Loose Commitment in Medium-Scale Macroeconomic Models: Theory and an Application

Debortoli, Davide; Maih, Junior; Nunes, Ricardo Working Paper Loose Commitment in Medium-Scale Macroeconomic Models: Theory and an Application Working Paper, No. 2010/25 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Debortoli, Davide; Maih, Junior; Nunes, Ricardo (2010) : Loose Commitment in Medium-Scale Macroeconomic Models: Theory and an Application, Working Paper, No. 2010/25, ISBN 978-82-7553-582-3, Norges Bank, Oslo, https://hdl.handle.net/11250/2497436 This Version is available at: https://hdl.handle.net/10419/209970 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no 2010 | 25 Loose commitment in medium-scale macroeconomic models: Theory and an application Working Paper Monetary Policy Department By Davide Debortoli, Junior Maih and Ricardo Nunes Working papers fra Norges Bank, fra 1992/1 til 2009/2 kan bestilles over e-post: [email protected] Fra 1999 og fremover er publikasjonene tilgjengelig på www.norges-bank.no Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. Hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. Synspunkter og konklusjoner i arbeidene står for forfatternes regning. Working papers from Norges Bank, from 1992/1 to 2009/2 can be ordered by e-mail: [email protected] Working papers from 1999 onwards are available on www.norges-bank.no Norges Bank’s working papers present research projects and reports (not usually in their final form) and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in working papers are the responsibility of the authors alone. ISSN 1502-8143 (online) ISBN 978-82-7553-5- (online) Loose Commitment in Medium-Scale Macroeconomic Models: Theory and an Application∗ Davide Debortoli UC San Diego Junior Maih Norges Bank Ricardo Nunes Federal Reserve Board First version: June 2008; This version: December 2010 Abstract This paper proposes a method and a toolkit for solving optimal policy with imperfect commitment in linear quadratic models. As opposed to the existing literature, our method can be employed in mediumand large-scale models typically used in monetary policy. We apply our method to the Smets and Wouters (2007) model, where we show that imperfect commitment has relevant implications for the interest rate setting, the sources of business cycle fluctuations, and welfare. JEL classification: C32, E58, E61. Keywords: Commitment, Discretion, Linear-Quadratic ∗We are grateful to seminar participants at the Conference on Computing in Economics and Finance Paris 2008. Any remaining errors are our own. The views expressed in the paper are those of the authors and do not necessarily reflect those of the Board of Governors, the Federal Reserve System, or the Norges Bank. Email: davide.deb[email protected], [email protected], ricardo.p.n[email protected]v 1 1 Introduction In the modern macroeconomic literature, economic outcomes result from the interactions between policymakers and rational firms and households. A common feature of these models is that economic decisions (e.g. consumption, hours worked, prices) depend on expectations about future policies (e.g. taxes, interest rates, tariffs). As shown by Kydland and Prescott (1977) optimal policy plans in this class of models are subject to time-inconsistency. The modern literature has taken different approaches to address this problem. One possibility is to assume that policymakers can fully commit – a single optimization is undertaken and the chosen policies are then implemented in all subsequent periods. This approach is known as full-commitment or simply commitment. An alternative, often referred to as discretion or nocommitment, assumes that policymakers cannot commit and that policy plans always need to be time-consistent. Although many types of timeconsistent equilibria can be studied, one of the most common approaches is to solve for Markov-perfect equilibria, where policy functions only depend on payoff relevant state variables. Both the full-commitment and discretion approaches are to some extent unrealistic. Commitment does not match the observation that governments and other institutions have defaulted on past promises. Discretion rules out the possibility that governments achieve the benefits of making and keeping a promise, despite the ex-post incentive to renege. Roberds (1987) developed an approach – recently extended by Schaumburg and Tambalotti (2007) and Debortoli and Nunes (2010a) – which escapes the “commitment vs discretion” dichotomy. Policymakers are endowed with a commitment technology, but with some exogenous and common knowledge probability they may succumb to the temptation to revise their plans. This approach