The Cost Channel in a New Keynesian Model

The Cost Channel in a New Keynesian Model: Evidence and Implications Federico Ravenna and Carl E. Walsh∗ First draft: October 2002 This draft: January 2003

Abstract In the standard new Keynesian framework, an optimizing policy maker does not face a trade-off between stabilizing the gap between actual output and output under flexible prices and stabilizing the inflation rate. An ad hoc, exogenous cost-push shock is typically added to the inflation equation to generate a meaningful policy problem. In this paper, we show that a cost-push shock arises endogenously when a cost channel for monetary policy is introduced into the new Keynesian model. A cost channel arises when firms’ marginal cost depends directly on the nominal rate of interest. Besides providing additional empirical evidence for a cost channel, we explore its implications for monetary policy trade-offs, the objectives of monetary policy, and the effects of shocks on the economy under optimal discretionary and optimal commitment policies. We show that the presence of a cost channel alters the optimal policy problem in important ways. For example, both the output gap and inflation are allowed to fluctuate in response to productivity and fiscal shocks under optimal monetary policy. ∗

Department of Economics, University of California, Santa Cruz, CA 95064, [email protected], [email protected]. The authors thank Alina Carare for helpful comments.

1

1

Introduction

In the standard new Keynesian framework, an optimizing policy maker does not face a tradeoff between stabilizing the gap between actual output and output under flexible prices and stabilizing the inflation rate. The result that the optimal policy problem has a trivial solution is widely recognized as a shortcoming of this modeling framework. Clarida, Gal´ı and Gertler (1999) show that the introduction of an ad hoc, exogenous cost-push shock in the inflation equation allows the new Keynesian model to generate a meaningful policy problem. In this paper, we show that a cost-push shock arises endogenously in the presence of a cost channel for monetary policy. A cost channel arises when firms’ marginal cost depends directly on the nominal rate of interest. Barth and Ramey (2001) present evidence for the cost channel, and Christiano, Eichenbaum, and Evans (2001) have incorporated a cost channel into a model estimated using U.S. aggregate data. Besides providing additional empirical evidence for a cost channel of monetary policy, we explore its implications for monetary policy trade-offs, the objectives of monetary policy, and the effects of shocks on the economy under optimal discretionary and optimal commitment policies. We derive the appropriate welfare-based loss function that should be the policy-maker’s objective in a cost-channel economy and show it is possible to express the loss function in terms of the gap between output and a measure of potential output that is invariant to assumptions on monetary policy in the flexible-price equilibrium. As a consequence, the optimal policy implications can be directly compared with the standard new Keynesian results. As we show, the presence of a cost channel alters the standard policy conclusions in important ways. If a cost channel exists, any shock to the economy—whether a productivity, government spending, or preference shock—generates a trade-off between stabilizing inflation and stabilizing the output gap. In the standard new Keynesian model of Clarida, Gal´ı and Gertler (1999), henceforth CGG, the optimal response to these shocks guarantees that neither inflation nor the output gap deviate from their steady state values. In contrast, these shocks lead to inflation and output gap fluctuations under optimal policy (either commitment or discretion) when a cost channel is present. An adverse productivity shock, for example, leads to a fall in the output gap and a rise in inflation under the optimal monetary policy. Hence, if we assume the central bank

2

behaves optimally, observing a rise in the inflation rate does not imply that a cost push-shock has hit the economy; an adverse productivity shock would generate the same inflation behavior. Conversely, observing a positive productivity shock coupled with constant inflation would imply that the central bank is not following the optimal policy. We also show that the optimal policy does not fully insulate the output gap and inflation from fiscal shocks. This finding is independent of the presence of the cost channel. A conclusion from most recent analyzes of monetary policy is that shocks to the expectational IS curve should be neutralized so that they do not affect the output gap. We show that when the objective function is derived as a second-order approximation to the representative agents utility function, neutralizing IS shocks is not the optimal policy because such shocks can affect welfare even when the output gap and inflation remain equal to zero. The rest of the paper is organized as follows. In section 2, the model is set out and the equilibrium under flexible prices and under sticky prices is derived. Section 3 estimates a new Keynesian inflation-adjustment equation and tests for the presence of a cost channel. We find that we cannot reject the hypothesis that a cost channel is present. Hence, in section 4 we analyze the consequences of the cost channel for optimal policy. We derive a second order approximation to the utility of the representative agent in the model, and use this to define optimal policy objectives. Then, we analyze optimal under discretion and under commitment and show how previous results are modified when monetary policy operates through the cost channel. Section 5 numerically simulates the model to illustrate how the economy responds to various shocks under optimal policy. Finally, conclusions are contained in section 6.

2

The basic model

Several theoretical models can generate a cost-side effect of monetary policy. Models which incorporate a balance-sheet or credit channel of monetary policy imply movements in interest rates directly affect firms’ ability to produce (Bernanke and Gertler, 1989). Christiano and Eichenbaum (1992) introduce the cost of working capital into the production side of their model, assuming that factors of production have to be paid before the proceeds from the sale of output are received. Barth and Ramey (2001), using data for trade credit from the US Flow of Funds, 3

report that over the period 1995 to 2000 net working capital (inventories plus trade receivables, net of trade payables) averaged 11 months of sales—an amount comparable to the investment in fixed capital. The basic framework we use to illustrate the cost channel is a cash-in-advance model with sticky prices that is similar to the model employed by Christiano, Eichenbaum, and Evans (2001). We simplify their model by ignoring capital and habit persistence in consumption. In order to capture the role of demand shocks, we follow McCallum and Nelson (1999) and introduce both a taste shock to the marginal utility of consumption and stochastic shocks to government purchases. The model economy consists of households, firms, the government, and financial intermediaries interacting in asset, goods, and labor markets. The goods market is characterized by monopolistic competition, and the adjustment of prices follows the by now standard treatment based on Calvo (1983). Derivations of the basic new Keynesian model can be found in Woodford (2000), Clarida, Gal´ı, and Gertler (1999), and Gal´ı (2002). We focus here on those aspects of the model that differ from the standard specification. The preferences of the representative household are defined over a composite consumption good Ct , a taste shock ξ t , and leisure 1 − Nt . Households maximize the expected present discounted value of utility: Et

∞ X

β

i

i=0

"

# 1+η 1−σ ξ t Ct+i Nt+i −χ . 1−σ 1+η

(1)

The composite consumption good consists of differentiated products produced by monopolistically competitive final goods producers (firms). There is a continuum of such firms of measure 1. Ct is defined as Ct =

·Z

1

θ−1 θ

cjt dj

0

θ ¸ θ−1

,

θ > 1.

Given prices pjt for the final goods, this preference specification implies the household’s demand for good j is cjt =

µ

pjt Pt

4

¶−θ

Ct ,

(2)

where the aggregate price index Pt is defined as

Pt =

·Z

1

p1−θ jt dj 0

1 ¸ 1−θ

.

Households enter period t with cash holdings of Mt . They receive their wage income paid as cash at the start of the period. They use this cash to make deposits Dt at the financial intermediary. Their remaining cash balances of Mt + Wt Nt − Dt are available to purchase consumption goods subject to a cash-in-advance constraint: Pt Ct ≤ Mt − Dt + Wt Nt . At the end of the period, households receive profit income from the financial intermediary and firms and the principle plus interest on their deposits at the intermediary. Consequently, cash carried over to period t + 1 is Mt+1 = Mt − Dt + Wt Nt − Pt Ct + Rt Dt + Πt − Tt , where Rt is the gross nominal interest rate, Πt is equal to aggregate profits from intermediaries and firms, and Tt are (lump-sum) taxes. In addition to the demand functions for the individual goods, the following first order conditions must hold in an equilibrium with a positive nominal interest rate: ξ t Ct−σ

= Et

µ

Rt Pt Pt+1

¶

−σ ξ t+1 Ct+1

(3)

χNtη Wt −σ = Pt ξ t Ct

(4)

Pt Ct = Mt − Dt + Wt Nt .

(5)

Equilibrium in the goods market requires that Yt = Ct + Gt , where Gt are government purchases. We assume the government purchases individual goods in the same proportions as households and that aggregate government purchases are proportional to output; Gt = (1−γ t )Yt , where γ t is stochastic and bounded between zero and one. The aggregate resource constraint 5

then takes the form Yt = Ct + Gt = Ct + (1 − γ t )Yt , or Ct = γ t Yt .

(6)

Following the literature on staggered price setting, we adopt a Calvo specification in which the probability a firm optimally adjusts its price each period is given by 1 − ω. The fraction ω of firms that do not optimally adjust simply update their previous price by the steady-state inflation rate. If firm j sets its price at time t, it will do so to maximize expected profits, subject to the demand curve it faces, given by (2), and the production technology cjt = At Njt , where Njt is employment by firm j in period t. At is an aggregate productivity factor. The firm must borrow an amount Wt Nt from intermediaries at the gross nominal interest rate Rt , so the nominal cost of labor is Rt Wt . The real marginal cost of the firm is equal to Rt ϕjt ≡

³

Wt Pt

At

´

= Rt Sjt .

(7)

where St ≡ Wt Njt /Pt cjt is labor’s share of income. When prices are flexible, real marginal cost is equal to the (constant) mark up θ/(θ − 1), and Rt

³

Wt Pt

At

´

=

θ . θ−1

(8)

As is well known (see Gal´ı and Gertler 1999, Sbordone 2002), this model leads to an inflation adjustment equation of the form π t = βEt π t+1 + κˆ ϕt ,

(9)

where π t is the deviation of inflation around the steady-state rate of π ¯ and ϕ ˆ t is the percentage deviation of real marginal cost around its steady-state value of θ/(θ − 1). (A hat ˆ notation will be used to denote percentage deviations around steady-state values.) The parameter κ is given 6

by κ=

(1 − ω) (1 − ωβ) . ω

The intermediary receives deposits and a cash injection Xt from the monetary authority. These funds are lent to firms at a gross nominal interest rate Rt . Intermediaries operate costlessly in a competitive environment, so profits in the intermediary industry are Rt (Dt + Xt ) − Rt Dt = Rt Xt = Πit . Letting Gt+1 denote the gross growth rate of money from t to t + 1, the cash injection can be expressed as Xt = (Mt+1 − Mt ) = (Gt+1 − 1) Mt . Equilibrium in the market for loans implies that Wt Ntd = Dt + Xt , where Ntd is aggregate labor demand by firms.

