Learning about Monetary Policy Rules when the Cost Channel Matters

Transcription

1 Learning about Monetary Policy Rules when the Cost Channel Matters Luis-Gonzalo Llosa,a, Vicente Tuesta c,b a Economics Department, University of California, Los Angeles, CA, USA. b Emerging Markets Research, Deutsche Bank, Lima, Peru. c CENTRUM Católica, Pontificia Universidad Católica del Perú, Lima, Peru. Abstract We study how monetary policy may affect determinacy and expectational stability (E-stability) of rational expectations equilibrium when the cost channel of monetary policy matters. Focusing on instrumental Taylor-type rules and optimal target rules, we show that standard policies can induce indeterminacy and expectational instability when the cost channel is present. A naïve application of the traditional Taylor principle could be misleading, and expectations-based reaction function under discretion does not always induce determinate and E-stable equilibrium. This result contrasts with the findings of Bullard and Mitra (2002) and Evans and Honkapohja (2003) for the standard new Keynesian model. The ability of the central bank to commit to an optimal policy is an antidote to these problems. Key words: Learning, Monetary Policy Rules, Cost Channel, Indeterminacy. JEL classification: E4, E5, F3, F4. Introduction There is recent and growing empirical evidence showing that the cost channel of monetary policy, when the interest rate directly affects a firm s price setting behavior, has important implications in both inflation dynamics and the design of optimal monetary policy. Ravenna and Walsh (2006) and Chowdhury, Hoffmann, and Schabert (2006) have provided empirical evidence for the cost channel in the United States and the euro area, respectively. Barth and Ramey (200) found a significant cost channel effect on U.S. data at industry level. Christiano, Eichenbaum, and Evans (2005) estimated a dynamic stochastic general equilibrium model of the U.S. economy and found that monetary policy operates also through the supply side. From the normative point of view, Ravenna and Walsh (2006) showed that a trade-off between stabilizing inflation and output arises endogenously as a consequence of the cost channel. At the same time, in recent literature, economists have begun an ongoing evaluation of the stability under adaptive learning of rational expectations equilibrium (REE) in new Keynesian models. Using the standard new Keynesian framework, Bullard and Mitra (2002) found that determinacy and learnability of a variety of instrument rules is guaranteed if the traditional Taylor principle is satisfied, that is, the interest rate reacts more than one-for-one to inflation (also referred as active rules). In the same framework, Evans and Honkapohja (2003, 2006) showed that optimal target rules (under discretion or commitment) render the REE always unstable under learning if policymakers attempt to implement it using an interest-rate reaction function derived under the rational expectations (RE) assumption as a function of exogenous and lagged endogenous variables (referred to as fundamentals-based reaction function). Evans and Honkapohja (2003, 2006) proposed an alternative implementation of the optimal rule by relaxing the assumption of rational Corresponding author. Tel addresses: luisllosa@ucla.edu (Luis-Gonzalo Llosa), vtuesta@pucp.edu.pe (Vicente Tuesta) As Bullard (2006) pointed out, since adaptive learning is a minimal deviation from rational expectations, its stability should be viewed as an additional minimal criterion, besides determinacy (i.e. unique and non explosive solution), that a REE should meet. Preprint submitted to Elsevier May 3, 2009

2 expectations on private agents, (referred to as expectations-based reaction function), and they found that this type of function (which reacts optimally to private sector expectations) can always induce determinacy and learnability. Partial economic intuition given by Evans and Honkapohja (2003, 2006) is that their proposed expectations-based reaction function always satisfies the Taylor principle. This paper examines the effects of the cost channel coupled with a variety of instrument and optimal target rules on the determinacy (i.e. a unique nonexplosive solution) and learnability conditions of the REE. In particular, we study local determinacy and E-stability properties of the REE in the cost channel model proposed by Ravenna and Walsh (2006). 2 In this sense, our work extends Bullard and Mitra (2002) and Evans and Honkapohja (2003, 2006) standard-economy results to a cost-channel framework. We perform the analysis of instrumental Taylor-type rules under two specifications: contemporaneous data specification, which reacts to current variables, and forward expectations specification (also referred as forward-looking or forecast-based rules), which reacts to one-period-ahead expectations. In the case of target rules, in the fashion of Evans and Honkapohja (2003, 2006), we analyze fundamentals-based (FB- RF) and expectations-based (EB-RF) reaction functions, under both discretion and commitment. We also analyze the corresponding specific target rules under discretion and commitment. In general, our results highlight an important link between the cost channel and both determinacy and learnability of REE. The nature of the policy adopted by the monetary authorities might change this link in important ways. The main findings of our analysis can be summarized as follows: i) Under instrumental rules, standard policies recommended to guarantee determinacy and E-stability in the standard economy (see Bullard and Mitra, 2002) may not be effective or could even be counterproductive if the cost channel is present. For instance, even if the nominal interest rate is adjusted according to the traditional Taylor Principle a determinate and E-stable REE is not necessarily attainable. ii) A discretionary optimal policy does not solve completely the indeterminacy and expectational instability problems as is the case in the standard model, see Evans and Honkapohja (2003). On one hand, the FB-RF implies that the optimal equilibrium is indeterminate and unstable in the learning dynamics, a result that coincides with those of Evans and Honkapohja (2003). On the other hand, the EB-RF that performs well on both grounds in the standard economy does not always lead to determinacy and stability under learning in the cost channel model. iii) A commitment optimal policy can fix the indeterminacy and expectational instability problems found under optimal discretion or instrumental rules. Moreover, it is possible to reach a determinate and E-stable optimal REE not only through the EB-RF, as in the standard model, but also through the FB- RF, 3 which is never E-stable in the standard model (see Evans and Honkapohja, 2006). Nevertheless, the FB-RF is not robust to alternative calibrations and could be less appealing given the difficulties that its implementation involves. Thus, problems of instability under learning and indeterminacy when the cost channel is active can be safely mitigated by committing to the EB-RF. iv) As in the standard model (Evans and Honkapohja, 2003, 2006), the specific target rules under commitment and discretion share the same determinacy and E-stability properties of their corresponding expectations-based reaction functions. Our paper contributes to an important strand of the literature that deals with stability issues when the cost channel matters. Brückner and Schabert (2003) focused on determinacy and pointed out that the cost channel introduces an additional upper bound to the inflation reaction in the Taylor rule. Surico (2008) found that if a central bank assigns positive weight to output fluctuations a model with a cost channel 2 Evans and Honkapohja (999, 200) developed the criterion of Expectational Stability (or E-stability): the conditions under which agents are able to learn (through least squares) the reduced form dynamics under the assumption of rational expectations. E-stability, therefore, provides a robustness criterion: if agents make small mistakes in expectations relative to those consistent with the associated REE, then a policy rule that is E-stable ensures such mistakes are corrected over time. Even though learnability is a more general concept than E-stability, throughout the paper we will use both terms interchangeably. 3 This finding concurs with those of Duffy and Xiao (2007). They found that if one includes the interest rate deviations in the objective, E-stability can be achieved without requiring the central bank to react to private sector expectations, by having the interest rate react suitably to contemporaneously observed aggregate output and inflation. 2

