Interest-Rate Rules in a New Keynesian Framework with Investment

Pavlova, Elena Book Interest-Rate Rules in a New Keynesian Framework with Investment Schriften zur Wirtschaftstheorie und Wirtschaftspolitik, No. 44 Provided in Cooperation with: Peter Lang International Academic Publishers Suggested Citation: Pavlova, Elena (2012) : Interest-Rate Rules in a New Keynesian Framework with Investment, Schriften zur Wirtschaftstheorie und Wirtschaftspolitik, No. 44, ISBN 978-3-653-01444-0, Peter Lang International Academic Publishers, Frankfurt a. M., https://doi.org/10.3726/978-3-653-01444-0 This Version is available at: https://hdl.handle.net/10419/178445 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/4.0/ Peter Lang Internationaler Verlag der Wissenschaften Lang elena Pavlova · Interest-rate rules in a new Keynesian Framework with Investment 44 Sc h r i f t e n z u r Wi r t S c h a f t S t h e o r i e u n d Wi r t S c h a f t S p o l i t i k 44 elena Pavlova Interest-rate rules in a new Keynesian Framework with Investment The last decades have witnessed major progress in both monetary policy theory and practice, with broad academic consensus on the desirability of monetary policy rules and ongoing research on their exact specification. Typically, the analysis is carried out in a New Keynesian framework with nominal rigidities and constant capital stock. The latter represents a constraint that this study seeks to overcome by introducing a model with investment and capital adjustment costs. The work assesses different interest-rate rule specifications with respect to the target variables included, based on two criteria: determinacy of rational-expectations equilibrium and convergence to steady state after a shock. The study concludes that rules with both an inflation and an output gap target ensure a unique rational-expectations equilibrium and a less distressful adjustment of the economy after the occurrence of shocks. Elena Pavlova studied Economics at the University of National and World Economy in Sofia. During her studies, she joined as a Researcher the Centre for Economic Development, a leading economic policy think tank in Bulgaria. In 2004–2008 she was a Teaching and Research Assistant at the Institute for Theoretical Economics at the Helmut Schmidt University Hamburg. Since 2008 the author works as an Economist at the European Commission in Brussels. Rüze=12,2pt www.peterlang.de ISBN 978-3-631-61128-9 SWWP 44-Pavlova-261128-HCA5-AK.indd 1 24.02.11 14:00:11 Uhr HKS 41 HKS 04 Peter Lang Internationaler Verlag der Wissenschaften Lang elena Pavlova · Interest-rate rules in a new Keynesian Framework with Investment 44 Sc h r i f t e n z u r Wi r t S c h a f t S t h e o r i e u n d Wi r t S c h a f t S p o l i t i k 44 elena Pavlova Interest-rate rules in a new Keynesian Framework with Investment The last decades have witnessed major progress in both monetary policy theory and practice, with broad academic consensus on the desirability of monetary policy rules and ongoing research on their exact specification. Typically, the analysis is carried out in a New Keynesian framework with nominal rigidities and constant capital stock. The latter represents a constraint that this study seeks to overcome by introducing a model with investment and capital adjustment costs. The work assesses different interest-rate rule specifications with respect to the target variables included, based on two criteria: determinacy of rational-expectations equilibrium and convergence to steady state after a shock. The study concludes that rules with both an inflation and an output gap target ensure a unique rational-expectations equilibrium and a less distressful adjustment of the economy after the occurrence of shocks. Elena Pavlova studied Economics at the University of National and World Economy in Sofia. During her studies, she joined as a Researcher the Centre for Economic Development, a leading economic policy think tank in Bulgaria. In 2004–2008 she was a Teaching and Research Assistant at the Institute for Theoretical Economics at the Helmut Schmidt University Hamburg. Since 2008 the author works as an Economist at the European Commission in Brussels. Rüze=12,2pt www.peterlang.de SWWP 44-Pavlova-261128-HCA5-AK.indd 1 24.02.11 14:00:11 Uhr HKS 41 HKS 04 Interest-Rate Rules in a New Keynesian Framework with Investment Peter Lang Frankfurt am Main ∙ Berlin ∙ Bern ∙ Bruxelles ∙ new York ∙ Oxford ∙ Wien Sc h r i f t e n z u r Wi r t S c h a f t S t h e o r i e u n d Wi r t S c h a f t S p o l i t i k Herausgegeben von Klaus Beckmann, Michael Berlemann, rolf Hasse, Jörn Kruse, Franco reither, Wolf Schäfer, thomas Straubhaar und Klaus W. Zimmermann Band 44 Peter Lang Internationaler Verlag der Wissenschaften elena Pavlova Interest-rate rules in a new Keynesian Framework with Investment Bibliographic Information published by the Deutsche Nationalbibliothek Gratefully acknowledging financial support by Helmut Schmidt University / University of the Federal Armed Forces Hamburg. D 705 ISSN 1433-1519 ISBN 978-3-631-61128-9 © Peter Lang GmbH Internationaler Verlag der Wissenschaften Frankfurt am Main 2011 www.peterlang.de The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the internet at http://dnb.d-nb.de. Open Access: The online version of this publication is published on www.peterlang.com and www.econstor.eu under the international Creative Commons License CC-BY 4.0. Learn more on how you can use and share this work: http://creativecommons.org/licenses/by/4.0. This book is available Open Access thanks to the kind support of ZBW – Leibniz-Informationszentrum Wirtschaft. ISBN 978-3-653-01444-0 (eBook) Acknowledgements This book is based on research carried out during my activity as Teaching and Research Assistant at the Institute for Theoretical Economics at the Helmut Schmidt University. It has been accepted as PhD thesis by the Helmut Schmidt University. I am exclusively responsible for any errors and omissions. Preparing and successfully completing a PhD thesis has been an intensive learning experience for me, which required intellectual curiosity, innovative thinking, hard work and persistence. My efforts have been supported by a number of colleagues and friends. In the first place, I would like to express my deep gratitude to my academic advisor Prof. Dr. Franco Reither, who has offered continuous and most valuable guidance and support during all stages of the process in the form of in-depth thematic exchanges, targeted insights and excellent cooperation. In addition, I would like to thank my second advisor, Prof. Dr. Barbara Dluhosch and Prof. Dr. Klaus Zimmermann for their open attitude and the interesting discussions. I would also like to acknowledge the moral support and assistance of all my colleagues at the Helmut Schmidt University, in particular Lars Bennöhr and Sabine Michels. Finally, I would like to thank my family for their motivating influence and their understanding. Their appreciation and firm belief in my dreams have been the basis for any professional success so far. This is why I would like to dedicate this book to my parents and to my partner Peter. Brussels, February 2011 Elena Pavlova List of Symbols Chapter III Latin symbols A coefficient matrix Bcolumn vector of constant parameters E expectation operator gstochastic shock (IS relation) Iidentity matrix i nominal interest rate mnumber of non-predetermined variables m number of eigenvalues of the coefficient matrix outside the unit circle r equilibrium (steady-state) real interest rate tarbitrary point in time ustochastic shock (AS relation) X vector of predetermined variables Y vector of non-predetermined variables y actual output y potential/steady-state output  y output gap Z vector of exogenous variables Greek symbols D price stickiness parameter J coefficient matrix T eigenvalue * S O effective inflation response coefficient (monetary policy rule) y O output gap response coefficient (monetary policy rule) S inflation rate * S inflation target M interest-rate elasticity of the output gap : information set Other symbols  set of real numbers n  n-dimensional Euclidian space Chapter IV Latin symbols A(labour-augmenting) technology shock a matrix coefficient bone-period government bonds Ccapital adjustment costs C steady-state unit adjustment cost 1 C steady-state marginal adjustment cost cconsumption c  deviation of consumption from its steady-state level E expectation operator Fcoefficient matrix Gcoefficient matrix ggovernment consumption of goods and services Hcoefficient matrix i nominal interest rate i steady-state nominal interest rate inv investment inv steady-state investment m inv deviation of investment from its steady-state level Jcoefficient matrix kcapital k steady-state capital  k deviation of capital from its steady-state level lleisure m real money balances mpk marginal product of capital mpk steady-state marginal product of capital mpl marginal product of labour n labour input in the production function A n d aggregate labour n labour demand P A price of the household’s product P aggregate price level rreal interest rate r steady-state real interest rate rmc real marginal cost rmc steady-state real marginal cost n rmc deviation of real marginal cost from its steady-state level W A household’s nominal wage W aggregate nominal wage 14 y actual output y potential/steady-state output  y A Y output gap aggregate demand n y natural rate of output Greek symbols PF D elasticity of substitution between capital and labour E discount factor i H monetary policy unit shock G capital depreciation rate K elasticity of total adjustment costs with respect to investment P K probability that households cannot adjust their price W K probability that households cannot adjust their nominal wage 12 ,44 adjustment-cost function parameters T elasticity of substitution between differentiated goods W T elasticity of substitution between differentiated labour units - Lagrange multiplier O Lagrange multiplier i O interest-rate smoothing coefficient (interest-rate rule) S O inflation response coefficient (interest-rate rule) * S O effective inflation response coefficient (interest-rate rule) y O output gap response coefficient (interest-rate rule) * y O effective output gap response coefficient (monetary policy rule) ; semi-elasticity of investment with respect to the real asset’s premium [ scale parameter (investment adjustment costs) S inflation rate S steady-state inflation rate A U autocorrelation coefficient (technology shock) X U autocorrelation coefficient (consumption preference shock) 1 intertemporal elasticity of substitution in consumption V  9 consumption preference shock W leisure relative risk aversion coefficient X consumption preference shock Z real wage rate Y Lagrange multiplier c Z steady-state share in output of consumption g Z steady-state share in output of government expenditure inv Z steady-state share in output of investment 15 I. Introduction At the beginning of the 21st century, after the wide-scale collapse of centrally planned economies, the consensus perception prevails that prosperity and economic growth are generated by private enterprise and free markets. Nevertheless, government policy has maintained its role as a major factor, responsible for creating the necessary conditions for promoting enterprise and growth. In this context, monetary policy has emerged as an important means for achieving these goals1. The arguments for this statement are twofold, concerning both how quickly and how accurately the intervention takes effect on the market. In the first place, unlike fiscal policy, which often serves multiple (sometimes conflicting) goals and may be subject to political influences and lengthy legislative decision-making and approval procedures, monetary policy conducted by an independent central bank can be adjusted relatively quickly to respond to the latest macroeconomic developments. Furthermore, the impact of monetary impulses especially on the financial markets under a sufficient degree of central bank credibility takes place immediately. Sometimes the financial market response even precedes the actual central bank intervention, as market participants anticipate the envisaged measures and act accordingly in advance. The last decades have witnessed major transformations pertaining to both monetary policy theory and practice. Since the Bretton Woods collapse central banks exposed not only to a higher degree of freedom, but also to the need to define clear monetary policy goals and communicate them to the public. In the last two decades, a growing number of central banks (such as the Bank of England, Bank of Canada, the Reserve Bank of New Zealand and the Swedish Riksbank) have opted for systematic policy behaviour by means of introducing inflation-targeting. On the theoretical side, major advances have been made in the last two decades. One facet of the new consensus on monetary policy is that low, stable inflation is crucial for market-driven growth and that the monetary policy stance in the medium to long run is the major determinant of inflation. After a long period of focusing on the impact of non-monetary factors on the business cycle, empirical studies since the late 1980s have argued that monetary policy significantly influences the short-term course of the economy. Another facet is the strengthened focus on monetary policy design and the interest for optimal rule-based monetary policy in particular. Recent macroeconomic research features nominal rigidities and output fluctuations and focuses on the stabilization role of monetary policy, by allowing the monetary authorities to choose from a 1 Or, as Bernanke et al. (1999) argue: “… of all the government’s tools for influencing the economy, monetary policy has proven to be the most flexible instrument for achieving medium-term stabilization objectives.” variety of monetary policy rule specifications2 in terms of policy instruments, target variables and size of the response coefficients assigned to the target variables. Since the results obtained in the literature when assessing the different monetary policy rule specifications are to a large extent model-dependent, the choice of a macroeconomic framework based on sufficiently realistic assumptions is crucial for the analysis of the implications of different rule specifications. Building on the arguments of the New Classical Critique3 in the 1970s, New Keynesian models that incorporate rational expectations, as well as microeconomic foundations, have been developed. The optimizing behaviour on the part of households and firms, as well as the intertemporal methodology of New Keynesian models with nominal rigidities enable detailed study of the monetary transmission mechanism and optimal monetary policy design. However, with respect to investment and capital, most of these models (e.g. Woodford (1995), Rotemberg and Woodford (1999) and McCallum and Nelson (1999b)) abstract from investment (constant-capital specification). One reason is that introducing endogenous capital and investment to a model with sticky prices may lead to multiple rational-expectations equilibria under certain monetary policy rule specifications4. Moreover, the exclusion of capital is often justified on the grounds that capital does not exhibit substantial volatility at business cycle frequencies (e.g. McCallum and Nelson, 1997). However, such an approach is clearly unsatisfactory, as it leaves out an important monetary transmission channel and shock propagation mechanism. In the following chapters, the analysis is carried out within a New Keynesian framework with endogenous capital, sticky prices and wages and capital adjustment costs. The purpose of this study is to assess different interest rate rule specifications with respect to the degree of activeness (measured by the inflation response coefficient) and the target variables included, based on two criteria: (i) the existence of a determinate rational expectations equilibrium and (ii) the characteristics of the convergence path towards steady state after a shock occurs. In particular, policy rule specifications that yield determinacy of rational expectations equilibrium and in addition involve quantitatively smaller deviations and fast, monotonic convergence after a shock occurs would be preferred. The results obtained confirm that the introduction of endogenous capital and investment has important implications for the monetary policy outcomes. A stronger than one-on-one nominal interest rate response to inflation in the policy rule (i.e. 2 The most famous example in recent years being the Taylor rule as in Taylor (1993). 3 The New Classical Critique focused on the use of conventional methods of econometric policy evaluation (Lucas (1976)) and of optimal control (Kydland and Prescott (1977)). In general, according to the real-business cycle theory, monetary policy has no relevance for economic welfare when rational expectations of economic agents are assumed. 4 E.g. forward-looking rules (see Huang and Meng (2007)). 18 19 adherence to the Taylor principle) does not per se guarantee the best outcomes in terms of equilibrium uniqueness and responses to shocks. Under endogenous capital and investment, the inclusion of an output target in the policy rule is crucial under both criteria. The study is organised as follows. In Chapter II, I summarise the main issues in monetary policy theory and practice, including the possible rule specifications (Section 1). In Section 2, I give an overview of the criteria for assessing the performance of monetary policy rules. The focus is on determinacy of rational expectations equilibrium and the response to shocks, explicitly used for the analysis in Chapters III and IV. In the last subsection I present formally the Taylor principle, which since 1999 has been a benchmark for formulating rulebased policy and forms the basis for the distinction between “active” and “passive” policy rules made in the subsequent chapters. Section 3 provides a preliminary summary. In Chapter III I derive the New Keynesian framework with sticky prices and wages, endogenous capital and investment and capital adjustment costs and study the system’s determinacy properties under different values assigned to the inflation and output gap response coefficients in the interest-rate rule. In particular, Section 1 provides an overview to the New Keynesian framework, while Section 2 presents the main approaches to modelling capital and investment in the literature. Then in Section 3, I concentrate the model with endogenous capital and adjustment costs, by examining the household optimisation problem and the resulting first-order conditions, deriving the “IS block” equations, the aggregate supply and real-wage relation and adding an interest rate rule to the system. As a next step, in Section 4 I complete the calibration of the model, so as to permit quantitative analysis of its properties. In addition, I provide some numerical analysis of the systems’ determinacy properties under different rule specifications. The findings from Section 4 in Chapter III are then considered when assessing the shock impulse responses under different monetary policy specifications in Chapter IV. Active and passive rules in three possible specifications for each class are tested in this chapter: (i) rules with a sole inflation target; (ii) rules with an inflation and output target and (iii) rules with inflation and output gap target and interest-rate smoothing. Results are obtained for three types of shocks: (i) a monetary policy unit shock; (ii) a technology unit shock and (iii) a consumption preference shock. The results are summarised in Section 4. Chapter V summarises the main findings and concludes. II. Monetary policy design and criteria for assessing monetary policy rules In this chapter, I provide a more general perspective to the main theoretical and practical issues in monetary policy design, which are relevant for the assessment of the several interest rate rule specifications in the subsequent parts. The issues covered in Section 1 include the advantages and implications of systematic policy behaviour (as opposed to discretionary measures) and choice of instruments/target variables that enter the rule specification. In addition, Section 2 presents the main criteria that will be used in the subsequent determinacy and impulse response analysis in Chapters III and IV. The Taylor principle, which forms the basis for the classification of monetary policy rules in terms of the degree of their “activeness” (measured by the size of the inflation response coefficient), is presented formally in the last subsection. Later on, in Section 4 in the subsequent chapter the main findings will be centred on the question whether adherence to the Taylor principle guarantees a determinate rational-expectations equilibrium when endogenous capital with adjustment costs is introduced to a New Keynesian model with staggered price-and wage-setting. 