Firm-Specific Labor, Trend Inflation, and Equilibrium Stability. Takushi Kurozumi and Willem Van Zandweghe December 2012 RWP 12-09

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1 Firm-Specific Labor, Trend Inflation, and Equilibrium Stability Takushi Kurozumi and Willem Van Zandweghe December 212 RWP 12-9

2 Firm-Specific Labor, Trend Inflation, and Equilibrium Stability Takushi Kurozumi Willem Van Zandweghe December 212 Abstract In a Calvo sticky price model based on micro evidence that each period a fraction of prices is kept unchanged, we examine implications of firm-specific labor for determinacy and expectational stability (E-stability) of rational expectations equilibrium under interest rate policy. Firm-specific labor causes higher trend inflation to be more likely to induce not only indeterminacy but also E-instability. The latter is in contrast with the result of E- stability in the case of homogeneous labor analyzed in recent research. Moreover, under the same calibration of structural model parameters, indeterminacy and E-instability are much more likely in the case of firm-specific labor than in the case of homogeneous labor. The recent argument a decline in trend inflation along with the Fed s change from a passive to an active policy response to inflation explains much of the U.S. economy s shift from indeterminacy during the Great Inflation era to determinacy during the Great Moderation era depends crucially on the assumption of firm-specific labor. JEL Classification: E31, E52 Keywords: Firm-specific labor, Trend inflation, Sticky prices, Determinacy, Expectational stability The authors are grateful for comments and discussions to Olivier Coibion, Andrew Foerster, and participants at the Federal Reserve Bank of Kansas City and the Midwest Macroeconomics Meetings 212. The views expressed herein are those of the authors and should not be interpreted as those of the Bank of Japan, the Federal Reserve Bank of Kansas City or the Federal Reserve System. Bank of Japan, Nihonbashi Hongokucho, Chuo-ku, Tokyo , Japan. Tel.: ; fax: address: Federal Reserve Bank of Kansas City, 1 Memorial Drive, Kansas City, MO 64198, USA. Tel.: ; fax: address: 1

3 1 Introduction Recent literature has studied implications of positive trend inflation rates for macroeconomic stability using sticky price models based on micro evidence that each period a fraction of prices is kept unchanged. Ascari and Ropele (29), Hornstein and Wolman (25), and Kiley (27) analyze determinacy of equilibrium under the Taylor (1993) rule and show that higher trend inflation is more likely to induce indeterminacy. Moreover, Coibion and Gorodnichenko (211) argue that a decline in trend inflation along with an increase in the Fed s policy response to inflation accounts for much of the U.S. economy s shift from indeterminacy during the Great Inflation era to determinacy during the Great Moderation era, against Clarida et al. (2) and Lubik and Schorfheide (24) who all attribute this shift solely to the Fed s change from a passive to an active policy response to inflation. 1 In a Calvo (1983) sticky price model, Ascari and Ropele (29) consider homogeneous labor, whereas Coibion and Gorodnichenko (211) introduce firm-specific labor. 2 This difference in the specification of labor yields two differences in inflation dynamics represented by a general formulation of the New Keynesian Phillips curve (NKPC). First, like firm-specific capital analyzed in Altig et al. (211), Eichenbaum and Fisher (27), Sveen and Weinke (25, 27) and Woodford (25), firm-specific labor introduces strategic complementarity or real rigidity. 3 As a consequence, the long-run inflation elasticity of output implied by the NKPC is smaller in the case of firm-specific labor than in the case of homogeneous labor, when the trend inflation rate is greater than a certain threshold that is positive but close to zero. 4 Second, price distortion 1 Boivin and Giannoni (26) show, using counterfactual simulations, that the shift to the Great Moderation cannot be explained solely, or even primarily, by a change in shocks to the U.S. economy, and conclude that in order to explain this shift, it is crucial for the U.S. monetary policy to have changed the way it has, along with the shocks. Kimura and Kurozumi (21) offer theoretical support for the good policy hypothesis about the shift to the Great Moderation using a Calvo model with endogenous price stickiness. Specifically, they show that a more aggressive monetary policy response to inflation makes firms less likely to reset prices and gives the resulting New Keynesian Phillips curve a flatter slope and a smaller disturbance, as observed during the Great Moderation era, and that such a policy response can stabilize both inflation and the output gap by exploiting the feedback effects of this policy response on firms price-setting. 2 Hornstein and Wolman (25) and Kiley (27) use a Taylor (198) sticky price model. 3 See also Levin et al. (27) for implications of strategic complementarity for sticky price models. 4 By contrast, when the trend inflation rate is lower than this threshold (e.g., the zero trend inflation rate), the long-run inflation elasticity of output is larger in the case of firm-specific labor. 2