3 is more prone to multiple equilibria (indeterminacy) relative to the standard one. 4 Hence, our paper s contribution to the existing literature is twofold. First, we evaluate whether or not a minimal state variable (MSV) solution is E-stable in the case of indeterminacy. We have shown that E-stability depends on type of instrument rule (contemporaneous or forward-looking). Second, we evaluate determinacy and E-stability for a set of interest rate rules that aim to implement optimal discretionary policy and optimal policy with commitment. Overall, we find that the very strong results of Evans and Honkapohja (2003, 2006) do not carry over to models with the cost channel under optimal discretionary policy. However, their results do extend to models with a cost channel under optimal policy with commitment. The rest of the paper is organized as follows. Section 2 outlines Ravenna and Walsh (2006) model and discusses its main differences with respect to the standard model. Section 3 describes the analysis of determinacy and learning under instrumental and target rules. Section 4 concludes the paper. 2. The Simple Environment In this section we present the log-linearized version of Ravenna and Walsh (2006) model. The model can be summarized by the following equations (see Equations 27 and 28 in Ravenna and Walsh, 2006): π t = κ x x t + βe t π t+ + δκi t + µ t () x t = E t x t+ σ (i t E t π t+ ) (2) κ [( θβ) ( θ) /θ] and κ x κ (η + σ). Variable x t is the output gap, π t inflation and i t is the percentage point deviation of the nominal interest rate around its steady state value. In the model, µ t represents the traditional cost-push shock and E t symbolizes the standard expectation operator. We implicitly base our analysis of learning and monetary policy on the Euler equation approach as Honkapohja, Mitra, and Evans (2003) suggested. Therefore, throughout the paper we assume that our systems are valid under both rational expectations and learning. In this sense, the expectation operation is taken to describe aggregate behavior regardless of the precise nature of agents expectations formation. 5 Equation () is a short run aggregate supply (AS) curve that relates inflation to the output gap and the nominal interest rate. The parameter β denotes the discount factor and κ x captures the sensitivity of inflation to movements in the output gap that depends on deep parameters such as the degree of price stickiness captured by θ and the inverse of the elasticity of the labor supply η. Equation (2) is a IS curve that relates the output gap inversely to the domestic interest rate and positively to the expected future output gap. In this equation /σ represents the intertemporal elasticity of substitution. Note that the previous two-equation system differs from the standard new Keynesian model (see Woodford, 2003) due to the presence of the nominal interest i t in the staggered price equation, that is, the cost channel of monetary policy. The existence of the cost channel is justified if firms must borrow working capital from intermediaries, for further details see Ravenna and Walsh (2006). Just for comparison we define δ which is a dichotomous parameter that takes the value of when there is cost channel and 0 where there is not (standard model). We assume that µ t evolves according to an exogenous first order autoregressive process µ t = ρµ t + ε t where ε t is an independent, identically distributed (i.i.d.) noise with variances σ 2 ε and 0 ρ < is the correlation parameter. We supplement equations () and (2) with a policy rule for the interest rate i t that represents the behavior of the monetary authority. In the next sections we provide, in most of the cases, analytical results regarding the effects of the cost channel and the alternative policy rule specifications on determinacy and learnability conditions. In order 4 Other papers shed lights on the effects of other supply-side mechanisms on the determinacy and E-stability. Kurozomi (2006) proved that even a small degree of non-separability between consumption and money balances in the utility function causes the traditional Taylor principle to be much more likely to induce indeterminacy or E-instability. 5 Recently, Preston (2005) has proposed an interesting reformulation of intertemporal behavior under learning in which agents are assumed to incorporate a subjective version of their intertemporal budget constraint into their behavior under learning. In this paper, we abstract from this approach. 3