1. Monetary policy issues In the 1960s and 1970s “activist” monetary policies, aimed at achieving “full employment” have been widely discussed and implemented5. The rise of such policies has been motivated by the conviction that there exists a stable long-run trade-off between inflation and unemployment, captured by the Phillips curve6. According to this view, in the long run the monetary authority can attain a permanent reduction in the unemployment rate by allowing for a higher rate of inflation. At the same time, estimates of these (allegedly) stable relations between inflation and unemployment have been computed by using large econometric models that assigned precise quantitative dimensions to the policy trade-off. The actual experience with the activist pursuit of full employment by monetary policy means has contradicted the policy-makers’ anticipated outcome. Not only did the business cycle fluctuations not disappear in the 1970s, but the worldwide recessions of 1973-74 and 1981-82 were among the most severe in the second half of the twentieth century, characterised by high unemployment and inflation (“stagflation”). Thus, in spite of not attaining the aspired policy objective, the “sacrifice” in terms of higher inflation rate has still been realised: the late 1960s and most of the 1970s were characterised by rising and variable rates of inflation in many countries. 5 For a more detailed discussion of monetary policy strategies in the 1960s and 1970s, see Mishkin (2006). 6 See Phillips (1958) and Samuelson and Solow (1960). Parallel to the practical experience with activist policies, there has also been an academic critique of using monetary policy as a tool to obtain full employment at the expense of a higher inflation rate. Among the most influential arguments are Milton Friedman’s monetary critique concerning the uncertain outcomes of monetary policy interventions7, the Lucas critique of the optimal control paradigm for monetary policy8, the conclusion reached by Friedman (1968) and Phelps (1968) that there is no long-run trade-off between inflation and unemployment, as well as the warning against the perils of time inconsistency under discretionary policy delivered by Kydland and Prescott (1977), Calvo (1978) and Barro and Gordon (1983). Without denying the significant impact of monetary policy on the economy, Friedman argued that monetary policy is a tool that cannot be used with precision for stabilisation purposes. In “The Role of Monetary Policy” he explicitly warns against “the belief that the state of employment itself should be the proximate criterion of policy”, adding that …I fear that… the pendulum may well have swung too far, that… we are in danger of assigning to monetary policy a larger role than it can perform, in danger of asking it to accomplish tasks that it cannot achieve, and, as a result, in danger of preventing it from making the contribution that it is capable of making (Friedman, 1968, p. 5). Friedman emphasised that, due to long and variable lags of monetary policy, a too strong policy response might have a destabilising effect on the economy. This forms the basis for the subsequent discussion on the appropriate degree of activism of monetary policy. As argued by Blanchard and Fischer (1993), under long and variable lags very strong policy responses to shocks could create instrument instability. Instrument instability arises when the current effects of changes in the monetary policy instrument are small and the lagged effects large, so that large changes in the policy instrument are required to offset the effects of a recent shock, creating the need for even larger changes later on. An even more powerful argument for more moderate policy responses has to do with uncertainty about the structural coefficients in the model, i.e. to what extent the variability of the instrument increases the variability of the target variables. Although Friedman’s criticism emphasises the technical difficulties in controlling the policy outcomes, it does not fundamentally rule out activist central bank behaviour. If the pitfalls of the active pursuit of short-run output stabilisation had been predominately of instrumental nature, implementing more elaborate methods such as the (at that time increasingly popular) techniques of optimal control would have been sufficient to compensate for lags between policy measures and their effects. Thus, for policy activism to be ruled out as an appropriate central bank strategy, a formal methodological critique of the proposed 7 See Friedman and Schwartz (1963). 8 See Lucas (1976). 22 iour of some monetary authorities can be described by an explicit rule20, the central banks of some major economies nowadays apply explicit inflation targets (inter alia, the ECB, the Bank of England and Bank of Canada). In the subsequent exposition, I will concentrate on interest-rate rules for conducing monetary policy. Although other definitions of a monetary policy rule could be possible, for the purpose of this work I adopt the definition of Taylor (1998): “…a monetary policy rule is defined as a description—expressed algebraically, numerically or graphically-of how the instruments of policy, such as the monetary base or the federal funds rate, change in response to economic variables” (Taylor, 1998, p.3). Furthermore, for the subsequent analysis it is necessary to provide a definition of instruments, targets, goals/objectives and indicators of monetary policy. From a practical perspective, Borio (1997) and Ho (2008) distinguish between the strategic and tactical level of the monetary authorities’ pursuit of policy goals (see Figure 2.1). At the strategic level, certain macroeconomic goals or final objectives are pursued, such as price stability, long-term growth or employment. By contrast, the tactical (operating) level covers the choice of instruments and operating objectives/targets. The latter are variables which can be influenced quite fast and closely by the central bank21. Examples of monetary policy instruments include the official interest rates (e.g. on standing facilities), market operations (e.g. repo tenders, FX operations), reserve requirements and, in the past, direct controls (e.g. ceilings on loans or on bank depositor loan rates). According to McCallum (2001b), instruments are variables that the central bank can control quickly and directly or at least accurately, the usual contenders being short-term interest rates, the monetary base or a measure of bank reserves. Goals/Objectives are variables that enter the central bank objective function. In case the ultimate goals are not promptly observable, the central bank employs a two-step procedure, attempting to hit target variables that are treated as surrogates for the ultimate objectives. Examples for target variables are the monetary aggregates M1, M2 and nominal GDP. Indicators are information variables, for which the central bank does not try to hit specified paths. 20 For example, the Taylor rule (Taylor (1993)) seems to have described the actions of the Fed remarkably accurately during most of the 80s and 90s of the past century. 21 For example, short-term financial market interest rates, exchange rates, etc. 29 Figure 2.1: The monetary policy framework (Borio (1997) 1.2.2. Choice of instruments A long-standing discussion in monetary policy research has concentrated on the issue whether the monetary authority should use money22 or the interest rate to target inflation and/or output. Money supply as a policy instrument gained increasing popularity in the 1960s and 1970s. In fact, in his policy proposals Friedman went even further and argued for a rule involving money supply not only as an instrument, but as a policy target as well. Actually, Friedman (1960) acknowledged that a constant money growth rule does not correspond to optimal policy. Still, arguments such as imperfect knowledge about the “true” objectives inflation and output, the possibly inaccurate estimation of the unobserved natu- 22 In the form of a monetary aggregate. 30 ral rates of interest, output or employment and their dynamics, as well as the difficulties with obtaining real-time measures of policy objectives seemed to support choosing a rule expressed in terms of money supply only. Of course, the above arguments to a greater extent address the question whether the central bank should target an intermediate variable or its final objectives directly and provides less insight to the choice of instruments. A further contribution by Poole (1970) analyses the conditions under which pegging money or interest rates are appropriate. For a static IS-LM framework, Poole suggested that if disturbances originate primarily in the money demand, fixing the level of the interest rates is appropriate. For the case of goods market shocks, the money stock should be pegged. Despite the simplicity of the framework and the lack of supply side in particular, the Poole model provides an essential insight to understanding the choice between money and interest rates as monetary policy instruments. Taylor (1995, 1998) suggests that a constant money growth rule will generally induce an interest rate response to inflation and output similar in form as in the case of interest rate policy rules, but not necessarily similar in size. For example, for very high or negative inflation rate, the resulting variability of inflation expectations can render interest rate rules less efficient than controlling the money supply23. That is why, even when pursuing an interest rate rule, it is advisable for the central bank to still monitor money supply. Recent models, such as Casares and McCallum (2006), McCallum and Nelson (1999b) and Rotemberg and Woodford (1998a) all include an interest rate instrument, whereby the money supply is endogenous since the central bank must vary it in order to sustain its desired interest rate level. In these models the path for money growth followed by the monetary authority in the long run under an interest rate instrument exactly coincides with the one that would be followed under a money supply instrument. In line with the choice of instrument in these studies and the recent central bank practice, the monetary policy rules that I introduce in the next chapter include the nominal interest rate as a policy instrument. 1.2.3. Choice of target variables A second issue in policy design (apart from the choice of the instrument) concerns the choice of target variables. In this subsection, the variables entering the rule specifications in Chapter III (inflation, the output gap and the lagged interest rate) will be briefly discussed from a theoretical point of view. 23 As Taylor (1995) shows, in such circumstances interest rate rules can break down completely. 31 Inflation-targeting There has been consensus in the literature on the fact that in the recent years a growing number of countries has opted for some form of inflation targeting24 combined with central bank independence25. Inflation-targeting is subject to differing definitions in the literature. For example, according to Svensson (2002) inflation targeting involves stabilizing inflation around an inflation target26. With respect to the target variables selected, it could be distinguished between inflation targeting in a narrower sense27 (involving a central bank reaction to inflation deviations from a target level only) and a broader definition where inflation and a measure of real economic activity are both considered. Adhering to the former perspective, Bernanke et al. (1999) define inflation-targeting as a framework, rather than a rule for monetary policy28. According to them, inflation-targeting is characterised by public announcement of official quantitative targets (or target ranges) for the inflation rate over one or more time horizons, and by explicit acknowledgement that low, stable inflation is monetary policy’s primary long-run goal. Arestis and Sawyer (2003)29 offer a definition of inflation targeting complemented by certain institutional requirements. Thus, inflation targeting involves more than just targeting the rate of inflation as an objective of economic policy and implies: (i) setting a numerical target range for the inflation objecttive; (ii) using monetary policy as an instrument to achieve the target by adjusting the nominal interest rate; (iii) central bank independence; and (iv) monetary policy only targeting the inflation rate with the possible effects of monetary policy on other objectives ignored, with the exception of short-term effects. 24 Snowdon and Vane (2005) distinguish between four types of monetary regimes that have been implemented since the middle of the 20th century: exchange rate targeting (e.g. in the UK, 1990-1992), monetary targeting (e.g. in the UK, 1976-1987), explicit inflation targeting (UK, 1992-to date) and implicit inflation targeting (USA).The difference between explicit and implicit inflation targeting pertains to whether the central bank announces an exact inflation target. In this sense, the ECB policy would qualify as “explicit inflation targeting”. 25 For more detailed contributions on modern central bank practice issues, see Alesina and Summers (1993), Fischer (1995a, 1995b, 1996), Bernanke and Mishkin (1992, 1997), Bernanke and Woodford (1997), Bernanke et al. (1999), King (1997), Svensson (1997a 1999b, 2000), Mishkin (1999, 2000), Bernanke and Woodford (2006). 26 P. 6. 27 The inflation-targeting rules referred to in Chapter IV abide by this narrower definition of the term. 28 In the terms of Bernanke et al. (1999) inflation-targeting as a framework for monetary policy implies that the inherent discipline of a rule is extended by maintaining some degree of flexibility. 29 See p. 2. 32 The broader definition of inflation targeting with reference to the target variables entering the central bank’s loss function (or monetary policy rule) is denoted as “flexible” inflation targeting30. Svensson (1997b, 2002) argues that in the central bank practice inflation targeting is flexible31, as it also involves some concern about the stability of the real economy. The latter means that monetary policy should contribute to the welfare of the representative citizen. However, as this objective is not operational, stabilising output around potential output is in-cluded in the central bank reaction function32. There exists a general consensus in the recent empirical literature that maintaining a low and stable rate of inflation is an appropriate monetary policy objective33. Yet the justification of such a policy emphasis from a theoretical point of view may not be straightforward. According to the real-business-cycle models of the 1980s relative prices rather than the absolute level of prices are relevant for the allocation of resources in the economy34. Traditional Keynesian models, by contrast, postulate that variations in the growth rates of prices and wages induce variations in output and employment35. However, this inflationoutput relation has typically been considered as an argument in favour of achieving output and employment goals rather than a justification for establishing price stability as a primary objective of monetary policy. With the introduction of rational expectations to a framework with nominal rigidities, keeping inflation low and less volatile “locks in” expectations about future inflation and helps to contain the possible inflationary impact of macroeconomic shocks. Furthermore, in the short run and with sluggish price adjustment inflationary impulses can have a destabilising impact on output and employment. In this line, Woodford (2003) argues that, since instability of the general level of prices causes substantial real distortions – leading to inefficient variation both in aggregate employment and output and in the sectoral composition of economic activity –price stability is important and should therefore be the primary aim of monetary policy. In addition, the theoretical literature so far suggests a distinction between the costs of anticipated and unanticipated inflation. Anticipated inflation causes loss of social welfare because it promotes economising on real money balances, generates costs for frequent price adjustments and increases relative price uncer- 30 As opposed to “strict” inflation targeting, with low and stable inflation being the only goal of monetary policy (i.e. when the reaction coefficient to inflation solely is different from zero). 31 See pp. 6-7. 32 For further analysis of the output target, see Subsection 3.2.3.2. 33 See, for example, Fischer (1996) and Mishkin (2000). 34 Real business cycle authors support „real“ as opposed to “monetary” theories of fluctuations. For example, Kydland and Prescott (1982) and Prescott (1986) construct models that include real variables only. 35 For a thorough discussion, see Leeson (1994, 1997a, 1997b). 