2.1

The flexible-price equilibrium

The flexible-price equilibrium for output, employment, the real wage, and the real rate of interest is obtained by jointly solving equations (3), (4), (8), and the production function. Let superscript f denote the flexible price equilibrium. In the flexible-price equilibrium, firms equate the real wage, include interest costs, to the marginal product of labor divided by the markup Φ = θ/(θ − 1) > 1:

Rft wtf =

At . Φ

Households equate the real wage to the marginal rate of substitution between leisure and consumption:

VN χNtη = = wtf . Uc ξ t Ct−σ

7

¿From the production function, Yt = At Nt and the resource constraint, Ct = γ t Yt , labor market equilibrium in the flex-price equilibrium requires that χ ³

³

Ytf At

´η

ξ t γ t Ytf Rearranging to solve for Ytf , Ytf

´t−σ =

·

1+η ξ t γ −σ t At = χΦRtf

At . ΦRtf

1 ¸ σ+η

.

(10)

Using an over bar to denote steady-state values, the steady-state level of output is ·

γ¯ −σ Y¯ = ¯ χΦR

1 ¸ σ+η

.

(11)

Even with flexible prices, output is distorted by the presence of monopolistic competition and by a positive nominal rate of interest. The first of these distortions would be eliminated if Φ = 1; the second distortion would be eliminated if the nominal interest rate were zero (R = 1). Hence, the efficient flexible-price output level is obtained by setting R = Φ = 1 in equation (10):

Yt∗

·

1+η ξ t γ −σ t At = χ

1 ¸ σ+η

.

Expressed in terms of percentage deviations around the steady-state, the flexible-price equilibrium output level is Yˆtf =

µ

1 σ+η

¶h i ˆ tf . (1 + η)Aˆt − σˆ γ t + ˆξ t − R

(12)

When only productivity disturbances are present, as in most new Keynesian models, equation (12) reduces to Yˆtf = (1 + η)Aˆt / (σ + η). In the present model, the flexible-price output level is also affected by fiscal shocks (ˆ γ t ), taste shocks (ˆξ t ), and the nominal interest rate. Both fiscal and taste shocks would affect Yˆtf even in the absence of a cost channel because they affect labor supply. From (6), Cˆt = γˆ t + Yˆt . A positive γˆ t increases the share of output going to consumption; this lowers the marginal utility of consumption and reduces household labor

8

supply. As a consequence, flexible-price output falls. At the same time, consumption rises. As a result, the equilibrium real interest rate must rise. The fact that output and consumption move in opposite directions in response to a fiscal shock in the flexible price equilibrium contrasts with the situation with either a productivity shock or a taste shock. For example, a positive taste shock ˆξ t increases the marginal utility of consumption and therefore increases labor supply; flexible-price output rises, and so does consumption. Because of the cost channel, flexible-price output is not independent of the nominal rate of interest. A rise in the nominal interest reduces labor demand and causes a fall in the equilibrium level of flexible-price output. The effects of the cost channel, fiscal shocks, and taste shocks on output operate through their impact on labor supply. In the case of an inelastic labor ˆ γˆ nor ˆξ t affect the flexible-price level of output. supply (the limit as η → ∞), neither R, Using the resource constraint, the Euler condition (3) can be linearized around the steady state and solved for the deviation of the flexible-price equilibrium value of the real interest rate around the steady state: ³ ´ ³ ´ ¡ ¢ f rˆtf = σ Et Yˆt+1 − Yˆtf − Etˆξ t+1 − ˆξ t + σ Et γˆ t+1 − γˆ t µ ¶h ³ ´ ³ ´ ¢i ¡ 1 = σ Et Aˆt+1 − Aˆt − η Et ˆξ t+1 − ˆξ t + ση Et γˆ t+1 − γˆ t . σ+η

(13)

Equation (13) shows that the real interest rate in the flexible-price equilibrium will be affected by productivity, taste, and fiscal shocks, unless the shocks follow random walk processes.

2.2

Equilibrium with sticky prices

When prices are sticky (ω < 1), inflation adjustment is given by equation (9). The difference between the model developed here and that of Gal´ı and Gertler (1999) is that, from (7), real marginal cost now depends on the nominal interest rate: ˆ t + sˆt , ϕ ˆt ≈ R where sˆ = (w ˆt − pˆt ) − (ˆ yt − n ˆ t ) is the log deviation of labor’s share of output around the steady-

ˆ t is the percentage point deviation of the nominal interest rate around its steady-state state and R 9

value. Hence, in the presence of a cost channel, ´ ³ ˆ π t = βEt π t+1 + κ Rt + sˆt .

(14)

The linearized versions of (3) and (4) can be used to express the sticky-price model in terms of the gap between output and the flexible-price output level, a nominal interest rate gap and a real interest rate gap: ³ ´ µ 1 ¶ h³ ´ i f f ˆ ˆ t − Et π t+1 − rˆtf ˆ ˆ ˆ Yt − Yt = Et Yt+1 − Yt+1 − R σ ³ ´ ³ ´ ˆt − R ˆ tf , π t = βEt π t+1 + κ(σ + η) Yˆt − Yˆtf + κ R

(15)

(16)

where rˆtf is the flexible-price real interest rate. This two equation system differs from the ˆt − R ˆ tf in standard new Keynesian model due to the presence of the nominal interest rate gap R the inflation adjustment equation. While we deal explicitly with the policy implications of this difference in section 4, two ˆ f in the conclusions can be briefly stated. First, since Yˆ f is conditional on the monetary policy R flexible-price equilibrium (see equation 12), the policy-maker should not necessarily minimize the output-gap Yˆt − Yˆtf . There is no reason why the flexible-price equilibrium obtained under

ˆ f should welfare-dominate the sticky price equilibrium obtained an arbitrary monetary policy R ˆ Second, it may be possible to stabilize both the output under a different monetary policy R. ˆt − R ˆ tf in equation (16), gap Yˆt − Yˆtf and the inflation rate, despite the presence of the term R

ˆ tf are not independent variables. However, we show in section 4 that a policy since Yˆtf and R that stabilizes both would not be welfare-maximizing. Before exploring the policy implications of the presence of the nominal interest rate gap in equation (16) further, we first turn to the aggregate empirical evidence for the cost channel.

3

Empirical evidence

This section provides empirical evidence on the relevance of the cost channel in the estimation of the forward-looking Phillips curve. In the baseline version of the Calvo sticky price model 10

without variable capital or a cost channel, real marginal cost is proportional to the output gap, making (14) similar to a standard Phillips curve. Gal´ı and Gertler (1999) provide evidence that testing the forward-looking pricing equation using conventional measures of the output gap can be misleading, while the model performs well when the correct definition of real marginal cost is used. To assess the empirical evidence for the cost channel, we first generalize the model by dropping the simple production function assumed in section 2 and instead letting the production function for the monopolistically competitive firm j be given by Yt (j) = At Kt (j)αk Nt (j)(1−αk ) , 0 < αk < 1,

(17)

where At is the technology level, Kt capital, and Nt labor. Real marginal costs will now differ across firms if their production levels differ. Sbordone (2002) shows that inflation can be related to average real marginal costs according to π t = βEt π t+1 + κ ˜ϕ ˆ t, where

·

¸ (1 − ω)(1 − βω) κ ˜= τ, ω

(18)

(19)

τ ≡ (1 − αk )/[1 + αk (θ − 1)] and ϕt is given by labor’s average share St divided by 1 − αk . If the cost channel is introduced, firm j faces a total nominal production cost of Rt Wt Nt (j)+ Rtk Kt (j). The inflation dynamics are still given by equation (18) and are a function of average real marginal cost defined as ϕt =

Rt Wt /Pt Rt St = M P Nt (1 − αk )

(20)

ˆ t + sˆt . which implies ϕ ˆt = R We estimate equation (18) for the US over the sample 1960:1 - 2001:1 using quarterly data. The econometric specification nests the definition of marginal cost given by (20) and allows a test of the hypothesis that movements in the nominal interest rate affect inflation dynamics via the cost channel.

11

The estimation procedure follows Gal´ı and Gertler (1999) and Gal´ı, Gertler, L´opezSalido (2001). We obtain estimates of the deep parameters ω and β conditional on αk and θ. The production function implies that αk = 1 −

St Φ

, where Φ denotes the steady-state average

markup of price over nominal marginal cost (equal to θ/(θ − 1)). As in Gal´ı, Gertler and L´opezSalido, we assume a labor share of 2/3 and an average markup of 1.1. Given these values, estimates of ω and β can be identified by estimating the pricing equation (18). We can rewrite (18) in terms of realized variables to obtain πt = κ ˜ϕ ˆ t + βπ t+1 + ζ t , where ζ t is a linear combination of the forecast error χt = −β [π t+1 − Et (π t+1 )] and a random variable ut . If ut is taken to represent a measurement error, it is reasonable to expect it has an iid distribution. We will later address the issue of alternative interpretations for ut . Let zt be a vector of variables within the firm’s information set Ωt that are orthogonal to ζ t . Then (18) implies the orthogonality condition

Et [(π t − κ ˜ϕ ˆ t − β π t+1 ) zt ] = 0. If we express the orthogonality condition in terms of the deep parameters, and use the definition of real marginal cost (20), we can write a testable equation, which nests the case of the baseline pricing model and the case of the cost channel model, as ˆ t ) − ωβ π t+1 ) zt } = 0 st + αR Et {(ω π t − [(1 − ω)(1 − βω)τ ](ˆ

(21)

ˆ t respectively the log-deviation from steady state values of the labor income with sˆt and R share and the riskless nominal interest rate. For α = 0, (21) gives the standard Calvo pricing model, tested in Gal´ı, Gertler, L´opez-Salido (2001). To find empirical support for the baseline cost channel model with the wage bill paid in advance, we should expect estimates of α to be not significantly different from 1 1 . Given our identifying assumptions and the orthogonality 1

Our specification assumes firms must pay their entire wage bill at the start of the period. ³If workers´ receive ˆ t + sˆt . This a fraction φ < 1 of their wages at the start of the period, eq. (14) becomes πt = βEt πt+1 + κ φR

12

condition (21) we obtain estimates of α, β, and ω using a GMM estimator.