4 to gain an insight, we illustrate the results by using a calibrated case. Table summarizes the baseline parameterization. We let δ take two possible values: 0 or, where the former characterizes standard model, whereas the latter characterizes the model with the cost channel. Parameters η and σ are taken from Ravenna and Walsh (2006). As is common in the literature on the Calvo (983) pricing technology, we let the probability of not adjusting prices, θ = We set β to be equal to 0.99, which implies an annualized real interest rate of 4%. As in Bullard and Mitra (2002) we calibrate the policy reaction parameters for non-negative values. Table : Baseline Parameterization δ Dichotomous parameter for the cost channel 0 or θ Probability of not adjusting prices 0.75 β Discount factor 0.99 σ Coefficient of risk aversion.5 η Inverse of the elasticity of labor supply φ π Reaction to inflation 0 φ π φ x Reaction to output gap 0 φ x κ Implied slope of the Phillips curve Notice that under our baseline parameterization, β + κ >. Throughout the paper we assume that this condition holds based on two reasons. First, β+κ > is the only case that is empirically plausible. Condition β + κ < holds if we allow for very small values of κ. Yet, note that for κ to be small enough, we require high values of θ (probability of not adjusting prices), which would contradict macro and micro estimates of the frequency of price adjustment. 6 Second, this assumption avoids a number of technical complications that distract from the main argument of the paper Determinacy and E-stability 3.. Instrumental Rules 3... Contemporaneous data in the Taylor Rule We first assume a simple Taylor type rule (Taylor, 993) in which the central bank reacts to price inflation and the output gap i t = φ π π t + φ x x t (3) where φ π and φ x are non-negative and measure the degree of responsiveness of the policy interest rate to inflation and output gap, respectively. Substituting the policy rule (3) into () and (2), we can write the model involving the two endogenous variables x t and π t y t = Γ + ΩE t y t+ + kµ t (4) w t = ρw t + ε t where y t = [π t, x t ], w t = µ t, Γ = 0, and [ σβ + κx + βφ Ω = ψ x + δκφ x σ (κ x + δκφ x ) βφ π κφ π σ ( δκφ π ) ] (5) 6 For example, using our baseline calibration, β + κ < requires θ > 0.90, which implies that firms fix their prices for about three years on average. 7 We recall this issue when analyzing optimal policy under commitment. 4

5 with ψ = (σ + φ x + κ x φ π δκσφ π ). Determinacy is analyzed by asking under which conditions Ω has both of its eigenvalues inside the unit circle. 8 Surico (2008) has provided the necessary and sufficient conditions for determinacy, 9 ( ) β δκ φ κ x + φ π x > (6) 2σ ( + β) + ( + β + δκ) φ x + (κ x 2δκσ) φ π + κ x > 0. (7) Condition 6 can be interpreted as a generalization of the long-run Taylor principle that guarantees both determinacy and E-stability in the standard model (Bullard and Mitra, 2002; Woodford, 2003). The difference between the conventional long-run Taylor principle and this generalized version relies on κ, which measures the impact of the interest rate on the inflation rate through the cost channel. In line with Woodford (2003), this generalized version has the following economic interpretation: each percentage point of permanently higher inflation implies a permanent change in the output gap of ( β δκ) /κ x percentage points. 0 Under the standard case (δ = 0), any increment in the steady-state inflation leads to a higher output gap whereas under the cost channel (δ = ), it leads to a permanent reduction in the output gap. The left-hand-side of Inequality (6) determines the long-run increase of the interest rate given by the Taylor rule for each unit of increment in the steady-state inflation rate. Note that under the cost channel the traditional Taylor principle, φ π >, does not imply its long-run version as in the standard case. Inequality (7) is a second necessary condition for determinacy. In the standard model, such a condition is redundant and hence does not impose any constraint on the policy parameters. Nevertheless, when the cost channel is active, Condition (7) may impose additional restrictions for determinacy. Assuming δ =, it is straightforward to note that Condition (7) is binding if the inverse of the intertemporal elasticity of the substitution is greater than the inverse of the elasticity of the labor supply, σ > η. Thus, even in the case in which the central bank does not respond to the output gap, the traditional Taylor principle, φ π >, does not guarantee determinacy. To show this result, note that if φ x = 0, φ π is two-sided constrained: < φ π < [2σ ( + β) + κ (η + σ)] /κ (σ η). Condition σ > η reflects the importance of the cost channel relative to the standard demand channel. In the standard economy, any inflationary pressure can be controlled by an increment in the nominal interest rate. The only requirement is that the nominal interest rate reacts more than inflation expectations such that the real interest rate increases, which in turn lowers the output gap and stabilizes inflation. For a given intertemporal elasticity of substitution /σ, the reduction of the inflation rate is proportional to the inverse of the elasticity of the labor supply, η. The reason is that an increment in the real interest rate boosts the labor supply and reduces real wages and, hence, the marginal cost. Therefore, the higher the labor supply elasticity (i.e. lower η) the weaker the standard transmission mechanism of monetary policy. In the cost-channel model such a mechanism coexists with the borrowing-cost mechanism of the interest rate. For a given small enough value of η and if the response of the nominal rate to inflation is too high, the cost channel dominates the standard mechanism and then expectations of higher inflation become self-fulfilling. 2 McCallum (2007) has shown that determinacy is a sufficient (though not necessary) condition for E- stability for a broad class of models, including the one in this paper. Hence, the generalized long-run Taylor principle and Condition (7) are sufficient for E-stability, that is, all RE solutions of (4) have the property of E-stability. Nevertheless, we further need to check whether indeterminate equilibria are E-stable or not. To 8 For details see Blanchard and Kahn (980). 9 Surico (2008) studied a model that allows that only a fraction of firms is subject to the cost channel. Furthermore, the instrument rule analyzed in that paper is endowed with interest rate smoothing. In the following conditions, we have canceled both extensions. A formal proof of those conditions is available upon request from the authors. 0 This elasticity was derived using long-run versions of () and (2), i.e. π t = E t π t+ = π, x t = E t x t+ = x and i t = i. In the long run, (2) implies i = π and thus () reduces to κ xx = ( β δκ) π. This result concurs those of Kurozomi (2006). The author stressed that the traditional Taylor principle does not always imply its long run version unless the central bank does not target the output gap or the degree of non-separability between consumption and money balances in the utility function is small enough. 2 A similar argument is provided by Surico (2008). 5