33 tainty. Unanticipated inflation, on the other hand, increases relative price variability and costs of information gathering and leads to income redistribution. Finally, as Blanchard and Fischer (1993)36 put it: “Its presence as the only macroeconomic variable in addition to output in the loss function reflects in part the fact that, right or wrong, inflation is perceived as costly by people and is costly for policymakers to ignore.” An essential practical issue in policy design concerns the specification of the inflation target. Commonly under inflation-targeting the rule followed by the monetary authority includes a weighted measure of the deviation of the inflation rate from its target value. Concerning the choice of an appropriate target, Taylor (1986) emphasises that the policy objective is to minimize fluctuations around the target, regardless of what the actual value of the target is. Output-targeting Svensson (1997a) and Rudebusch and Svensson (1998) show that it is advisable that the monetary policy instrument responds to the determinants of the target variables rather than the target variables themselves. Thus, even under a primary price stability goal, it is generally appropriate to respond to both current inflation and the output gap, since both are determinants of future inflation. The relevance of simple policy rules that, as proposed by Taylor (1993), include both an inflation and an output target for recent central bank practice can be substantiated by the following statement by Federal Reserve Board Governor Yellen, made in January 1995: “Now, if you take the case of the FOMC, it seems to me that a reaction function in which the real funds rate changes by roughly equal amounts in response to deviations of inflation from a target of 2 percent and to deviations of actual from potential output describes tolerably well what this Committee has done since 1986. This policy, which fits the behavior of this Committee, is an example of the type of hybrid rule that would be preferable in my view, if we wanted a rule. I think the Greenspan Fed has done very well by following such a rule, and I think that is what sensible central banks do.” (Federal Reserve Board, 1995, pp. 43-44) Different definitions of the output gap with respect to the reference term are common in the literature. According to Galí (2002) the output gap is defined as “the deviation of output from its equilibrium level in the absence of nominal rigidities”. Woodford (1999) on the other hand, distinguishes in his proposed model between “potential output” and “natural rate of output” as reference values for the deviation of actual output. The former term represents steady state (or long-term equilibrium) output in the presence of nominal rigidities and market frictions, while the second term denotes the “equilibrium level of output under perfectly flexible prices”. The natural rate of output can is relatively time- 36 Chapter 11, p.569. 34 invariant and, at least in the short run, cannot be influenced by economic policy actions. Walsh (2003) provides an alternative definition of the output policy objective - a “speed-limit policy” that targets growth in demand relative to growth in potential, i.e. output gap changes. He finds that targeting the change in the output gap introduces inertia into monetary policy under central bank discretion, as the lagged output gap becomes an endogenous state variable. The final outcome is that targeting the output gap change is superior to inflation targeting unless inflation adjustment is prevailing backward-looking. In contrast, McCallum (2001a) argues that the output gap is unobservable and instead assesses the implications of using a trend-type measure. His results show that highly undesirable consequences in terms of higher inflation variability arise in case policy responds strongly to the measured gap. However, this result is obtained in a framework with constant capital. By contrast, the results obtained in Chapter IV in a model with endogenous capital and adjustment costs reveal that including the output gap in the monetary policy rule plays an important stabilising role in the occurrence of shocks. The central bank’s output objective is often included as a quadratic term on output deviations in the loss function, so that deviations of output from the target (natural or potential) level are symmetrically penalised. This feature is subject to critique by some authors37 who argue that in sticky-prices Keynesian models in the presence of market frictions (e.g. monopoly power by firms), the equilibrium level of output is too low, in which case a negative output gap should be penalised more heavily by the central bank. Whereas this argument rests on the subtleties of defining target output38, a more general critique point could be elaborated in terms of whether a positive and a negative gap actually impose the same welfare loss to the economy. Another critique on interest rate rules including an output gap target (such as, for instance, the Taylor rule as in Taylor (1993)) arises from measurement difficulties. In the first place, no real-time data on the value of current output are available to policy-makers. Thus, the operational usefulness of the output gap target is limited by the availability of timely and reliable estimates39. This shortcoming can be alleviated to a certain extend by assuming that the current output gap is equal to the expectation in the previous period. Secondly, as McCallum and Nelson (1999b) point out40, “…there is considerable uncertainty regarding 37 See Blanchard and Fischer (1993), Chapter 11. 38 In Blanchard and Fischer (1993) the full-employment level of output denotes potential level of output when certain distortions have been taken account of, instead of the level of output under flexible prices and perfect competition. 39 This argument has been used by Orphanides and van Norden (2004) to describe the difficulties when using the output gap to predict inflation within the Phillips curve relation. 40 P. 5. 35 the realised value of real GDP even at the end of the quarter in actual economies”. The reason is that empirical estimates of the output gap are in general subject to significant and highly persistent revisions. The third source of practical difficulties when measuring the output gap target pertains to correctly estimating the value of potential output41. Kuttner (1994) and McCallum (1997) argue that monetary policy decisions are in practice complicated by the risk of output gap mismeasurement. Smets (1998) analyses the effect of measurement error in the output gap on efficient policy rules in a simple estimated model of the US economy. The conclusion is that output gap uncertainty can have a significant impact on the efficient response coefficients in instrument rules (such as the Taylor rule) by reducing the response to the current estimated output gap relative to current inflation. Orphanides (1998)42 shows that output gap real-time measurement errors “lead to a significant deterioration of feasible policy outcomes and cause efficient policies to be less activist”. Interest-rate smoothing A policy rule with the short-term interest rate as instrument can be designed to involve interest-rate smoothing. Here it should be differentiated between smoothing in the sense of lowering the variance of the level of interest rates, as opposed to lowering the variability of interest-rate changes. According to Woodford (1999) reducing short-term interest rate level variability can be justified under the assumption that the distortions associated with positive nominal interest rates are described by a convex function of the interest rate. In this case, for any average level of nominal interest rates, a lower variance in them will reduce the size of the distortions. The case for lowering interest rate level variability could easily be illustrated by analysing the implications of significant interest rate changes in the two extreme scenarios of a low and a high average level of nominal interest rates respectively. A policy consistent with a low average rate of inflation and nominal interest rates faces the zero nominal interest rate bound and therefore cannot apply large interest rate reductions to combat deflationary shocks. On the other hand, when the levels of inflation and nominal interest rates are already relative high, in order to avoid distortions (as private sector’s resources are wasted on attempts to economise on cash balances), the monetary authority should avoid significant further increases in interest rates in response to inflationary shocks. Interest-rate smoothing in the sense of seeking to minimise the variability of interest rate changes is a widely observed phenomenon in actual central bank 41 The difficulties associated with measurement of potential (and, by analogy, the natural rate of) output and resource utilization and their implications for monetary policy and macroeconomic stabilization have a long tradition in modern economic literature, beginning with Friedman (1947, 1953). 42 See also Orphanides et al. (2000). 36 practice. From a theoretical point of view, minimising the variability of interestrate changes by adjusting official interest rates in a sequence of relatively small steps in the same direction means that central bank behaviour depends not only on current states and current forecasts of future conditions, but also on past conditions and commitments. Sack (1998)43, for example, estimates the optimal federal funds rate policy given the structural form of the US economy and compares it to actual historical data. He finds that in the absence of parameter uncertainty, the calculated optimal policy responds more aggressively to changes in the economy than the observed policy, resulting in a substantially higher volatility of the funds rate than observed. He explains lower variability of interest rates in actual policy with the existence of parameter uncertainty, which limits the willingness of the Fed to deviate from the policy rule that has been previously implemented. There are several possible explanations of interest-rate smoothing that have been discussed in the recent literature on central bank practice. First, as Lowe and Ellis (1997) point out, policy-makers are averse to frequent changes in the direction of interest rate movements as it may undermine confidence in the central bank and therefore its ability to influence private sector expectations and behaviour in a desirable manner. Second, it can be argued that the nature of the decision-making process on monetary policy leads to conservatism that is at the heart of interest-rate smoothing44. According to this line of argument, central banks are not able to gain broad political support for prospective interest rate changes until sufficient evidence has been gathered. Because the evidence needed accumulates slowly, interest rates tend to be changed gradually. A third motive for interest-rate smoothing, as discussed by Sellon and Roley (1995), concerns the fact that a predictable path for short-term interest rates confers the central bank greater influence over long-term bond yields, and consequently over future output and inflation. Similarly, Lowe and Ellis (1997) provide the explanation that central banks tend to modify interest rates gradually in order to be able to assess the policy impact on longer rates and adjust the direction and pace of changes accordingly. 2. Criteria for assessing monetary policy rules After having presented some main theoretical insights concerning monetary policy design (which are taken into account when specifying the policy rules to be assessed in the next two chapters), I will discuss several criteria for assessing rule-based monetary policy performance, including operationality/simplicity, local determinacy of rational expectations equilibrium and its implications for response to shocks, as well as adherence to the Taylor principle. In the next 43 For further empirical evidence of interest-rate smoothing, see Clarida et al. (1998) and Rudebusch (1995). 44 Also in Lowe and Ellis (1997). 37 chapters, I will assess several interest-rate rule specifications in terms of whether they induce a locally determinate rational expectations equilibrium and desirable responses to a number of shocks45. By definition, the operationality/simplicity criterion is fulfilled for all the interest-rate rule specifications assessed in Chapters III and IV. In the next parts, adherence to the Taylor principle is discussed from a broader perspective in terms of critically examining whether within the model chosen fulfilling this criterion actually guarantees local determinacy of rational expectations equilibrium46. 2.1. Operationality/Simplicity In a series of studies on monetary policy rules, McCallum (1988, 1989, 1993, 1994) has emphasised operationality47 as a crucial property when deciding on a policy strategy. The operationality criterion limits consideration to policy rules (i) that are expressed in terms of instrument variables that could be controlled on a high-frequency basis by the monetary authority and (ii) that require only information that could actually be possessed by this authority. The use of simple instrument rules to specify rule-based monetary policy behaviour has a long tradition in the literature. Wicksell (1898) and Henderson and McKibbin (1993) suggested simple instrument rules with the interest rate as the instrument. Meltzer (1987) and McCallum (1988) proposed simple instrument rules with the monetary base as the instrument. The most prominent simple instrument rule is the Taylor rule (Taylor (1993)), incorporating an interest-rate instrument responding to the inflation and output gaps. Recent discussions of Taylor rules include Clarida et al. (1999), Hetzel (2000), Kozicki (1999), Woodford (2001b). A certain degree of disaccord pertains to the definition of a simple rule in the existing literature. Earlier contributions, such as Blanchard and Fischer (1993) use the term “simple rules” to refer to non-activist rules48. Schmitt-Grohe and Uribe (2004) define simple rules in terms of restricting attention to rules whereby policy variables are set as a function of a small number of easily observable macroeconomic indicators. In compliance with this criterion they propose studying central bank interest-rate feedback rules that include measures of inflation and output. Svensson (1997b, 1999a, 2002) offers a more detailed classification of monetary policy rules. According to him, a rule-based monetary policy procedure could take the form of either an instrument or a targeting rule. An instru- 45 „Desirable responses to shocks“ here refers to quantitatively modest and short-lived model variable deviations from steady state as a result of a shock. 46 This means that solely the fact that a policy rule is active is not seen as a positive trait. The determinacy results in Section 4 in Chapter III support this critical perspective. 47 In the studies mentioned, McCallum treats simple rules as being operational as well. 48 See pp. 581-583. 38 and relatively small and short-lived initial variable deviations from steady state values are also considered to be advantageous63. 2.3. The Taylor principle Taylor (1999) derives a relationship between the stability of inflation and the size of the interest rate coefficient on inflation in the policy rule. He shows that it is crucial to set the interest rate response coefficient on inflation above a critical “stability threshold” of one. The interest rate rule is given by64   * ttt t ir y S y SOSS O     , (2.6) where t is the nominal interest rate in t,t i S denotes the inflation rate over the previous four quarters, * S is the central bank’s inflation target65,  t y is the output gap66,ris the equilibrium (steady-state) real interest rate67, S O an y d O are policy parameters, denoting the central bank’s response to inflation and output gap deviations from target68. The total response to inflation in (2.6) is given by *1 SS OO !  *,0 y S OO . It is assumed that the monetary policy stance is counter-cyclical, i.e. . Taylor (1999) combines (2.6) with backward-looking IS- and AS- specifications given by  () tt tt y ir MS    g (2.7) and  1 1 tt tt y u SD S   , (2.8) where M , D >0 are reduced-form parameters that depend on the policy parameters; t and t u are serially uncorrelated stochastic shocks with zero mean. By substituting equation (2.6) in equation (2.7), an aggregate demand (AD) relation between inflation and output gap can be derived g 69  1 11 t t yy t y g S MO S MO MO    . (2.9) 63 The reason for this is quite straightforward: considerable deviations from the steadystate level and/or long-lasting adjustment all induce losses and uncertainty in the economy. 64 The rule specified by Taylor (1999) is a more general form of the Taylor (1993) one. 65 Taylor (1993) sets the inflation target to be equal to 2 percent per annum. In Taylor (1999) only t S enters the policy rule, i.e. *0 S . 66 In Taylor (1993) the output gap denotes the percent deviation of real GDP from a target (trend real GDP). Taylor (1999) defines the output gap as the percentage deviation of real GDP from potential GDP. For the purpose of this work, the latter definition will be used. 67 The steady-state real interest rate is estimated in Taylor (1993) using US data and set at 2 percent per annum. 68 In Taylor (1993), 0.5 y S OO g69 The stochastic term is left out here. t 45 The slope of the AD curve  /1 y S M OMO  is determined by the choice of the policy parameters S O and y O . For 0 S O ! <0 S (i.e. ), the aggregate demand curve is downward-sloping *1 S O ! 70 and for O it is upward-sloping. Figure 2.2 reveals graphically the stability properties of a Taylor rule for an effective inflation response coefficient greater or smaller than unity (the upper two panels) and the resulting AD relations for these two cases (the two lower panels). The horizontal lines in the two lower panels represent the aggregate supply (price adjustment) relation. The zero slope of the AS line is determined by the fact that in equation (2.8) current-period inflation depends in the previous-period output gap, rather than on the contemporaneous one. Changes in t  y are thus transmitted to the inflation dynamics with a time lag. The intersection of the two solid lines (the AS and the AD line) in the two lower panels represents a situation when t yy (actual output is equal to the potential output), i.e.  y0 t . The AS line eventually moves up when the output gap is positive and vice versa. A positive supply shock also shifts the line upwards. Stable case Unstable case Figure 2.2: Stable and unstable monetary policy rules 70 As . 0 y O ! 46 The two left-hand panels in Figure 2.2 present a rule with and the two right-hand panels one with . The case on the left is stable because an upward shift in the AS line (a positive inflation shock) results in a decline of the output gap below zero, which leads to a downward adjustment of the inflation rate, represented by a downward shift of the price adjustment (AS) line. The case on the right is unstable, as, by contrast to the previous case, a positively sloped AD line implies that an upward shock to inflation leads to a positive output gap and contributes to further increases in inflation. This explosive property of the system has as consequence that supply-side shocks will tend to have a permanent, self-accelerating effect on inflation, bringing the system farther away from its equilibrium. *1 S O ! *<1 S O Algebraically, the above results can be substantiated as follows. The stability question can be expressed in terms of whether shocks will have a permanent, self-accelerating effect on inflation, i.e. 1 1 t t d d S S  L ? (2.10) whereby only under 1 / tt dd SS 1 does the system converge to steady state after a shock occurs. Equivalently, iterating (2.10) one period forward yields 11 t t d d S S  L ? (2.10’) and 1/ tt dd SS 1 as a condition for stability respectively. The policydependence of the stability property of the system becomes evident after one last transformation, this time of the AS relation. One period forward, equation (2.8) becomes  1tt t1t y u SDS   . (2.8’) Then,  11 tt tt d dd SD dy SS   . (2.8”) Substituting  //1 t t dy d S y SMO MO   from the AD relation (2.9) into (2.8”) yields 111 t ty d d S S DMO SMO  . (2.8”’) Then, for stability, ! 1 1y S DMO MO ! . (2.11) Equivalently, ! 