has been labeled quasi-commitment or loose commitment. Several questions can be addressed with the loose commitment approach. What are the gains of achieving more credibility? How does the possibility of future re-optimizations affect current outcomes and promises? What are the consequences of revising policy plans? How do occasional re-optimizations affect the shock propagation, volatilities, and cross-correlations between relevant variables? To answer these questions and derive the associated positive and normative implications, one must depart from the frameworks of commitment and discretion and consider instead loose commitment. Nevertheless, given some technical difficulties, the loose commitment ap2 proach has so far been limited to relatively simple and stylized models. The goal of this paper is to overcome this limitation. We propose a simple and relatively general algorithm to solve for the optimal policy plan under loose commitment in linear quadratic frameworks. We show how these types of problems reduce to solving a system of linear difference equations, and do not present any additional challenge with respect to the commitment or discretion cases. Assuming plans’ revisions to be stochastic events, rather than endogenous decisions, is clearly a simplification analogous in spirit to the Calvo pricing model. While more complex credibility settings can be easily imagined (e.g. an endogenous timing of re-optimizations), such complexity may become prohibitive in mediumand large-scale models. In those type of models, the tractable though simplified approach employed here is particularly valuable. The paper is related to the literature on optimal monetary policy in linear quadratic frameworks. Solution algorithms for full-commitment, together with a discussion about the computational aspects, have been developed by Currie and Levine (1993) and S¨oderlind (1999), among others. Methods to solve for (Markov-perfect) time-consistent equilibria are described in Backus and Driffill (1985), S¨oderlind (1999), and Dennis (2007).1The main contribution of the paper is to extend these methodologies to address problems under loose commitment. To illustrate the benefits of our approach, the methodology is then applied to analyze the effects of commitment in the medium-scale model of Smets and Wouters (2007), which has arguably become one of the benchmark models in the dynamic stochastic general equilibrium literature.2 The paper continues as follows. In section 2 we introduce the general formulation of the model. In section 3 we study the optimal policy problem and describe the solution algorithm. Section 4 discusses the role of commitment in the Smets and Wouters (2007) model and section 5 concludes. We provide as supplementary material a collection of codes and documentation that implement our algorithm in a variety of models. 1See also Klein et al. (2005) and Judd (2004) for more general non-linear problems. 2We have also tested our methodology with bigger models used for monetary policy analysis, such as the Norwegian Economy Model (NEMO) of the Norges Bank. 3 2 General form of the models Consider a general linear model, whose structural equations can be cast in the form A−1yt−1+A0yt+A1Etyt+1 +Bvt= 0,∀t(1) where ytindicates a vector of endogenous variables and vtis a vector of serially uncorrelated exogenous disturbances with zero mean and Evtv0 t= Σv. As it is well known, models with more lags and leads, lagged expectations, constants, and serially correlated shocks can be accommodated into this formulation by expanding the ytvector. The policymaker is assumed to have a quadratic loss function ∞ X t=0 βty0 tWyt(2) A purely quadratic objective function is consistent with a second-order Taylor expansion of general time-separable utility functions around an efficient steady-state (see e.g. Woodford (2003a)).3Stochastic targets and preference shocks can also be incorporated by suitably expanding the vector yt.4 3 Optimal policy under Loose Commitment In a loose commitment setting it is assumed that policymakers do have access to a commitment technology, but at the same time they occasionally revise their plans. More formally, suppose that the occurrence of a re-optimization is driven by a two-state Markov stochastic process ηt=n1 with Prob. γ 0 with Prob. 1 −γ(3) 3In the presence of steady-state distortions, a purely quadratic objective can be obtained using a simple linear combinations of the structural equations approximated to a second-order. However, as shown by Debortoli and Nunes (2006), this requires imposing the so-called “timeless perspective assumption”, which contrasts with the loose commitment settings considered in this paper. For an alternative approach, see Schmitt-Grohe and Uribe (2005). 