3.1

Model estimates

Our instrument vector zt includes four lags of the unit labor costs, GDP deflator inflation, a commodity price index inflation, the term spread, the nominal interest rate, wage inflation, and a measure of the output gap2 . This vector zt is labeled ‘instrument set A’ in the tables. Table 1 reports the estimates using a nonlinear instrumental variables two-stage GMM estimator and the specification of the orthogonality condition as in equation (21). All standard errors are Newey-West corrected to take into account residual serial correlation3 . The estimates for ω and β are reasonably close in the restricted and unrestricted models, and also close to the Gal´ı and Gertler (1999) estimates of 0.475 and 0.837. The estimate of the discount factor actually increases toward a value closer to the 0.98 − 0.99 range used in standard calibrations exercises. Note that the implied estimate of κ is positive and significant, and that the implied average duration of posted price is between 2 and 3 quarters in both cases. When α is not constrained to 0, its point estimate of 1.276 is not significantly different from 1, as verified by a Wald test of the null H0 = 1. The estimate of α has a higher standard error than the estimates of β and ω, yet it is significant at the 1% confidence level when the significance is tested with the Wald statistic. The difference between the values of the maximized criterion function for the restricted and unrestricted model can be used to perform the equivalent of a likelihood ratio test for the null hypothesis that α = 0. This test, known in the literature as D-test (see Matyas, 1999), rejects the null at a p-value below 0.1%. The Hansen test confirms that we cannot reject the overidentifying restrictions, although it is well known that this test has low power against model misspecifications. specification would justify estimates of α < 1. However, when φ < 1, firms pay the fraction φ of the interest tax on wages while households pay the remainder. As a consequence, the labor market equilibrium condition used to express marginal cost in terms of an output gap is unaffected by the value of φ and (15) and (16) do not change. Therefore, our results on optimal monetary policy in section 4 are unaffected. 2 This is the same set of instruments used by Gali and Gertler (1999), except for the addition of the nominal interest rate. Also, we use the Hodrick-Prescott filter measure of output gap rather than detrended output since the former can far better accommodate the surge in potential output during the second half of the 1990s. The results do not change significantly using the Gali and Gertler (1999) set of instruments. 3 The residual correlation is evidence of some misspecification of the model, since if εt is a forecast error and ut is assumed ιid no residual serial correlation should be present. See Hamilton (1994).

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3.2

Robustness to Small Sample Bias

GMM guarantees a consistent estimate of the unknown parameter vector but not unbiasedness. Small sample bias of a number of GMM estimators can be large, and it is not obvious which estimator is appropriate in different situations (see Florens, Jondeau and Le Bihan, 2001, for a discussion). A first issue relevant for the small sample properties of GMM estimators is the choice of instruments. While asymptotically any subset zt ∈ Ωt should give the same GMM estimates, this is not necessarily true in small samples. Kocherlakota (1990) and Tauchen (1986) suggests that increasing the number of instruments can increase the bias of the estimates while reducing its variance. Hall (1993) discusses instrument selection in small sample. In the linear case, a good instrument should not only be uncorrelated with ζ t , but also be strongly correlated with the regressors. The instrument relevance for the results reported has been checked by running F-test to verify the predictive power of the instruments. Other test criteria, like Theil’s U-test and sequential elimination of instruments using the correlation matrix, would lead over some samples to a different instrument set. Table 1 also reports the estimation results for the instrument set used in Gal´ı, Gertler, L´opez-Salido (2001), to which four lags of the nominal interest rate are added. This smaller zt vector, labeled ‘instrument set B’ on the tables, includes: two lags of the unit labor costs, wage inflation, a measure of the output gap, and four lags of GDP deflator inflation and the nominal interest rate. The difference between the unrestricted and restricted model estimates of ω increases relative to the previous estimate. Using the new instrument set, the estimate of α has a p-value of only 11%. The D-test though can reject the hypothesis that α = 0 at a confidence level below 1%. The Hansen test cannot reject the overidentifying restrictions. We considered other vectors zt , and it is clear that estimates of all three parameters are considerably volatile as the instrument set is changed, the more so for the parameter estimated with the larger variance, α. A possible explanation would be in the misspecification of the measurement error ut , or of the marginal cost measure. If ut is serially correlated (as might be the case if it represent a cost-push shock, see Clarida, Gal´ı and Gertler 2001a) the lagged

14

variables chosen as instruments would be correlated with the error term, and would not represent valid instruments. Also, if firms used imported intermediate goods in production, variables like the exchange rate would enter into the definition of real marginal costs, which would then not be appropriately measured by equation (20). A second issue related to the nonlinear GMM small sample bias is the estimates’ sensitivity to the orthogonality condition specification. Table 2 reports the estimates for the specification: ˆ t ) − β π t+1 ) zt } = 0 Et {(ω π t − [(1 − ω)(1 − βω)τ ω −1 ]( st + αR

(22)

The estimate of ω in all cases increases considerably, implying an average price duration between 4 and 6 quarters. The point estimate of α using the instrument set A is significant at the 5% confidence level but very high (11.831). The D-test also rejects the null hypothesis of α = 0. Since the variance of the estimate is also fairly high, the hypothesis that α = 1 can be rejected at the 5% confidence level, but not at the 10% level. Using the instrument set B, evidence on the significance of α is mixed. The restricted model cannot be rejected according to the Wald test, while is strongly rejected by the D-test. It has to be noted that the GMM estimates are much more sensitive to slight variations in the regressors or the sample when using the orthogonality condition (22). In some instances, the nonlinear estimator would not converge at all using gradient methods. A third issue, the choice of the GMM estimator itself, has been widely explored in the literature as a way to try to correct for the small sample bias. The standard two-stage GMM estimator minimizes the scalar gT (ϑ)WT gT (ϑ) where gT (ϑ) is the sample equivalent of the orthogonality condition and W is the GMM weighting matrix. The optimal feasible estimator is obtained for WT = (S˜T )−1 where S˜T is the estimate of the asymptotic covariance matrix. Usually an estimate for ST is obtained by an initial estimate of ϑ (using the W matrix that yields the conventional IV estimator). The optimal weighting matrix can then be built, and the GMM estimates are obtained by minimizing the criterion function. Hansen (1982) suggests a second approach: the parameters and the weighting matrix can be estimated recursively until (i)

˜ (ϑ

(i−1)

˜ −ϑ

) is smaller than a convergence criterion. This iterative GMM estimate has the

same asymptotic distribution as the two-stage estimate.

15

The estimates of α in table 1 and 2 are indeed very sensitive to the initial weighting matrix chosen in the first stage. Table 3 reports the estimates using the orthogonality condition (21) and the iterative GMM procedure, which has the advantage of being independent with (1)

respect to WT . On the other hand, as Florens, Jondeau and Le Bihan (2001) show, it may have worse small sample properties. Iterative estimates confirm the two-stage estimates when instrument set A is used. The α estimate is significant, and we cannot reject the null hypothesis that α = 1. When instrument set B is used, the point estimate increases to 5.282 from the value 1.915 obtained with the two-stage GMM. But the estimate is now highly significant. The null of α = 1 can be rejected at the 10% confidence level, but not at the 5% level. Note also that using the instrument set B the estimate for ω is now much more variable between the restricted (ω = 0.38) and the unrestricted model (ω = 0.719). In summary, the empirical evidence is supportive of a direct interest rate effect on inflation, consistent with the presence of a cost channel through which marginal cost depends on both real wages relative to marginal productivity and the nominal rate of interest. Given this evidence, we proceed in the following section to explore the policy implications of the cost channel.

4

Optimal Monetary Policy

In this section, we explore the policy implications of our model. First, we show that the presence of the fiscal shock γˆ t implies a wedge between the output gap Yˆt − Yˆtf and the appropriate gap variable in the central bank’s loss function. This conclusion is independent of the presence of a cost channel. Second, we derive optimal policies and show that the cost channel leads to policy trade-offs between output and inflation even in the absence of the ad hoc cost shock that is typically added to the new Keynesian model to generate such trade-offs. Third, we provide simulation results under optimal policy in a calibrated version of the model.

4.1

Policy objectives

Following Woodford (2000), we obtain our policy objective function by taking a second-order approximation to the utility of the representative agent. Our derivation differs from Woodford’s 16

due to the presence of the cost channel, the taste shock in the utility function, and the demand shock arising from government expenditures. The appendix shows that the present discounted value of the utility of the representative household can be approximated by ∞ X t=0

¯ −Ω β Ut ≈ U t

where Lt =

π 2t

∞ X

β t Lt ,

t=0

³ ´2 ∗ ˆ ˆ + λ Y t − Zt − z ,

(23)

(1 + η) Aˆt + ˆξ t + (1 − σ) γˆ t Zˆt = , σ+η

(24)

and z ∗ is the gap between the flexible-price steady-state equilibrium output and the efficient output level. The parameter λ in (23) is given by ·

(1 − ω)(1 − ωβ) λ= ω

¸µ

σ+η θ

¶

.