0 S O !, (2.11’) 47 and ! *1 S O !. (2.11”) The relationship between the stability of inflation and the size of the inflation response coefficient in the central bank’s monetary policy rule has been reaffirmed by empirical analysis, such as Clarida et al. (2000), Judd and Rudebusch (1998) and Wright (1998). Benhabib et al. (1999) also argue that real determinacy could be attributed to the degree of activeness of monetary policy and the inflation measure that enters the central bank’s interest rule. They find that under sticky prices and an active monetary policy stance (i.e. a policy that aggressively fights inflation by raising the nominal interest rate by more than the registered increase in inflation), a forward-looking component in the intermediate target is more likely to lead to indeterminacy than a backward-looking component. The assumption that the demand for money also plays a role in the monetary-transmission mechanism and that productivity is affected by the cost of funds leads to novel results concerning equilibrium determinacy. A further point is recognising the difference between local and global determinacy of equilibrium. An active policy stance may appear to lead to macroeconomic stability as it ensures locally unique equilibrium, but in fact be destabilizing because it is associated with global indeterminacy and equilibria in which the economy converges to a cycle71. Clarida et al. (2000) provide empirical analysis of US monetary policy, based on a baseline sticky-prices model72 with a modified Taylor rule where the Fed responds to expected future deviations of inflation and the output from their target values, instead of to their current values. Their findings support the results of and Kerr and King (1996) concerning the destabilising impact of an excessively weak reaction of the policy instrument to an increase in expected inflation. This is the case of an effective policy reaction coefficient on the inflation gap . Values of *1 S O d* S O below unity lead to equilibrium indeterminacy and monotone divergence or fluctuations around the steady-state values of inflation and output, resulting from self-fulfilling changes in expectations. The rise of self-fulfilling changes in expected inflation can be explained by the fact that with , a rise in anticipated inflation is accompanied by a decline in the real interest rate which stimulates aggregate demand and causes a rise in inflation. Thus, due to the accommodating stance of monetary policy, the initial rise in expected inflation becomes “self-confirmed”. As shown by Clarida et al. (2000), the unity threshold value of *1 S O  * S O is obtained only in the absence of a systematic policy response to output variations (i.e. 0 y O ). For values of the policy reaction coefficient on the output gap 0 y O !, the lower bound for S O decreases be- 71 For further analysis, see Benhabib et al. (2001). 72 For similar models, see King and Wolman (1996), Woodford (1996, 1998) and Yun (1996). 48 low unity, although the deviation from unity is quantitatively negligible, and is independent of whether an interesting-smoothing parameter (i.e. the lagged nominal interest rate) enters the rule specification. As far as the upper bound for the unique equilibrium defined by the range of values of * S O is concerned, Bernanke and Woodford (1997) find that an excessive response to variations in expected inflation may also lead to indeterminacy. 3. Preliminary summary Since the middle of the last century, the design, transmission channels and outcomes of monetary policy have been the focus of extensive research. The practical experience with “activist” monetary policies in the 1960s and 1970s, motivated by the conviction that in the long run the monetary authority can attain a permanent reduction in the unemployment rate by allowing for a higher rate of inflation, has been disappointing. Parallel to the practical experience with activeist policies, there has also been an academic critique of using monetary policy as a tool to obtain full employment at the expense of a higher inflation rate. Among the most influential arguments are Milton Friedman’s monetary critique concerning the uncertain outcomes of monetary policy interventions, the Lucas critique of the optimal control paradigm for monetary policy, the conclusion reached by Friedman (1968) and Phelps (1968) that there is no long-run tradeoff between inflation and unemployment, as well as the warning against the perils of time inconsistency under discretionary policy delivered by Kydland and Prescott (1977), Calvo (1978) and Barro and Gordon (1983). The experience with time-inconsistent discretionary policy has not only fuelled the debate on the benefits of systematic policy behaviour vs. discretion, but has also given an impetus for extensive research on how monetary policy rules should be designed and assessed. In terms of policy instrument choice, in the last two decades setting the nominal interest rate has generally prevailed over directly controlling money supply in both theory and practice. However, selecting the target variables and their response coefficients in the policy rule currently remains a controversial issue, since the estimated outcomes are to a great extent model-dependent. In order to ensure a comprehensive assessment of the different rule specifications, it is essential to consider not only the aggregate demand, but also the aggregate supply monetary transmission channel, which requires using a comprehensive macroeconomic framework incorporating endogenous capital (as the one used in Chapter III). Under such a framework, the determinacy and shock response properties of different rule specifications, as well as the relevance of the Taylor principle will be examined and assessed in the next two chapters. 49 III. A New Keynesian model with endogenous capital with adjustment costs One of the main findings concerning rule-based monetary policy design in Chapter II has been that the assessment of different rule specifications in terms of the criteria presented in Section 2 (determinacy of rational-expectations equilibrium and response to shocks), as well as the evaluation of the relevance of the Taylor principle for fulfilling these criteria are to a great extent dependent on the model used and the inclusion of a supply-side transmission channel in particular. Therefore, in this chapter I focus on deriving a New Keynesian model with endogenous capital and adjustment costs and proceed with examining the system’s determinacy properties under different rule specifications. In terms of the general applicability of the Taylor principle, the findings are non-trivial. In the model with endogenous capital an inflation response coefficient greater than unity is not sufficient per se for ensuring a unique rational-expectations equilibrium. Apart from a small value interval of the inflation response coefficient above unity, implying a moderately strong policy reaction, determinacy under an “active” rule requires some degree of output gap response. Moreover, even within the inflation coefficient interval yielding determinacy under a sole inflation target in the rule, adding an output gap term is still associated with a unique equilibrium. In terms of “passive” rules (with an inflation response coefficient below one), the Taylor principle is only partially valid as well. Indeed, if inflation is the only target variable entering the rule, a smaller than one-on-one nominal interest rate response to inflation deviations yields indeterminacy of rational-expectations equilibrium. However, introducing a sufficiently large output gap response can lead to a unique equilibrium even under a passive rule73. Thus, modelling endogenous capital provides important new insights and an extension to the baseline formulation of the Taylor principle, namely by adding the requirement of introducing an output gap response in order to guarantee uniqueness of the system’s rational-expectations equilibrium. The chapter is organised as follows. In Sections 1 and 2, I give an overview to the baseline New Keynesian framework and possible approaches to modelling capital and investment. The model with endogenous capital and adjustment costs, which is the basis for the determinacy and impulse response analysis to follow in the next chapters, is presented in Section 3. In this section, I first examine the households’ optimisation problem. The resulting first-order conditions describe the aggregate demand side of the model. Then I present the producers’ optimisation and derive the aggregate supply curve and the real-wage equation and add an interest rate rule to the system. Dynamics of the whole economy are 73 In fact, as it is evident in the analysis in Section 4, the output gap response required for determinacy increses as the inflation response coefficient decreases. thus fully characterised by combining equilibrium conditions from both the demand and the supply side. As a next step, in Section 4 these log-linearised equilibrium conditions are used to study the determinacy property of interest rate rules. Then, in Section 4 I present the responses generated under different interest-rate-rule specifications to three types of shocks. Section 5 provides a preliminary summary of results. 1. The New Keynesian framework: an overview Over the past decade numerous examples of small-scale monetary business cycle optimising models featuring nominal rigidities have appeared in the literature. They are generally known as New Keynesian models. Both their theoretical appeal as micro-founded models, and their ability to explain the short-run effects of monetary policy, have contributed to their popularity among researchers. Taylor (2000) describes New Keynesian models as “…dynamic, stochastic, economy wide models with forward-looking behaviour and some rigidities that make them useful for policy evaluation”. This kind of models are also sometimes referred to as “Dynamic New Keynesian” (Bernanke et al. (1998)) or New Neoclassical Synthesis (Goodfriend and King (1997)). New Keynesian models typically integrate standard Keynesian elements (imperfect competition, nominal rigidities in price- and wage-setting) into a dynamic general equilibrium framework with rational expectations of market participants. One substantial improvement in recent research in comparison to the traditional Keynesian framework consists in stronger theoretical and microeconomic foundations. Behavioural functions for aggregate variables are derived from optimal individual behavior of households and firms with simultaneous clearing of all markets. Thus, these models are an appropriate tool for analysing the connection between interest rates, inflation and the business cycle, as well as for comparing the impact of alternative monetary policies. An important feature of New Keynesian models is the inclusion of rational expectations of market participants. Muth (1961) first formulated the rational expectations hypothesis, which requires that the subjective expectation of economic actors (households and firms) regarding a particular variable be equal to the objective expectation for that variable conditional on the information set available74. In the following decade the idea has been further developed, among others, by Lucas (1972, 1973), Sargent (1973), Sargent and Wallace (1975, 1976) and Barro (1976). Another essential characteristic of New Keynesian models concerns the nature of inflation dynamics under monopolistic competition reflected in the New Keynesian Phillips Curve. Under the widely adopted staggered price specifica- 74 In the original paper, Muth suggested that „...expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory” (1961, p. 316). 52 tion as in Calvo (1983), inflation has forward-looking character as a result of the assumption that firms face constraints on the frequency of price-adjustment. This means that previously adjusted prices are likely to remain effective for longer than one period, i.e. current price-setting decisions (and therefore current inflation) are based on expectations about future cost and demand developments. Another determinant of inflation dynamics are mark-up variations (or real marginal cost variations) that arise from the monopolistic firms’ repeated attempts to adjust actual to desired mark-ups. Roberts (1998) suggests that the aggregate supply equation fits better to empirical data if the rational expectations assumption is replaced by a partially backward-looking model of expected inflation. The output gap is an endogenous variable in the New Keynesian models, related to the ex ante real interest rate and expected output gap in the aggregate demand relation. Frequently the variable enters as an inflation fluctuations determinant the aggregate supply relation75 and as a policy target the central bank reaction function. The empirical relevance of New Keynesian models has often been criticised. The dependence of optimising New Keynesian models on a forward-looking decision making process limits their capacity to capture some of the business cycle regularities observed in the data. For instance, most optimising models are not very successful in replicating the delay in the responses of output and inflation to a monetary shock. In particular, an optimising model should explain why, rather than immediately, responses of both output and inflation to a monetary impulse reach their maximal impact several quarters after the shock. This phenomenon has been widely investigated in recent papers using optimising models incorporating frictions in price-setting and/or wage-setting, e.g. in Chari et al. (2000), Christiano et al. (2001) and Giannoni and Woodford (2003). The canonical New Keynesian model, as well as most of its standard generalisations, abstracts from investment in order to maintain simplicity76. One possible explanation is the emphasis on short-run analysis of macroeconomic stabilisation processes that allows abstracting from long-term capital accumulation implications. Moreover, the exclusion of capital is often justified on the grounds that the capital stock is not characterised by substantial volatility at business cycle frequencies and empirically there is a very small correlation between capital 75 As mentioned in the last paragraph, the output gap is sometimes substituted at the place of real marginal cost as inflation determinant. In fact, the Phillips curve relation derived from staggered price-setting as in Calvo (1983) involves the deviation of real marginal cost from its steady-state value. As Galí and Gertler (1999) and Clarida et al. (1999) show, certain assumptions about technology, preferences and the labour market structure can be made that infer a proportionate relation between real marginal cost and the output gap. 76 This view has been expressed by McCallum and Nelson (1999b). Examples of New Keynesian models with constant (exogenous) capital and investment include, among others, Kerr and King (1996), Bernanke and Woodford (1997) and Clarida et al. (2000). 53 and aggregate output measures (see McCallum and Nelson (1999a)). Difficulties with empirical measures of the capital stock also discourage developing models that involve capital accumulation. However, Dennis (2004) argues that abstracting from investment may imply that an important shock propagation mechanism that may have important implications for the design and implementation of an optimal monetary policy is omitted. Woodford (2003) provides a further critique on models with constant capital: “while this has kept our analysis of the effects of interest rates on aggregate demand quite simple, one may doubt the accuracy of the conclusions obtained, given the obvious importance of variations in investment spending both in business fluctuations and in the transmission mechanism for monetary policy in particular.” Casares and McCallum (2006) provide a further argument for the inclusion of endogenous capital and investment, as it enables not only studying issues relating to capital formation and growth, but also provides an endogenous explanation for the empirically observed contrasting variability of consumption and investment spending. When introducing investment within a New Keynesian framework, a crucial choice to make involves modelling the speed of capital stock adjustment in the occurrence of shocks. The case when capital adjusts relatively fast can be represented by modelling endogenous investment with an economy-wide rental market77 as in Hairault and Portier (1993), Kimball (1995), Yun (1996), King and Watson (1996), King and Wolman (1996) and Chari et al. (2000). A further option consists in introducing a certain degree of inertia in capital accumulation in the model, which, as an additional advantage, seems to match better empirical data on capital stock dynamics. This can be achieved by introducing assumptions that prevent the capital stock from immediately responding to shocks. As already mentioned in Section 1, three possible assumptions about investment and the capital stock could generate an inertial response on part of capital: capital accumulation adjustment costs, a time-to-build requirement and firm-specific capital. As shown by Casares and McCallum (2006), an appropriate possibility to endogenise investment in a less complex manner is to incorporate endogenous investment with capital adjustment costs under sluggish price adjustment in a dynamic model of the IS-LM type with optimising behaviour78. This is approach chosen in the next section for deriving the New Keynesian model that is later used for examining the determinacy and shock response properties of different 77 My results obtained under such a specification without adjustment costs (not included here) confirm the intuition that variable responses to shocks in such a case are unrealistically large. This makes the choice of such a modelling option quite unappealing for policy analysis. 78 Such models have been used by Woodford (1995), Kerr and King (1996), Rotemberg and Woodford (1998a, 1998b), Clarida et al. (1999) and Galí and Gertler (1999). 54 where P F D is the elasticity of substitution between capital and labour, t is a (labour-augmenting) technology shock, and t denote the household’s labour demand and capital stock at time t. Sales of the household’s specialised output are constrained by the demand function A d t nk At tA t P YP T  §· ¨¸ ©¹ , (3.3) where , t and denote aggregate demand, the price of the household’s product and the aggregate price level. The elasticity of substitution across differentiated consumption goods is denoted by A t YP A t P T . In addition, the representative household supplies differentiated labour services on the monopolistically competitive labour market. The quantity demanded is given by W At tA t W nW T  §· ¨¸ ©¹ , (3.4) where aggregate labour is denoted by , t is the household’s nominal wage, is the aggregate nominal wage and W A t nW A t W T is the elasticity of substitution for the differentiated labour services. At the same time, the representative household buys time units of a Dixit-Stiglitz composite labour input at the real wage rate . The household’s budget constraint in tis given by / A ttt WP A Z 93 11 11 11 (1 ) (1 ) (1 ) W AdA tt ttttt tttttttt AA tt PW Ytxnn ckkmmrb PW TT ZGS    ½ §· § · °°        ®¾ ¨¸ ¨ ¸ ©¹ © ¹ °° ¯¿ 1t b  1 (3.5) where is the inflation rate,  1 / AA ttt PP S  1t b denotes one-period government bonds purchased in t with real price 1 (1 ) t r  k ,t stands for lump-sum taxes (net of transfers) paid by the household and t is the capital stock and tx G is the capital depreciation rate. Next, I introduce two equilibrium conditions for the representative household’s production and labour supply. Production is equal to the quantity that is demanded as in (,) dA t tt t t A t P fAn k Y P T  §· ¨¸ ©¹ . (3.6) Labour supply is determined by the labour demand in the monopolistic competition labour market and is equal to 1 W At tA t W nW T  §·  ¨¸ ©¹ t l . (3.7) 93 If constant labour input is assumed, (3.5) is reduced to   111 11 /.