4 has an influence on the inflation dynamics in the case of homogeneous labor as long as the elasticity of labor supply is finite and the trend inflation rate is non-zero, but not in the case of firm-specific labor. Thus, when labor is homogeneous, the law of motion of price distortion adds lagged price distortion to the set of relevant model state variables. Despite these differences, the existing literature lacks a comparison of sticky price models with homogeneous labor and with firm-specific labor in terms of equilibrium determinacy or macroeconomic stability. The present paper fills this gap using a Calvo sticky price model with firm-specific labor based on Coibion and Gorodnichenko (28, 211) and its associated model with homogeneous labor, which is a stochastic version of the baseline model of Ascari and Ropele (29). We also examine expectational stability (E-stability) of fundamental rational expectations equilibrium (REE) in the model with firm-specific labor, and compare it with that in the model with homogeneous labor. 5 As McCallum (27) indicates, E-stability is very closely linked with least-squares learnability (i.e., stability under least-squares learning) and this learnability is arguably a necessary property for an REE to be plausible as equilibrium for the model at hand. In a broad class of linear models with expectations (including the log-linearized model of the present paper), a non-explosive fundamental REE is least-squares learnable if it is E- stable; otherwise, it is not least-squares learnable (Evans and Honkapohja, 21). Therefore, E-stability is an essential condition for an REE to be regarded as plausible. We establish the necessary and sufficient conditions for determinacy of REE and for E- stability of fundamental REE when labor is firm-specific. Using these analytical conditions and a plausible calibration of structural model parameters, we show that firm-specific labor causes higher trend inflation to be more likely to induce not only indeterminacy of REE but also E-instability of fundamental REE. Moreover, we find that under the same calibration, indeterminacy and E-instability are much more likely in the case of firm-specific labor than in the case of homogeneous labor. When labor is firm-specific, higher trend inflation is more likely to generate indeterminacy, as shown in Coibion and Gorodnichenko (211). This result is qualitatively consistent with that of Ascari and Ropele (29) who study the case of homogeneous labor, but firm-specific labor is much more likely to cause indeterminacy under the same calibration of structural 5 The term fundamental refers to Evans and Honkapohja s (21) minimal-state-variable (MSV) solutions to linear rational expectations models to distinguish them from McCallum s (1983) original MSV solution. 3

5 model parameters. 6 This is because firm-specific labor introduces strategic complementarity, which is a source of indeterminacy as is the case with firm-specific capital analyzed in Sveen and Weinke (25, 27). Specifically, two key conditions for determinacy are less likely to be satisfied in the model with firm-specific labor under the same calibration. One condition is the long-run version of the Taylor principle: in the long run the interest rate should be raised by more than the increase in inflation. This Taylor principle is less likely to be met as the longrun inflation elasticity of output is smaller. When the trend inflation rate is greater than a certain threshold that is positive but close to zero, the strategic complementarity incorporated by firm-specific labor makes the elasticity smaller and thereby causes the long-run version of the Taylor principle to be less likely to be satisfied. As a consequence, a large policy response to current or expected future output induces indeterminacy in the case of firm-specific labor, even if such a response may ensure determinacy under the same calibration in the case of homogeneous labor. The other condition for determinacy causes a small policy response to current or expected future output to induce indeterminacy under positive trend inflation rates. This condition is also less likely to be met in the case of firm-specific labor than in the case of homogeneous labor, when the same calibration is used in these two cases. Our result of E-instability in the case of firm-specific labor is in contrast with the result of E-stability in the case of homogeneous labor. In the latter case, E-stability of fundamental REE is likely even when trend inflation is high, as shown in Kurozumi (211). This difference is due to two factors. First, the long-run version of the Taylor principle is a necessary condition for E- stability of fundamental REE as well. Therefore, as with the result regarding determinacy, the different result regarding E-stability arises because the long-run version of the Taylor principle is less likely to be satisfied in the case of firm-specific labor than in the case of homogeneous labor. Second, when labor is homogeneous, the NKPC depends on price distortion as long as the elasticity of labor supply is finite and the trend inflation rate is non-zero, and hence the law of motion of price distortion adds lagged price distortion to the set of relevant model state 6 Kurozumi (29) shows that the indeterminacy result of Ascari and Ropele (29) is overturned when price stickiness is endogenously determined in a Calvo model along the lines of the literature such as Ball et al. (1988), Romer (199), Kiley (2), Devereux and Yetman (22), and Levin and Yun (27). This is because the longrun inflation elasticity of output implied by the NKPC declines substantially with higher trend inflation in the case of exogenously given price stickiness, whereas in the case of endogenous price stickiness the decline in the elasticity is mitigated as higher trend inflation leads to a higher probability of price adjustment. 4