6 study the stability of REE under adaptive learning, we follow Evans and Honkapohja (200, chap. 0) and assume that agents utilize a perceived law of motion (PLM) for y t that corresponds to the minimal state variable (MSV) solution (McCallum, 983) to the system (4): y t = a + cµ t. The PLM can be written as: y t = a + cµ t. Using this PLM, agents form expectations of y t+ : E t y t+ = a + cρµ t. Substituting these expectations into (4) delivers a T-mapping from the PLM to the actual law of motion (ALM): y t = T a (a) + T c (c)µ t. The rational expectations solution consists of values such that a = T a (a) and c = T c (c). The answer to the question of whether the system in (4) is stable under learning is given by the principle of E-stability, which comes from analyzing the local asymptotic stability of the following matrix differential equation d (a, c) /dτ = T (a, c) (a, c) evaluated at the REE solution (a, c). Specifically, the REE solution of the system in (4) is E-stable or learnable if the real parts of all eigenvalues of DT a (a) = Ω and DT c (c) = ρ Ω are lower than. Proposition summarizes the necessary and sufficient conditions for E-stability. Proposition. Under contemporaneous data interest rate rules, the necessary and sufficient condition for an MSV solution (0, c) to be E-stable is that ( ) β δκ φ x + φ π > (8) Proof. See appendix A. κ x 6 Standard Model (δ=0) 6 Cost Channel Model (δ=) φ x 3 φ x φ π φ π Determinate and E stable Indeterminate and E unstable Figure : Regions of determinacy and expectational stability for contemporaneous data policy rules. Left panel corresponds to the standard model (δ = 0). Right panel corresponds to the cost channel model (δ = ). The condition for E-stability given in Proposition is identical to the generalized long run Taylor principle defined above. Therefore, determinacy is sufficient for E-stability. Yet, if the condition for determinacy in Inequality (7) is not redundant, that is, σ > η, indeterminate equilibria may be learnable. In order to gain an insight, Figure depicts determinacy and E-stable regions as functions of both φ π and φ x, with the rest of the parameters set at their baseline values given in Table. The figure on the left side depicts the standard case (δ = 0) whereas the figure on the right shows the cost channel case (δ = ). The main effect of the cost channel is to rotate the line describing the border between the determinate and E-stable region and indeterminate and E-unstable region. That border is given by the generalized long-run Taylor principle (equations 6 and 8). Under the cost channel, the set of parameter values in the policy rule that are consistent with determinacy and learnability are a subset of those for the standard model. Note 6

7 that given σ > η and provided a null response to the output gap, φ π must lie between and 44 (not shown in the graph) to guarantee a determinate equilibrium Forward expectations in the Taylor rule Forward expectations Taylor rules adopt the following form i t = φ π E t π t+ + φ x E t x t+ (9) where φ π and φ x are non-negative. We reduce the system of equations (), (2) and (9) to two equations involving the endogenous variables x t and π t. The reduced system takes the form of (4), where Ω is defined by Ω = [ ] δκσφπ κ x (φ π ) + σβ κ x σ κ x φ x + δκσφ x (0) σ (φ π ) σ φ x The following proposition summarizes the necessary and sufficient conditions for a rational expectations equilibrium to be determinate. Proposition 2. Under interest rate rules with forward expectations the necessary and sufficient conditions for determinacy are that Proof. See appendix B. (β + δκ) φ x δκσφ π < σ ( + β) () δκσφ π (β + δκ) φ x < σ ( β) (2) ( + β + δκ) φ x + (κ x 2δκσ) φ π < 2σ ( + β) + κ x (3) ( ) β δκ φ x + φ π > (4) κ x Propositions 2 shows that the cost channel alters the conditions for determinacy relative to those for the standard model. Condition (4) is the generalized long-run Taylor principle as discussed in Section 3... Hence, the discussion about whether the traditional Taylor principle implies its long-run version applies (see the previous section). Yet, a Taylor rule with forward expectations imposes additional constraints. To show the effects of the cost channel, we let φ x = 0; then Condition (2) implies that the inflation reaction is bounded from above, that is, φ π < ( β) /δκ, whereas condition (4) implies the traditional Taylor principle, that is, φ π >. 3 Under the cost channel (δ = ) determinacy is not attainable because the former upper bound is below. The latter is a remarkable result since the idea that the traditional Taylor principle or active policy leads to determinacy is a celebrated result in the literature. The analysis of E-stability is analogous to that of the previous section. The following proposition provides the conditions for E-stability of the MSV solution. Proposition 3. Under interest rate rules with forward expectations, the necessary and sufficient conditions for an MSV solution (0, c) to be E-stable are that Proof. See appendix C. (κ x κδσ) φ π + φ x + σ ( β) > κ x (5) ( ) β δκ φ x + φ π > (6) κ x 3 Notice that under the baseline case (δ = 0), ( β) /δκ goes to infinite and thus the condition (2) does not bind. Yet, Bullard and Mitra (2002) noted that under forward expectations specification φ π has an upper bound given by condition (3) which is always greater than. Hence, the traditional Taylor principle is a necessary but not sufficient condition for determinacy. 7