(1) (1)(1)(1) AA ttt t tt tt t t t t tt YPP tinv n ck km m rb b T ZGS       1t . 61 The household’s optimality conditions consist of (3.5) - (3.7), together with94 1(, ,, , ) 0 ttttt t Ucml X9 O  (3.8)  1 21 (, ,, , ) 1 0 ttttt t tt t Ucml E X9 O E O S   ª   ¬ (, ,, , ) 0Ucml 1º ¼ (3.9) 3ttttt t X9 9  (3.10) 1(,) d tt tt tt t Af An k OZ -  0 1º ¼ (3.11) 11211 (1 ) ( , ) 0 d ttt tt ttt EEfAnk OEO G E -  ª    ¬ (3.12) 1 1 (1 ) 0 tt tt rE OEO     (3.13) 1(1) (1 ) /( ) /( ) 0 AAAA tt t t tt t t YPP YPP TT T T OT -T    (3.14) 1(1) (1 ) /( ) /( ) 0 WW W W AAAA ttt W t t ttW t t nWW nWW TT T T OZ T Y T    (3.15) whereby t O ,t - and t Y are the Lagrange multipliers to (3.5), (3.6) and (3.7) respectively. The marginal product of labour and capital are denoted by and : t mpl t mpk 1tt (,) d tt f An k m pl and 2(,) d tt t t f An k mpk . Equations (3.5)- (3.15) determine the paths of t,t,t, , 1t,1t,t,t W, t cml d t nk bP O ,t - and t Y given t S ,t Z ,t, , and . For general equilibrium, there are two market clearing conditions: rA t PA t Wt tx A t n ¦d t n (3.16) t tA t M mP , (3.17) the identity 1 1 A t tA t P P S   (3.18) and the government’s budget constraint 11 1 (1 ) (1 ) tt t t t tt gtxm m rb b S       1t , (3.19) whereby t M and t g denote the nominal money supply and government consumption of goods and services per household, while t r is the real interest rate. Without nominal rigidities, and t W. Altogether, equations (3.5)- (3.19) determine the paths of , , , , , , , , , A tt PP cm A tW d nkblA nPW O , - , Y , Z , and r S in response to the exogenous paths of t M,t and t tx . Alternatively, it can be assumed that the government sets the path of t instead of t or t. By analogy, the central bank can implement monetary policy by determining the nominal interest rate t, instead of using g bgtx it Mas an instrument. The latter option will be pursued further in Subsection 3.5. 94 Here (.) i f denotes the partial derivative of the function (.) f with respect to its i-th argument. 62 3.2. The “IS sector” Having presented the model’s micro-foundations in Subsection 3.1, some of the “IS sector” relations will be derived in this subsection, including the consumption equation and the overall resource constraint. The equations relating to capital and investment will be presented in Subsection 3.3. In order to derive the consumption relation for the “IS block”, (3.8) and (3.13) can be combined to yield > @ 111111 (, ,, , ) ( , , , , )(1 ) ttttt t t t t t t t Ucml EUc m l r 1 X9 E X 9   (3.20) Then, combining (3.8), (3.9) and (3.13) yields 21 (, ,, , ) (, ,, , ) 1 t ttttt ttttt t i U cml Ucml i X9 X9  (3.21) with 1 1(1)(1 tttt irE) S     . The period utility function is approximated by95 11 ( , , , , ) exp( ) (1 )exp( ) 1(1) tt ttttt t t cm Ucml 1 1 t l VJW X9 X 9 VJ  b  b /  W    (3.22) or 11 exp( ) ( ) (1 ) tt t t t t cEexp c r VV XXE   ªº ¬¼ , (3.23) with 1( , , , , ) exp( ) ttttt tt Ucml c V X9 X  b ,2( , , , , ) (1 )exp( ) ttttt tt Ucml m J X9 9  b , and 01b ,,, 0 VJW /! . Equation (3.23) can be transformed to yield 11 11 () ( tt tt tt cEc rr E ) t VVXX      , (3.24) whereby the “hat” variables denote logarithmic fractional deviations from the respective steady-state values, i.e.  log / tt cc c. Assuming that the consumption preference shock follows the AR (1) process 1t tt XX XUX H  and also substituting 1tt tt riE S   in (3.24) yields the following consumption equation: 11 11 ()(1 tt tttt cEc iE r X )t VSVU     X . (3.25) Next, the log-linear approximation to the overall resource constraint can be written as l l l m tctgtinv ycg in ZZ Z   t v, (3.26) where the value of the coefficient inv Z depends on the share of investment in total output in steady-state, i.e. (1 ( , )) / inv inv f inv k y Z  . By analogy, the steadystate shares in output of consumption and government expenditure are given by / ccy Z and / ggy Z . 95 The period utility function is separable in terms of consumption and real money balances. 63 3.3. Capital accumulation adjustment costs In order to specify realistic movements of capital and investment, it is necessary to add investment adjustments costs to the model. As already discussed in Section 1, some plausible ways of achieving this would include adopting the assumption that capital investment (or capital good “installation”) creates certain costs or adding exogenous “time to build” constraints. Here I choose to endogenise the sluggishness of capital stock adjustments and adopt the adjustmentcost specification as in Hayashi (1982) and Casares and McCallum (2006). Gross investment is given by: 1(1 ) tt inv k kt G   (3.27) Adjustment costs take the form (,) t tt t t inv C inv k inv f k § ¨ ©¹ · ¸ t (3.28) where the unit capital installation cost depends on the investment-capital ratio according to / t inv k (,) tt t tt C inv k inv f inv k § ¨ ©¹ · ¸, (3.28’) with and 2'(.) 0f!'(.) ''(.) 0ff G  (,)( d tt t !. Within this specification, total adjustment cost of investment in t varies with both t inv and t. The production function with adjustment costs tt k ,) f An k C inv kis assumed to be homogeneous of degree 1, implying constant returns to scale96, i.e. the size of the plant has no influence on the steady-state ratio of adjustment cost to output (,)/Cinvk y. If the functional form  1 / tt t kinvk  /2 t nvfi 4 4 is assumed, total adjustment costs are given by 2 2 1 1 (,) t tt t inv C inv k k 4 4 4 . (3.29) If adjustment costs for investment are introduced, the household’s budget constraint (3.5) from Subsection 3.1 becomes 96 Another possibility is to specify adjustment costs as in Abel (1983), where total adjustment cost of investment in tvaries only with gross investment according to t () t Cinv inv K [ where the scale parameter 0 [ ! represents adjustment costs and 1 K ! is the elasticity of total adjustment costs with respect to investment. Then, the values of the parameters [ and K imply increasing marginal adjustment costs as result of a rise in gross investment. Under a homogeneous production function of degree 1, subtracting adjustment costs from the production function (,)( t d tt t ) f An k C inv implies decreasing returns to scale. 64 11 11 11 (,) (1) (1 ) (1) W AdA tt ttttt tttttttttt AA tt PW Ytxnn Cinvkckkmmrb PW TT ZGS    ½ §· § · °°         ®¾ ¨¸ ¨ ¸ ©¹ © ¹ °° ¯¿ 1t b . (3.5’) After introducing capital adjustment costs of the form (3.29), the capital stock first-order optimality condition in t+197 takes the form > @ > @ > @ 112111 1(,) 1 ( ,) ttttt ttttt C inv k E C inv k E mpk 10 OEOG E-       ) ) t 1 t . (3.12’) Marginal adjustment costs in tand t+1 are given by 1t CC and . After substituting for next period’s real marginal cost (, tt inv k 12 11 (, tt CCinvk  111 / ttt rmc A mpl Z   from (3.11), (3.12’) takes the form  1111 (1 ) (1 ) tt tt t tt tt C E C E rmc mpk OEOGEO     1 r  . (3.12’’) Substituting from (3.13) into (3.12’’) yields 1 1(1 ) tt t t E EO O     11 1 11 tt t t t t t E C E rmc mpk rC G      1, (3.30) where is the return on the financial asset (the opportunity cost for investment). The right-hand side denotes the expected net marginal return on investment in the real asset. After substituting t C and 1tt 1r EC, the marginal cost in tand the marginal cost expected in t+1, from (3.29), (3.30) becomes  22 2 1 11 12 12 1 1 11 12 1(1)(1) 1 1(1) tt tt tt t tt t t t E inv E inv Ermcmpk kk r inv k GG 44    4 §· §·  44 44  ¨¸ ¨¸ ©¹ ©¹  §· 4 4  ¨¸ ©¹ (3.31) Using the log approximation, (3.31) yields the following semi-log-linear expectational investment equation98: m m n   11 1 11 22 1* 1 1( (1 ) (1 ) tt t tt ttt rmc mpk inv E inv E rmc E mpk r k CC G G GG GG       4 4 1) t, (3.32) where m / tt inv inv inv , with 2 112 (1)C G 4 4 4  . Steady-state adjustment costs are denoted by 1 C;rmc ,mpk and inv are the steady-state values of real marginal cost, marginal product of capital and investment respectively. The gap between the expected net return on capital and the return on the financial asset is denoted by 1tt t Empk r G . Equation (3.32) shows that current investment depends not only on current, but also on the expected future premiums on investment in real  97 See equation (3.12) in Subsection 3.1. 98 For steady-state analysis, see Appendix. 65 assets because of the inclusion of the forward-looking term m 1t t Einv. For simplicity, a parameter ; denoting the semi-elasticity of investment with respect to the real asset’s premium in (3.32) can be introduced as 1 2 1/(1 ) C G ;  4 . Thus, investment behaviour is described by m m n   11 t tt rk 11 1 1** tt t ttt inv E inv rmc mpk E rmc E mpk G G GG   ;   . (3.32’) Thus, under endogenous capital and investment adjustment costs, the model consists of an “IS sector” in the form 11 1() (1) tt tt cEc rr X t VVU     X (3.25) m m n   11 1 1** 11 tt t tttt inv E inv rmc mpk E rmc E mpk r kt t G G GG   ;   (3.32’) n l    1 PF tt t t t PF rmc k y A D ZD     (3.33) l l ( tt mpk mpk y k ) t (3.34) m l m 1(1 ) t tt kk GG   inv (3.35) l l l m tctgtinv ycg in ZZ Z   t v (3.26) where (3.25) describes consumption decisions made by the households as derived in Subsection 3.2, (3.32’) represents investment behaviour by firms, (3.33) and (3.34) are log-linear approximations to real marginal cost and the marginal product of capital for the Cobb-Douglas production function (3.2), (3.35) is a log-linearisation around the steady state of the investment specification (3.27) and (3.26) is the overall resource constraint with investment from Subsection 3.2. The technology shock t and the consumption preference shock At X are modelled as AR(1) processes, with 1t tAt AA A U H   and 1t tt XX XUX H  . 3.4. Inflation and real wage equations under sticky prices and wages In this section, I derive the inflation and real wage relations under sticky prices and wages, which will be used in the subsequent determinacy and impulse response analysis. For illustration, the corresponding equations under flexible prices and wages are presented in Appendix. As a first step, I assume that P K is the fixed probability that households cannot adjust their price as in Calvo (1983), so that the first-order condition on price-setting (3.14) becomes 1(1) 0 (1 ) /( ) /( ) 0 ii A A A A t P ti ti t ti ti ti t ti i EYPPYPP TT T T EK O T - T f      ªº  ¬¼ ¦ , (3.14’) 66 which can be log-linearised to yield an equation describing sluggish price adjustment99  n 1 (1 )(1 )(1 ) (1 ) PP PF t ttt PPFPF Ermc E KKD SSE S S KDDT       . (3.36) Similarly to prices, for nominal wages it may be assumed that they cannot be adjusted with a fixed probability W K , so that the nominal wage first-order condition (3.15) is transformed into 1(1) (1 ) /( ) /( ) 0 WW W W ii A A A A t W ti ti ti W t ti ti ti W t ti EnWWnWW TT T T EK O Z T Y T f      ªº  ¬¼ ¦ . (3.15’) 0i Log-linearising (3.15’) yields an expression for the dynamic real wage behaviour under sticky prices of the form l l l    11 1 11 11 1 1 1 tt t t tt tt a EEd E E ZZ Z SS SS E EE E          E (3.37) with  (1 )(1 ) / 1 / WWWW an  t dl EK K K T W ªº    ¬¼ and denoting the log deviations from steady state of the ratio of the average leisure-consumption marginal rate of substitution over the real wage or the left-hand side in (A3.10)100. 3.5. Interest-rate rule specifications This subsection concludes the derivation of the New Keynesian model with endogenous capital and capital adjustment costs by adding a monetary policy rule. For the sake of realism, monetary policy is assumed to be implemented through a nominal interest rate instrument, so that money supply becomes an endogenous variable101. Policy behaviour can be specified in terms of a Taylor rule as in Taylor (1993):   * t ttt y t ir y S i SOSS O H      , (3.38) ,0 y S OO !, where t is the nominal interest rate, t i S denotes the inflation rate,  t y is the deviation of actual output from its steady-state (potential) level, r is the steady- 99 For a more detailed derivation of the New Keynesian Phillips curve, see Sbordone (2002) Sveen and Weinke (2004) and Woodford (2005). 100 See Appendix. Sbordone (2001) offers a detailed derivation of this real wage equation under sticky wages. 101 The money-demand relation is derived in Appendix. However, with the nominal interest rate chosen as a monetary policy instrument and with a separable period utility function (3.22), the LM equation can be excluded from the model relations that will be calibrated in Subsection 4.1 and used for the subsequent determinacy and impulse response analysis. 67 state real interest rate and * S is the central bank’s target inflation rate102. The term t i H stands for is a monetary policy unit shock103. The reaction coefficients S O and y O determine how strongly the monetary authority stresses inflation and output stabilisation respectively. The response to inflation deviations in (3.38) is actually for *11 SS OO  ! 0 S O !. Thus, in the classification of Leeper (1991), the Taylor rule as in (3.38) is an “active” monetary policy rule, describing a more than one-on-one increase in the policy instrument as a result of deviation of the actual inflation rate from the target (steady-state) rate. Another possible specification of policy behaviour is given by a rule proposed by Casares and McCallum (2006)  l   1t yt i t i ii(1 ) (1 ) S ti t yii OOS  1) SO  OH  ªº      ¬¼  )( i , (3.39) and *(1 SS OOO *  ;(1 y) iy OOO  , (3.40) and whereby * S O * y O are the effective inflation and output gap response coefficients implied by the policy rule. Additionally, the real interest rate equals the difference between the nominal interest rate and next period’s expected inflation in accordance with the Fisher equation, i.e. 1t. (3.41) tt ri t E S   Equations (3.25), (3.26), (3.32’) and (3.33)-(3.35), together with (3.36), (3.37), (3.39) or (3.38) and (3.41) determine time paths for the ten endogenous variables in the model: l t c, m t inv , , , t mpk n t rmc m 1t k, l t y , , t rt S , l t Z and . t i 4. Determinacy analysis After the New Keynesian model with endogenous capital and adjustment costs has been derived in the previous section, I proceed with studying the system’s determinacy properties under several policy rule specifications. Since, when deriving the model’s reduced forms, the ten system equations yield quite complex coefficients, it is not possible to use the analytical approach to assessing the system’s determinacy properties. Thus, as a second-best solution, the eigenvalues 102 In the following subsections instead of the target rate * S the steady-state inflation rate S enters the Taylor rule, reflecting the assumption that the central bank correctly assesses the steady-state inflation rate and uses it as a target value. 103 The term t i H is not included in the original Taylor (1993) version, but is added here in order to enable analysis of the impact of a monetary policy shock on the system under different policy specifications. The monetary policy shock term captures nominal interest rate changes that are not a result of the central bank’s response to the target variables as prescribed by the rule. The monetary policy unit shock is modelled as an upward blip of 1 percent in the shock term t i H in the policy rules (3.38) and (3.39) that is transmitted to the policy instrument, causing it to rise. 68 of the system are examined after substituting the coefficients with the numerical values provided in Subsection 4.1. Then, I offer some preliminary insights to the relevance of the Taylor principle using the calibration in the previous subsection for active and passive rules with and without an output target. Finally, in the last subsection a more global perspective to determinacy outcomes under a wide range of inflation and output gap response coefficients is provided. 4.1. Calibration In order to enable quantitative analysis of the model’s properties it should be specified in numerical terms. Table 3.1 presents the values of the model parameters104, used for the determinacy analysis in Subsection 4.2 and the impulse responses in Chapter IV. In addition, some of the calibration choices should be considered in detail. For equation (3.25), 5 V implies an intertemporal elasticity of substitution in consumption as in Hall (1988) and Fuhrer (2000). The value of the steady-state real interest rate 10.2 V  0.005r 5363 corresponds to an annual value of 2 percent. In equation (3.32’), 1 4 and 2 yield 3.144 2 10.05C G 4 4 , implying that the unit adjustment cost in steady state equals 5 percent of investment. The marginal adjustment cost in steady state is then 2 112 (1) 0.2C G 4 4 4  1. The steady-state values of real marginal cost are derived from the relation for the steady-state marginal product of capital (A3.6)105. In equations (3.25) and (3.33), the autocorrelation coefficients assigned to the preference and technology shocks 1t tt XX XUX H   and 1tAt AAt A U H   are 0.3 X U and 0.95 A U . Table 3.1: Parameter values (in order of appearance) “IS sector”: 5 V Equations (3.25), (3.26), (3.32’), (3.33)- (3.35) 0.3 X U 0.005r 0.025 G 153634 23.144 1.5; 0.83rmc 0.04mpk 0.36 PF D 104 The parameter values chosen are based on the calibration in Casares and McCallum (2006). 105 See Appendix. 69 0.95 A U 0.78 c Z 0 g Z 0.22 inv Z Aggregate supply (3.36): 0.005 S 0.99 E 6 T 0.75 P K Real wage equation (3.37): 4 W T 0.75 W K 2 W /0.nl 5 Monetary policy rule (3.38), active 0.5 S O *1.5 S O *0.5 yy OO Monetary policy rule (3.38), passive *0.5 S O *0.5 yy OO * ; 0.1 yy OO 106 Monetary policy rule (3.39): 1.5 S O *0.3 S O 0.1 y O *0.02 y O 0.8 i O In the aggregate supply equation (3.36) 0.005 S corresponds to an annual value of steady-state inflation of 2 percent. The degree of price and wage stickiness is represented by the probability that the representative household is not able to adjust its price or wage; 0.75 PW K K 6 implies that prices and wages are reset once a year on average107. The value of the elasticity of substitution between differentiated consumption goods T is frequently assigned in the businesscycle literature. Thus some simple calculations show that in (3.36) the coeffi- 106 In Chapter III, Subsection 4.2 the value has been used for the determinacy analysis. Based on the finding that such a value of the output gap coefficient cannot yield a determinate rational-expectations equilibrium, the value has been chosen for the impulse-response analysis in Chapter IV, Subsections 3.2 and 3.3. *0.1 yy OO *0.5 yy OO 107 As in Erceg et al. (1999). 