4In the companion code, models with more lags, leads, constants etc., are automatically transformed to be consistent with the formulation in equations (1) and (2). 4 At any given point in time if ηt= 1, previous commitments are honored. This event occur with probability 0 ≤γ≤1. Instead, if ηt= 0 the planner makes a new plan. This formulation nests both the full-commitment and discretion approaches as limiting cases where γ= 1 and γ= 0, respectively. More importantly, this formulation also spans the continuum between those two extremes. Considering stochastic re-optimizations is a necessary simplification to address large scale models. Such an assumption also seems justified if the timing of plans revisions is uncorrelated with the state of the economy. Possible candidates for such events are changes in the dominating view within a central bank due to time-varying composition of its decision-making committee or varying intensity of outside pressures by politicians and the financial industry.5Alternatively, our approach can be interpreted as the reduced form of a model in which commitment to a policy is sustained by the threat of a punishment in case of re-optimization. If the punishment requires a priori coordination among private agents and in some random periods cannot be implemented, then such a model may bear similarities with our approach.6 Following Schaumburg and Tambalotti (2007) and Debortoli and Nunes (2010a), the policymaker’s problem can be written as y0 −1Py−1+d= min {yt}∞ t=0 E−1 ∞ X t=0 (βγ)t[y0 tWyt+β(1 −γ) (y0 tPyt+d)] (4) s.t. A−1yt−1+A0yt+γA1Etyt+1 + (1 −γ)A1Etyr t+1 +Bvt= 0.∀t≥0 The objective function is given by an infinite sum discounted at the rate βγ, summarizing the history in which re-optimizations never occur. Each term 5In the case of the United States, the reserve bank presidents serve one-year terms as voting members of the FOMC on a rotating basis, except for the president of the New York Fed. Furthermore, substantial turnover among the reserve bank presidents and the members of the Board of Governors arises due to retirement and outside options. With the (up to) seven members of the Board of Governors being nominated by the U.S. President and confirmed by the U.S. Senate, the composition of views in the FOMC may be affected by the views of the political party in power at the time of the appointment. Chappell et al. (1993) and Berger and Woitek (2005) find evidence of such effects in the U.S. and Germany, respectively. 6Such a framework would build on the seminal contributions of Chari and Kehoe (1990), and Kehoe and Levine (1993). A related approach using a model of imperfect information is described in Sleet (2001). Most of these frameworks model the private sector as a representative household therefore avoiding the coordination problem. 5 in the summation is composed by two parts. The first part is the period loss function. The second part indicates the value the policymaker obtains if a re-optimization occurs in the next period. The policymaker faces a sequence of constraints, where in any period t expectations of future variables are an average between two terms. The first term (yt+1), with weight γ, relates to the allocations prevailing when current plans are honored. The second term yr t+1, with weight (1 −γ), refers to the choices made in period t+ 1 if a re-optimization occurs (i.e. if ηt+1 = 0). Analogously to the Markov-perfect literature, we assume that expectations about choices following a re-optimization only depend on state-variables. Etyr t+1 =˜ Hyt.(5) The policymaker cannot decide directly on the allocations implemented if a re-optimization occurs and therefore the matrix ˜ His taken as given. For any ˜ H, the policymaker problem can be solved using recursive methods. We follow the approach of Kydland and Prescott (1980) and Marcet and Marimon (2009), and write the Lagrangean associated with the policymaker’s problem L ≡ E−1 ∞ X t=0 (βγ)t(y0 t[W+ (1 −γ)βP ]yt+λ0 t−1β−1A1yt+(6) λ0 thA−1yt−1+³A0+ (1 −γ)A1˜ H´yt+Bvti) λ−1= 0 ˜ H, y−1given. This Lagrangean can be written recursively after expanding the state of the economy to include the Lagrange multiplier vector λt−1as co-states. The solution to the problem is then characterized by a time-invariant policy function ·yt λt¸=·Hyy Hyλ Hλy Hλλ ¸·yt−1 λt−1¸+·Gy Gλ¸vt,(7) where the matrices Hand Gdepend on the unknown matrix ˜ H. When a re-optimization occurs in a given period t, the vector λt−1must be reset to zero. This result has been formally proved by Debortoli and Nunes (2010a). According to the formulation in (6), this result has an intuitive 6 the