According to (23), the appropriate welfare gap measure in the policy maker’s loss function, Yˆt − Zˆt − z ∗ , differs from the gap between output and the flexible price equilibrium output

level, Yˆt − Yˆtf . The difference between these two gaps can be seen by writing the welfare gap as ´ µ ³ f ∗ ˆ ˆ ˆ ˆ Yt − Zt − z = Yt − Yt −

1 σ+η

¶³ ´ ˆ ft + γˆ t − z ∗ . R

(25)

The last term on the right, z ∗ , is the gap between steady-state output and the efficient output ¯ − 1)/[¯ ¯ + η)]. It depends on the presence of level. The appendix shows that z ∗ = (¯ γ ΦR γ ΦR(σ

monopolistic competition via the markup Φ−1 , the fiscal tax γ¯ , and the monetary distortion

¯ > 1). Because our main focus is on generated by a non-zero average nominal rate of interest (R stabilization policies, we will follow the literature in assuming that fiscal subsidies ensure these two efficiency distortion are eliminated so that z ∗ = 0. With z ∗ = 0, the welfare gap consists of two terms. The first term on the right, Yˆt − Yˆtf , is the output gap expression that arises in the standard new Keynesian model. As Woodford (2000) has shown, welfare loss is associated with output fluctuations around the flexible-price equilibrium level of output. Marginal cost

17

is proportional to this same output gap, so Yˆt − Yˆtf also appears in the basic new Keynesian inflation equation. Hence, a policy designed to keep output equal to the flexible-price output level succeeds in stabilizing inflation and maximizing the welfare of the representative agent. As the second term in (25) suggestions, in the presence of a cost channel and government spending shocks, this result no longer holds. Because γˆ t acts like a tax on households and reduces welfare, even if prices are flexible or the central bank keeps Yˆt = Yˆtf , it may be optimal to offset fluctuations in γˆ t be allowing Yˆt − Yˆtf to fluctuate, even though this also leads to inflation fluctuations. Finally, fluctuations in the nominal interest rate directly affect household utility via the cost channel. Comparing Zˆt in (24) and Yˆtf in (12), we can define the output level which would obtain ˆ tf = 0 for all t, as4 in the flexible-price equilibrium, conditional on the policy rule R (1 + η)Aˆt − σˆ γ t + ˆξ t Yˆt∗ = . σ+η Then, the welfare gap (25) becomes (with z ∗ = 0) Yˆt − Zˆt = Yˆt − Yˆt∗ −

µ

1 σ+η

¶

γˆ t .

This also means that we can re-express real marginal cost as ³

´ ∗ ˆ ˆ ˆt. ϕ ˆ t = (σ + η) Yt − Yt + R

(26)

If we define xt ≡ Yˆt − Yˆt∗ as the output gap—the gap between output and the flexible-price output under a constant nominal interest rate—the policy problem can be written as ∞

1 X t max − Et β 2 t=0 subject to

(

· µ 2 π t + λ xt −

1 σ+η

¶

γˆ t

µ ¶ 1 ˆ t − Et π t+1 ) + gt xt = Et xt+1 − (R σ

¸2 )

(27)

(28)

ˆ t = 0 corresponds to an interest rate peg in the flexible-price equilibrium, not a zero nominal Note that R interest rate. 4

18

and ˆt, π t = βEt π t+1 + κ(σ + η)xt + κR where gt ≡

µ

1+η σ+η

¶·

(Et Aˆt+1 − Aˆt ) −

³η´ σ

(Et ˆξ t+1 − ˆξ t ) +

(29) ¸

η (Et γˆ t+1 − γˆ t ) 1+η

(30)

is a composite, expgenous disturbance term that depends on productivity, taste, and fiscal shocks. Before discussing the details of optimal policy, it is useful to highlight the way the policy problem differs from that in the standard new Keynesian model. First, the objective function is written in terms of the inflation rate, output gap and the disturbance term γ t ; second, the cost channel implies movements in the nominal interest rate affect the inflation equation in a manner similar to an ad hoc cost-push shock. In a standard new Keynesian model without a cost channel, setting ϕ ˆ t = 0 is the optimal policy. This policy ensures output is equal to its flexible-price level and both inflation and the output gap are stabilized. When the cost channel ˆ t = 0 will not deliver the flexible price equilibrium. The is present, setting ϕ ˆ t = κ(σ + η)xt + κR policy that sets xt = 0 requires that the real interest rate move in response to realizations of gt ˆ t are not consistent with zero inflation since changes (see 28). But with xt = 0, movements of R ˆ t directly affect real marginal cost. Alternatively, this point can be made by noting that in R ˆ t = 0, or R ˆ t = −(σ + η)xt . a policy of maintaining zero inflation requires that κ(σ + η)xt + κR But using (28), such a policy implies xt = (σgt − Et xt+1 ) /η and so xt = 0 for all t if and only if gt ≡ 0 for all t. Thus, zero inflation and a zero output gap can be maintained only if there are no shocks to the expectational IS curve, which in this model is equivalent to assuming a constant real interest rate rˆtf in every period5 . Another way to put the argument is that if monetary policy is engaged in fixing the inefficiency arising from price stickiness, it cannot at the same time match the policy adopted in the flexible price equilibrium. 5

Note that we have assumed there are no cost shocks that directly enter the inflation adjustment equation. A trade-off between inflation and output gap stabilization arises in the standard model only in the presence of such cost shocks. However, such shocks are ad hoc in that the underlying theory does not lead to any cost shock appearing in the equation for inflation.

19

4.2

Optimal discretionary policy

To highlight the relevance of the cost channel, in the following we will assume that productivity shocks are the only disturbance. That is, we assume γ t ≡ ˆξ t ≡ 0 for all t.6 Note that in this case, the appropriate output gap in the welfare function is simply xt , the same gap variable that appears in the inflation adjustment equation. If it feasible, the optimal policy would set both xt and π t equal to zero. This corresponds to the standard implication for monetary policy in the face of real productivity shocks—allow output to fluctuate as in the flexible price equilibrium while maintaining zero inflation. Assume the productivity shock follows an AR(1) process given by Aˆt = ρa Aˆt−1 + at . ³ ´ 1+η (ρa − 1)Aˆt in (28). It will prove useful to introduce the Then, with γ t ≡ ˆξ t ≡ 0, gt = σ+η index variable δ defined as

  0 if there is no cost channel δ=  1 if there is a cost channel

and write the inflation adjustment equation as

ˆt, π t = βEt π t+1 + κ(σ + η)xt + δκR

(31)

ˆ t , and the implied paths for The problem of the central banker is to choose a path for R xt and π t , to maximize 1 Ut ≡ − Et 2

(

∞ X £ i=0

λx2t+i + π 2t+i

¤

)

subject to (28) and (31). Let χt and ψ t denote the Lagrangian multipliers associated with each of these constraints at time t. Under optimal discretion, the first order conditions for the central The assumption on taste shocks is without loss of generality since both ˆξ and Aˆ enter the policy problem through their impact on the composite disturbance gt . The fiscal shock enters via gt and through the welfare gap measure in the loss function. We consist the policy implications of γˆ below. 6

20

bank’s problem are: ∂Ut = −λxt − κ(σ + η)ψ t + χt = 0; ∂xt ∂Ut = −π t + ψ t = 0; ∂π t µ ¶ ∂Ut 1 = −δκψ t + χt = 0. ˆt σ ∂R The only first order condition that changes relative to the standard sticky-price model of CGG is the last one, implying that in our model χt 6= 0. It is the result that χt = 0 in the standard model (δ = 0) that allows the IS curve to be ignored and optimal policy derived as if the central bank controlled xt directly. When the cost channel is present, the IS curve matters. Eliminating ψ t and χt from the first order conditions, xt = −

½

κ [σ(1 − δ) + η] λ

¾

πt.

(32)

In the absence of the cost channel, xt = − [κ(σ + η)/λ] π t ; in the presence of the cost channel, xt = −κη/λ. Since κη/λ < κ [σ + η] /λ, optimal policy will be less aggressive in trading off output gap movements for inflation stability. Intuitively, stabilizing inflation has become more ˆ t is increased, xt decreases, and this serves to reduce inflation, but costly; for example, as R the direct inflation effect of the rise in the nominal interest rate partly offsets the deflationary impact of tighter monetary policy. The equilibrium behavior of inflation is found by substituting equation (32) into the constraints (28) and (31, and solving the resulting two-equation system. This yields π t = µEt π t+1 + νgt ,

(33)

where µ=

λ (β + δκ) − δκ2 σ [σ(1 − δ) + η] δκλσ and ν = . 2 λ + κ2 [σ(1 − δ) + η] λ + κ2 [σ(1 − δ) + η]2

If |µ| ∈ [0, 1), the inflation equation has a stationary solution; this occurs only for values of

21

ˆ where7 λ 1. In the case of the cost channel (δ = 1), this critical value of λ ˆ can be If λ > λ, written as κ2 (σ + η)η/ (β + κ − 1).8 In the absence of a cost channel (δ = 0), the disturbance gt has no effect on inflation; v = 0; optimal policy keeps inflation and the output gap equal to zero and actual output fluctuates in line with the impact of productivity on the flexible-price output level. When the cost channel is present, however, v > 0 in (33) and it is optimal to let inflation fall in response to a productivity shock. (Recall that for 0 < ρa < 1, a positive realization of Aˆt implies a negative realization of gt .) In turn, from (32), the output gap rises. Thus, actual output expands more than the flexible-price equilibrium output level in response to a positive productivity shock. The responses of output and inflation under the optimal discretionary policy in the face of a positive productivity shock appear similar to the experience of the U.S. in the 1990s—in the face of a positive productivity shock, output expanded above most estimates of trend growth, while inflation declined. The impact of gt on inflation is increasing in λ; if the central bank places no weight on output gap stabilization (λ = 0), then it adjusts the nominal interest rate to offset the impact of output movements on inflation, keeping inflation equal to zero. When output gap stabilization is also desirable, the central bank must accept some fluctuation in both π and x in the face of g disturbances. The definition of the disturbance term gt given in equation (30) shows that we would reach the same conclusions if fluctuations in gt were due to the taste shock ˆξ t . The situation with respect to government spending shocks γ t is somewhat different, however. That is because γˆ t affects output and inflation through the composite disturbance gt , as do Aˆt and ζˆt , but γˆ t shocks also alter the definition of the gap variable in the loss function (see 27). In the presence of fiscal shocks, the central bank’s policy problem under discretion is to 7 This assumes β + κ > 1 as would be true for a wide range of parametrizations. For δ = 0, the sign of the ˆ since β < 1. inequality condition changes to λ > λ 8 ˆ suppose σ = η = 1, β = 0.99, and ω = 0.5. Then λ ˆ = 1.03; if To give some sense of the magnitude of λ, ˆ ω = 0.67, λ falls to 0.35. In McCallum and Nelson (2000), Jensen (2002), and Walsh (2002), a value of λ = 0.25 ˆ Using the theoretically consistent value of λ obtained in is used, so this value falls below the critical level of λ. ˆ the appendix, λ = 0.09 when ω = 0.5 and 0.03 when ω = 0.67, well below the critical value λ.

22

minimize 1 2

(

·

π 2t + λ xt −

µ

1 σ+η

¶

γˆ t

¸2 )

subject to (28) and (29). The first order conditions imply µ

λ πt = − κ [σ(1 − δ) + η]

¶·

xt −

µ

1 σ+η

¶

¸

γˆ t .

