherian property within a textbook New Keynesian model. Garín, Lester and Sims (2018) argue, in particular, that a modest, empirically plausible degree of backward-looking behavior in the NKPC (through rule-of-thumb price setters) can eliminate Neo-Fisherism under strict inflation targeting. Differently from their finding, however, we find that the backward-looking elements of the model play a markedly smaller role in determining whether the model exhibits Neo- Fisherism when the model is calibrated to the U.S. economy. We do not view these findings as necessarily conflicting because our calibrated model has two features different from their model: (i) a Taylor-type rule and (ii) an indexation to the inflation target in firms pricesetting process. How do these features help the model to be likely to exhibit Neo-Fisherism? The intuition behind the short-run positive comovement between the nominal interest rate and inflation can be understood from the Fisher equation (1). Under strict inflation targeting, inflation and expected inflation are largely stabilized around the target. When the target is raised, the real interest rate decreases contemporaneously. It follows then, from (1), the nominal interest rate is more likely to contemporaneously decrease, following the movement of the real rate. Under a Taylor-type rule (flexible inflation targeting), however, agents expect inflation to be less stabilized, which implies that expected inflation jumps more following an inflationtarget increase. This in turn enhances the possibility of a short-run comovement between the nominal interest rate and inflation. Therefore, as the inflation reaction coefficient in a Taylortype rule gets larger, the model is less likely to exhibit Neo-Fisherism. Our calibrated model is likely to exhibit Neo-Fisherism in spite of the presence of substantial backward-looking elements arising from the price indexation to past inflation and habit formation. 3 Hence, in a way, assuming strict inflation targeting understates the role of the forward-looking elements and overstates the role of the backward-looking elements in conventional New Keynesian 3 Previous studies (e.g., Ireland (2007), Cogley, Primiceri and Sargent (2010), and Castelnuovo (2012)) find that a highly persistent change in the inflation target leads to the short-run positive comovement between inflation and the nominal interest rate based on estimated models with backward-looking elements for the postwar U.S. data. 4

6 models in forming inflation expectations. The additional forward-looking effect arising from our specification of a partial indexation to the inflation target further enlarges the region of the parameter space for Neo-Fisherism. Also, we show that these findings are robust to empirically plausible parameterizations of the model. The rest of this paper is organized as follows. Section 2 presents several versions of a prototypical New Keynesian model in which the closed-form analytical solutions are available. We also introduce a generalized NKPC where firms explicitly take into account the inflation target in their price-setting process. Section 3 analytically studies the relationship between Neo-Fisherism and several key structural parameters related to backward- and forwardlooking elements of the model. Section 4 considers a more general model calibrated to the U.S. economy, which we use to investigate the short-run comovement between the inflation target, inflation, and the nominal interest rate for a range of empirically plausible parameter values. Section 5 concludes. 2 Prototypical New Keynesian models We present a prototypical New Keynesian model along the line of a textbook model in Galí (2015) to examine whether raising the inflation target is associated with an increase in the nominal interest rate, as well as inflation. This simple model has a rich enough propagation mechanism for our purpose and it allows us to derive a closed-form analytical solution in Section 3. In particular, we consider a generalized version of a New Keynesian Phillips curve (NKPC) in which firms explicitly take into account the change in the inflation target when setting their prices. 4 This generalized NKPC produces a stronger forward-looking effect, which enlarges the region of the parameter space in which raising an inflation target leads to the increase in the nominal interest rate. We first introduce the generalized NKPC and then complete the model with an IS curve and two different monetary policy rules: strict inflation targeting and a Taylor-type rule. Adopting a strict inflation-targeting rule is useful in understanding how backward-looking 4 Section 4 provides empirical support for the specification in the U.S. economy. For the euro area, Fève, Matheron and Sahuc (2010) find gradual changes in the inflation target have been a major driving force of business cycle fluctuations using a similar specification. 5

7 elements reduce the possibility of Neo-Fisherism using analytical solutions, but a larger inflation reaction coefficient in a Taylor-type rule overstates the role of the backward-looking elements in breaking down Neo-Fisherism. We will discuss these Neo-Fisherian properties in Section A generalized New Keynesian Phillips curve with an inflationtarget adjustment As in Calvo (1983) and Yun (1996), only a (1 θ) [0, 1) fraction of the firms are allowed to optimally adjust their prices at any given period. Similar to Fève, Matheron and Sahuc (2010), firms that are not allowed to adjust optimally, with probability θ, simply index their prices to a weighted average of the gross inflation target at time t, Π t, and steady-state gross inflation, Π: P t (i) = P t 1 (i)π ρ t Π 1 ρ, (2) where ρ [0, 1] is interpretable as the degree of indexation to the current inflation target. We will discuss how the inflation target evolves over time later. Each optimizing firm i chooses an identical optimal nominal price, Pt, to maximize the expected discounted sum of profits s=0 [ Q t,ts θ s Pt Π ρ ( Π1 ρ ) ] s ts Yts (i) W ts (i)l ts (i) (3) where Q t,ts is the nominal stochastic discount factor between t and t s. The resulting first-order condition of the firms optimal pricing problem and the associated aggregate-price level equation make up the pricing block of the model. Taking the first-order approximation of these equations around the steady state leads to a generalized version of the New Keynesian Phillips curve (NKPC) equation: π t ρπ t = βe t [ πt1 ρπ t1] κyt, (4) where π t denotes inflation deviation from its steady state and y t denotes the output gap, 6