6 variables. However, this is not the case when labor is firm-specific. For the REE in question, E- stability examines whether an associated equilibrium in which agents form expectations under adaptive learning reaches over time the REE. For such expectation formation, lagged price distortion is useful information in the model with homogeneous labor. Particularly, it helps agents form inflation expectations, since price distortion affects inflation dynamics. Therefore, E-stability is likely in the case of homogeneous labor. By contrast, in the case of firm-specific labor, price distortion is absent in the set of relevant model state variables and hence higher trend inflation is much more likely to induce E-instability. 7 Our results demonstrate that indeterminacy and E-instability are much more likely in the model with firm-specific labor than in the model with homogeneous labor. Specifically, the argument of Coibion and Gorodnichenko (211) who use the former model to show that a decline in trend inflation plays a key role in the U.S. economy s shift from indeterminacy during the Great Inflation era to determinacy during the Great Moderation era depends crucially on the assumption of firm-specific labor. 8 The remainder of the paper proceeds as follows. Section 2 presents a Calvo sticky price model with firm-specific labor. In this model, Section 3 derives conditions for equilibrium determinacy and for E-stability of fundamental REE and investigates implications of these conditions. Section 4 compares the results regarding determinacy and E-stability with those in the case of homogeneous labor. Finally, Section 5 concludes. 7 Kobayashi and Muto (211) use a Calvo sticky price model with homogeneous labor based on Sbordone (27) and Cogley and Sbordone (28), but their model follows Sbordone (27) to assume that real marginal cost does not depend on price distortion. Due to this assumption, price distortion does not appear in their NKPC and hence it is absent in the set of relevant model state variables. Consequently, they reach the conclusion that when trend inflation is high, E-instability is likely. This is qualitatively consistent with our result of E-instability obtained in the case of firm-specific labor. 8 Ascari et al. (211) show, conducting counterfactual exercises, that the impact of variations in trend inflation on the likelihood of equilibrium indeterminacy is both modest and limited to the second half of the 197s, suggesting that the Fed s change from a passive to an active policy response to inflation is likely to have been the main driver leading the U.S. economy to a unique equilibrium during the Great Moderation era. 5

7 2 A Calvo sticky price model with firm-specific labor The model is a Calvo sticky price model based on Coibion and Gorodnichenko (28, 211). In the model economy there are a representative household, a final-good firm, a continuum of intermediate-good firms, and a monetary authority. Key features of the model are that the household s members supply labor specific to intermediate-good firms, while each period a fraction of intermediate-good firms keeps prices of their differentiated products unchanged. The behavior of each economic agent is described in turn. 2.1 Household The representative household consumes C t final goods, supplies {N t (i)} labor specific to each intermediate-good firm i [, 1], and purchases S t one-period riskless bonds so as to maximize the utility function E β [ln t 1 C t 1 + 1/η t= 1 ] (N t (i)) 1+1/η di exp(ε t ) subject to the budget constraint P t C t + S t = 1 W t (i)n t (i) di + R t 1 S t 1 + T t, where E t is the rational expectation operator conditional on information available in period t, β (, 1) is the subjective discount factor, η is the elasticity of labor supply, ε t is a preference shock governed by a first-order autoregression process with a persistence parameter ρ [, 1), P t is the price of final goods, W t (i) is the wage paid by intermediate-good firm i, R t is the gross interest rate on bonds, and T t consists of lump-sum transfers and firm profits. Combining first-order conditions for utility maximization with respect to consumption, labor supply, and bond holdings yields where Π t = P t /P t 1 denotes gross inflation. W t (i) = C t (N t (i)) 1/η, (1) P t ( ) Ct exp(ε t+1 ) R t 1 = βe t, (2) C t+1 exp(ε t ) Π t+1 6

8 2.2 Firms The representative final-good firm produces homogeneous goods Y t under perfect competition by choosing a combination of intermediate inputs {Y t (i)} so as to maximize profit P t Y t subject to the CES production technology 1 P t (i)y t (i) dj [ 1 θ/(θ 1) Y t = (Y t (i)) di] (θ 1)/θ, where P t (i) is the price of intermediate good i and θ > 1 is the price elasticity of demand for each intermediate good. The first-order condition for profit maximization yields the final-good firm s demand for intermediate good i, while perfect competition in the final-good market leads to The final-good market clearing condition is given by ( ) Pt (i) θ Y t (i) = Y t, (3) P t [ 1 1/(1 θ) P t = (P t (i)) di] 1 θ. (4) Y t = C t. (5) Each intermediate-good firm i produces one kind of differentiated goods Y t (i) under monopolistic competition. Firm i s production function is given by Y t (i) = A t (N t (i)) α, (6) where α (, 1] is the labor elasticity of output and the technology A t follows the process ln A t = g + ln A t 1, (7) where g is the rate of technological change. The first-order condition for minimization of production cost determines firm i s marginal cost MC t (i) = W t(i)n t (i). (8) αy t (i) 7