total gains obtained from discretion to full-commitment. Therefore, any affine transformation of the central bank’s objective function would leave Figure 1 unchanged. As expected, higher credibility leads to higher welfare.14 More importantly, the figure suggests that if a central bank has low credibility to start with, a partial enhancement of its credibility will not deliver much of the welfare gains that credibility can potentially offer. Similarly, a central bank with high credibility should be especially cautious. It will face severe welfare losses if its credibility is deemed to have been minimally affected. These results contrast with those obtained by Schaumburg and Tambalotti (2007) using a more stylized monetary policy model.15 Figure 1: Welfare 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Alternative objective commitment probability − γ relative welfare 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Benchmark objective commitment probability − γ relative welfare Notes: The figure plots the relative welfare gains of increasing credibility from full discretion to a degree of commitment γ: (Vγ−Vγ=0)/(Vγ=1 −Vγ=0). The panel on the left corresponds to the benchmark objective function, whereas the right panel corresponds to the alternative welfare function with the interest rate level. The welfare measure corresponds to conditional welfare and the results are robust to unconditioning on the shocks. Credibility may also affect the relative contribution of inflation and outputgap volatilities to the overall welfare loss. A higher credibility level translates into better management of the policy trade-offs because forward guidance is 14This result is formally proven in Debortoli and Nunes (2010a). 15The result is instead consistent with those of Debortoli and Nunes (2010a). Also, as discussed there, the shape of the relative welfare gains change with the commitment metric. Here, we are considering and comparing results in the literature along the probability of commitment metric. 13 more effective as a policy tool. Therefore one might conjecture that higher credibility would reduce the volatilities of all welfare relevant variables. Figure 2 exemplifies that such a conjecture does not always hold. The figure shows that for a given relative weight in the objective function, a loss in credibility leads to a rise in inflation volatility but a reduction in outputgap volatility. The reason is that stabilizing inflation is the most important welfare objective. A central bank with high credibility can achieve a higher welfare by promising to stabilize inflation even if doing so implies more output-gap volatility. Figure 2: Credibility and volatility 0.1 0.2 0.3 0.4 0.5 0.6 4.6 4.8 5 5.2 5.4 5.6 5.8 6 σπ σy 0.5wπ 0.5wπ 1wπ 1wπ 2wπ 2wπ 4.5 5 5.5 6 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 σy σr 0.5wy 0.5wy 1wy 1wy 2wy 2wy γ = 0.5 γ = 1 Notes: the figure plots the volatilities of inflation, output-gap, and interest rate for different credibility levels. The left and right panel change the weight on inflation and output-gap, respectively. The two panels plot several weights from half to double of the benchmark value. The solid and dashed lines consider the probability of commitment to be 0.5 and 1, respectively. Figure 2 also discriminates among the points in the policy frontiers associated with doubling or halving wπor wyrelative to the baseline calibration. Doubling or halving the mentioned parameters leads to changes that are not extreme in magnitude. Thus, results are not overly sensitive to the calibration of the objective function. Also, the finding that a loss in credibility increases inflation volatility but reduces output-gap volatility holds for those extreme calibrations as well. 14 4.2 Loose commitment and simple interest rate rules The optimal policy under loose commitment can be implemented through targeting rules or through an appropriately defined interest rate rule.16 In DSGE monetary policy models it is instead common to adopt simple reducedform interest rate rules to describe the central bank’s behavior. Clearly, such behavior is affected by the degree of commitment γ. An open question is to see how changes in γare captured by the parameters of a simple rule. To address this question, we perform a Monte-Carlo exercise taking our model as the pseudo-true data generating process but estimating the interest rate rule it=φiit−1+φππt+φyyt+²t,(25) where ²tis assumed to be i.i.d. and normally distributed. Table 1 presents the regression results. The coefficient estimates are similar to those found in the literature using actual data. In most cases, the coefficient on output-gap is small (and in