8 which is the log deviation of output from its natural level. 5 The slope of the NKPC is given by κ (1 θβ)(1 θ)(ση), where β (0, 1) is the discount factor, σ > 0 is the inverse elasticity θ of intertemporal substitution, and η 0 is the inverse Frisch elasticity of labor supply. We assume that the natural level of output is constant so that output deviation is the same as the output gap. Additional details on the derivation of (4) are presented in Appendix A. 2.2 IS curve and monetary policy The log-linearized version of the prototypical New Keynesian model yields the following representation for the IS curve: y t = E t y t1 σ 1 (i t E t π t1 ) (5) where i t is the nominal interest rate deviation from its steady state. 6 For monetary policy, the inflation target is adjusted as follows: π t = φ π π t 1 δ π,t, (6) where πt is the inflation target deviation from steady-state inflation, 0 < φ π 1, and δ π,t 0 when the central bank newly adjusts the inflation target. 7 When φ π = 1, the inflation target is adjusted permanently, and it is equivalent to shifting its long-run target (steadystate inflation). We consider two different types of monetary policy: (i) strict inflation targeting and (ii) a Taylor-type rule. Under strict inflation targeting, the monetary policy authority conducts monetary policy in such a way to set inflation to its target: π t = π t. (7) 5 The indexation rule in (2) implies that we do not have to assume a zero inflation steady state for the log-linearization. 6 See Galí (2015) for the derivation. For now, without any loss of generality, we assume away the technology shock and the preference shock. 7 This autoregressive specification follows that in Smets and Wouters (2003), Cogley, Primiceri and Sargent (2010), Del Negro, Giannoni and Schorfheide (2015), and Bhattarai, Lee and Park (2016), which assume a stationary, but highly persistent process. 7

9 Under a Taylor-type rule, the authority adjusts the nominal interest rate according to i t = ψ π (π t π t ) ψ y y t. (8) Note that strict inflation targeting is a special case of the Taylor-type rule in that the policymaker puts a high weight on inflation such that ψ π in (8). 3 Analytical results based on prototypical NK Models In this section, we first analytically show the relationship between Neo-Fisherism and key structural parameters including those related to backward-looking elements using a strict inflation-targeting rule as well as IS curves and NKPCs, similar to Garín, Lester and Sims (2018). Strict inflation targeting helps to find the closed-form solutions to models with the backward-looking elements. We then consider a Taylor-type rule and examine how monetary policy coefficients such as reactions to the inflation gap and output affect the possibility of Neo-Fisherism. 3.1 Strict inflation targeting A purely forward-looking NK Phillips curve We first consider a complete model with the generalized purely forward-looking NKPC (4), the IS curve (5), the inflation target adjustment rule (6), and the strict inflation targeting rule (7). The closed-form solution for the nominal interest rate can be expressed as a function of the inflation target: i t = [ φ π (1 ρ)(1 φ π ) (1 βφ π ) ] σ 1 κ [ = φ π π t }{{} E tπ t1 (1 ρ)(1 φ π ) (1 βφ π ) σ 1 κ }{{} r t πt (9) ] πt. (10) Whether the nominal interest rate increases in the inflation target is determined by two terms in (9): φ π and (1 ρ)(1 φ π ) (1 βφ π ). The first term φ σ 1 κ π is related to expected 8