70 l Jare given by  0.89, 0.99, 1.05, 1.11, 2.03 0.1 The eigenvalues of , of which the first and the second eigenvalues of the system are within the unit circle113. Thus, a passive rule with and y, cannot yield a determinate rational expectations equilibrium when both inflation and output enter the interest rate rule. * S O * O 0.5 4.2.3. Interest-rate rule response coefficient values and determinacy: a global perspective The previous numeric examples reveal differing stability properties of the system when certain numeric values of the monetary policy response coefficients are assumed. Subsection 4.2.1 shows that adherence to the Taylor principle alone does not guarantee determinacy of rational expectations equilibrium under endogenous capital, as the output gap coefficient also plays an important role. Therefore, it is necessary to examine the results for a greater number of parameter values within a plausible interval, in order to identify the stability regions for the values of and . * S O * y O More generally, the system’s coefficient matrix can be represented in terms of both policy parameters from the interest rate rule as: **** **** 1.99 0.33 0.98 0.32 0.04 0.04 0.03 0.001 0 1000 0.1 0.11 1.01 0.01 0.02 1.76 0.02 1.72 0.02 0.2 0.2 0.16 1.002 0 0.1 0.11 0 0.01 2.03 yy y yy y S S OOOO OOOO ªº    «» «» «»  «»    «» «»  ¬¼ 0 . Fig. 3.1 shows in three-dimensional space the determinacy results for values of the policy response parameters 114 and from 0 to 2 * S O * y O . The vertical axis represents the unit circle. Whenever only one eigenvalue of the system is smaller than unity, the respective combination of values of and * S O * y O is marked in blue. Alternatively, the cases when the parameter values generate less or more than one eigenvalue within the unit circle are denoted by blank space. The intersection of the and * S O * y O axes with the unit circle (1.0) plane is represented from a twodimensional perspective on Fig. 3.2, where the black regions stand for indeterminacy of rational expectations equilibrium (none or two or more eigenvalues within the unit circle) and the white areas correspond to parameter values inducing a unique convergence path with a single eigenvalue smaller than unity. Figures 3.1 and 3.2 reveal a picture that is consistent with the numeric results in the previous section. With endogenous capital, the introduction of output-targeting is crucial for the determinacy of rational expectations equilibrium. 113 The exact value of the second eigenvalue of the system is 0.99341, i.e. very close to unity, but still within the unit circle. 114 For the purpose of the analysis, the values of * S O and * y O are being altered with a 0.05 step. 77 What is more, adherence to the Taylor principle alone does not necessary guarantee a unique convergence path of the system when capital is endogenous. Only within a very limited value interval of the inflation reaction coefficient between 1.0 and 1.3 does an active rule with no output gap response yield determinacy. For all other values of * S O above 1.3 and under 1.0, setting leads to more than one eigenvalue within the unit circle, i.e. to multiple rational expectations equilibria. Still, bearing in mind the above mentioned findings on outputtargeting, an active policy in terms of inflation (i.e. ) is more likely to lead to a unique equilibrium, as the required intensity of the output response in this case is significantly smaller than under a passive rule. For example, for *0 y O *1 S O ! *2 S O , any * y O higher than 0.1 yields determinacy; for values of the inflation coefficient significantly smaller than unity, a much stronger output gap response is needed, e.g. an inflation parameter requires an output response of an equal size (). * S 0.5 O * y O 0.5 * * y O S O Figure 3.1: Determinacy regions (3D) * S O * y O Figure 3.2: Determinacy regions (2D) 78 Apart from the requirement that some small degree of output-targeting should be introduced when * S O is above 1.3 in order to guarantee determinacy of rational expectations equilibrium, the above findings in general reaffirm adherence to the Taylor principle as a policy recommendation. Except for a small indeterminacy region, active policy in fact generates determinacy and even unconditionally so for the interval . The second, more controversial, conclusion refers to the implications of passive policy. Contrary to the results of Taylor (1993), in a model specification with endogenous capital * 1 S O 1.3 *1 S O  does not necessarily induce indeterminacy of rational expectations equilibrium. With a sufficient (everincreasing as * S O decreases) degree of an output gap response, a determinate equilibrium may still be obtained even if . * O 1 S Generally, the results plotted on Figures 3.1 and 3.2 show that an active interest rate rule might be more advantageous as it is associated with a wider range of response coefficients’ values inducing a unique equilibrium. Determinacy of rational expectations equilibrium is a highly desirable quality, as it assures that, after some disturbance has occurred, the system will converge to steady state following a unique, predictable path. Still, the exact unique adjustment path of the economy and especially the size and the persistence of deviations from steady state also play a crucial role in monetary policy decision-making. For example, a monetary policy stance that guarantees a faster convergence to steady state with smaller and less fluctuating deviations in variables such as interest rates, inflation, the output gap, consumption and investment (to name a few) will clearly be preferred to scenarios generating more uncertainty and distress in the economy. The adjustment paths induced by several different interest rate rule specifications will be presented and discussed in the next chapter. 5. Preliminary summary of results In Sections 1 and 2, I gave an overview to the baseline New Keynesian framework and possible approaches to modelling capital and investment. Then, in Section 3 I derived a model with endogenous capital, sticky prices and wages and capital adjustment costs. As a next step, different specifications of the interest rate rule with respect to the degree of activeness of the rule (measured by the inflation coefficient) and the target variables included were assessed based on the existence of a determinate rational expectations equilibrium. The numerical experiments pertaining to the specification of the central bank’s interest rate rule were illustrated by Figures 3.1 and 3.2 that plot determinacy and indeterminacy regions for each combination of values for the inflation and the output gap reaction coefficients in the interval > @ 0,2 and * S O * y O . The determinacy analysis in Section 4 led to the conclusion that, under the assumption of endogenous capital and investment with capital adjustment costs and 79 115 calibration of the model parameters consistent with the existing literature , adherence to the Taylor principle alone does not guarantee determinacy of rational expectations equilibrium under endogenous capital, as the output gap coefficient also plays an important role. Even under an active rule, some degree of outputtargeting is needed to guarantee determinacy (except for the very limited value interval of the inflation reaction coefficient between 1.0 and 1.3). Secondly and more interestingly, a passive interest rate rule does not necessarily yield indeterminacy of rational expectations equilibrium. With a significantly strong response of monetary policy to the output gap (ever-increasing as * S O decreases), a determinate equilibrium may still be obtained even if the inflation response coefficient is smaller than unity. The general conclusion from the numerical experiments in Subsection 4.2.3 that an output target improves the performance of policy rules irrespective of the degree of the inflation response is a useful starting point for assessing the results from the shock impulse responses in the next chapter. 115 See, for example, Casares and McCallum (2006). 80 IV. Shock impulse responses In Chapter III, I studied the determinacy properties of different interest rate rule specifications with respect to the degree of activeness of the rule (measured by the inflation coefficient) and the target variables included were assessed within a New Keynesian framework with endogenous capital and adjustment costs. Based on the results obtained, I will now assess the characteristics of the convergence path back to steady state after a shock occurs, implied by active and passive rules under three possible specifications for each class: (i) rules with a sole inflation target; (ii) rules with an inflation and output target and (iii) rules with inflation and output gap target and interest-rate smoothing. The types of shocks entering the impulse responses are threefold: (i) a monetary policy unit shock; (ii) a technology unit shock and (iii) a consumption preference shock. In the subsequent analysis, policy rule specifications that yield determinacy of rational-expectations equilibrium (REE) and in addition involve quantitatively smaller deviations and fast, monotonic convergence path after a shock occurs would be preferred. The chapter is organised as follows. Section 1 provides some general preliminary insights to the adjustment mechanisms in the system as a result of three types of shocks (monetary policy unit shock, technology unit shock and consumption preference unit shock) and identifies the two main transmission channels in the model (the real interest rate and the real marginal cost channel). Then in Sections 2 and 3 the adjustment paths of consumption, investment, the capital stock, real marginal cost, the output gap, the marginal product of capital, inflation, real wages and the nominal and real interest rate are traced for the case of active and passive rule specifications. Section 4 summarises the impulse response results. 1. Some preliminary remarks on the adjustment mechanisms in the system In this subsection, I will offer some general preliminary insights to the adjustment mechanisms in the system as a result of three types of shocks (monetary policy unit shock, technology unit shock and consumption preference unit shock). As it will be further revealed in more detail, the deviations registered as a result of the occurrence of shocks within each of the two main group of rules (active vs. passive) share the same sign. The differences observed under the different interest-rate rule specifications within each group pertain rather to the magnitude of the responses and the period convergence to the steady-state level takes. More generally, a comparison between active and passive specifications reveals the crucial role of the real-rate response that is the determinant of differing dynamics of the remaining variables. A second initial transmission channel 82 pertains to real marginal cost (i.e. in the case of a technology shock)116. A closer look at the ten system equations reveals a common pattern of adjustment dynamics that can be anticipated before assigning numerical values to the reaction coefficients in the monetary policy specification. 1.1. Monetary policy unit shock In accordance with the theoretical framework presented, a monetary policy unit shock is modelled as an upward blip of 1 percent in the shock term t i H in the policy rules (3.38) and (3.39) that is transmitted to the policy instrument, causing it to rise. The monetary policy shock term captures nominal interest rate changes that are not a result of the central bank’s response to the target variables as prescribed by the rule. Under rational expectations, expected inflation will sink. Investment, consumption and output gap dynamics as evident by (3.32’), (3.25) and (3.26) are all triggered by the impulse to the real interest rate. The Fisher equation 1tt tt riE S   postulates that the dynamics of the real interest rate is approximated by the difference between the deviations of expected inflation and the nominal interest rate. Due to the expected rise in the real interest rate as a result of the shock under both active and passive policy117, negative spikes of differing magnitude depending on the policy rule specification selected occur in investment, consumption and the output gap immediately after the shock, followed by a gradual return to the steady-state level118. The stronger deviation in investment compared to consumption is straightforward in the case of a monetary policy shock as the real interest rate enters (3.32’) with a coefficient 1.5; and (3.25) with a coefficient 10.2 V  . Since according to equation (3.26) the output gap is a weighted average of consumption and investment, its immediate response can be expected to have an intermediate value. The magnitude of the output gap response reveals the significance of the endogenous capital assumption for the model’s quantitative results: under constant capital, the output gap response to a monetary policy unit shock would still be negative, but of a considerably smaller magnitude, as it would only incorporate the relatively more moderate fall in consumption. Still, because of assigning a higher weight to consumption ( 0.78 c Z ) than to investment ( 0.22 inv Z ) in (3.26), the adjustment path of the output gap can be anticipated to match the dynamics in consumption more closely than that of investment. 116 Real marginal cost dynamics determine the inflation response. Under sticky prices, the real marginal cost channel plays a quantitatively modest role, as firms cannot adjust their prices immediately after the shock occurs. Changes in expected inflation, however, play a crucial role and are transmitted nearly one-to-one to current inflation (as 0.99 E ). 117 The rise in the real interest rate as a result of the positive blip in the nominal interest rate can only be circumvented if an immediate positive spike of comparable magnitude in inflation expectations occurs. 118 As the shock is assumed to be of temporary nature. 83 The impact on investment is transmitted to capital with a one-period lag as evident from (3.35). The value assigned to the rate of depreciation 0.025 G in the capital accumulation equation determines the significantly smaller maximum response of capital compared to the initial deviation in investment. The speed of adjustment of the capital stock to its steady-state level can be anticipated as being slower than that of investment. This result is a logical consequence of the fact that capital is a predetermined variable as specified in equation (3.35). By contrast, investment according to equation (3.32’) does not depend on the past realisations of any of its determinants and can therefore respond immediately and in full magnitude as soon as the shock occurs. In comparison to the response of capital and the output gap the magnitude of the deviation of the marginal product of capital will be of a considerably smaller magnitude, due to the relatively small value of the steady-state marginal product of capital ( 0.04mpk ) that enters equation (3.34) as a coefficient. The assumption of a steady-state real interest rate of 2 percent p.a., depreciation rate of 0.025, steady-state unit adjustment cost and steady-state marginal adjustment cost of 0.05 and 0.21 respectively in (A3.6) 1 (1 ) (1 ) /mpk C C rmc UG  imply a relatively weaker response of marginal product of capital to deviations in output and capital. The assumption of sticky wages in (3.37), modelled by the inclusion of a fixed probability W K (0.75 W K ) that nominal wages cannot be adjusted in the current quarter (which implies that on average wages are re-set once a year) and the partial history-dependence of the variable determines the sluggish adjustment of the real wage. The transmission of the shock to real wages occurs through the adjustment of the inflation expectations 1tt E SS  and through the dynamics of the inflation differential t SS . Quantitatively, with 0.99 E , the coefficient on the inflation differential  1/ 1 E  is only slightly higher than the coefficient on expected inflation  /1 E E . As the two terms enter (3.37) with opposite signs, the inflation transmission channel plays a role only as long as the changes in expected inflation are not countervailed by a proportional blip in actual inflation. In the initial period after the shock such a development can be assumed, with a subsequent disappearance of the effect leading to a gradual convergence of the real wage to steady state. The responses of capital, the output gap and real wage determine the dynamics of real marginal cost, as evident from equation (3.33). Compared to the output gap dynamics, the path of real marginal cost is expected to show a lengthier adjustment process due to its partial dependency on the dynamics of the capital stock. A more moderate maximum response compared to output seems also plausible for the same reason. A monetary policy unit shock can be anticipated to have a quantitatively small and non-persistent impact on inflation. As evident from equation (3.36), the monetary impulse is transmitted by real marginal cost. Under sticky prices, 84 even with a strong response of real marginal cost to the shock, inflation remains hardly affected, as the value of the fixed probability that households cannot adjust their price 0.75 P K (i.e. priced adjusted once a year on average) implies that the coefficient (1 )(1 )(1 ) / (1 ) PP PFPPFPF E KKDKDDT   equals 0.02. Thus, the initial negative blip in inflation is significantly smaller than the unit shock. Until the effect subsides completely, the path of inflation will continue to mimic the deviation of real marginal cost on a smaller scale. Although the direct impact of the monetary policy unit shock on the nominal interest rate is straightforward, the final adjusted outcome is worth further discussion. As far as the nominal interest rate is concerned, its final response depends essentially on three factors. Firstly, the choice of an active or a passive rule with respect to inflation plays quite logically a significant role. Secondly, under output-gap targeting, the response of the policy instrument depends on the change in inflation in (3.38) and (3.39) and of the output gap. Since the impulse to inflation is expected to be quite moderate, the effect of decline in actual output will prevail. A fall in actual output would motivate decreasing the nominal interest rate, thus (at least partly) offsetting the initial shock impulse. Thirdly, the magnitude of the interest-rate change depends on whether interest-rate smoothing is added to the monetary policy rule and on the size of the interestrate coefficient i O . In the calibration of in Chapter III, Subsection 4.1 the choice of 0.8 i O implies that under interest-rate smoothing the path of the nominal interest rate is to a significant extent history-dependent. Thus the maximum response of the nominal interest rate will remain modest and its adjustment will be characterised by a lengthy, graduate convergence to steady state. 