some cases not significant), the coefficient on inflation is plausible, and there is a considerable degree of interest rate smoothing.17 Most of the motive for interest-rate smoothing comes from commitment. Commitment implies that past policies matter for current allocations, thus introducing history dependence.18 As a result, when commitment is high, the estimated values of φiare high even under the alternative loss function, where per se there is no interest-rate smoothing motive. Overall, the coefficient φiis more plausible for relatively loose commitment settings rather than with full-commitment. Simple interest rate rules have been widely adopted to study the central bank behavior across different periods of time. In that respect, our exercise shows that a change in the interest rate parameters (φi, φπ, φy) should not be necessarily interpreted as a change in the central bank’s preferences. Even if preferences remain unaltered, the reduced form interest rate parameters may change because of a loss of credibility. 16Evans and Honkapohja (2003) discuss how interest rate rules can implement the optimal policy plan, while targeting rules are discussed by Giannoni and Woodford (2010) in a general framework and by Debortoli and Nunes (2010b) in a loose commitment setting. 17For comparability with some studies the coefficient on inflation and output-gap should be adjusted as φπ/(1 −φi) and φy/(1 −φi), respectively. 18For example, an optimal policy plan under full-commitment displays history dependence even when all the disturbances are i.i.d. and in the absence of natural state variables. See e.g. Gal´ı (2008, ch. 5). 15 Table 1: Interest rate regressions Benchmark Loss Function Alternative Loss Function 1 0.9 0.5 0 1 0.9 0.5 0 φπ0.241 0.207 1.204 1.914 0.175 0.057 0.725 2.334 (0.047) (0.103) (0.141) (0.048) (0.043) (0.138) (0.312) (0.072) φy0.002 -0.003 0.059 0.105 0.002 -0.010 -0.030 0.12 (0.003) (0.007) (0.014) (0.005) (0.002) (0.009) (0.033) (0.008) φi0.971 0.926 0.875 0.75 0.972 0.843 0.503 0.159 (0.022) (0.033) (0.038) (0.015) (0.022) (0.06) (0.062) (0.027) R20.923 0.865 0.843 0.977 0.921 0.759 0.416 0.930 Notes: The table displays the coefficients and standard deviations corresponding to estimating equation (25) in the original model. The Monte-Carlo exercise is comprised of 1000 estimations of 200 periods each (roughly corresponding to the size of actual samples). The average standard deviations across simulations are reported in parenthesis. The last row displays the R2. The panel on the left and the right correspond to the benchmark and alternative welfare functions, respectively. In general the simple rule (25) captures fairly well the interest rate behavior, as signaled by the high value of the R2. The R2is plausible but lower at intermediate degrees of commitment. The reason is that re-optimizations imply a non-linear change in the policy setting that the linear regression is not capturing well. The re-optimization uncertainty vanishes with fullcommitment or discretion, and therefore those two cases can be better described by a linear rule. Also, the R2is lower for the alternative specification of the loss function. In that case, the absence of an interest rate smoothing motive in the objective function causes the interest rate to change more abruptly when re-optimizations occur. This suggests that our benchmark loss function is more consistent with available estimates of the central bank behavior. 4.3 Business cycle properties under loose commitment We now analyze the effects of commitment on business cycle properties. Impulse responses to different shocks are reported in Figures 3-5. The probability of commitment is set to γ=.9, implying that policy reoptimizations occur on average every 10 quarters. The solid line considers the specific history where re-optimizations do not occur over the reported horizon (ηt= 1,∀t). 16 On impact, the sign of the responses does not change with the commitment assumption. However, for each of the shocks considered, after about 6 quarters the response of the nominal interest rate does not lie between full-commitment (dashed line) and discretion (dash-dotted line). These differences arise because of the uncertainty about future re-optimizations, a feature unique to loose commitment settings. For example, the interest rate response to a positive wage markup shock shown in Figure 3 peaks after about 10 quarters – as opposed to a negligible response at a similar horizon both under full-commitment and discretion. In turn, the output-gap response is more prolonged, while both price and wage inflation are close to the values prevailing under commitment. Intuitively, the promise of a