10 inflation because from (6) and (7) expected inflation is given by E t [π t1 ] = E t [π t1] = φ π π t. In addition, the Fisher equation i t = E t π t1 r t implies that the second term is related to the real interest rate. Also, (10) shows that when the inflation target is raised, the real interest rate is always negative or equal to zero, while expected inflation is always positive. Thus, the necessary and sufficient condition for the positive comovement between the nominal interest rate and inflation is given by φ π > (1 ρ)(1 φ π ) (1 βφ π ). (11) σ 1 κ It is straightforward to verify that the bigger ρ is, the coefficient on the inflation target in (9) is more likely to be positive. This analytical expression thus implies that all else equal, the more firms take into account the inflation target explicitly in the price-setting process, the more likely the nominal rate needs to increase when the inflation target is raised. When ρ = 1, i.e. prices are fully indexed to the inflation target, the solution is given by i t = φ π π t and the nominal interest rate always increases in the inflation target the real interest rate is always zero (i t = E t π t1 ) and output is unchanged. When ρ = 0 instead, the expression (9) collapses to an analytical expression for a textbook New Keynesian model in Garín, Lester and Sims (2018). We can further verify the effect of other structural parameters on the sign of the changes in the nominal rate. The nominal rate is more likely to increase (i) the more persistent is the inflation target adjustment (φ π ), (ii) the more flexible are prices (κ or θ ), and (iii) the more sensitive is output to the real interest rate (σ 1 ). These structural parameters all are related to the extent to which monetary policy can affect inflation. To numerically check the sign of the coefficient on the inflation target in (9) and assess the impact of structural parameters on the Neo-Fisherian relationship, we calibrate the model with θ = 0.7, β = 0.99, σ = 1, and η = 1 throughout this section, unless noted otherwise. Figure 1 graphically illustrates the sign of nominal interest rates for the parameter space of (ρ, φ π ) with two different values of θ. Under θ = 0.7 (benchmark), or equivalently κ = 0.26, when ρ = 0 the parameter value for φ π associated with Neo-Fisherism requires a value greater than 0.6. However, as ρ increases, the region of the parameter space associated with Neo-Fisherism expands massively. We then consider an alternative value of θ = 0.8, which 9

11 Figure 1: Neo-Fisherian region: Strict inflation targeting with the generalized purely forwardlooking NKPC * ( * t persistence) * ( * t persistence) (indexation to * t ) (a) θ = (indexation to * t ) (b) θ = 0.80 Notes: The sign of indicates a pair of (ρ, φ π ) values associated with the positive comovement between inflation and the nominal interest rate conditional on a change in the inflation target. The Calvo parameter θ is inversely related to the slope of the NKPC. implies a flatter NKPC slope of κ = As suggested by the closed-form expression for i t in (9), the flatter NKPC slope reduces the Neo-Fisherian region. Similar to the benchmark θ = 0.7 case, however, Neo-Fisherism is more likely as the value of ρ increases A hybrid NK Phillips curve We now introduce a more general indexation mechanism, for firms that are not allowed to adjust prices optimally, involving a combination of past inflation, the inflation target, and steady-state inflation: P t (i) = P t 1 (i)π ρ ( Π1 τ ) t Π τ 1 ρ t 1. (12) 8 The flattening of the Phillips curve in the U.S. and other advanced economies since the early 1980s has been documented in various studies, for example, Roberts (2006), Kuttner and Robinson (2010), and Blanchard (2016). 10

12 Here, τ can be interpreted as the degree of indexation to the first lag of inflation as in Christiano, Eichenbaum and Evans (2005). There are two relevant special cases of the price indexation mechanism in (12). First, when τ = 0 it is equivalent to the price-setting mechanism in the previous section with indexation to the weighted average of the inflation target and the steady-state inflation rate, and there is no dependence of past inflation in the NKPC. Second, when ρ = 0, we have the standard hybrid NKPC resulting from the indexation mechanism with the steady-state inflation rate and past inflation. The resulting NKPC based on the indexation rule (12) is given by π t τ(1 ρ)π t 1 ρπ t = βe t [ πt1 τ(1 ρ)π t ρπ t1] κyt (13) or, equivalently where π t = γ b π t 1 γ f E t π t1 κy t ρδ ( π t βe t π t1 ) (14) γ b γ f δ and κ τ(1 ρ) 1 βτ(1 ρ), β 1 βτ(1 ρ), 1 1 βτ(1 ρ), κ 1 βτ(1 ρ) (1 θβ)(1 θ)(σ η) =. θ(1 βτ(1 ρ)) As noted above, when τ = 0 the NKPC in (13) reduces to (4). Assuming the same IS curve (5), the evolution of the inflation target (6), and the strict inflation-targeting rule (7), we can obtain the analytical solution for the nominal interest rate: i t = [( φ π (1 φ π )(1 φ π γ ) f) γ b σ 1 κ γ b σ 1 κ π t 1. ( ρδ(1 φπ )(1 βφ π ) σ 1 κ )] π t (15) We confirm the parameter space for the positive increase of the nominal interest rate by 11

13 Figure 2: Neo-Fisherian region: Strict inflation targeting with the generalized hybrid NKPC * ( * t persistence) * ( * t persistence) (indexation to * t ) (a) τ = (indexation to * t ) (b) τ = 0.50 Notes: The sign of indicates a pair of (ρ, φ π ) values associated with the positive comovement between inflation and the nominal interest rate conditional on a change in the inflation target. The structural parameter τ is related to the extent to which firms take into account past inflation in setting their prices. checking the coefficient on π t, since the coefficient on π t 1 is always greater than or equal to zero. After some simple algebra one can show that the coefficient on π t simplified to [ ] (1 φ)(1 βφ) (1 (1 φ)β)τ φ (1 ρ) σ 1 κ in (15) can be which can be positive or negative. It is straightforward to verify that the coefficient (16) decreases as τ increases. That is, as firms take into account past inflation more in setting their prices, the nominal interest rate is more likely to be negative on impact in response to a change in the inflation target. Meanwhile, indexation to the inflation target ρ mitigates the role of past inflation in reducing the parameter space for Neo-Fisherism as we have shown in the previous subsection. We can also find a set of the structural parameter pairs (ρ, φ π ) which lead to a positive response of the nominal interest rate on impact (i.e., the positive sign of (16)) for different values of τ. Figure 2 presents the Neo-Fisherian regions for τ = 0.25 and (16) Panel (a), where τ = 0.25, shows that the Neo-Fisherian region shrinks in comparison to Figure 1 (a) where 12