8 The last proposition shows that E-stability is also affected by the presence of the cost channel. In fact, the effects of the cost channel go beyond the possible disconnection between the traditional Taylor principle and its long-run version as we stressed earlier. The cost channel imposes a new E-stability condition, (5), which in turn impose new constraints over the policy parameters. 4 To illustrate this result we let φ x = 0 and δ =, then Condition (5) reduces to the following condition φ π > + σ (β + κ ) /κη. Therefore, in contrast to the standard model, E-stability needs a reaction to inflation expectations above the one prescribed by the Taylor principle. 6 Standard Model (δ=0) 6 Cost Channel Model (δ=) φ x 3 φ x φ π φ π Determinate and E stable Indeterminate and E stable Indeterminate and E unstable Figure 2: Regions of determinacy and expectational stability for forward expectations policy rules. Left panel corresponds to the standard model (δ = 0). Right panel corresponds to the cost channel model (δ = ). Figure 2 plots the intersections of the regions of determinacy and learnability. A forward expectations Taylor rule described by φ π > and a relatively small response to output gap guarantees a determinate and learnable equilibrium in the standard-economy model. Moreover, a passive reaction to inflation may also promote stability if it is accompanied by a sufficient reaction to the output gap. On the contrary, when the cost channel is present, an active (but moderate) response to inflation expectations (i.e. φ π > ) and a small response to the output gap do not guarantee a determinate and E-stable equilibrium. Therefore, determinacy and E-stability are only attainable if the central bank reacts modestly to both output gap and inflation expectations. Overall, the parameters region that induces both determinacy and learnability shrinks Target Rules Discretionary Policy We now take the standard formulation of the central bank s loss function similar to the one derived in Ravenna and Walsh (2006) from first principles L 0 = (/2)E 0 i=0 β i [ λx 2 t+i + π 2 ] t+i (7) 4 Under the standard model (δ = 0), (5) collapses to φ x /κ x +φ π +σ ( β) /κ x >, which is implied by the long-run Taylor principle. Therefore, as emphasized by Bullard and Mitra (2002), the long-run Taylor principle is necessary and sufficient for E-stability in the standard model. 8

9 where λ is the relative weight of output deviations. Following Evans and Honkapohja (2003, 2006) and Giannoni and Woodford (2003), we treat λ as a free positive parameter. Thus, our results can be compared to those of the existing literature. When necessary, we also evaluate microfounded λ based on the optimal loss function derived in Ravenna and Walsh (2006). Optimal monetary policy under discretion implies minimizing Equation (7) subject to versions of Equations () and (2), modified to take into account the central bank s lack of commitment. It is straightforward to obtain the optimal First Order Conditions (FOC) that shows the trade-off between stabilizing domestic inflation and the output gap, which reads: π t = λ κ o x t (8) where κ o κ x δκσ κ [η (δ ) σ]. Note that when δ = 0, κ o = κ x = κ (η + σ), we return to the standard trade-off found in Clarida, Galí, and Gertler (2000) whereas when δ = we have κ o = κη. Note also that the cost channel entails larger volatility of inflation since κ (η + σ) > κη. If one does not concern oneself with how the policy is implemented, then one might view Equation (8) as the specific targeting rule defining discretionary policy. Combining () and (2) with this optimal rule the model reduces to π t = ω 0 E t π t+ + ω u t (9) where ω 0 = (λ (β + δκ) δκσκ o ) / ( λ + κ 2 o). It is straightforward to show that when δ = 0 both determinacy and E-stability immediately follow. However, when the cost channel matters both determinacy and E- stability cannot be taken for granted. Proposition 4. Under the specific targeting rule defining discretionary policy the necessary and sufficient condition for a rational expectations equilibrium to be determinate is that Case I: if σ > η or Case II: if σ η Proof. See Appendix D. κ 2 η (σ η) (β + κ + ) < λ < κ2 η (η + σ) (β + κ ) λ < κ2 η (η + σ) (β + κ ) Proposition 5. Under the specific targeting rule defining discretionary policy the necessary and sufficient condition for a MSV solution (0, c) to be E-stable is that Proof. See Appendix E. λ < κ2 η (η + σ) (β + κ ) Therefore, the optimal specific targeting rule given by (8) does not always lead to determinacy and learnability for all parameter values as in the standard model. To illustrate to what extent the presence of the cost channel affects determinacy and E-stability we use our baseline parameterization. The optimal discretion rule leads to a determinate REE as long as λ ranges between and 0.24 and E-stability follows if λ is lower than For robustness, we also evaluate the thresholds under alternative parameterizations provided in Woodford (999), Clarida et. al. (2000) and McCallum and Nelson (999). Table 2 summarizes the alternative parameterizations and the resulting boundaries for λ. Note that under Woodford (999) 5 Ravenna and Walsh (2006) derived a microfounded optimal λ as a function of deep parameters. For our baseline calibration, this optimal λ is which falls inside the determinate and E-stable region. 9

10 and Clarida et. al. (2000) the slope coefficients of the Phillips curve (κ x ) are significantly smaller, hence the effect of the output gap over inflation through the Phillips curve is relatively less significant and the cost channel effect becomes more important. This explains intuitively why the permissible parameter ranges for λ that guarantee both determinacy and E-stability shrink under the calibrations of Woodford (999) and Clarida et. al. (2000) Table 2: Critical Values for λ under Specific Targeting Rule - Discretion Baseline W (999) CGG (2000) MN (999) σ /0.64 κ κ x Determinacy 0.008<λ< λ< λ< <λ< E-stability λ< λ< λ< λ< Note: W=Woodford; CGG=Clarida, Gali and Gertler; MN=McCallum and Nelson Fundamentals-based reaction function. Evans and Honkapohja (2003) discussed several forms of implementing the optimal plan given by (8). A first form is called fundamentals-based reaction function (FB-RF) and implies that the central bank assumes that private agents have perfect rational expectations with the REE taking the form of the MSV solution. Under such assumptions, the FB-RF reacts only to fundamental shocks. 6 i t = φ µ µ t (20) We reduce the system of equations (), (2) and (20) involving the endogenous variables x t and π t. The reduced system takes the form of (4), where Ω is defined by [ β + Ω = σ κ ] x κ x (2) σ Note that Ω is independent from δ and it is exactly the same matrix analyzed by Evans and Honkapohja (2003). Thus, irrespective of whether the cost channel is present or not, the FB-RF always leads to indeterminacy and instability under learning. In fact, Evans and Honkapohja proved that any linear policy rule of the form of (20) induces both indeterminacy and E-instability. Expectations-based reaction function. Evans and Honkapohja (2003) proposed a second form, referred to as expectations-based reaction function (EB-RF) which is derived under the assumption that agents may not possess RE and that their expectations can be observed by the central bank. The EB-RF can be viewed as a way of using an explicit interest-rate rule to implement the optimal policy FOC for all possible expectations. This rule is obtained by solving i t from the structural equations () and (2) and the optimal condition (8) where the coefficients are i t = φ π E t π t+ + φ x E t x t+ + φ µ µ t (22) φ π = (λ + σκ oβ + κ o κ x ) (λ + κ 2 ; φ x = σ (λ + κ oκ x ) o) (λ + κ 2 o) and φ µ = (σκ o) (λ + κ 2 o) Notice that when δ = 0 the EB-RF collapses to the one proposed by Evans and Honkapohja (2003). 7 Evans and Honkapohja (2003) pointed out that the traditional Taylor principle holds under their EB-RF, that is, φ π >. We highlight that this is also the case for our EB-RF (since κ o < κ x ). In addition, we 6 Evans and Honkapohja (2003) considered a more general case in which fiscal shocks appear in the fundamentals-based reaction function. It is straightforward to show that the result of this section applies to the general case. 7 In the standard model the optimal coefficients are φ π = ( λ + σκ x β + κ 2 ) ( ) x / λ + κ 2 x ; φx = σ and φ µ = σκ x / ( λ + κ 2 ) x. 0