1.2. Technology unit shock In accordance with the theoretical framework presented, a technology unit shock is modelled as an upward blip of 1 percent in the shock term t A H in the AR(1) process 1t tAt A AA U H   that enters the real marginal cost condition (3.33). The initial transmission of the technology unit shock occurs through the immediate and quantitatively large effect on real marginal cost in (3.33). This negative initial impulse is then transmitted to inflation as evident from (3.36). The assumption of sticky prices, reflected in the reaction coefficient 0(1 )(1 )(1 )/ (1 ) 1 PP PFPPFPF E KKDKDDT       determines a relatively modest negative deviation of inflation from steady state. The negative spike in inflation induces a partial response of the real wage. The initial impulse on the real wage bears the opposite sign of the inflation deviation- thus, real wage is characterised by a positive deviation from its steadystate value as a result of the labour-augmenting technology unit shock. The response is initially triggered by the inflation expectations and then sustained by the inflation differential and the history-dependence on own past realisations of the real wage. The latter property implies a more gradual adjustment in time. 85 As a result of the negative blip in actual inflation, the nominal interest rate is reduced in accordance with the policy rule. Of course, under a passive rule and/or under interest-rate smoothing the response of the policy instrument is quantitatively smaller as under an active rule and/or no history-dependence of the nominal interest rate. As far as the impact of output-targeting is concerned, the results in the case of a technology shock differ from those under a monetary policy shock. A technology unit shock typically induces a positive spike in actual output, thus ceteris paribus requiring an increase in the nominal interest rate. A negative blip in the policy instrument as a response to the shock can therefore be expected with certainty only in the case when inflation is the only target variable in the policy rule. Under inflation- and output-targeting the path of the nominal interest rate is more difficult to predict. Since under a rule of the form (3.38) or (3.39) a fall in inflation requires reducing the nominal interest rate, whereas increased actual output implies raising it, it can be expected that under inflation- and output-targeting the monetary policy response will be weaker under both active and passive policy than in the case of inflation-targeting only. The sign of the initial impulse depends on the magnitude of the inflation and the output responses, as well as the values of the reaction coefficients S O and y O . The responses of investment, consumption, the output gap, capital and the marginal product of capital to a technology unit shock all are triggered by the dynamics of the real interest rate. Again, differing magnitudes of the maximum responses of investment and consumption can be anticipated due to the values of the coefficients assigned to the real interest rate in (3.32’) and (3.25), 1.5; 119 and 10.2 V  respectively. Since according to equation (3.26) the output gap is a weighted average of consumption and investment, its immediate response will have an intermediate maximum value. Again, the response of the real interest rate is determined by the Fisher equation 1tt tt riE S   . Because, as explained above, the nominal interest rate response cannot be predicted with certainly, the initial impulse on the real interest rate can also vary depending on the interest-rate rule specification entering the system. What can be concluded even without explicit knowledge of the numerical results, however, is that under quantitatively too weak a response of the nominal interest rate that is not sufficient to offset the change in the inflation expectations determined by (3.36) the real interest rate can actually increase. Alternatively, under a sufficiently strong response of the nominal interest rate the deviations of both the nominal and the real interest rate will bear the same sign. Thus, a more moderate response to inflation (passive rule) in the case of a technology shock acts counter-cyclically; a 119 The value of ;, the semi-elasticity of investment in the investment function (4.27’), is yielded by substituting for the depreciation rate G , the scale parameter of the adjustment-cost function 2 4 and marginal adjustment cost in steady state 1 C in 1 2 1/(1 ) C G ; 4 . 86 stronger response to inflation (active rule), on the other hand, reinforces the effect of the shock. As far as capital and the marginal product of capital are concerned, the technology shock is channelled through its impact on investment and the output gap respectively. It can therefore be expected that the paths of capital and the marginal product of capital will be characterised by a weaker initial response and a lengthier convergence process. In addition, the deviations in investment are channelled with a time lag to the dynamics of capital. In comparison to the response of capital and the output gap the magnitude of the deviation of the marginal product of capital is quite modest, due to the inclusion of steady-state unit and marginal adjustment costs. 1.3. Consumption preference unit shock In accordance with the theoretical framework presented, a consumption preference unit shock is modelled as an upward blip of 1 percent in the shock term t X H in the AR(1) process 1t tt XX XUX H   that enters the consumption equation (3.25). The consumption preference shock denotes a shift of the household’s preferences towards consumption in the utility function (3.1). The initial transmission of the shock occurs through the immediate effect on consumption as in (3.25), causing a positive blip observed immediately after the shock. Under the parameter values chosen, the initial impulse to consumption can be assessed as being quite moderate (0.14 times the shock). Through its effect on consumption, the preference shock is channelled to the output gap (see equation (3.26)) which is also expected to register a positive spike. The magnitude of the output reaction can be predicted as smaller than the initial impulse to consumption in (3.25), based on the value of the consumption coefficient 0.78 c Z in the output identity (3.26). For a clear-cut prediction of the sign of the output deviation, the sign and size of the investment response should also be taken into account. In order to determine the adjustment path of investment as a result of the consumption preference shock, the response of the real interest rate should be given detailed consideration. For a prediction about the real-interest rate response, the adjustment of the nominal interest rate and of expected inflation are to be examined. If the consumption-enhancing effect prevails in (3.26) and a positive output gap is registered, the nominal interest rate should be increased in accordance with the policy rules (3.38) or (3.39) if output-targeting is pursued. As far as inflation is concerned, according to (3.36) the triggering impulse is given by the adjustment of real marginal cost. The history-dependence of capital in (3.35) and the real wage in (3.33) imply that at least initially after the shock the path of the real marginal cost will mimic the dynamics of the output gap on a smaller scale, given by the coefficient  0/1 1 PF PF DD . Thus, if actual output is characterised by a positive deviation from steady state, real marginal cost will 93 equation (3.26) the output gap is a weighted average of consumption and investment, its immediate response has an intermediate largest positive and negative value of about 0.1 and –0.1 times the size of the technology shock respecttively. The largest positive and negative responses of capital (approximately 0.02 and –0.08 times the size of the shock respectively) are significantly smaller than the strongest deviations in investment. In addition, the deviations in investment are channelled with a time lag to the dynamics of capital. It takes the capital stock a whole 35 quarters until it reaches its largest negative deviation. Once again, capital is also characterised by a longer-lasting adjustment to its steadystate level than investmenteven after 60 quarters, capital remains under its steady-state level. In comparison to the response of capital and the output gap the magnitude of the deviation of the marginal product of capital (about 0.003 times the shock in both directions) is again quite modest, due to the inclusion of steady-state unit and marginal adjustment costs. Both the sluggish adjustment of marginal product of capital over more than 60 quarters and the undershooting path registered after the forth quarter following the shock can be traced back to the different adjustment dynamics of capital and output gap discussed above. Finally, the impulse response of the real wage is worth some consideration. As a result of the labour-augmenting technology shock, the variable reaches a quantitatively significant126 maximum positive deviation from steady state of nearly 0.5 times the shock after 12 quarters and converges monotonically to its steady-state value thereafter. The response is initially triggered by the inflation differential and then sustained by the dynamics of inflation expectations and the history-dependence on own past realisations of the real wage. ment-cost function 2 4 and marginal adjustment cost in steady state 1 C in 1 2 1/(1 ) C G ; 4 . 126 The magnitude of the impulse response of real wage is significant especially compared to the magnitude of the responses of the other model variables. 94 95 Figure 4.2: Responses to a technology unit shock under an active rule with inflation-targeting only In conclusion, the analysis of the impact of a technology unit shock in a model with sticky prices and wages, endogenous capital and adjustment costs shows that the only long-lasting deviations from steady-state values concern the capital stock that takes more than 60 quarters to converge. Compared to the impact of a monetary policy shock convergence to the steady-state values generally appears to be a longer-lasting process. For most variables (except inflation, real marginal cost and real wage) the magnitude of the observed deviations is relatively smaller than in the case of a monetary policy unit shock. The reason for the milder impact of a technology unit shock is that in the model it induces a more moderate response of the nominal interest rate cuts as a reaction to the inflation differential that quantitatively exceed the latter and thus generate a moderate real-interest rate response. Through the real-interest-rate channel investment and consumption report initial positive deviations significantly smaller than the size of the shock. 2.1.3. Consumption preference unit shock In accordance with the theoretical framework presented, a consumption preference unit shock is modelled as an upward blip of 1 percent in the shock term t X H in the AR(1) process 1t tt XX XUX H   that enters the consumption equation (3.25). Figure 4.3 reports impulse responses to a consumption preference unit shock for the model specification with sticky prices and wages, endogenous capital and adjustment costs. Variable deviations, caused by a technology unit shock, are of a very small magnitude and in each case of a much lower value than the shock itself. Another distinction in the case of a preference shock is that the adjustment to the steadystate values for all variables occurs much faster than in the case of a technology or a monetary policy shock. Within approximately 16 quarters after the initial impulse all variables apart from capital and the real wage have returned to their 96 steady-state values. The strongest deviations as a result of the shock are reported for consumption, the output gap and real marginal cost. The initial transmission of the shock occurs through the immediate effect on consumption as in (3.25). A positive blip of 0.2 percent is observed immediately after the shock, followed by a fast return to the steady-state level within the first 4 quarters. Through its effect on consumption, the preference shock is channelled to the output gap (see equation (3.26)) which also reports a sharp rise by 0.15 percent, followed by a rapid convergence to steady state. The positive output gap transmits the impulse to the real marginal cost which shows an initial positive spike of almost 0.09 times the shock, followed by a gradual convergence to steady state within the first 4 quarters. The positive initial effect on real marginal cost is transmitted to inflation as evident from (3.36), whereby the maximum impact on inflation occurs immediately after the shock and is quantitatively small (an increase of 0.002 times the shock). The impulse disappears in less than 4 quarters. As a result of the inflation differential, the nominal interest rate is increased in accordance with the policy rule (3.38), but the response is of a smaller magnitude (0.003 times the shock at its peak). After the first quarter, the nominal interest rate is gradually decreased and reaches its steady-state value in the fourteenth quarter. The path of the real interest rate mimics that of the nominal interest rate. The rise in the real interest rate releases an increase in investment of a very small magnitude, followed by a gradual adjustment within 12 quarters. Capital initially responds to the increase in investment by a gradual increase, reaching a maximum positive deviation from its steady-state level 10 quarters after the shock. After that, capital converges to steady state after approximately 60 quarters. The deviations of capital and the output gap from steady state, entering the term l l () tt y k in equation (3.34) determine the responses of marginal product of capital. Due to the combined impact of the output gap and the capital fluctuations, the maximum response of marginal product of capital (0.006 times the shock) is stronger than that of capital. Finally, as in (3.37) the real wage response is driven by the positive inflation differential and then sustained by the dynamics of inflation expectations and the history-dependence on own past realisations of the real wage. Unlike inflation, the real wage takes about 2 quarters to reach its highest negative deviation of 0.002 times the shock. 97 98 Figure 4.3: Responses to a consumption preference unit shock under an active rule with inflation-targeting only In conclusion, a consumption preference unit shock has quantitatively a relatively modest effect on the model variables. Its initial impact on consumption, the output gap and investment is alleviated by the increase in the real interest rate (caused by the nominal interest rate hike). The moderate responses of all other variables are determined by the magnitude of the deviations in the two latter variables in particular. 2.2. The case of inflation- and output-targeting This subsection shows the impulse responses of l t c, m t inv , l t y , n t rmc , l t k, n t mpk , l t Z ,t S ,t r and t i as for three shocks: the random component of monetary policy t i H , the technology shock t A H and the preference shock t X H when monetary policy is defined in terms of a baseline Taylor rule of the form (3.38). Here no interestrate smoothing enters the monetary policy rule and the effective response coefficients to inflation and output deviations take the values of 1.5 and 0.5 respectively. 2.2.1. Monetary policy unit shock In accordance with the theoretical framework presented, a monetary policy unit shock is modelled as an upward blip of 1 percent in the shock term t i H in the policy rule (3.38). Figure 4.4 reports impulse responses to a monetary policy unit shock for the model specification with sticky prices and wages, endogenous capital and adjustment costs. Variable deviations, caused by a monetary policy unit shock, are characterised by a relatively fast convergence to the steady state. Only as far as capital is concerned the shock impulse is still present after 60 quarters. The strongest deviations are reported for investment, the output gap, the nominal and the real interest rate. Negative spikes in investment, consumption and the output gap are observed immediately after the shock, followed by a gradual return to the steady-state 99 level. Here the investment response is characterised by the sharpest fall (2 times as large as the shock). Compared to investment, consumption shows a more moderate response (about 0.3 times the shock) and a fast adjustment to its steady-state value. The output gap’s immediate response has an intermediate value of about 0.7 times the size of the monetary policy unit shock. It takes investment, consumption and the output gap less than 8 quarters until the effect of the monetary policy unit shock subsides completely. The maximum negative response of capital (approximately 0.1 times the size of the shock) is significantly smaller than the initial fall in investment. Contrary to the immediate impact of the shock on investment, it takes about 6 quarters until capital reaches its maximum negative response. Moreover, capital is also characterised by a longer-lasting adjustment to its steady-state level than investmenteven after 60 quarters, capital remains slightly under its steady-state level. These results are a logical consequence of the fact that capital is a predetermined variable as specified in equation (3.35). Investment according to equation (3.32’), however, does not depend on the past realisations of any of its determinants and can therefore respond immediately and in full magnitude as soon as the shock occurs. As in the case of inflation-targeting only, once again the marginal product of capital responds moderately (-0.02 times the shock). The sluggish adjustment of the real wage over the entire 60 quarters is determined by the partial history-dependence of the variable as seen from (3.37) and by the assumption of sticky wages. The responses of capital, the output gap and real wage determine the dynamics of real marginal cost, as evident from equation (3.33). Compared to the output gap dynamics, the path of real marginal cost reaches a more moderate maximum negative value (0.4 times the shock), but does not adjust monotonically to its steady-state value. Instead, 4 quarters after the shock real marginal cost overshoots the steady-state value, reaching a maximum positive value of 0.05 about 7 quarters after the initial impulse. After that, real marginal cost converges monotonically to the steady—state level. While the initial negative impulse to real marginal cost is mainly a result of the immediate decline of the output gap after the shock, the overshooting part is essentially explained by the slow and lasting response of capital and the real wage. A monetary policy unit shock has a quantitatively small and non-persistent impact on inflation. As evident from equation (3.36), the monetary impulse is transmitted by real marginal cost. Under sticky prices, despite the strong response of real marginal cost to the shock, inflation remains hardly affected. Thus, the initial negative blip in inflation is less than 0.005 times the shock. Until the effect subsides completely after 16 quarters, the path of inflation includes overshooting dynamics (a maximum of 0.006 in the fourth quarter) before steady-state value is reached. According to the policy rule (3.38), the nominal interest rate response depends on the unit shock itself and on the responses of the target variables (infla- 100 tion differential and output gap). The monetary authority’s reaction to the unit shock involves an increase in the nominal interest rate of 0.7 times the shock, followed by a gradual convergence to the steady-state level. The effect of the monetary impulse disappears completely after 10 quarters. The Fisher equation 1tt tt riE S   postulates that the dynamics of the real interest rate is approximated by the difference between the paths of expected inflation and the nominal interest rate. Due to the quantitatively weak response of actual and expected inflation, the real interest rate response (a positive blip with a maximum value of about 0.7 percent, followed by a monotonic decrease) matches the nominal interest rate dynamics. 101 Figure 4.4: Responses to a monetary policy unit shock under an active rule with inflation- and output-targeting In conclusion, the analysis of the impact of a monetary policy unit shock in a model with sticky prices and wages, endogenous capital and adjustment costs and a standard Taylor specification for monetary policy shows that the only long-lasting deviations from steady-state values concern the capital stock and the real wage, which take at least 60 quarters to converge. Consumption, investment, the output gap, the real and the nominal interest rate, on the other hand, all return to their steady-state values within a maximum of 10 quarters. 