14 Figure 3: Neo-Fisherian region: Strict inflation targeting with the hybrid IS curve * ( * t persistence) * ( * t persistence) (indexation to * t ) (a) h = (indexation to * t ) (b) h = 0.50 Notes: The sign of indicates a pair of (ρ, φ π ) values associated with the positive comovement between inflation and the nominal interest rate conditional on a change in the inflation target. The structural parameter h is the degree of habit formation in consumption. τ = 0. Note that other parameters are set to the same values. When τ = 0.50, the indexation parameter to the inflation target ρ needs to be roughly greater than 0.5 to ensure Neo-Fisherism with high values of φ π. Thus, the effect of τ appears to be significant in determining the response of the nominal interest rate on impact to the change in the inflation target Habit formation: Hybrid IS curve What we have learned from the hybrid NKPC case is that allowing for a backward-looking component may limit the Neo-Fisherism region s parameter space. This observation implies that introducing a backward-looking component into the IS curve in (5) may yield a similar effect. Here, we do so by assuming internal habit formation in consumption as in Fuhrer (2000), 9 Also, note that the coefficient on π t 1 in (15), τ(1 ϱ)/κ, increases in τ and this lagged effect can alter the shape of the impulse response of the nominal interest rate conditional on the inflation target shock over time. 13

15 which leads to the following hybrid IS curve: y t = 1 1 h E ty t1 h 1 h y t 1 σ 1 1 h 1 h (i t E t π t1 ), (17) where h [0, 1) is the parameter that governs the degree of habit formation. When completing the model with the generalized version of NKPC (4), evolution of the inflation target (6), and the strict inflation-targeting rule (7), the analytical solution for the nominal interest rate is given by i t = [ φ (1 ρ)(1 φ 1 h )1 h 1 h ] (1 βφ) σ 1 κ πt h (1 ρ) 1 h (1 βφ) σ 1 κ π t 1. (18) The solution for the nominal interest rate (18) shows that as h becomes higher, the nominal interest rate is less likely to increase for a given increase in the inflation target. Figure 3 confirms this implication with two different values of h = 0.25 and A Taylor-type rule and the inflation target adjustment We now consider a Taylor-type rule (8), which is a more realistic setting to study the possibility of Neo-Fisherism. The complete model consists of the IS curve (5), the generalized purely forward-looking NKPC (4), the evolution of the inflation target (6), and the Taylortype rule (8). The closed-form expressions for output, inflation, and the nominal interest rate are given by y t = π t = σ 1 (1 βφ π ) [(1 ρ)ψ π ρφ π ] σ 1 κ(ψ π φ π ) (1 βφ π )(1 φ π σ 1 ψ y ) π t (19) σ 1 κψ π ρ(1 βφ π )(1 φ π σ 1 ψ y ) σ 1 κ(ψ π φ π ) (1 βφ π )(1 φ π σ 1 ψ y ) π t (20) and i t = ψ π [σ 1 κφ π (1 ρ)(1 φ π )(1 βφ π )] ψ y ρ(1 βφ π )σ 1 φ π π σ 1 κ(ψ π φ π ) (1 βφ π )(1 φ π σ 1 t. (21) ψ y ) To ensure equilibrium determinacy in the model above, we assume that ψ π > 1 so that the denominator for all the variables in the solution, σ 1 κ(ψ π φ) (1 βφ)(1 φ σ 1 ψ y ), 14