9 In the face of the final-good firm s demand (3) and marginal cost (8), intermediate-good firms set prices of their products on a staggered basis as in Calvo (1983). Each period a fraction λ (, 1) of firms keeps previous-period prices unchanged, while the remaining fraction 1 λ of firms sets the price B t (i) so as to maximize the profit function E t j= ( ) λ j Bt (i) θ Q t,t+j Y t+j (B t (i) MC t+j(i)), P t+j where Q t,t+j = β j P t C t exp(ε t+j )/[P t+j C t+j exp(ε t )] is the stochastic discount factor for a unit of money between period t and period t + j. For this profit function to be well-defined, the following assumption is imposed throughout the paper. Assumption 1 The two inequalities λ Π θ 1 < 1 and βλ Π θ(1+1/η)/α < 1 hold, where Π is gross trend (or steady-state) inflation. Moreover, the trend inflation rate is non-negative, i.e., Π 1. Using eqs. (1), (3), (5), (6), and (8), the first-order condition for staggered price-setting leads to { ( ) [ 1 θ E t (βλ) j Bt exp(ε t+j ) θ ( ) ] 1 Y θ γ } t+j Bt =, (9) P t+j θ 1 α A t+j P t+j j= where B t is the price set by firms that reoptimize prices in period t and the composite parameter γ is given by γ = (1 + 1/η)/α. Moreover, the final goods price equation (4) can be reduced to (P t ) 1 θ = (1 λ)(b t ) 1 θ + λ(p t 1 ) 1 θ. (1) 2.3 Monetary authority The monetary authority conducts interest rate policy according to a Taylor (1993) rule. This rule adjusts the interest rate R t in response to deviations of either contemporaneous or expected future inflation and output from their trend levels ln R t = ln R + ϕ π (ln E t Π t+i ln Π) + ϕ y [ln E t (Y t+i /A t+i ) ln y], i =, 1, (11) where R is the gross steady-state interest rate, y is the steady-state level of detrended output y t = Y t /A t, and ϕ π, ϕ y are the degrees of policy responses to inflation and output. The interest rate policy is referred to as outcome-based if i = and forecast-based if i = 1. 8

10 2.4 Log-linearized equilibrium conditions Under Assumption 1, detrending and log-linearizing the equilibrium conditions (2), (5), (9), (1), and (11) leads to ŷ t = E t ŷ t+1 ( ˆR t E t ˆΠt+1 ) + ε t E t ε t+1, (12) ˆΠ t βλ Π θγ E t ˆΠt+1 = β(e t ˆΠt+1 βλ Π θγ E t ˆΠt+2 ) + β( Π 1+θ(γ 1) 1)(1 λ Π θ 1 ) [θγe t ˆΠt+1 (ε t E t ε t+1 )] 1 + θ(γ 1) + γ(1 λ Π θ 1 )(1 βλ Π θγ ) λ Π θ 1 (ŷ t βλ [1 + θ(γ 1)] Π θ 1 E t ŷ t+1 ), (13) ˆR t = ϕ π E t ˆΠt+i + ϕ y E t ŷ t+i, i =, 1, (14) where all hatted variables represent log-deviations from steady-state values. Eq. (13) presents a general formulation of the NKPC, since under the zero trend inflation rate (i.e., Π = 1), this equation is rewritten as ˆΠ t βλe t ˆΠt+1 = β(e t ˆΠt+1 βλe t ˆΠt+2 ) + which can be reduced to 2.5 Calibration ˆΠ t = βe t ˆΠt+1 + γ(1 λ)(1 βλ) (ŷ t βλe t ŷ t+1 ), λ[1 + θ(γ 1)] γ(1 λ)(1 βλ) ŷ t. λ[1 + θ(γ 1)] The ensuing analysis uses a plausible calibration of structural model parameters to illustrate conditions for determinacy and E-stability. The benchmark calibration for the quarterly model is summarized in Table 1. In line with Coibion and Gorodnichenko (211), we set the subjective discount factor at β =.99, the elasticity of labor supply at η = 1, the price elasticity of demand for differentiated intermediate goods at θ = 1, the labor elasticity of output at α = 1, and the probability of no price adjustment at λ =.55. Then, we have γ = (1 + 1/η)/α = 2. We also choose the persistence of preference shocks at ρ =.35 similarly to Woodford (23). Note that to meet Assumption 1 under this calibration, the annualized trend inflation rate needs to be less than 12 percent. 9

11 3 Analysis of determinacy and E-stability This section establishes the necessary and sufficient conditions for determinacy of REE and for E-stability of fundamental REE in the model presented in the preceding section, and then investigates implications of these conditions using the calibration of model parameters presented in Table Determinacy conditions For the analysis of equilibrium determinacy, the log-linearized equilibrium conditions (12) (14) can be reduced to a system of the form x t = AE t x t+1 + Bε t, (15) where x t = [ ˆΠ t ŷ t E t ˆΠt+1 ] and the coefficient matrix A is given in Appendix A. 9 In this system all variables in x t are non-predetermined. Hence, the REE is determinate if and only if all eigenvalues of the matrix A are inside the unit circle. Thus, the next two propositions can be obtained. Proposition 1 Suppose that the interest rate policy is outcome-based, that is, i = in eq. (14), and Assumption 1 holds. Then, the REE is determinate if and only if the next two inequalities are satisfied. ϕ π + ϕ y ϵ y > 1, (16) F (ϕ π, ϕ y, Π; β, η, θ, α, λ) <, (17) where ϵ y is given by ϵ y = λ Π θ 1 {(1 β)(1 βλ Π θγ )[1 + θ(γ 1)] βθγ( Π 1+θ(γ 1) 1)(1 λ Π θ 1 )} γ(1 λ Π θ 1 )(1 βλ Π θ 1 )(1 βλ Π θγ ) and F ( ) is given in Appendix B. Proof See Appendix B. Proposition 2 Suppose that the interest rate policy is forecast-based, that is, i = 1 in eq. (14), and Assumption 1 holds. Then, the REE is determinate if and only if the condition (16) and 9 The form of the coefficient vector B is omitted, since it is not needed in what follows. 1