11 6 Standard Model (δ=0) 6 Cost Channel Model (δ=) φ x 3 φ x φ π φ π Determinate and E stable Indeterminate and E stable Indeterminate and E unstable Figure 3: Regions of determinacy and expectational stability for forward expectations policy rules. Left panel corresponds to the standard model (δ = 0). Right panel corresponds to the cost channel model (δ = ). The dotted lines plot the optimal parameters φ π and φ x under the expectations-based rule derived under discretion. notice that, given κ o < κ x, the optimal reaction to output gap expectations under the cost channel is bigger than the one under the standard case. The reduced form of the model under (22) takes the form of (4), where Ω is defined by [ ] (β + κδ) λ δσλκ Ω = ψ (23) κ o (β + δκ) δσκκ o with ψ = (λ + κ o κ x δκσκ o ).This system is determinate if and only if matrix Ω has both eigenvalues inside the unit circle and E-stable if the real parts of all eigenvalues of Ω are lower than. One of the roots of Ω is zero and the other one is exactly equal to ω 0, defined in (9). Therefore, the system under the EB-RF has the same determinacy and E-stability properties as the system under the specific targeting rule under discretion (see Propositions 4 and 5). Proposition 6. The EB-RF (22) shares the same determinacy and E-stability properties as the specific targeting rule defining discretionary policy, (8). Hence, the EB-RF does not lead to determinacy and stability under learning for all structural parameter values. By contrasting the performance of the FB-RF with that of the EB-RF, Evans and Honkapohja (2003) concluded that the endogenous response to inflation expectations is the key to providing a stabilizing role to the latter under both rational expectations and adaptive learning for all structural parameters. Evans and Honkapohja (2003) provided an intuition for this result based on the fact the EB-RF satisfies the Taylor principle and hence succeeds in guiding the expectations and the economy to the optimal REE. 8 Nonetheless, as we stressed previously when the cost channel is active the economy may display indeterminacy and/or expectational instability even if the Taylor principle holds. Hence, a crucial question is which condition of Propositions 3 and 4 does not hold under the EB-RF and thus generates the indeterminacy and E-instability 8 Evans and Honkapohja (2003) stressed that the EB-RF with the specified policy coefficients succeeds in guiding the expectations of private agents and the economy to the optimal REE. The authors were aware that if instrumental rules of the form of their EB-RF, i.e., forward expectation Taylor rule, are embedded with too large values of φ π, then indeterminacy will follow; see Evans and Honkapohja (2003).

12 result. Figure 3 plots the determinacy and E-stable regions of the Taylor rule with forward expectations (9) as in Figure 2. The dotted lines plot the combination of EB-RF s optimal coefficients under different values of λ (ranging from 0 to ). For the standard model, the dotted line always stays in the determinate and E-stable area. For the cost channel model, the dotted line crosses the long-run Taylor principle and falls inside the indeterminate and E-unstable area. We check this result by substituting the optimal coefficients φ π and φ x into the long-run Taylor principle. The resulting expression collapses to the condition shown in Proposition 5. 9 To sum up, our results suggest that the Evans and Honkapohja (2003) proposal to solve the instability of the FB-RF by conditioning optimally on private sector expectations can be misleading when the cost channel matters. As the next section will prove, the ability to commit to an optimal policy is an antidote to this problem Commitment Policy The Lagrangian of the policy problem is the following: L 0 = E 0 t=0 β t { 2 [ λx 2 t + π 2 t ] + ϕ,t [ xt x t+ + σ (i t π t+ ) ] +ϕ 2,t [π t βπ t+ κ x x t δκi t ] } ϕ,0 π 0 (24) The FOCs with respect to π t, x t and i t are respectively: π t σβ ϕ,t + ϕ 2,t ϕ 2,t = 0 (25) λx t + ϕ,t β ϕ,t κ x ϕ 2,t = 0 (26) σ ϕ,t δκϕ 2,t = 0 (27) and π 0 = π 0. Combining (25), (26), (27) we get the following set of equations: 20 ϕ 2,t = ( + β δκ)ϕ 2,t π t (28) x t = λ β σδκϕ 2,t + λ (κ x σδκ) ϕ 2,t (29) Analogously to the discretionary case, equations (28) and (29) provided the specific targeting rule for the commitment case. Combining equations (), (2) with equations (28) and (29), we can obtain the following reduced form, y t = Γ + ΩE t y t+ + Φy t + kµ t (30) where y t = [π t, ϕ 2,t ], A = 0, and [ βλ (β + δo ) σβδ Ω = ϑ o κ o βλ (β + δ o ) σβδ o κ o ] [ 0 σκo δ ; Φ = ϑ o + ( σ 2 δ 2 ) ( o + βκ 2 o β δ o + ) 0 βλ ( β δ o + ) σκ o δ o ] 9 An interesting implication of this result is that there exists a conflict between the desirable properties of an optimal discretionary rule in terms of the volatility that it entails and the learnability and determinacy criteria. Indeed, note that optimal condition (8) shows that the cost channel increases the trade-off between stabilizing inflation and the output gap (κ 0 < κ x ) and simultaneously it implies smaller optimal reaction to expected inflation with respect to the standard model (see Figure 3). Yet, the latter might induce undesirable properties in terms of both learnability and determinacy. Hence, the achievement of determinacy and learnability under the cost channel would imply a bigger reaction to inflation expectations - as if the cost channel were absent - at the cost of larger macroeconomic volatility. 20 Note that unlike the standard case, analyzed by Evans and Honkapohja (2006), we cannot eliminate the lagrange multipliers in order to get a tractable system. Note also that if δ = 0, the FOCs are reduced to κ x π t = λ (x t x t ) which is the optimal FOC under the standard model (Evans and Honkapohja, 2006). 2