16 is always positive. 10 This implies that the coefficient of the solution for output in (19) is always positive, i.e., there is a comovement between output and the inflation target. The conditions for the comovement between inflation and the nominal interest rate under a Taylor-type rule are directly comparable to that under strict inflation targeting in Section 3.1.1, because both models share the same IS curve, NKPC, and inflation-target adjustment process. It is straightforward to confirm that, all else equal, the effects of ρ, κ, σ, and φ π on the possibility of Neo-Fisherism are the same as under strict inflation targeting. For example, as φ π increases, it is more likely for the coefficients of the solution for the nominal interest rate in (21) and the solution for inflation in (20) to be positive. Additional conditions for the Taylor-type rule are summarized as follows. Proposition 1 Assume 0 < ρ < 1 for a generalized NKPC (4). Under the Taylor typerule (8), with the IS curve (5) and the generalized NKPC, the sufficient condition for the comovement between inflation, the inflation target, and the nominal interest rate is φ π > (1 ρ)(1 φ π ) (1 βφ π ), (22) σ 1 κ while it is the necessary and sufficient condition for the comovement under strict inflation targeting. Proof. For the strict inflation-targeting rule, see Section Regarding the Taylor-type rule, it is straightforward to show that inflation moves in the same direction as the inflation target based on the solution for inflation in (20). In addition, the solution for the nominal interest rate (21) shows that the model can exhibit the comovement as long as the numerator in (21) is positive because the denominator in (21) is always positive. Because ψ y > 0, 0 < β < 1, σ > 0, and 0 < φ π < 1, the second term in the numerator ψ y ρ(1 βφ π )σ 1 φ π is also always positive. Therefore, if σ 1 κφ π > (1 ρ)(1 φ π )(1 βφ π ), then the numerator is always positive. Proposition 2 Assume 0 < ρ < 1 for a generalized NKPC (4). Under the Taylor-type rule (8) with the IS curve (5) and the generalized NKPC, if the sufficient condition (22) is not 10 For details on the determinacy condition for a Taylor-type rule, see Bullard and Mitra (2002). 15

17 satisfied, the model is more likely to exhibit a comovement between inflation, the inflation target, and the nominal interest rate as (i) the policy response to the inflation gap in the Taylor-type rule is weaker and (ii) the policy response to the output gap in the Taylor-type rule is stronger. Proof. Inflation and the inflation target always move in the same direction as the coefficient in the solution for π t in (20) is always positive. From (21), the necessary and sufficient condition for the comovement between the inflation target (and hence, inflation) and the nominal interest rate under the Taylor-type rule is ψ π [σ 1 κφ π (1 ρ)(1 φ π )(1 βφ π )] ψ y ρ(1 βφ π )σ 1 φ π > 0. It can be rearranged as ψ y ρ(1 βφ π )σ 1 φ π ψ π σ 1 κ [ > φ π (1 ρ)(1 φ π )(1 βφ π ) ]. (23) σ 1 κ If the sufficient condition (22) is not satisfied, the RHS in (23) is always positive. Therefore, to satisfy the condition, the LHS in (23) should be always positive. Because ρ(1 βφ π )σ 1 φ π σ 1 κ in the LHS in (23) is always positive, the LHS increases in ψ y and decreases in ψ π. What is the intuition behind Proposition 2? Consider the extreme case of strict inflation targeting, i.e., ψ π. We know from (10) that under this policy, the real interest rate always decreases in response to a given increase in the inflation target. Output also increases on impact. From the Fisher equation i t = E t π t1 r t, whether the nominal interest rate increases on impact depends on whether the increase in expected inflation is large enough. Suppose now ψ π is much lower and finite. In this case, expected inflation increases more on impact compared to that under strict inflation targeting because economic agents expect inflation to be less stabilized. This higher expected inflation makes it more likely for the nominal interest rate to increase, based on the Fisher equation. Note that the real interest rate still decreases on impact even when ψ π is finite. Regarding the output feedback coefficient, a larger ψ y means that in response to higher output caused by an increase in the inflation target, the monetary authority increases the nominal interest rate by more (based on the Taylor rule). All else equal, this would make Neo-Fisherism more likely. 16

18 Figure 4: Neo-Fisherian region: Taylor-type rule with the generalized purely forward-looking NKPC * ( * t persistence) * ( * t persistence) (indexation to * t ) (a) ψ π = (indexation to * t ) (b) ψ π = 3.00 Notes: The sign of indicates a pair of (ρ, φ π ) values associated with the positive comovement between inflation and the nominal interest rate conditional on a change in the inflation target. The parameter ψ π is the reaction coefficient to the inflation gap in the Taylor-type rule. We examine how the inflation feedback coefficient ψ π affects the Neo-Fisherism region for the parameter space of (ρ, φ π ) with two sets of policy coefficients: (a) φ π = 1.5 and φ y = 0.5/4 and (b) φ π = 3.0 and φ y = 0.5/4. The benchmark case (a) assumes the standard Taylor-rule coefficients as in Taylor (1993). We also consider an alternative value of ψ π = 3.0 in case (b) following Schmitt-Grohé and Uribe (2007), who argue that an inflation feedback parameter larger than 3.0 would be difficult for policymakers to communicate to the public. Figure 4 shows a larger ρ or φ π value expands the Neo-Fisherian region, as in the strict inflation-targeting case, and the Neo-Fisherian region for ψ π = 1.5 is marginally larger than for the alternative value of ψ π = 3.0, which confirms Proposition 2. This finding that the Neo-Fisherian region shrinks as ψ π gets larger naturally raises a question: Would the importance of backward-looking elements highlighted using the strict inflation-targeting rule still apply under a Taylor-type rule? We answer this question in the next section. 17