12 the next two inequalities are satisfied. ϕ π + (ϕ y 2)ω < 1, (18) G(ϕ π, ϕ y, Π; β, η, θ, α, λ) <, (19) where ω is given by ω = λ Π θ 1 (1 + β)(1 + βλ Π θγ )[1 + θ(γ 1)] + βθγ( Π 1+θ(γ 1) 1)(1 λ Π θ 1 ) γ(1 + βλ Π θ 1 )(1 λ Π θ 1 )(1 βλ Π θγ ) > and G( ) is given in Appendix C. Proof See Appendix C. The condition (16) can be interpreted as the long-run version of the Taylor principle. The NKPC (13) implies that each percentage point of permanently higher inflation yields ϵ y percentage points of permanently higher output. Hence, ϵ y represents the long-run inflation elasticity of output. Then, ϕ π + ϕ y ϵ y shows the permanent increase in the interest rate by the interest rate policy (14) in response to each unit permanent increase in inflation. Therefore, the condition (16) suggests that in the long run the interest rate should be raised by more than the increase in inflation. Note that the long-run version of the Taylor principle (16) is less likely to be satisfied for the interest rate policy coefficients ϕ π, ϕ y as the long-run inflation elasticity of output ϵ y is smaller. 3.2 E-stability conditions We turn next to the analysis of E-stability of fundamental REE. Following the literature on learning in macroeconomics (e.g., Bullard and Mitra, 22; Evans and Honkapohja, 21), we use the so-called Euler equation approach suggested by Honkapohja et al. (211): the rational expectation operator E t is replaced with a possibly non-rational expectation operator Ê t in the system of eqs. (12) (14). Then, the system can be reduced to z t = CÊtz t+1 + D[1 ]Êtz t+2 + F ε t, (2) where z t = [ˆΠ t ŷ t ] and the coefficient matrices C, D are given in Appendix A. 1 In this system, fundamental REE is given by z t = c + Γε t = (I ρc ρ 2 D[1 ]) 1 F ε t, (21) 1 The form of the coefficient vector F is omitted, since it is not needed in what follows. 11

13 where I denotes a conformable identity matrix. Following Section 1.3 of Evans and Honkapohja (21), we investigate E-stability of the fundamental REE (21). Corresponding to this REE, all agents are assumed to be endowed with a perceived law of motion (PLM) of z t z t = c + Γε t. (22) Using forecasts from this PLM to substitute Êtz t+1 and Êtz t+2 out of the system (2) leads to the actual law of motion (ALM) of z t z t = (C + D[1 ])c + [ρ(c + ρd[1 ])Γ + F ]ε t. (23) Then, the mapping T from the PLM (22) to the ALM (23) can be defined by T (c, Γ) = ((C + D[1 ])c, ρ(c + ρd[1 ])Γ + F ). For the fundamental REE ( c, Γ) to be E-stable, the matrix differential equation d dτ (c, Γ z) = T (c, Γ) (c, Γ) must have local asymptotic stability at the REE, where τ denotes a notional time. Hence, the fundamental REE ( c, Γ) is E-stable if and only if all eigenvalues of two matrices, DT c ( c, Γ) = C + D[1 ] and DT Γ ( c, Γ) = ρ(c + ρd[1 ]), have real parts less than unity. Thus the following two propositions can be obtained. Proposition 3 Suppose that the interest rate policy is outcome-based, that is, i = in eq. (14), and Assumption 1 holds. Then, the fundamental REE is E-stable if and only if the long-run version of the Taylor principle (16) and the next two inequalities are satisfied. ϕ π + λ Π θ 1 {[2 β βλ Π θγ (1 β)][1 + θ(γ 1)] βθγ( Π 1+θ(γ 1) 1)(1 λ Π θ 1 )} γ(1 λ Π θ 1 )(2 βλ Π θ 1 )(1 βλ Π θγ (ϕ y + 1) ) > γ(1 λ Π θ 1 )(1 βλ Π θγ ) + λ Π θ 1 [1 + θ(γ 1)] γ(1 λ Π θ 1 )(2 βλ Π θ 1 )(1 βλ Π θγ, (24) ) ϕ π + λ Π θ 1 {(1 ρβ)(1 ρβλ Π θγ )[1 + θ(γ 1)] ρβθγ( Π 1+θ(γ 1) 1)(1 λ Π θ 1 )} γ(1 λ Π θ 1 )(1 ρβλ Π θ 1 )(1 βλ Π θγ (ϕ y + 1 ρ) ) > ρ. (25) Proof See Appendix D. 12