(34)

In the presence of a cost channel (δ = 1) this becomes µ

λ πt = − κη

¶· µ xt −

1 σ+η

¶

¸

γˆ t .

This must then be jointly solved with (28) and (29) to obtain the equilibrium expressions for inflation and the output gap. Ccomparing (34) to (32), shows that the presence of a fiscal shock serves to shift the combination of the output gap and inflation consistent with the central bank’s first order condition. That is, if γˆt = 0, the central bank ensures that κηπ t + λxt = 0. When, γˆ t 6= 0, the central bank ensures that κηπ t + λxt = [λ/(σ + η)] γˆ t ; for a given rate of inflation, the central bank accepts a higher output gap in the face of a positive fiscal shock. Recalling the definition of γˆt , a positive realization corresponds to a reduction in the share of output going to government purchases. We postpone further discussion of the optimal policy under discretion until numerical solutions are obtained in section 5.

4.3

Optimal commitment policies

As emphasizes by Woodford (2000), optimal commitment policies in forward-looking models involve inertial behavior. However, we initially consider the case of optimal commitment to a non-inertial policy that makes inflation and the output gap functions of the current shock gt . We then turn to the fully optimal commitment policy in the following subsection.

23

4.3.1

Optimal commitment to a non-inertial policy

Assume the policy rule is function of the shock gt and results in the output gap being given by xt = φgt . Ignoring the composite nature of gt , assume that it follows an AR(1) process with coefficient ρ. If the central banker can commit to the policy defined by φ (where φ is chosen to maximize the welfare function, and is thus the optimal policy within the family of non-inertial ˆ t from policy functions we are looking at), then Et xt+1 = ρxt , and this result, together with R the IS equation, can be used to write the inflation equation as

π t = (β + δκ)Et π t+1 + [κ (η + σ) − δκσ(1 − ρ)]xt + δκσgt . Iterating forward: π t = [κ (η + σ) − δκσ(1 − ρ)][1 + (δκ + β)ρ + (δκ + β)2 ρ2 ...]xt +δκσ[1 + (δκ + β)ρ + (δκ + β)2 ρ2 ...]gt If ρ < 1/(β + δκ), a stationary solution exists and is given by ·

· ¸ ¸ κ (η + σ) − δκσ(1 − ρ) δκσ δκσ πt = xt + gt = Bxt + gt . 1 − ρ(δκ + β) 1 − ρ(δκ + β) 1 − ρ(δκ + β)

(35)

To understand the gain from commitment in the presence of the cost channel, it is useful to use the IS relationship to eliminate the nominal interest rate from the inflation adjustment equation. Doing so reveals that inflation and the output gap must satisfy π t = (β + δκ) Et π t+1 + κ [σ(1 − δ) + η] xt + δσκEt xt+1 + δκσgt

(36)

for all t. Note that if δ = 0, this is simply the standard new Keynesian inflation adjustment equation. However, when δ = 1 and the cost channel is present, current inflation depends on the expected future output gap as well as expected future inflation. Inflation also depends on the disturbance in the IS curve gt . A rise in the expected future output gap, given the current gap and expected future inflation, implies a rise in the nominal interest rate. This interest rate increase raises marginal costs and so has a positive impact on current inflation. 24

πt = (β + δκ)Et π t+1 + κ[σ(1 − δ) + η]xt + δκσ(Et xt+1 + gt ).

(37)

For δ = 1 the coefficient on xt in (36) is smaller than in the standard infation adjustment equation (κη versus κ(σ + η)). A negative output gap has a smaller effect in reducing inflation because it affects π t via two separate channels: through the standard (negative) real marginal cost channel and through the (positive) impact of xt on Rt and therefore on π t via the cost channel. Ceteris paribus, the cost channel reduces the deflationary impact of a contractionary policy. The effect of the expected future output gap on current inflation increases the gains from commitment. Compare now the coefficient on xt in equations (35) and (37). Under commitment, a 1% fall in the output gap generates a larger reduction in π t than under discretion. For δ = 1, the gain from commitment is larger than in the standard new Keynesian model. The intuition for this result is as follows. Commitment alters the xt coefficient in two ways. First, as the central bank responds to a positive shock gt , the optimal policy would generate a negative output gap. Under commitment the private sector expects the contractionary policy to be enforced also in the following periods when gt is serially correlated (ρ 6= 0). Expactions of a negative Et xt+1 directly serves to reduce current inflation (see equation 36). in the standard model, Et xt+1 affects current inflation only through its impact on Et πt+1 . This direct channel when δ = 1 means that a commitment policy that can influence both Et π t+1 and Et xt+1 will have a larger impact on current inflation than would be the case in the standard model with δ = 0. Thus, the gains to a commitment policy are increased when the cost channel is present. Et xt+1 has a direct impact on π t because, certaris paribus, a lower expected future output gap reduces the nominal interest rate associated with any given current gap. A lower nominal interest rate directly reduces current inflation through the cost channel. Under the optimal non-inertia commitment policy, central banker chooses a value for φ, ˆ t , to maximize the welfare measure Ut subject to the and the implied paths for xt , π t and R constraints (35) and xt = φgt . Let the c (d) subscript indicate the solution under commitment

25

(discretion). The first order conditions for the optimal non-inertialcommitment policy imply xct

1 =− λ

½

κ[σ(1 − δ(1 − ρ)) + η] 1 − ρ(δκ + β)

¾

π ct .

(38)

As in the standard new Keynesian model, a central banker that can commit takes advantage of the improved inflation-output gap trade-off and is more aggressive against inflation than a central banker who implements policy under discretion. For δ = 1 a given positive shock gt will cause a smaller movement in inflation under the commitment policy than under the discretionary policy: λκσ [1 − ρ(κ + β)]λκσ gt = π ct < π dt = gt 2 2 λ[1 − ρ(κ + β)] + κ[η + σρ] λ[1 − ρ(κ + β)] + κη[κη + κσρ] and a larger movement in the output gap: [1 − ρ(κ + β)]λκσ λκσ κ(σρ + η) κη gt gt = |xct | > |xdt | = 2 2 λ[1 − ρ(κ + β)] λ[1 − ρ(κ + β)] + [κη + κσρ] λ λ[1 − ρ(κ + β)] + κη[κη + κσρ] Condition (38) also implies that for δ = 1 a Rogoff (1985) conservative central banker optimizing under discretion a welfare function with preference weight given by: λC =

η[1 − ρ(κ + β)] λ 0. At time ˆ t such that t, the central bank sets π t , xt , and R πt + ψt = 0 · µ λ xt −

(39)

¶ ¸ 1 γˆ t + χt − κ(σ + η)ψ t = 0 σ+η µ ¶ 1 χt − δκψ t = 0, σ

while for t + i > t, ¡ ¢ π t+i + ψ t+i − ψ t+i−1 − χt+i−1

· µ λ xt+i −

1 σ+η

¶

¸

µ ¶ 1 =0 σ

(40) (41)

(42)

γˆ t+i + χt+i − β −1 χt+i−1 − κ(σ + η)ψ t+i = 0

(43)

µ ¶ 1 χt+i − δκψ t+i = 0. σ

(44)

If δ = 0 (no cost channel), χt+i = 0 for all i and these first order conditions reduce to the case considered by Woodford (2000), Clarida, Gal´ı, and Gertler (1999), or McCallum and Nelson (2000). When δ = 1, the nature of the time inconsistency inherent in this problem shows up in the comparisons of (39) to (42) and (40) to (43). These first order conditions for time t can be rewritten as

µ

λ πt = − κ [σ(1 − δ) + η]

¶·

xt −

µ

1 σ+η

¶

γˆ t

¸

and for t + i, i > 0, as π t+i = −ψ t+i + (1 + κδ) ψ t+i−1 27

(45)

·

λ xt+i −

µ

1 σ+η

¶

¸

γˆ t+i = β −1 σκδψ t+i−1 + κ [σ(1 − δ) + η] ψ t+i

(46)

Svensson and Woodford (1999) have described a policy that implements equations (45) and (46) for all t as the timeless perspective precommitment policy. Rather than attempt to obtain analytic results, we turn in the next section to numerical methods to compare optimal policies under discretion and commitment.

5

Simulations

Optimal discretionary and fully optimal commitment policies are found by calibrating the model and numerically solving the model using the approach described by S¨oderlind (1999) and Jensen (2001). The basic parameter values we use are shown in Table 4. The values for σ and η are fairly standard. The discount factor, β, is set equal to 0.99, appropriate for interpreting the time interval as one quarter. The value of ω is consistent with the empirical findings of Gal´ı and Gertler (1999) and those reported in section 3. A value of 11 for θ implies a steady-state markup of 1.1. For the impulse responses, we report the impact of 1-unit innovations to the various shocks and we allow the shocks to be highly serially correlated, ρa = ρξ = ργ = 0.9.9 Optimal policy also depends on the value of λ. Given the parameter values in table 4, the underlying theory implies λ = (1 − βω)(1 − ω)(σ + η)/(ωθ) = 0.0195. Following McCallum and Nelson (2000), Jensen (2001), and Walsh (2002), we also report results for λ = 0.25. Table 4: Baseline parameter values σ

η

β

ω

1.5

1

0.99 0.75

θ

ρa

ρξ

11

0.9 0.9

ργ 0.9

Figure 1 shows the response of the output gap and inflation to a 1-unit, positive productivity shock Aˆt under the fully optimal commitment policy when λ = 0.0195. The output gap 9

The autocorrelation coefficients play a different role from the one outlined in Clarida, Gali and Gertler ξ t ), (Et γˆ t+1 − γˆt ). Larger ξ t+1 − ˆ (1999). The IS curve shock gt is a function of the difference (Et Aˆt+1 − Aˆt ), (Et ˆ values of ρ map into a smaller shock gt .