19 4 Comovement between inflation and the nominal interest rate in the U.S. economy In the previous section, we show analytically that including backward-looking elements habit formation in consumption and price indexation to past inflation may break down Neo-Fisherism under strict inflation targeting, but also find that the strict inflation-targeting rule may exaggerate their roles in doing so. In this section, we jointly consider the backward-looking elements as well as a Taylor-type rule and examine whether these more general models exhibit Neo-Fisherism. The closedform solution to models with both backward-looking elements and a Taylor-type rule is, in general, not available. We therefore numerically study the possibility of Neo-Fisherism in models with these features calibrated to the U.S. economy. 4.1 Model and calibration The model considered in this section consists of (i) IS curve with habit formation in (17), (ii) the generalized NKPC with backward- and forward-looking elements in (13), (iii) the inflation target adjustment in (6), and a Taylor-type rule with interest rate smoothing, i t = φ i i t 1 (1 φ i ) [ψ π (π t π t ) ψ y y t ]. (24) We calibrate the deep structural parameters of the model to the U.S. economy with the time unit of one quarter. Table 1 presents the structural parameter values. The quarterly discount rate β and the Frisch elasticity of labor supply η are set to 0.99 and 1, respectively. The values of other parameters are obtained from the posterior means based on the estimated model for the U.S. data with the addition of four economic shocks. 11 More details about the model with the structural shocks and Bayesian estimations are presented in Appendix B. For the IS curve in (17), the habit formation parameter and the inverse elasticity of 11 The four shocks are a monetary policy shock added to the Taylor-type rule, an inflation target shock to the evolution of the inflation target, a cost-push shock added to the generalized NKPC, and a preference shock added to the IS curve. The sample period for the estimation ranges from 1982:Q4 to 2009:Q2, excluding the passive monetary policy period and the Volcker-disinflation period following Lubik and Schorfheide (2004) and the zero-lower bound period, in which the unconventional monetary policy has been conducted. 18

20 Table 1: Model calibration Parameter Value Description β 0.99 quarterly discount rate η 1.00 Frisch elasticity of labor supply σ 1.29 preference parameter h 0.31 habit formation κ 0.30 NKPC slope τ 0.44 indexation to past inflation ρ 0.56 indexation to the inflation target ψ π 2.63 inflation coefficient in the interest rate rule ψ y 0.05 output coefficient in the interest rate rule φ i 0.71 smoothing parameter in the interest rate rule Note: The calibrated values of the structural parameters except for β and η are from the estimates for the U.S. economy with the sample period of 1982:Q1 to 2009:Q2. intertemporal substitution are h = 0.41 and σ = 1.29, respectively. The deep parameter values for the NKPC in (13) are τ = 0.44, ρ = 0.56, and θ = These values imply that the slope of the NKPC is 0.30, and the firms price-setting behavior is governed by the indexation weights of 0.19 for past inflation, 0.25 for the steady-state inflation rate, and 0.56 for the inflation target. The Taylor-rule coefficients are ψ π = 2.63, ψ y = 0.05, and φ i = The inflation reaction coefficient ψ π is greater than the conventional value of 1.5, but it is within the range of compelling values described in Schmitt-Grohé and Uribe (2007) and also consistent with the estimates found in the literature (e.g., Lubik and Schorfheide (2004)). In addition, the inflation target persistence parameter is set to φ π = 0.99, which is also the value assumed for the postwar U.S. economy in various studies in the literature (e.g., Cogley, Primiceri and Sargent (2010), Del Negro, Giannoni and Schorfheide (2015), and Bhattarai, Lee and Park (2016)). We will later consider alternative values of φ π to assess the effect of φ π on responses of the nominal interest rate to the inflation target shock. 4.2 Impulse response functions to inflation-target adjustment To assess how changes in the inflation target affect various variables in the calibrated model, Figure 5 plots the impulse responses to a 1% per-annum inflation target shock for several 19

21 Figure 5: Impulse responses to an inflation-target shock for different values of φ π Notes: This figure plots the impulse responses of selective variables to a 1% per annum inflation target shock for different values of inflation target persistence, φ π. All other parameter values are set as in Table 1. different values of φ π. Other parameter values are set to those presented in Table 1. In addition to φ π = 0.99, we also consider two alternative values: φ π = 0.63, which corresponds to the cut-off value (or the lower bound) for the model to exhibit Neo-Fisherism, and φ π = 0.96, for which the half-life of the inflation-target adjustment is roughly 17 quarters. The latter value implies that when the monetary authority raises the inflation target by 2% for example, it will remain to be 1 % or bigger than the steady-state inflation rate for about 4 years, which we believe to be a reasonable policy scenario. When φ π = 0.99, inflation increases on impact by slightly less than 2% per annum in response to a 1% increase in the target. This positive inflation gap, defined as the gap between inflation and the inflation target, is due to higher expected inflation, caused by the 20