13 with ϑ = ( βλ + σ 2 δ 2 o + βκ 2 o), δo = δκ and κ o κ x δκσ. The MSV solution of (30) can be written as a function of the lagged lagrange multiplier, ϕ 2,t, and the cost-push shock, µ t, π t = b π ϕ 2,t + c π µ t (3) ϕ 2,t = b ϕ ϕ 2,t + c ϕ µ t. (32) After replacing (3) and (32) (and their respective expected values) into (30), we obtain the following polynomial characterizing b ϕ, βb 2 ϕ γb ϕ + = 0 (33) where γ = [ ( σ 2 δ 2 o + βκ 2 o + λβ ) +λ (β + δ o ) 2 ]/ [λ (β + δ o ) σκ o δ o ]. The product of the roots is given by β which is positive and greater than. Thus both roots have the same sign and at least one lies inside the unit circle. In the standard model, γ = + β + κ 2 x/λ and thus both roots are always positive (see Evans and Honkapohja, 2006). In contrast, with the cost channel (δ = ), we have positive roots if γ > 0, that is, λ > σκ 2 η/(β + κ) and negative roots if λ < σκ 2 η/(β + κ). That is, for high (low) values of λ both roots are positive (negative) and hence the sum of roots increases with λ. We are not able to establish tractable conditions under which exactly one root lies inside the unit circle. Nevertheless, we establish the solutions of the polynomial as λ 0 and λ + and appeal to the continuity of the roots of Equation (33). On one hand, λ 0 and δ = imply that the stable solution is equal to either σ/βη or η/σ. Hence, the system is always determinate as λ 0. On the other hand, λ + and δ = imply that the only root that is stable is /(β + κ); the other root is + κ/β and hence always explosive. 2 Therefore, exactly one root lies inside the unit circle and thus optimal policy under commitment guarantees determinacy. 22 The stable solution is denoted by b ϕ and the rest of coefficients of the MSV are given: b π = β κδ + ( b ϕ ) ; cπ = c ϕ c ϕ = βλ [( σ 2 δ 2 o + βκ 2 o + βλ ) + σκ o δ o (δ o + β) (λ (β + δ o ) σκ o δ o ) β ( ρ b π )] We refer to this equilibrium as the optimal REE under commitment. As we did for the discretionary case, before proceeding with the analysis of implementable optimal rules we ask whether the specific targeting rule defined by (28) and (29) delivers an E-stable equilibrium. The next proposition summarizes our results. Proposition 7. Under the specific targeting rule defining commitment policy, (28) and (29), the optimal REE is both determinate and stable under learning for all structural parameter values. Proof. See Appendix F. Fundamentals-based reaction function. We replace the RRE solution of the form of (3) and (32), as well as their respective expectations into the structural relationships () and (2) after using (29). Then, we solve for i t, and the resulting equation is the FB-RF under commitment, where φ ϕ and φ µ are given by φ ϕ = (β ) ( i t = φ ϕ ϕ 2,t + φ µ µ t (34) b 2 πβ 2 λ 2 δ o κ x σ 2 ( b ϕ βκ o + δ o σ ) +b π βλσ (( + b ϕ ) βκo βδ o σ + δ o ( κ x + σ) ) ) 2 In the standard model (δ = 0), λ + implies that one root b ϕ (the other root is /β) and thus the system is always determinate. 22 Notice that if β + κ <, an optimal target rule under commitment policy delivers explosive solutions for high enough values of λ. Yet, as we emphasize in Section 2, β + κ < is not empirically plausible since it implies extremely low frequencies of price adjustment. We leave the issue of explosiveness for future research. 3