deeper and longer recession dampens inflation expectations and helps achieve a higher welfare. When the central bank reoptimizes (line with crosses), it reneges upon past promises. It then reduces the interest rate, causing inflation to increase and the output-gap to decrease. The bottom right panel shows that the welfare gain of reoptimizing in a given quarter – a measure of the time-inconsistency at each moment in time – is maximum after roughly 9 quarters. The central bank is fulfilling the promise of a deep recession, which becomes especially costly at that time because inflation is already below target and the output-gap is at its lowest level. Similar reasoning also applies to productivity and government spending shocks.19 In response to the latter shocks – as well as to other demand-type shocks – the output-gap and the two measures of inflation are well stabilized. This occurs regardless of the degree of commitment, and as long as the central bank sets its policy optimally. This suggests that commitment would not be very important if these shocks were the main sources of business cycle fluctuations.20 Also, the time-inconsistency problem, measured by the gains from re-optimizations (bottom right panel), is much smaller in response to technology and government spending shocks than in response to wage markup shocks. Table 2 shows how commitment affects the second moments for some relevant variables. The correlation of output with the two measures of inflation is positive under full-commitment, it becomes negative at intermediate de19The responses to other shocks also present the same features and are omitted for brevity, but are available upon request. 20However, this result is not obvious in the current model. The presence of both price and wage rigidities implies a trade-off between inflation and output stabilization, and thus a scope for commitment, even in response to demand and technology shocks. 17 Figure 3: Impulse responses to a wage markup shock 0 5 10 15 20 −1.5 −1 −0.5 0Output gap 0 5 10 15 20 −0.02 0 0.02 0.04 0.06 0.08 Price inflation 0 5 10 15 20 −0.2 0 0.2 0.4 0.6 Wage inflation 0 5 10 15 20 −0.2 −0.1 0 0.1 0.2 Interest rate 0 5 10 15 20 0 0.05 0.1 0.15 0.2 Wage markup shock 0 5 10 15 0 0.005 0.01 0.015 Gains from reoptimization Full−Commitment Loose commitment (γ=.9) Discretion Reoptimization Notes: The figure plots the impulse responses to a one standard deviation shock, under different commitment settings. The solid line refers to a particular history where the probability of commitment γ=.9 and re-optimizations do not occur (ηt= 1,∀t). The line with crosses refers to a particular history where the probability of commitment γ=.9 and a single re-optimization occurs after 10 quarters (η10 = 0, ηt= 1,∀t6= 10). For any quarter, the gains from re-optimization are computed as the welfare difference between keeping the announced plan vs reoptimizing in that particular quarter. 18 Figure 4: Impulse responses to a productivity shock 0 5 10 15 20 −0.1 0 0.1 0.2 0.3 Output gap 0 5 10 15 20 −0.04 −0.02 0 Price inflation 0 5 10 15 20 −0.02 0 0.02 0.04 0.06 0.08 Wage inflation 0 5 10 15 20 −0.2 −0.15 −0.1 −0.05 0 Interest rate 0 5 10 15 20 0 0.1 0.2 0.3 0.4 Productivity 0 5 10 15 0 0.5 1x 10−3 Gains from reoptimization Full−Commitment Loose−Commitment (γ=.9) Discretion Reoptimization Notes: The figure plots the impulse responses to a one standard deviation shock, under different commitment settings. The solid line refers to a particular history where the probability of commitment γ=.9 and re-optimizations do not occur (ηt= 1,∀t). The line with crosses refers to a particular history where the probability of commitment γ=.9 and a single re-optimization occurs after 10 quarters (η10 = 0, ηt= 1,∀t6= 10). For any quarter, the gains from re-optimization are computed as the welfare difference between keeping the announced plan vs reoptimizing in that particular quarter. 19 Figure 5: Impulse responses to a government spending shock 0 5 10 15 20 −0.05 0 0.05 0.1 0.15 Output gap 0 5 10 15 20 −1 0 1 2 3x 10−3 Price inflation 0 5 10 15 20 −10 −5 0 5x 10−3 Wage inflation 0 5 10 15 20 0 0.02 0.04 0.06 Interest rate 0 5 10 15 20 0 0.2 0.4 0.6 0.8 Gov’t expenditure shock 0 5 10 15 0 0.5 1 1.5 x 10−5 Gains from reoptimization Full−commitment Loose Commitment (γ=.9) Discretion Reoptimization Notes: The figure plots the impulse responses to a one standard deviation shock, under different commitment settings. The solid line refers to a particular history where the probability of commitment γ=.9 and re-optimizations do not occur (ηt= 1,∀t). The line with crosses refers to a particular history where the probability of commitment γ=.9 and a single re-optimization occurs after 10 quarters (η10 = 0, ηt= 1,∀t6= 10). For any quarter, the gains from re-optimization are computed as the welfare difference between keeping the announced plan vs reoptimizing in that particular quarter. 