22 persistent (but temporary) increase in the inflation target. Inflation remains elevated well above the initial target even after 20 quarters and so does the nominal interest rate. Given the inflation target increase, the monetary authority needs to contemporaneously raise the nominal interest rate by about 0.7% per annum on impact. With a prolonged period of higher nominal interest rates, higher expected inflation leads to only a short period of lower real interest rates, resulting in a comparable period of higher output levels. Similar patterns emerge when we set φ π = 0.96 instead. The slightly lower persistence of the inflation-target adjustment leads to a slightly lower increase in inflation, expected inflation, and the nominal interest rate, while the impulse responses of output and the real interest rate appear to be similar compared to the case of φ π = When φ π = 0.63 instead, the nominal interest rate is unchanged on impact, but it increases slightly in the next several quarters. In this case, a much lower increase in expected inflation leads inflation to increase by only 1% per annum on impact. The calculated half-life of the inflation-target shock when φ π = 0.63 is only about 1.5 quarters. Thus, as long as the half-life of the inflation-target shock is greater than 1.5 quarters, there is a short-run comovement between the nominal interest rate and inflation. We note that an inflation target shock is not equivalent to a conventional monetary policy shock in a Taylor-type rule. While a contractionary monetary policy shock similarly calls the monetary authority to raise the nominal interest rate, Figure 6 shows that the responses of output, inflation, and the real interest rates are markedly different. Here, a comparable rise in the nominal interest rate under a contractionary monetary policy shock raises the real interest rate and lowers inflation and output. 4.3 Neo-Fisherism and the degree of forward-lookingness (backwardlookingness) We now study how the comovement between inflation and the nominal interest rate is affected by different values of ρ, τ, and h, in a manner similar to the analytical study in the previous section. All three parameters affect the degree of forward-lookingness (backwardlookingness) of the model. In addition, we study how the comovement depends on the 21

23 Figure 6: Impulse response functions to an inflation-target shock and a contractionary monetary policy shock Note: This figure plots the impulse response functions to (i) a 1% inflation target shock with φ π = 0.96 and (ii) a contractionary monetary policy shock whose magnitude is set to generate the same response of i t on impact (i.e., period 0) as that on impact to the 1% inflation target shock. All other parameters are set to those presented in Table 1. inflation feedback coefficient in the Taylor-type rule. As shown in Figure 5 and in the analytical study, an inflation-target shock generates a hump-shaped nominal interest rate response. This implies that if the response of the nominal interest rate to the shock is positive on impact, the model would exhibit Neo-Fisherism in the short run. For this reason, the analysis below focuses on the contemporaneous responses of the nominal interest rate to the inflation-target shock. In generating all the results below, we assume a 1% per-annum shock. 22

24 Figure 7: Contemporaneous response of i t to an inflation-target shock as a function of φ π and ρ Notes: This figure plots the contemporaneous (period-0) response of the nominal interest rate, i t, to a 1% per annum inflation target shock for various values of inflation target persistence, φ π, and degree of indexation to inflation target, ρ. All other parameter values are set to those presented in Table 1. 23

25 4.3.1 Indexation to the inflation target Figure 7 plots the contemporaneous response of the nominal interest rate i t as a function of φ π, the parameter governing the inflation target persistence, for a range of values of ρ, the degree of price indexation to the inflation target. In generating the figure, we set all other parameter values to those reported in Table 1. The contemporaneous response of i t increases everywhere in φ π across different values of ρ. When ρ = 0, the response is positive for any φ π > 0.77, for which the half-life of the inflation target shock is calculated to be roughly 2.7 quarters only. The monetary authority is thus not required to maintain a higher inflation target for an extended period for the economy to exhibit Neo-Fisherism, even without an indexation to the inflation target. As the degree of indexation to the inflation target gets higher, the cut-off value decreases monotonically. 12 For example, when ρ = 0.56 in the benchmark calibration (not shown) the cut-off value is φ π = This implies that, as in the analytical study in the previous section, the Neo-Fisherian region in the (ρ, φ π ) parameter space expands as the extent to which firms take the inflation target adjustment into account in their pricing mechanism increases. When the degree of indexation to the inflation target is at the highest with ρ = 1, the contemporaneous response of i t is everywhere non-negative Indexation to past inflation Figure 8 plots the contemporaneous response of i t as a function of φ π, for a range of values of τ, the degree of indexation to past inflation, instead. Here, we set ρ = 0.56, i.e., at our benchmark value. As τ increases, the Neo-Fisherism cut-off value also increases. Hence, the result depicted in Figure 8 is consistent with the analytical finding in the previous section and in Garín, Lester and Sims (2018): it is more likely for the nominal interest rate to decrease on impact as the coefficient on the backward-looking inflation component in the NKPC gets larger. Differently and interestingly, however, we find that there is no possible value of τ in which the model does not exhibit Neo-Fisherism regardless of the persistence of the inflation target. Garín, Lester and Sims (2018) find that, under strict 12 Since the model is linear, the size of the inflation-target shock is immaterial for the cut-off value. 24