14 φ µ = ( βcπ κ o κ x λρ βκ o ( cϕ κ o κ x ρ + ( + b ϕ ) λ ( cπ + βc π ρ) ) σ δ o λ ( + c π ( + βρ) σ 2 + b π βλ (( + c π ) λ + βc ϕ κ o ρσ) ) with = λ ( βκ o κ x + b π β (β + δ o ) λ + ( ) + b ϕ βδo κ o σ + δ 2 oσ 2). Combining (34) with (), (2) and (29) we collapse a system of the form of (30), where y t = [π t, ϕ 2,t ], Γ = 0, and [ ( ) σ βκ Ω = ψ o (βσ + κ x ) βσ 2 δ o βκ 2 oκ x λ ] βλσ βκ o [ ( 0 (βλσ) δ Φ = ψ o σ (( ) βλφ ϕ + σκ x (βκo σδ o ) βκ o κ x σ ) β 2 ) ] κ o κ x λφ ϕ 0 σ ( ) βλφ ϕ + σ 2 δ o with ψ = (βκ o σδ o ), δ o δκ and κ o κ x δκσ. The MSV solution can be written as y t = a+by t +cµ t, where the REE is given by (a, b, c) = ( a, b, c ). Evans and Honkapohja (2006) performed a numerical analysis of determinacy and E-stability conditions for the FB-RF under the standard model. The authors stated that this rule leads to determinacy only for a subset of the parameter space and, more importantly, fails in generating an E-stable equilibrium. Remarkably, when the cost channel is present the FB-RF delivers an E-stable optimal REE under some specific parameterizations and determinacy follows for a wider range of parameter values than those reported in Evans and Honkapohja (2006). Table 3 reports the results. Table 3: Critical Values for λ under FB-RF Baseline W (999) CGG (2000) MN (999) Determinacy 0.058<λ <λ 0.474<λ 0 5 <λ E-stability 0.004<λ No No 0 5 <λ Note: Parameterizations are the same as in Table 2. W=Woodford; CGG=Clarida, Gali and Gertler; MN=McCallum and Nelson. Never E-stable. Critical values are less than 0 5. Proposition 8. Under the FB-RF (34), there are parameter regions in which the model is determinate and other regions in which it is indeterminate. Proposition 9 shows that in the model with the cost channel the strength of demand-side effects relative to the supply-side effects is crucial for E-stability. That is, if the inverse of the intertemporal elasticity of substitution (σ) is greater than the inverse of the intertemporal elasticity of labor supply (η) the REE is E-stable. The latter is only a sufficient condition for E-stability, but it explains why the calibrations of Woodford (999) and Clarida et. al. (2000) do not generate E-stability. Proposition 9. Under the FB-RF (34), there are parameter regions in which the model is stable under learning and other regions in which it is unstable under learning. Moreover, σ > βη is a sufficient condition for E-stability. Proof. See Appendix G. Expectations-based reaction function. After substituting (28) and (29) into the aggregate supply (Equation ), we can replace x t for ϕ 2,t. Then, we express the aggregate demand (Equation 2) in terms of ϕ 2,t by using equation (29). By solving i t from the resulting equations, we get the following expression for the EB-RF under commitment, i t = φ L ϕ 2,t + φ π E t π t+ + φ ϕ E t ϕ 2,t+ + φ µ µ t (35) where the coefficients are given by φ L = (βλ ) σ ( δ 2 oλσ β (β + δ o ) κ o λ δ 2 oκ x σ 2) φ π = β ( λ δ o σ 2 + κ o (κ o + βσ) ) φ ϕ = (λ ) (βκ o σ (κ o κ x + λ)) φ µ = ( ) σ (βκ o δ o σ) 4 )

15 where = δ 2 oσ 2 + β ( λ + κ 2 o). Evans and Honkapohja (2006) have shown that, in the standard model, the EB-RF under commitment succeeds in generating a determinate and E-stable optimal REE. Under (35), it can be shown that the reduced form for the vector y t = [π t, ϕ 2,t ] is the same as (30). Thus, the EB-RF shares the same determinacy and E-stability properties as the specific targeting rule defined by (28) and (29). Hence, recalling Proposition 7, the EB-RF guarantees determinacy and E-stability of the optimal REE under commitment. Therefore, a solution to the indeterminacy and expectational instability problems relies on whether the central bank is able to commit to an optimal policy. Proposition 0. The EB-RF, (35), shares the same determinacy and E-stability properties as the specific targeting rule defining commitment policy, (28) and (29). Hence, the EB-RF delivers an optimal REE that is both determinate and stable under learning for all structural parameter values. We summarize our main results under target rules as follows. First, a commitment optimal policy can fix the indeterminacy and expectational instability problem found under optimal discretion or instrumental rules. Second, under commitment the cost channel opens the possibility of reaching a determinate and E-stable optimal REE not only through the EB-RF but also through the FB-RF, the latter being counterproductive under the standard model. Yet, the FB-RF is not robust to alternative calibrations and could be less appealing given the difficulties that its implementation involves. Thus, problems of instability under learning and indeterminacy when the cost channel is active can be safely mitigated by committing to the above EB-RF Additional discussion Following Evans and Honkapohja (2003, 2006), the results presented throughout the paper are based upon the assumption of t dating in expectations, that is, current endogenous variables are excluded in an individual s information set. If instead one considers t dating in expectations, that is, current endogenous variables are included in an individual s information set, a problem with the simultaneous determination of expectations and current endogenous variables arises and thus E-stability conditions change accordingly. 24 In this section, we evaluate the E-stability properties of the policy rules under such an assumption. For those systems that do not involve lagged endogenous variables this additional exercise is unnecessary since E-stability conditions under both assumptions coincide. Therefore, the only systems that matter for this purpose are those derived under commitment. As stated in Proposition of McCallum (2007), determinacy is a sufficient (though not necessary) condition for E-stability under t-dating expectations for a broad class of models, including the one in this paper. Nevertheless, we further need to check whether an indeterminate equilibrium is E-stable or not. First we perform a numerical evaluation of the E-stability conditions under the FB-RF. Table 4 reports the ranges for λ that induce E-stability. Although determinacy conditions are not affected by the information assumption, we also report the critical values that induce determinacy. As in Evans and Honkapohja (2006), the instability problem under learning of this type of rule is less severe when we consider t dating in expectations. Table 4 also shows that under the FB-RF small values of λ generate indeterminate and learnable equilibria. 23 These results also suggest that endowing instrument rules with interest rate smoothing, which captures the importance of lagged endogenous variables, might alleviate the problems of indeterminacy and expectational instability when the cost channel is active. For a discussion of the benefits of interest rate smoothing in the standard new keynesian model, see Bullard and Mitra (2007). 24 E-stability is guaranteed if DT a (a, b) = (I Ωb) Ω, DT b (b) = [(I Ωb) Φ] [(I Ωb) Ω], and DT c (b, c) = ρ [(I Ωb) Ω] have the real parts of all of their eigenvalues lower than. See Evans and Honkapohja (200, Section 0.3) for further details. 5