14 Proposition 4 Suppose that the interest rate policy is forecast-based, that is, i = 1 in eq. (14), and Assumption 1 holds. Then, the fundamental REE is E-stable if and only if the long-run version of the Taylor principle (16) and the next two inequalities are satisfied. ϕ π + λ Π θ 1 [1 + θ(γ 1)] γ(1 λ Π θ 1 )(1 βλ Π θγ ) (ϕ y + 1 β) > 1 + βλ Π θ 1 {θγ ( Π1+θ(γ 1) 1 ) (1 λ Π θ 1 ) + (1 β)λ Π θγ [1 + θ(γ 1)]} γ(1 λ Π θ 1 )(1 βλ Π θγ, (26) ) ϕ π + λ Π θ 1 {(1 ρβ)(1 ρβλ Π θγ )[1 + θ(γ 1)] ρβθγ( Π 1+θ(γ 1) 1)(1 λ Π θ 1 )} γ(1 λ Π θ 1 )(1 ρβλ Π θ 1 )(1 βλ Π θγ ) ( ϕ y + 1 ρ ) ρ > 1. (27) Proof See Appendix E. 3.3 Implications of determinacy and E-stability conditions We now illustrate the conditions for determinacy and for E-stability given in Propositions 1 to 4 using the calibration of model parameters presented in Table 1. For the outcome-based interest rate policy (i.e., i = in eq. (14)), Fig. 1 shows regions of the policy coefficients (ϕ π, ϕ y ) that guarantee determinacy of REE or E-stability of fundamental REE in the cases of the annualized trend inflation rate of two, four, six, and eight percent. Fig. 2 displays similar regions for the forecast-based interest rate policy (i.e., i = 1 in eq. (14)). Note that the coefficients estimated by Taylor (1993) are (ϕ π, ϕ y ) = (1.5,.5/4) = (1.5,.125) which is marked by in each panel of these figures and thus it is reasonable to consider the range of ϕ π = 4.5 and ϕ y =.375. These two figures demonstrate that higher trend inflation is more likely to induce not only indeterminacy but also E-instability. Indeed, Taylor s estimates generate indeterminacy and E-instability once the annualized trend inflation rate is equal to or greater than four percent. Moreover, higher trend inflation is more likely to generate indeterminacy rather than E-instability, especially when the policy response to output is small. Indeed, in the case of the outcome-based policy with no response to output (i.e., ϕ y = ), determinacy requires a very active policy response to inflation under the annualized trend inflation rate of four percent, whereas E-stability requires a mildly active one. The forecast-based policy with no response to expected future output induces indeterminacy for any trend inflation rate considered unless the policy response to expected future inflation is extremely strong, whereas it ensures E-stability in the cases of the annualized 13

15 trend inflation rate of two and four percent as long as the policy response to expected future inflation is strong enough. When trend inflation increases, the long-run inflation elasticity of output ϵ y declines, as can be seen in Fig. 3. Therefore, the long-run version of the Taylor principle (16) is less likely to be satisfied for the interest rate policy coefficients ϕ π, ϕ y, as noted above. For instance, the coefficients estimated by Taylor (1993) (i.e. (ϕ π, ϕ y ) = (1.5,.125)) no longer satisfy the longrun version of the Taylor principle once the annualized trend inflation rate is equal to or greater than four percent. Moreover, when the trend inflation rate is greater than a certain threshold that makes the numerator of the long-run inflation elasticity of output ϵ y equal to zero (i.e.,.2 percent in annualized rate terms), the elasticity becomes negative. As a consequence, a large policy response to output induces indeterminacy and E-instability. A small policy response to output also makes indeterminacy and E-instability more likely under higher trend inflation. The indeterminacy region of the interest rate policy coefficients expands with higher trend inflation, because the condition (17) is less likely to be satisfied in the case of the outcome-based policy and the condition (19) is less likely to be met in the case of the forecast-based policy. 11 The E-instability region expands with higher trend inflation, because the condition (24) is less likely to be satisfied in the case of the outcome-based policy and the condition (26) is less likely to be met in the case of the forecast-based policy. 4 Comparison with the case of homogeneous labor This section compares the cases of firm-specific labor and homogeneous labor in terms of determinacy of REE and E-stability of fundamental REE. When labor is homogeneous, the utility function of the representative household is and the budget constraint is ) E β (ln t 1 C t 1 + 1/η N 1+1/η t exp(ε t ), t= P t C t + S t = W t N t + R t 1 S t 1 + T t, 11 In addition to the long-run version of the Taylor principle (16), the condition (18) imposes an upper bound on the size of the policy response to expected future output that ensures determinacy. However, this upper bound is substantially larger than the maximum value of the policy coefficient considered in Fig