28

    1 W V R W V  I C R  Z      

+P HNC V KQ P  π 

     

















Figure 1: Response to a productivity shock: optimal commitment, λ = 0.0195. rises, implying output increases more than flexible-price output does, and inflation falls. For the small weight on output gap stabilization implied by the model, inflation responds very little to the shock. In the absence of a cost channel, however, neither the output gap nor inflation would be affected by a productivity shock under either the optimal commitment policy or the optimal discretionary policy. Instead, actual output would simply track the behavior of the flexible price equilibrium output level in reaction to the productivity shock. Maintaining a zero output gap requires, however, a fall in the flexible-price real interest rate (since Yˆtf increases f more than Et Yˆt+1 ). With the cost channel, the accompanying fall in the nominal interest rate

reduces inflation. To limit this fluctuation in inflation, the optimal policy lets output rise above the flexible-price equilibrium level.

Figure 2 illustrates the response of inflation and the output gap to a productivity shock when λ takes on the much larger value of 0.25. Now, inflation falls more, and the output gap rises less. The nominal interest rate (not shown) falls when the positive productivity shock occurs. It then returns gradually to its steady-state value. Both figure 1 and figure 2 show the standard result that the optimal commitment policy results in a stationary price level; the fall

29

   1 W V R W V  I C R  Z 

  +P HNC V KQ P  π 

                   

















Figure 2: Response to a productivity shock: optimal commitent, λ = 0.25. in inflation is followed by a period of positive inflation.10

When policy is conducted under discretion, the central bank faces a worse policy trade-off because it cannot manipulate future expectations. In the standard new Keynesian model, this means that the central bank cannot act to stabilize current inflation by committing to a path for future inflation. This aspect operates in the model with a cost channel, but another factor is also at work. Because the central bank operating under discretion cannot raise expected future inflation as it would like to under the commitment policy, it must move the nominal interest rate more (in the case of a positive productivity shock, lowering it) to produce a given change in the real interest rate. This reduction in the nominal rate leads to a larger decline in inflation under discretion. Figure 3 and 4 show the response of the output gap and inflation to a positive productivity shock Aˆt under the optimal discretionary policy for λ = 0.0195 and λ = 0.25. The output gap rises while inflation falls, but both responses are much larger than occur under the commitment policy. In addition, the dynamic patterns of the responses are quite different than those under commitment. The gap and inflation decay monotonically back 10

Vestin (2000) has shown that price-level targeting under discretion can, in some circumstances, replicate the optimal commitment policy by ensure the price level is stationary.

30







1 W V R W V  I C R  Z 







+P HNC V KQ P  π 

   

   

















Figure 3: Response to productivity shock: optimal discretion, λ = 0.0195. to zero. Contrary to the standard New Keynesian model, falling prices can be a consequence of the optimal discretionary policy-maker reacting to a temporary increase in productivity.

Figure 5 compares the behavior of the nominal interest rate in response to a positive productivity shock under commitment and discretion. (The case λ = 0.25 is shown.) Because the discretionary central bank is unable to induce a rise in expected future inflation, it must lower the nominal rate much more than would occur under a commitment policy.

¿From equations (28) and (29) it is apparent that the behavior of the output gap and inflation in the face of a taste shock ξ will be proportional (with opposite sign) to the responses to a productivity shock. Both enter (28) and (29) only through the composite disturbance gt . Both affect welfare only through inflation and the output gap. Thus, Figures 1 - 4 can also be interpreted as reflecting the effects of a negative taste shock. Government fiscal shocks γ t pose policy trade-offs even in the absence of the cost channel since they create a wedge between the output gap and the “welfare gap” variable appearing in the loss function. The responses of the output gap and inflation to a positive γ t shock under the 31





1 W V R W V  I C R  Z 



   

   

   

+P HNC V KQ P  π 

   

  

   

















Figure 4: Response to a productivity shock: optimal discretion, λ = 0.25.



% Q O O KV O G P V 

   

& KU E TG V KQ P 

  

   

  

   

















Figure 5: Nominal interest rate behavior after a positive productivity shock

32

  1 W V R W V  I C R  Z  +P HNC V KQ P  π 

  0 Q O KP C N TC V G  4 

     

9 G NHC TG  I C R  ;   <    

' HHKE KG P V  Q W V R W V  ; 

        

















Figure 6: Response to fiscal shock: optimal commitment, λ = 0.0195. optimal commitment policy when λ = 0.0195 are shown in Figure 6. Also shown in the figure are the welfare gap measure and the efficient level of output. Figure 7 shows the same variables when λ = 0.25. The positive innovation to γ t causes households to reduce labor supply, and the efficient level of output falls. The output gap rises, but it rises less than Yˆ ∗ falls, implying that actual output declines. Inflation increases, while the welfare gap falls. The responses are qualitatively similar for the different values of the weight placed on the welfare gap measure in the objective function (λ).11

The effects of a fiscal shock (ˆ γ t ) under discretion are illustrated in Figures 8 and 9 corresponding to the two alternative values of λ. The basic responses are similar under both discretion and commitment. Under discretion, the output gap responds less (implying output falls more since Yˆ ∗ is independent of the policy regime). The major difference, however, occurs 11

It is noteworthy that the optimal response to fiscal shocks generates a negative correlation between inflation and deviations of output from its trend. Thus, optimal monetary policy could account for the negative coefficient obtained in New Keynesian Phillips curve estimates where the theoretical output gap is proxied by standard statistical measures of deviations of output from potential output. The standard model can explain such finding only by assuming the existence of cost shocks, or the adoption of sub-optimal policies.

33

  1 W V R W V  I C R  Z 



+P HNC V KQ P  π 

  0 Q O KP C N TC V G  4 

     

9 G NHC TG  I C R  ;   < 

  

' HHKE KG P V  Q W V R W V  ; 

        

















Figure 7: Response to a fiscal shock: optimal commitment, λ = 0.25.

 +P HNC V KQ P  π 





0 Q O KP C N TC V G  4 

1 W V R W V  I C R  Z 

   9 G NHC TG  I C R  ;   < 

  

  

' HHKE KG P V  Q W V R W V  ; 

  

  

  

















Figure 8: Response to a fiscal shock: optimal discretion, λ = 0.0195

34

  +P HNC V KQ P  π    0 Q O KP C N TC V G  4    1 W V R W V  I C R  Z 

    

9 G NHC TG  I C R  ;   < 

     

' HHKE KG P V  Q W V R W V  ;  















Figure 9: Response to a fiscal shock: optimal discretion, λ = 0.25 in the behavior of inflation, which is much more sensitive to the γˆ shock under discretion than under commitment. Figure 9 shows that on impact a 1% fiscal shock causes a jump of inflation of about 1.3% - over ten times larger than the increase in inflation following a productivity (or taste) shock of the same magnitude (see Figure 4). Thus the optimal policy under discretion can generate a prolonged rise in inflation following a persistent decline in government demand. The introduction of the γ t term in the objective function has implications also for the instrument ½ h ³ ´ i2 ¾ P∞ t 1 2 behavior. Since the policy-maker minimizes Ut = ˆt , for t=0 β Et π t + λ xt − σ+η γ γˆ t > 0 the optimal policy calls for a positive output gap xt . Given the efficient level of output

drops on impact and then increases back to the steady state, a positive and decreasing output gap implies a drop in the real interest rate. Under commitment, this is achieved by a rise in ˆ t . Under discretion, expected future expected inflation and a fall in the nominal interest rate R inflation rises significantly, and it is optimal to increase the nominal interest rate. The preceding discussion has focused on the effects of the shocks that are in some sense internal to the model. The productivity shock originated in the specification of the production function, the taste shock in the utility function, and the fiscal shock in the resource constraint. In the standard new Keynesian model, an ad hoc cost shock is added to the inflation adjustment

35

curve in order to generate an interesting policy problem. If such a cost shock is added to equation (31) and the responses under optimal commitment and discretion are derived, the general findings are similar to those highlighted by Woodford (2000).12 Under discretion, there is no inertia—the output gap and inflation return to baseline in the period immediately after a serially uncorrelated cost shock occurs. Commitment introduces inertia — the output gap and inflation take over ten periods to return to baseline. (These responses are not shown.)

6

Conclusions

In the new Keynesian model that has become a standard framework for investigating monetary policy issues, monetary policy operates on aggregate spending through an interest rate channel. For many purposes, the exact nature of the monetary policy transmission mechanism is unimportant—the critical factors for policy are the objective function of the central bank and the inflation-adjustment mechanism. The details of the channels through which interest rate changes affect spending are only relevant for determining the actual nominal interest rate behavior that is required to achieve the desired time paths of inflation and the output gap. In this paper, we have investigated the implications for optimal policy when monetary policy also affects the economy through a cost channel. If nominal interest rate movements directly affect real marginal cost, as the empirical evidence of Barth and Ramey (2001) and the evidence we reported in section 3 suggest, then monetary policy directly affects the inflation-adjustment equation. Previous new Keynesian models introduced an exogenous cost-push shock to generate a meaningful policy problem. Whether an exogenous shock or the cost channel generate the inflation-output gap stabilization trade-off has important consequences for the policy problem faced by the central bank. We derived the appropriate welfare-based loss function for the cost channel economy. The flexible-price level of output is not independent of monetary policy as in the standard model—therefore a reference ‘potential output’ for the economy is not uniquely defined. But since welfare can be expressed as a function of the distance between output and the level of output conditional on a constant interest rate monetary policy, the policy-maker loss can still 12

Impulse responses to cost shocks are shown in Jensen (2002).

36

be written in terms of inflation and a well-defined output gap. Interest rate changes necessary to stabilize the output gap lead to inflation fluctuations when a cost channel is present. This means that the output gap and inflation will fluctuate in response to productivity and demand disturbances even when the central bank is setting policy optimally. A positive productivity shock is shown to lead to a fall in inflation and a rise in the output gap under either the optimal commitment policy or the optimal discretionary policy. Thus, a period of above average productivity should also be associate with a rise in output above the flexible-price level (a rise in the output gap) and a decline in inflation. Inflationary episodes will arise in response to positive taste and fiscal shocks, or adverse productivity shocks, even if the central bank follows the optimal policy. If the cost channel is present, optimal monetary policy calls for gradually stabilizing the inflation rate around its steady state following any of the shocks considered—not only after an exogenous cost-push shock as in the standard framework. Finally, we also showed that an optimal policy, either under commitment or discretion, does not stabilize the output gap and inflation in the face of aggregate fiscal shocks. This result holds regardless of the presence of a cost channel for monetary policy. In earlier analyses, an ad hoc demand shock was often added to the expectational IS curve, and optimal policy would always move the interest rate to ensure these shocks were not allowed to affect the output gap. Because they did not affect the output gap, they also did not affect inflation. When these demand shocks arise from changes in fiscal policy that alter the share of output available for consumption, stabilizing their impact on the output gap is not an optimal policy. Instead, a positive fiscal shock reduces the flexible-price level of output. Under an optimal monetary policy, the output gap and inflation both rise, but the rise in the gap is less than the fall in the flexible-price level of output, so actual output declines.