26 Figure 8: Contemporaneous response of i t to an inflation-target shock as a function of φ π and τ Notes: This figure plots the contemporaneous (period-0) response of the nominal interest rate, i t, to a 1% per annum inflation target shock for various values of inflation target persistence, φ π, and degree of indexation to past inflation, τ. All other parameter values are set to those presented in Table 1. 25

27 inflation targeting, once the parameter governing the fraction of rule-of-thumb price-setters is greater than 0.15 a modest degree of backward-lookingness their model does not exhibit Neo-Fisherism for any value of φ π. 13 Further to this, as shown in Figure 8, the cut-off value only marginally increases across the range of backward-lookingness, from about 0.53 when τ = 0 (no indexation to past inflation), to about 0.72 when τ = 1 (full indexation to past inflation). This result indicates that the role of the parameter that governs the degree of backward-lookingness in the NKPC plays a markedly smaller role in determining whether the model exhibits Neo-Fisherism under a Taylor-type rule than under strict inflation targeting The inflation feedback coefficient in a Taylor-type rule In addition to the indexation to the inflation target, the discrepancy between our finding and that in Garín, Lester and Sims (2018) about the role of backward-looking components is potentially driven by the Taylor-type rule specification in place of strict inflation targeting. Strict inflation targeting is a special case of the Taylor-type rule as ψ π. As laid out in Proposition 2, when the inflation feedback coefficient in the Taylor-type rule, ψ π, is finite and much smaller (flexible instead of strict inflation targeting), the comovement between the inflation target, inflation, and the nominal interest rate is more likely. Since the U.S. Federal Reserve conducts a flexible instead of strict inflation targeting, it is important to further investigate the role of ψ π in generating Neo-Fisherism in our calibrated model. We conduct two additional experiments for this. First, we plot the region of the positive responses of i t on impact to the inflation target shock with the sign of in the parameter space of (ρ, φ π ) for τ = 0.25 and τ = 0.50 in Figure 9, while setting h = 0, σ = 1, and θ = 0.7 all other parameters, including the coefficients of Taylor-type rule, are set as in Table 1. This figure under the Taylor-type rule is thus directly comparable to Figure 2 under the strict inflation-targeting rule. The two model specifications share the same parameter values for 13 While we use a different modeling mechanism from that in Garín, Lester and Sims (2018) to incorporate the backward-looking component into the NKPC, the two mechanisms the rule-of-thumb price-setters mechanism of Galı and Gertler (1999) and the indexation mechanism of Christiano, Eichenbaum and Evans (2005) used for our analysis share a parallel. For example, we find that the case where ω, the fraction of rule-of-thumb price-setters, is 0.25 (shown in the top right panel of Figure 4 in Garín, Lester and Sims (2018)) is similar in terms of direction and magnitude to the case of τ = 0.5 under the indexation mechanism of Christiano, Eichenbaum and Evans (2005). Hence, ω = 0.15 corresponds to a value of τ < 0.5, which can be classified as a modest degree of backward-lookingness. 26

28 Figure 9: The response of i t to an inflation-target shock on impact under a Taylor-type rule * ( * t persistence) * ( * t persistence) (indexation to * ) t (a) τ = (indexation to * ) t (b) τ = 0.50 Notes: The sign of indicates a pair of (ρ, φ π ) values associated with the positive comovement between inflation and the nominal interest rate conditional on a change in the inflation target. The policy coefficients are set as ψ π = 2.63, ψ y = 0.05, and φ i = 0.71 while other structural parameters are set as β = 0.99, h = 0, σ = 1, θ = 0.7. Thus, this figure is comparable to Figure 2 under the strict inflation-targeting rule. the IS curve and the NKPC curve so that differing responses from the models should originate from the different specifications of the monetary-policy rule. Figure 9 clearly shows that the model under the Taylor-type rule exhibits Neo-Fisherism for a much wider range of values of ρ and φ π, for both values of τ in comparison to Figure 2. Under τ = 0.50, for example, even when there is no indexation to the inflation target (ρ = 0), the comovement occurs as long as φ π = 0.8 or larger. Recall that Figure 2 shows a modest degree of backward-looking behavior in the NKPC under strict inflation targeting significantly reduces the Neo-Fisherian region. This sharp contrast between the results under strict inflation targeting in Figure 2 and those under a Taylor-type rule in Figure 9 demonstrates that the limiting case of strict inflation targeting appears to exaggerate the effect of backward-looking elements on the response of the nominal interest rate on impact. To examine this possibility further, Figure 10 plots the contemporaneous responses of the nominal interest rate to an inflation target shock over ψ π (1, 20] for various combinations 27