16 where N t is the supply of homogeneous labor and W t is its wage. Combining first-order conditions for utility maximization with respect to consumption and labor supply yields W t P t = C t N 1/η t. (28) The first-order condition for intermediate-good firm i s minimization of production cost determines this firm s marginal cost MC t (i) = W tn t (i) αy t (i). (29) The labor market clearing condition is given by N t = 1 N t(i) di. Using this equation and eqs. (3), (5), (6), (28), and (29), the first-order condition for staggered price-setting leads to [ ( ) 1 θ E t (βλ) j Bt exp(ε t+j ) θ ( ) 1 γ ( ) ] θ/α Yt+j Bt d 1/η t+j =, (3) θ 1 α j= P t+j A t+j P t+j where d t = 1 (P t(i)/p t ) θ/α di denotes price distortion and evolves according to P θ/α t d t = (1 λ)b θ/α t + λp θ/α t 1 d t 1. (31) Under Assumption 1, detrending and log-linearizing the equilibrium conditions (1), (3), and (31) leads to ˆΠ t βλ Π θ/α E t ˆΠt+1 = β(e t ˆΠt+1 βλ Π θ/α E t ˆΠt+2 ) + β( Π 1+θ(1/α 1) 1)(1 λ Π θ 1 ) [θ/αe t ˆΠt+1 (ε t E t ε t+1 )] 1 + θ(1/α 1) + γ(1 λ Π θ 1 )(1 βλ Π θ/α ) λ Π θ 1 (ŷ t βλ [1 + θ(1/α 1)] Π θ 1 E t ŷ t+1 ) + (1/η)(1 λ Π θ 1 )(1 βλ Π θ/α ) λ Π θ 1 ( [1 + θ(1/α 1)] ˆd t βλ Π θ 1 E t ˆdt+1 ), (32) ˆd t = λ Π θ 1 ( Π 1+θ(1/α 1) 1)θ/α 1 λ Π θ 1 ˆΠt + λ Π θ/α ˆdt 1. (33) In line with Ascari and Ropele (29) and Kurozumi (211), the NKPC (32) now depends on price distortion and therefore eq. (33) adds lagged price distortion ˆd t 1 to the set of relevant state variables of the model with homogeneous labor, as long as the elasticity of labor supply is finite (i.e., η < ) and the trend inflation rate is non-zero (i.e., Π 1). Consequently, it seems impossible to analytically investigate determinacy and E-stability and we examine them numerically. 15

17 We now compare the models with firm-specific labor and with homogeneous labor in terms of determinacy of REE and E-stability of fundamental REE under the outcome-based interest rate policy (i.e., i = in eq. (14)). Fig. 4 illustrates regions of the outcome-based policy coefficients (ϕ π, ϕ y ) that guarantee determinacy of REE or E-stability of fundamental REE in the model with homogeneous labor when the annualized trend inflation rate is two, four, six, or eight percent, using the calibration of parameters presented in Table 1. This figure shows that for moderate rates of trend inflation (e.g., two, four, and six percent), both determinacy and E-stability are ensured as long as the policy coefficients satisfy the long-run version of the Taylor principle which is of the same form as the one (16) except that the long-run inflation elasticity of output is now given by 12 (1 β)(1 λ λ Π Π θ/α )(1 βλ Π θ/α )[1 + θ(1/α 1)] θ 1 θ/α( Π 1+θ(1/α 1) 1)[β(1 λ Π θ 1 )(1 λ Π θ/α ) + 1/η(1 βλ Π θ 1 )(1 βλ Π θ/α )] ϵ y = γ(1 λ Π θ 1 )(1 λ Π θ/α )(1 βλ Π θ 1 )(1 βλ Π θ/α. ) When trend inflation is high (e.g., eight percent), determinacy and E-stability are ensured basically as long as the long-run version of the Taylor principle is met, except for a very small region of the policy coefficients that induce indeterminacy of REE but generate E-stable fundamental REE. For instance, in the case of no policy response to output (i.e., ϕ y = ), a policy response to inflation ϕ π in the interval (1.1, 1.17) induces indeterminacy and E- stability. Therefore, regardless of the rate of trend inflation, the interest rate does not need to be adjusted much more than one-for-one with current inflation to satisfy the long-run version of the Taylor principle under the calibration. Hence, a low rate of trend inflation is not required to ensure determinacy and E-stability under the calibration, as long as the policy rate is adjusted one-for-one or slightly more strongly with contemporaneous inflation. For instance, Taylor (1993) s estimates (i.e. ϕ π, ϕ y ) = (1.5,.125)) satisfy the long-run version of the Taylor principle even at high trend inflation rates. The comparison of Figs. 1 and 4 thus demonstrates that under the same calibration of structural model parameters, the outcome-based interest rate policy is much more likely to guarantee determinacy of REE and E-stability of fundamental REE in the model with homogeneous labor than in the model with firm-specific labor. We turn next to the forecast-based interest rate policy (i.e., i = 1 in eq. (14)). Fig Note that in the case of an infinite elasticity of labor supply (i.e., η = ), the NKPC coincides in the models with firm-specific labor and with homogeneous labor, and hence not only the long-run inflation elasticity of output but also the long-run version of the Taylor principle is the same in these two models. 16