37

Appendix In this appendix, the approximation to the welfare of the representative household is derived. In doing so, we follow Woodford (1999). Our model differs from his in three ways. First, we assume the utility of the representative household depends on consumption and leisure, while Woodford assumes it depends on consumption and output, where the role of output is to capture the disutility of work. This change does not affect the results. The second, more substantive change, is that we allow explicitly for stochastic variation in the share of output going to the government. Finally, the steady-state values in our model are not independent of monetary policy (as they are in Woodford) because the level of the nominal interest rate in the steady-state affects equilibrium output and employment. We make use of the following notation: Notation ¯ X

Steady-state value

Xt∗

Efficient level

Xtf

Flex-price equilibrium level

˜t X

¯ Xt − X

¯ log Xt − log X

ˆt X Given this notation, Xt ¯ ≈ 1 + log X

µ

Xt ¯ X

¶

˜t = X ¯ Because one can always write X

· µ ¶¸2 1 Xt ˆ 2. ˆt + 1 X log ¯ + =1+X 2 2 t X

¡ Xt ¯ X

³ ´ ¢ ˜t ≈ X ¯ X ˆt + 1 X ˆ t2 . − 1 , it follows that X 2

Utility is assumed to be separable in consumption and leisure. We begin by approximating the utility of consumption. The second order Taylor expansion for U(Ct , ξ t ) is

¯ 1) + Uc (C, ¯ 1)C˜t + 1 Ucc (C, ¯ 1)C˜ 2 + U ξ (C, ¯ 1)˜ξ t + 1 Uξ,ξ ˜ξ 2 + Uc,ξ ˜ξ t C˜t . U (Ct , ξ t ) ≈ U (C, t t 2 2 The relationship between consumption and output is given by equation (6). This implies that · ³ ´2 ¸ C˜t ≈ γ¯ Y¯ γˆ t + Yˆt + 12 γˆ t + Yˆt . Given the utility function specification (1), the utility of 38

consumption becomes ´2 ¸ ³ 1 ¯ 1) + Uc (C, ¯ 1)¯ U(Ct , ξ t ) ≈ U (C, γ Y¯ γˆ t + Yˆt + γˆ + Yˆt 2 t · ´2 ¸2 1 1³ ¯ ¯ ˆ ˆ γ Y γˆ t + Yt + − σUc (C, 1)¯ γˆ + Yt 2 2 t · ³ ´2 ¸ 2 1 1 ¯ 1)ˆξ t + Uξ,ξ ˆξ + Uc (C)¯ ¯ γ Y¯ ˆξ t γˆ t + Yˆt + +U ξ (C, γˆ + Yˆt . t 2 2 t ·

Ignoring terms of order X i for i ≥ 2, · ³ ´ 1 ³ ´2 ¸ ˆ ¯ ¯ ¯ ˆ ˆ U (Ct , ξ t ) ≈ U (C, 1) + Uc (C, 1)¯ γ Y (1 + ξ t ) γˆ t + Yt + (1 − σ) γˆ t + Yt 2 ¯ 1)ˆξ t + 1 Uξ,ξ ˆξ 2 . +Uξ (C, t 2

(47)

The next step is to obtain an approximation for the disutility of work. The second order Taylor expansion for V (Nt ) is ¯ ) + VN (N) ¯ N ˜t + 1 VNN (N) ¯ N ˜2 V (Nt ) ≈ V (N t 2 where aggregate employment is ˜t = N

Z

1

n ˜ t (i)di.

0

For employment at firm i, · ¸ 1 2 ¯ n ˆ t (i) + n ˆ t (i) . n ˜ t (i) ≈ n 2 Each firm has a production technology given by yt (i) = At nt (i). Hence, n ˆ t (i) = yˆt (i) − Aˆt . We can then write

39

(48)

¸ 1 2 n ˆ t (i) + n ≈ n ˜ t (i)di = n ¯ ˆ t (i) di 2 0 0 ·Z 1 Z ´2 ¸ 1 1³ ˆ ˆ = y¯ yˆt (i) − At di . yˆt (i)di − At + 2 0 0 Z

˜t N

Z

1

1

·

Substituting this into (48), and ignoring terms of order X 2 and higher powers, ·Z

1

1 ¯ ) + VN (N)¯ ¯ y V (Nt ) ≈ V (N yˆt (i)di − Aˆt + 2 0 ·Z 1 ¸2 1 ¯ )¯ + VNN (N y2 yˆt (i)di − Aˆt . 2 0

Z

1 0

³

yˆt (i) − Aˆt

´2

¸ di (49)

Given the demand functions facing each individual firm, the aggregate output variable Yt is defined as Yt =

Z

1

yt (i)

θ−1 θ

di.

0

This implies

Hence,

·Z

Yˆt ≈

Z

yˆt (i)di

¸2

0

1

0

Note also that

Z

1

·

1 = Yˆt − 2

1 2

yˆt (i) di =

0

Therefore,

Aˆt

Z

1 0

·Z

µ

µ

1 0

¶ θ−1 vari yˆt (i). θ

θ−1 θ

1

yˆt (i)di

0

Z

In addition,

1 yˆt (i)di + 2

¶

¸2

¸2

vari yˆt (i)

≈ Yˆt2 .

+ vari yˆt (i).

yˆt (i)2 di ≈ Yˆt2 + vari yˆt (i).

1 yˆt (i)di ≈ Aˆt Yˆt − Aˆt 2

µ

40

θ−1 θ

¶

vari yˆt (i) ≈ Aˆt Yˆt .

Using these results, equation (49) becomes µ ¶ · ¸ θ − 1 1 ¯ + VN (N ¯ )¯ V (Nt ) ≈ V (N) y Yˆt − Aˆt − vari yˆt (i) 2 θ · ³ ¸ ´ 1 1 2 2 ¯ )¯ +VN (N y Yˆ + vari yˆt (i) − Aˆt Yˆt + Aˆt di 2 t 2 ³ ´2 1 ¯ y 2 Yˆt − Aˆt . + VNN (N)¯ 2

(50)

Combining terms, and using the utility function (1), µ ¶ · ³ ´2 ¸ 1 1 1 ¯ ¯ ˆ ˆ ˆ ˆ V (Nt ) ≈ V (N ) + VN (N)¯ y Yt − A t + vari yˆt (i) + (1 + η) Yt − At . 2 θ 2

(51)

Combining equations (47) and (51),

¯ 1) − V (N ¯) U (Ct , ξ t ) − V (Nt ) = U (C, · ³ ³ ´ 1 ´2 ¸ ¯ 1)¯ +Uc (C, γ Y¯ (1 + ˆξ t ) γˆ t + Yˆt + (1 − σ) γˆ t + Yˆt 2 2 1 ¯ 1)ˆξ t + Uξ,ξ ˆξ +U ξ (C, t 2 · µ ¶ ³ ´2 ¸ 1 1 1 ¯ ˆ ˆ ˆ ˆ −VN (N )¯ y Yt − At + (52) vari yˆt (i) + (1 + η) Yt − At 2 θ 2 Before simplifying this expression, note that the steady-state labor market equilibrium ¯ If we define Ω such that condition becomes V¯N /U¯c = w ¯ = 1/ΦR. 1−Ω≡

1 ¯, γ¯ ΦR

¯ y can be written as Uc (C)¯ ¯ γ Y¯ (1 − Ω). We will assume Ω is small so that terms such then VN (N)¯ ¡ ¢ ¯ −1 Yˆt2 = (1 − Ω) Yˆt2 become simply Yˆt2 .13 In this case, we can now write equation (52) as γ¯ ΦR

This is stronger than the corresponding assumption made by Woodford (1999). He assumes (Φ)−1 is small. The presence of γ¯ and a positive steady-state nominal interest rate increase the average distortions in the economy relative to the case in which the monopoly distortion is the only source of inefficiency. 13

41

as · ´ 1 ´2 ¸ ³ ³ ¯ 1) − V (N ¯ ) + Uc (C)¯ ¯ γ Y¯ (1 + ˆξ t ) γˆ t + Yˆt + (1 − σ) γˆ t + Yˆt U (Ct , ξ t ) − V (Nt ) ≈ U (C, 2 ¸ · ³ ´ 2 ¯ γ Y¯ (1 − Ω) Yˆt − Aˆt + 1 (1 + η) Yˆt − Aˆt −Uc (C)¯ 2 µ ¶ 1 ¯ γ Y¯ (1 − Ω) 1 vari yˆt (i) + U ξ (C, ¯ 1)ˆξ t + 1 Uξ,ξ ˆξ 2 . − Uc (C)¯ t 2 θ 2 Collecting terms, h i ˆ ¯ ¯ ¯ ¯ ˆ ˆ U (Ct , ξ t ) − V (Nt ) ≈ U (C, 1) − V (N) + Uc (C, 1)¯ γ Y ΩYt + ξ t Yt · ³ ´2 1 ³ ´2 ¸ 1 ˆ ¯ ¯ ˆ ˆ +Uc (C, 1)¯ (1 − σ) γˆ t + Yt − (1 + η) Yt − At γY 2 2 µ ¶ 1 ¯ 1)Y¯ 1 vari yˆt (i) − Uc (C, 2 θ i h ¯ 1)¯ ¯ 1)ˆξ t + 1 Uξ,ξ ˆξ 2t . +Uc (C, γ Y¯ (1 + ˆξ t )ˆ γ t + (1 − Ω)Aˆt + U ξ (C, 2 Define

"

(1 + η)Aˆt + ˆξ t + (1 − σ)ˆ γt Zˆt ≡ σ+η

#

and z∗ =

Ω . σ+η

Then the utility approximation can be written as ³ ´2 ¯ ) − 1 (σ + η) Uc (Y¯ )Y¯ Yˆt − Zˆt − z ∗ U (γ t Yt , ξ t ) − V (Nt ) ≈ U (Y¯ , 1) − V (N µ ¶ 2 1 1 vari yˆt (i) + t.i.s.p.. − Uc (Y¯ )Y¯ 2 θ where h i ¯ 1)¯ ¯ 1)ˆξ t t.i.s.p. = Uc (C, γ Y¯ (1 + ˆξ t )ˆ γ t + (1 − Ω)Aˆt + U ξ (C, 2 1 1 ¯ 1)Y¯ cZˆ 2 + Uξ,ξ ˆξ t − Uc (C, t 2 2

42

(53)

are terms independent of stabilization policy. Recalling that Yˆtf =

µ

1+η σ+η

¶

µ

1 σ+η

+

µ

Aˆt −

Zˆt can be written as Zˆt =

Yˆtf

¶³

1 σ+η

´ µ σ ¶ f ˆ ˆ Rt − ξ t − γˆ t , σ+η

¶³

´ f ˆ Rt + γˆ t .