20 grees of commitment. The reason is that under full-commitment output and inflation are positively correlated not only conditionally on demand shocks, but also conditionally on technology and markup shocks. In response to the latter shocks output and inflation move in opposite directions on impact, but after about 5 quarters they comove. Instead, with loose commitment, especially if a re-optimization has occurred, inflation and output move in opposite directions for a longer horizon. As a result, the correlation between inflation and output conditional on non-demand shocks, as well as the unconditional counterpart, changes sign with even a small departure from the full-commitment assumption.21 Even though it is beyond the scope of this paper to estimate the model and the degree of credibility, the loose commitment model matches some key moments relatively well. Table 2 shows that in the data the correlation between output and price inflation is mildly negative, whereas the correlation between output and wage inflation is mildly positive – a feature that the loose commitment model with γ= 0.9 matches quite well. In addition, the relative volatility of interest rates is also more plausible with limited commitment settings. Finally, loose commitment changes the relative contribution of alternative shocks to business cycle fluctuations, as summarized in Figure 6. This pattern is mostly evident for interest rate fluctuations. Under full-commitment about 55% of the fluctuations can be attributed to demand shocks. A small loss of credibility (γ=.9) is enough for this proportion to drop dramatically to about 16%. The contribution of wage and price markup shocks increases from 20% to 70%. The reason is that the interest-rate response to a demand shock does not change much with the degree of commitment. Instead, in response to markup shocks the interest rate barely responds under commitment, while it increases and remains high for a long period in limited commitment settings. For almost all the other variables, when commitment is lower, price markup shocks lose importance and wage markup shocks become more relevant. Hence, the variance decompositions and the earlier plots measuring time-inconsistency suggest that commitment is particularly important to stabilize wage markup shocks. In summary, loose commitment has important effects on price and wage inflation dynamics, and nominal interest rates – the main variables for which 21The conditional cross-correlations are omitted for brevity and are available upon request. 21 Table 2: Effects of loose commitment on second moments Model U.S. Data Full-Com. Loose Commitment Discr. (1970 - 2008) 0.9 0.5 Standard deviation (w.r.t. output) Output-gap 0.83 0.84 0.83 0.83 0.74 Price inflation 0.04 0.04 0.06 0.07 0.21 Wage inflation 0.08 0.08 0.08 0.09 0.26 Interest rate 0.09 0.15 0.21 0.18 0.29 Cross-correlations with output Output-gap 0.87 0.88 0.86 0.86 0.90 Price inflation 0.05 -0.17 -0.66 -0.70 -0.13 Wage inflation 0.21 0.13 -0.29 -0.38 0.05 Interest rate -0.34 -0.49 -0.56 -0.56 -0.32 Notes: The table displays several statistics for the output-gap, inflation, wage inflation, and the interest rate. The model statisitics are computed with 1000 simulations of 200 periods each. The sample regarding the real data goes from 1970:Q1 until 2008:Q3, where the latest data is determined by the beginning of the zero lower bound period. The output-gap data corresponds to the CBO measure. the central bank is responsible. The impulse responses to different shocks, as well as the interest rate volatility, is not necessarily in between fullcommitment and discretion. Finally, small departures from full-commitment change the sign of the correlation between output and inflation. In addition, the relative contribution of wage markup shocks to business cycle fluctuations increases dramatically, especially for interest rates and inflation. 5 Conclusions Imperfect commitment settings overcome the dichotomy between fullcommitment and discretion. In practice, policymakers have some degree of commitment that is not perfect – in some cases they keep a previously formulated policy plan whereas in other cases they reformulate those plans. Recent proposals of imperfect commitment settings were restricted to relatively simple and stylized models. The contribution of this paper is to propose a method and a toolkit that extends the applicability of loose commitment to mediumand large22