18 illustrates regions of the forecast-based policy coefficients (ϕ π, ϕ y ) that ensure determinacy of REE or E-stability of fundamental REE in the model with homogeneous labor when the annualized trend inflation rate is two, four, six, or eight percent, using the calibration of parameters presented in Table 1. The comparison of Figs. 2 and 5 demonstrates that under the same calibration of structural model parameters, the forecast-based interest rate policy is much more likely to guarantee determinacy of REE and E-stability of fundamental REE in the model with homogeneous labor than in the model with firm-specific labor, as is the case with the outcome-based interest rate policy. To satisfy the long-run version of the Taylor principle in the model with homogeneous labor, the interest rate does not need to be adjusted much more than one-for-one with expected future inflation, even at high rates of trend inflation. Hence, this Taylor principle is much more likely to be satisfied in the model with homogeneous labor than in the model with firm-specific labor. In addition to the long-run version of the Taylor principle, ensuring E-stability requires a sufficiently aggressive policy response to expected future output, while ensuring determinacy requires a more aggressive one. As a result, while the coefficients estimated by Taylor (1993) satisfy the long-run version of the Taylor principle even at high trend inflation rates, they do not ensure determinacy at high trend inflation rates because of an insufficiently strong response to expected future output. Nonetheless, under the calibration, these lower bounds on the policy response to expected future output are less likely to be binding in the model with homogeneous labor than in the model with firm-specific labor. The above results demonstrate that under the same calibration of structural model parameters, indeterminacy and E-instability are much more likely in the model with firm-specific labor than in the model with homogeneous labor analyzed in the recent literature such as Ascari and Ropele (29) and Kurozumi (211). Indeterminacy is much more likely in the model with firm-specific labor because this specification of labor introduces strategic complementarity or real rigidity, which causes indeterminacy like firm-specific capital analyzed in Sveen and Weinke (25, 27). As shown in Fig. 3, when the trend inflation rate is greater than a certain threshold (i.e.,.2 percent in annualized rate terms), the strategic complementarity makes the long-run inflation elasticity of output ϵ y take a smaller negative value and thereby causes the long-run version of the Taylor principle to be less likely to be satisfied. As a consequence, a large policy response to output induces indeterminacy in the model with firm-specific labor, even if such a response may ensure determinacy under the same calibration in the model with homogeneous labor. 17

19 Another condition for determinacy causes a small policy response to contemporaneous or expected future output to induce indeterminacy under positive trend inflation rates. This condition is also less likely to be met in the model with firm-specific labor than in the model with homogeneous labor, when the same calibration is used in these two models. Indeed, in the model with homogeneous labor, it is satisfied under the outcome-based interest rate policy for any trend inflation rate considered. When the interest rate policy is forecast-based, the minimum policy response to expected future output required to ensure determinacy increases substantially with higher trend inflation in the model with homogeneous labor, but it increases even more steeply with higher trend inflation in the model with firm-specific labor. E-instability is much more likely in the model with firm-specific labor than in the model with homogeneous labor for two reasons. First, the long-run version of the Taylor principle is a necessary condition for E-stability of fundamental REE as well. This Taylor principle is, as noted above, less likely to be satisfied in the model with firm-specific labor than in the model with homogeneous labor. Second, the NKPC depends on price distortion in the model with homogeneous labor as long as the elasticity of labor supply is finite and the trend inflation rate is non-zero, but not in the model with firm-specific labor. Then, in the former model, the law of motion of price distortion adds lagged price distortion to the set of relevant state variables. For the REE in question, E-stability examines whether an associated equilibrium in which agents form expectations under adaptive learning reaches over time the REE. For such expectation formation, lagged price distortion is useful information in the model with homogeneous labor. Particularly, it helps agents form inflation expectations, since price distortion affects inflation dynamics. Therefore, E-stability is likely in the model with homogeneous labor. By contrast, price distortion is absent in the set of relevant state variables of the model with firm-specific labor, and hence higher trend inflation is more likely to induce E-instability. We have demonstrated that indeterminacy and E-instability are much more likely in the model with firm-specific labor than in the model with homogeneous labor. It follows that the argument of Coibion and Gorodnichenko (211) who use the former model to show that a decline in trend inflation plays a key role in the U.S. economy s shift from indeterminacy during the Great Inflation era to determinacy during the Great Moderation era depends crucially on the assumption of firm-specific labor. Under the calibration of parameters presented in Table 1, which is also chosen by Coibion and Gorodnichenko, the model with homogeneous labor argues for Clarida et al. (2) and Lubik and Schorfheide (24), who all attribute the 18

20 U.S. economy s shift to the Great Moderation solely to the Fed s change from a passive to an active policy response to inflation. 5 Concluding remarks In a Calvo sticky price model based on micro evidence that each period a fraction of prices is kept unchanged, this paper has examined implications of firm-specific labor for determinacy and E-stability of REE under interest rate policy. It has shown that firm-specific labor causes higher trend inflation to be more likely to induce not only indeterminacy of REE but also E-instability of fundamental REE. The latter is in contrast with the result of E-stability in the case of homogeneous labor analyzed in recent research. Moreover, we have demonstrated that under the same calibration of structural model parameters, indeterminacy and E-instability are much more likely in the case of firm-specific labor than in the case of homogeneous labor. This shows that Coibion and Gorodnichenko s (211) argument a decline in trend inflation along with the Fed s change from a passive to an active policy response to inflation explains much of the U.S. economy s shift from indeterminacy during the Great Inflation era to determinacy during the Great Moderation era depends crucially on the assumption of firm-specific labor. Thus, future work will estimate the models with firm-specific labor and with homogeneous labor using the method of Lubik and Schorfheide (24) to empirically address the question of whether a decline in trend inflation is a cause of such a shift from indeterminacy to determinacy in the U.S. economy. 19

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