With the assumed utility function, log yt (i) = log Yt − θ (log pt (i) − log Pt ) so vari log yt (i) = θ2 vari log pt (i) The price adjustment mechanism involves a randomly chosen fraction 1−ω of all firms optimally adjusting price each period. Define P¯t ≡ Et log pt (i) and ∆t ≡ vari log pt (i). Then Woodford (2000) shows that ∆t ≈ ω∆t−1 +

µ

ω 1−ω

¶

π 2t ,

where 1 − ω is the fraction of firms that reset their price each period. If ∆−1 is the initial degree of price dispersion, then ∞ X t=0

·

¸X ∞ ω β ∆t = β t π2t + t.i.p., (1 − ω)(1 − ωβ) t=0 t

where t.i.p. denotes terms independent of monetary policy. Combining this with (53), the present discounted value of the utility of the representative household can be approximated by ∞ X t=0

¯ −Ω β Ut ≈ U t

43

∞ X t=0

β t Lt

where

³ ´2 Lt = π 2t + λ Yˆt − Zˆt − z ∗ ,

· ¸ ω 1 ¯ Ω = Uc Y θ, 2 (1 − ω)(1 − ωβ) and

·

(1 − ω)(1 − ωβ) λ= ω

¸µ

σ+η θ

¶

=

κ (σ + η) . θ

The parameter κ is the coefficient on real marginal costs in the inflation adjustment equation.

References [1] Barth, M. J. III and V. A. Ramey, “The Cost Channel of Monetary Transmission,” NBER Macroeconomic Annual 2001, Cambridge, MA: MIT Press, 199-239. [2] Bernanke, B. and Gertler, M., “Agency Costs, Net Worth and Business Fluctuations,” American Economic Review, 79 (1), 1989, 14-31. [3] Clarida, R., J. Gal´ı, and M. Gertler, “The Science of Monetary Policy: A New Keynesian Perspective,” Journal of Economic Literature, 37 (4), Dec. 1999, 1661-1707. [4] Christiano, Lawrence J. and Eichenbaum, Martin, “Liquidity Effects and the Monetary Transmission Mechanism,” American Economic Review, 82 (2), 1992, 346-53. [5] Christiano, Lawrence J., Martin Eichenbaum, and Charles Evans, “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” NBER Working Paper No. 8403, July 2001. [6] Erceg, C. J., D. W. Henderson, and A. T. Levin, “Optimal Monetary Policy with Staggered Wage and Price Contracts,” Journal of Monetary Economics, 46 (2), Oct. 2000, 281-313. [7] Florens, C., Jondeau, E. and Le Bihan, H., “Assessing GMM Estimates of the Federal Reserve Reaction Function,” mimeo, Banque de France, March 2001.. [8] Gal´ı, J., “New Perspectives on Monetary Policy, Inflation, and the Business Cycle,” NBER Working Paper No. 8767, Feb. 2002. 44

[9] Gal´ı, Jordi and Mark Gertler, “Inflation Dynamics: A Structural Econometric Investigation,” Journal of Monetary Economics, 1999, 44, 195-222. [10] Gal´ı, Jordi, Mark Gertler, and J. David L´opez-Salido, “European Inflation Dynamics,” European Economic Review, 45, 2001, 1237-1270. [11] Hall, A., “Some Aspects of GMM Estimation”, in Rao, C. and Maddala, G., eds., Handbook of Statistics, Vol. 11, Econometrics, Amsterdam: North-Holland, 1993. [12] Hamilton, James, Time Series Analysis, Princeton, Princeton University Press, 1994. [13] Hansen, L., “Large Sample Properties of Generalized Method of Moments Estimator,” Econometrica, 50 (4), 1982, 1029-1054. [14] Jensen, Henrik, “Targeting Nominal Income Growth or Inflation?” Working Paper, University of Copenhagen, American Economic Review, 92(4), Sept. 2002, 928-956. [15] Kocherlakota, N., “On Tests of Representative Consumer Asset Pricing Models,” Journal of Monetary Economics, 26(2), 1990, 285-304. [16] Matyas, Laszlo, Generalized Method of Moments Estimation, Cambridge: Cambridge University Press, 1999. [17] McCallum, Bennett T., “Should Monetary Policy Respond Strongly to Output Gaps?” American Economic Review, 91 (2), May 2001, 258-262. [18] McCallum, Bennett T. and Edward Nelson, “An Optimizing IS-LM Specification for Monetary Policy and Business Cycle Analysis,” Journal of Money, Credit, and Banking, 31 (3), August 1999, 296-316. [19] McCallum, Bennett T. and Edward Nelson, “Timeless Perspective vs. Discretionary Monetary Policy in Forward-Looking Models,” NBER Working Papers No. 7915, Sept. 2000. [20] Rogoff, K., “The Optimal Commitment to an Intermediate Monetary Target,” Quarterly Journal of Economics, 100(4), Nov. 1985, 1169-1189.

45

[21] Sbordone, A. M., “Prices and Unit Labor Costs: A New Test of Price Stickiness,” Journal of Monetary Economics, 49(2), Mar. 2002, 265-292. [22] Svensson, L. E. O. and M. Woodford, “Implementing Optimal Policy Through InflationForecast Targeting,” 1999. [23] Svensson, L. E. O. and M. Woodford, “Indicator Variables for Optimal Policy,” Sept. 2000. [24] S¨oderlind, P., “Solution and Estimation of RE Macromodels with Optimal Policy,” European Economic Review, 43 (1999), 813-823. [25] Tauchen, G., “Statistical Properties of GMM Estimators of Structural Parameters Obtained from Financial Market Data,” Journal of Business and Economic Statistics, 4 (4), 1986, 397-425. [26] Vestin, David, “Price-level targeting versus inflation targeting in a forward-looking model,” IIES, Stockholm University, May 2000. [27] Walsh, C. E., “Speed Limit Policies: The Output Gap and Optimal Monetary Policy,” 2002, American Economic Review, forthcoming. [28] Woodford, Michael, “Optimal Policy Inertia,” NBER Working Paper 7261, Aug. 1999a. [29] Woodford, Michael, Interest and Prices, Princeton University, Sept. 2000.

46

Table 1 Estimates of the New Phillips Curve - Specification (1)

ω

β

α

Restricted

0.512

0.895

0

Unrestricted

0.543

0.850 1.276

H0 : α = 1

D − test

Hansen test

0.311

13.659

11.059

0.572

7.240

8.226

Instrument set A (0.026)

(0.036)

(0.027)

(0.027)

(0.496)

[0.576]

[0.000]

[0.988]

[0.010]

Instrument set B Restricted

Unrestricted

0.546

0.921

0.611

0.879 1.915

(0.047)

(0.0612)

(0.033)

(0.034)

0

(1.210)

[0.449]

[0.007]

[0.60]

[0.114]

Note: Two stages GMM estimates of the structural parameters of equation (14) using orthogonality condition (21). Newey-West corrected standard errors reported in brackets, p-values in square brackets. The null hypothesis for the D-test is H0 : α = 0. Data sample is first quarter 1960 to first quarter 2001. Instrument set A includes four lags of: nonfarm business sector real unit labor cost, HP-filtered output gap, GDP deflator inflation, the CRB commodity price index inflation, 10 years - 3 months US government bonds spread, non-farm business sector hourly compensation inflation, 3-month T-bill interest rate. Instrument set B includes: two lags of non-farm business sector real unit labor cost, HP-filtered output gap, non-farm business sector hourly compensation inflation, and four lags of GDP deflator inflation and 3-month T-bill interest rate. Unless otherwise specified, all data are supplied by the Bureau of Labor Statistics, the Bureau of Economic Analysis, the Federal Reserve System FRED database.

47

Table 2 Estimates of the New Phillips Curve - Specification (2)

ω

β

α

H0 : α = 1

D − test

Hansen test

3.215

58.52

10.696

1.159

25.846

6.658

Instrument set A Restricted

Unrestricted

0.773 0.970 (0.056)

(0.017)

0

0.802 0.905 11.831 (0.048)

(0.021)

(6.040)

[0.072]

[0.000]

[0.991]

[0.050]

Instrument set B Restricted

Unrestricted

0.790 0.994 (0.102)

(0.021)

0

0.808 0.919 11.712 (0.078)

(0.030)

(9.946)

[0.281]

[0.000]

[0.757]

[0.238]

Note: Two stages GMM estimates of the structural parameters of equation (14) using orthogonality condition (22). Se notes to Table 1.

48

Table 3 Estimates of the New Phillips Curve (Recursive GMM) - Specification (1)

ω

β

α

0.93

0

H0 : α = 1

Hansen test

0.216

10.940

3.579

8.059

Instrument set A Restricted

0.476 (0.026)

(0.025)

Unrestricted

0.547

0.860 1.239

(0.038)

(0.024)

(0.51)

[0.641]

[0.989]

[0.016]

Instrument set B Restricted

0.382

0.817

Unrestricted

0.719

0.838 5.282

(0.036)

(0.049)

0

(0.054)

(0.028)

(2.263)

[0.058]

[0.622]

[0.019]

Note: Recursive GMM estimates of the structural parameters of equation (14) using orthogonality condition (21). See notes to Table 1.

49