2.2.2. Technology unit shock In accordance with the theoretical framework presented, a technology unit shock is modelled as an upward blip of 1 percent in the shock term t A H in the AR(1) process 1t tAt A AA U H   that enters the real marginal cost condition (3.33). Figure 4.5 reports impulse responses to a technology unit shock for the model specification with sticky prices and wages, endogenous capital and adjustment costs. Again, the strongest deviations are reported for real marginal cost, real wage and investment. The initial transmission of the shock occurs through the immediate effect on real marginal cost. A large negative spike of almost 0.9 percent is observed immediately after the shock, followed by a gradual return to the 102 steady-state level within the next 20 quarters. This negative initial impulse is then transmitted to inflation as evident from (3.36). As a result of the increased inflation differential, the nominal interest rate is reduced in accordance with the policy rule (3.38), reaching a minimum value of slightly above 0.1 times the shock. As the output gap also enters the monetary policy rule, the increase in actual output as a result of the technology shock implies that the actual nominal interest rate hike will eventually be of a smaller magnitude than under an active rule incorporating inflation-targeting only. Nevertheless, as a result of the large reaction coefficient assigned to inflation and the quantitatively small impact of the shock on the output gap , the path of the nominal interest rate mimics that of actual inflation. For the model specification chosen, a Taylor rule with no interest-rate smoothing induces a negative blip in the real interest rate of approximately 0.03 percent immediately after the shock occurs, followed by a gradual convergence to the steady-state value within the next 20 quarters. The real-interest rate response induced by the nominal interest rate fall additionally reinforces the positive deviations of investment, consumption and the output gap and acts procyclically. This effect can be offset by a stronger emphasis on output-targeting in the monetary policy rule, measured by the value of the coefficient y O . Investment, consumption, the output gap, capital and the marginal product of capital all reveal an initial positive response to the technology shock, followed by a countervailing negative deviation of a comparatively smaller magnitude before their steady-state values are reached. The strongest deviations in consumption, the output gap and the marginal product of capital all occur immediately after the shock, while it takes 9 quarters until the maximum impact of the shock on capital unfolds. The technology shock is channelled to consumption and investment through its impact on the real interest rate; the considerably differing magnitudes of the maximum responses of investment (0.23 times the shock) and consumption (0.03 times the shock) can be traced to the values of the coefficients assigned to the real interest rate in (3.32’) and (3.25), respectively. Again, since according to equation (3.26) the output gap is a weighted average of consumption and investment, its immediate response has an intermediate maximum value of about 0.08 times the size of the technology shock. The maximum positive response of capital (approximately 0.03 times the size of the shock) is significantly smaller than the strongest increase in investment. In addition, the deviation in investment is channelled with a time lag to the dynamics of capital. It takes the capital stock whole 9 quarters until it reaches its highest positive deviation from its steady-state level. Once again, capital is also characterised by a longer-lasting adjustment than investmenteven after 60 quarters, capital remains under its steady-state level. 109 nominal (and real) interest rate to a monetary unit shock when no lagged nominal interest rate enters the central bank’s rule (see Chapter IV, Subsections 2.1 and 2.2) is that under interest-rate smoothing the maximum response of both variables remains more moderate (0.5 times the shock). After the initial positive deviation, monotonic convergence to the steady-state value occurs within 16 quarters. Again, due to the quantitatively weak response of inflation, the real interest rate response (a positive blip with a maximum value of 0.5, followed by a monotonic decrease) matches the nominal interest rate dynamics. 110 Figure 4.7: Responses to a monetary policy unit shock under an active rule with inflation- and output-targeting and interest-rate smoothing In conclusion, the analysis of the impact of a monetary policy unit shock in a model with sticky prices and wages, endogenous capital and adjustment costs shows that the only long-lasting deviations from steady-state values concern the capital stock and the real wage, which take more than 60 quarters to converge. Consumption, the output gap, the real and the nominal interest rate, on the other hand, all return to their steady-state values within approximately 16 quarters. Compared to the results under a baseline Taylor rule in Subsection 2.2, all variables except the nominal and the real interest rate are characterised by larger deviations from their steady-state values when interest-rate smoothing is introduced. The history-dependence of the nominal and real interest rate explains the smoother adjustment paths of both variables, as well as the quantitatively smaller but longer-lasting impact of the shock. 2.3.2. Technology unit shock A technology unit shock is modelled as an upward blip of 1 percent in the shock term t A H in the AR(1) process 1t tAt A AA U H   that enters the real marginal cost condition (3.33). Figure 4.8 reports impulse responses to a technology unit shock for the model specification with sticky prices and wages, endogenous capital and adjustment costs. 111 Variable deviations, caused by a technology unit shock, are generally of a much smaller magnitude that those occurring as a result of a monetary policy unit shock and in each case of a lower value than the shock itself. Adjustment to the steady-state values of all variables (except for real marginal cost) is longerlasting than in the case of a monetary shock. An interesting result differing from the cases without interest-rate smoothing is the fact that the impact on investment, the output gap and consumption does not unfold immediately after the occurrence of the technology shock. Instead, the largest deviations from steady state are registered only after 2 quarters. The strongest deviations are reported for real marginal cost, investment and the real wage. The initial transmission of the shock occurs through the immediate effect on real marginal cost. A large negative spike of almost 0.9 percent is observed immediately after the shock, followed by a gradual return to the steady-state level within the next 20 quarters. This negative initial impulse is then transmitted to inflation as evident from (3.36). As a result of the inflation differential, the nominal interest rate is reduced in accordance with the policy rule (3.38), but the response is more gradual. The largest decrease in the nominal interest rate is registered after 3 quarters and is 0.09 times as large as the shock, followed by oscillatory convergence to the steady-state value within more than 60 quarters. The initial positive blip of the real interest rate in the first quarter is explained by the immediate sharp decrease in inflation (and expected inflation) and the initially quantitatively smaller decrease in the nominal rate. After that, the nominal interest rate effect prevails and path of the real interest rate mimics on a smaller scale that of the nominal interest rate. Investment, consumption, the output gap, capital and the marginal product of capital all reveal an initial positive response to the technology shock, followed by a countervailing negative deviation of a comparatively smaller magnitude before their steady-state values are reached. The strongest deviations in consumption, the output gap and the marginal product of capital all occur within the first four quarters, while it takes about 12 quarters until the maximum impact of the shock on capital unfolds. The technology shock is again channelled to consumption and investment through its impact on the real interest rate. The maximum response of investment (0.5 times the shock) is significantly stronger than that of consumption and consumption (0.08 times the shock). The output gap responds by a positive blip of about 0.17 times the size of the technology shock. The maximum positive response of capital (approximately 0.08 times the size of the shock) is significantly smaller than the strongest increase in investment. In addition, the deviations in investment are channelled with a time lag to the dynamics of capital. It takes the capital stock a whole 12 quarters until it reaches its highest positive deviation. Once again, capital is also characterised 112 by a longer-lasting adjustment to its steady-state level than investmenteven after 60 quarters, capital remains under its steady-state level. 113 Figure 4.8: Responses to a technology unit shock under an active rule with inflation- and output-targeting and interest-rate smoothing In comparison to the response of capital and the output gap the magnitude of the deviation of the marginal product of capital in both directions is again quite modest, due to the inclusion of steady-state unit and marginal adjustment costs. Both the sluggish adjustment of marginal product of capital over 40 quarters and the undershooting path registered after the eight quarter following the shock can be traced back to the different adjustment dynamics of capital and output gap discussed above. Finally, the real wage reaches a maximum positive deviation from steady state of 0.45 times the shock after 12 quarters and converges monotonically to its steady-state value thereafter. The response is initially triggered by the inflation differential and then sustained by the dynamics of inflation expectations and the history-dependence on own past realisations of the real wage. In conclusion, the analysis of the impact of a technology unit shock in a model with sticky prices and wages, endogenous capital and adjustment costs shows that the long-lasting deviations from steady-state values concern numerous model variables (the capital stock, real wage, inflation, the nominal interest rate and investment) that take more than 60 quarters to converge. Compared to the impact of a monetary policy shock convergence to the steady-state values generally appears to be a longer-lasting process. For all variables the magnitude of the observed deviations is relatively smaller than in the case of a monetary policy unit shock. A monetary policy unit shock under a Taylor rule with interest-rate smoothing generates the largest initial deviations in consumption, investment and the output gap of all active rule specifications considered. The largest negative response of the nominal interest rate is more modest here than in the two cases without interest-rate smoothing and, after the first quarter is quantitatively not sufficient to offset a pro-cyclical fall in the real interest rate. 2.3.3. Consumption preference unit shock A consumption preference unit shock is modelled as an upward blip of 1 percent in the shock term t X H in the AR(1) process 1t tt XX XUX H   that enters the con- 114 sumption equation (3.25). Figure 4.9 reports impulse responses to a consumption preference unit shock for the model specification with sticky prices and wages, endogenous capital and adjustment costs. Variable deviations, caused by a consumption preference unit shock, are again of a very small magnitude and in each case of a much lower value than the shock itself. Moreover, in the case of a preference shock is that the adjustment to the steady-state values for all variables occurs much faster than in the case of a technology or a monetary policy shock. Within approximately 16 quarters after the initial impulse all variables apart from capital, the real wage and the marginal product of capital have returned to their steady-state values. The strongest deviations as a result of the shock are reported for consumption, investment and the output gap. The initial transmission of the shock occurs through the immediate effect on consumption as in (3.25). A positive blip of 0.17 percent is observed immediately after the shock, followed by a fast return to the steady-state level within the first 4 quarters. Through its effect on consumption, the preference shock is channelled to the output gap (see equation (3.26)) which also reports a sharp rise by 0.09 percent, followed by a rapid convergence to steady state. As a result of the positive output gap, the nominal interest rate is increased in accordance with the policy rule (3.38), but the response is of a smaller magnitude (0.004 times the shock at its peak). After the first quarter, the nominal interest rate is gradually decreased and reaches its steady-state value in the fourteenth quarter. Because of the negligibly small impact of the shock on inflation, the path of the real interest rate mimics that of the nominal interest rate. The rise in the real interest rate releases an initial fall in investment of 0.2 times the shock, followed by a monotonic adjustment within 8 quarters. Capital initially responds to the negative blip in investment by a gradual decrease, reaching a maximum negative deviation from its steady-state level 8 quarters after the shock. After that, the negative dynamics is reversed and capital converges to steady state after approximately 60 quarters. The deviations of capital and the output gap from steady state determine the responses of marginal product of capital and real marginal cost respectively. Due to the combined impact of the output gap and the capital fluctuations, the maximum response of marginal product of capital (0.004 times the shock) is stronger than that of capital. The response in real marginal cost shows an initial positive spike of almost 0.5 times the shock. The positive initial effect on real marginal cost is transmitted to inflation as evident from (3.36), whereby the maximum impact on inflation occurs immediately after the shock and is quantitatively small (an increase of 0.001 percent). The impulse disappears in 12 quarters. Finally, as in (3.37) the real wage response takes about 12 quarters to reach its highest negative deviation of 0.003 115 times the shock. This negative deviation persists until about 60 quarters after the initial impulse. 116 Figure 4.9: Responses to a consumption preference unit shock under an active rule with inflation- and output-targeting and interest-rate smoothing In conclusion, a consumption preference unit shock has quantitatively a relatively modest effect on the model variables. Its initial impact on consumption, the output gap and investment is alleviated by the increase in the real interest rate (caused by the nominal interest rate hike). The moderate responses of all other variables are determined by the magnitude of the deviations in the two latter variables in particular. The results for a consumption preference unit shock for a Taylor rule with interest-rate smoothing are comparable to these under a baseline Taylor specification. The major difference pertains to the more moderate and gradual responses of the nominal and real interest rate. The induced negative deviation in investment is stronger than under a baseline Taylor rule; thus, the positive output gap impulse is smaller under interest-rate smoothing and the monetary policy stance acts stabilising. 3. Passive rule 3.1. The case of inflation-targeting only As it is evident from the determinacy analysis in Chapter III, Subsection 4.2.3, an interest-rate rule with *1 S O and *0 y O does not yield determinacy of rationalexpectations equilibrium for any value of the inflation coefficient under unity. Thus, the impulse response of each variable under a passive rule with a sole inflation target would reveal one adjustment path among several possible, generated by picking the smallest eigenvalue of the system within the unit circle. Still, alternative scenarios cannot be ruled out and therefore no reliable shock analysis can be carried out. 3.2. The case of inflation- and output-targeting This subsection shows the impulse responses of l t c, m t inv , l t y , n t rmc , l t k, n t mpk , l t Z ,t S ,t r and t i as for three shocks: the random component of monetary policy t i H , the technology shock t A H and the preference shock t X H . Monetary policy is 117 defined in terms of a passive Taylor rule with inflation and output coefficients both equal to 0.5128. 3.2.1. Monetary policy unit shock Figure 4.10 reports impulse responses to a monetary policy unit shock (an upward blip of 1 percent in the shock term t i H in the policy rule (3.38) for the model specification with sticky prices and wages, endogenous capital and adjustment costs. The variable responses to a monetary policy unit shock under a passive rule with inflation- and output-targeting do not differ substantially from the results under a baseline active Taylor rule. Compared with the case of a passive rule with a sole inflation target, more moderate negative spikes are observed in investment, consumption and the output gap immediately after the shock, followed by a gradual return to the steady-state level. Investment response reports by the sharpest fall (twice the shock). Compared to investment, consumption shows a more moderate response (0.25) and a faster adjustment to its steady-state value. As a weighted average of consumption and investment, the immediate response of the output gap has an intermediate value of about -0.7 times the size of the monetary policy shock. It takes both consumption and the output gap approximately 6 quarters until the effect of the monetary policy unit shock subsides completely. The maximum negative response of capital (approximately 0.1 times the size of the shock) is significantly smaller than the initial fall in investment. Contrary to the immediate impact of the shock on investment, it takes about 6 quarters until capital reaches its maximum negative response. As a predetermined variable, capital is also characterised by a longer-lasting adjustment to its steady-state level than investmenteven after 60 quarters, capital remains under its steady-state level. Again, in comparison to the relatively strong response of capital and the output gap the magnitude of the deviation of the marginal product of capital (- 0.03 times the shock at its minimum) is relatively weak. Both the sluggish adjustment of marginal product of capital over almost 60 quarters and the overshooting path registered after the forth quarter following the shock can be traced back to the different adjustment dynamics of capital and output gap discussed above. The sluggish adjustment of the real wage is determined by the partial history-dependence of the variable as seen from (3.37) and by the assumption of sticky wages. 128 For the values originally proposed by Casares and McCallum (2006), the effective response coefficient assigned to the output gap in (3.39) is so small that no significant difference to the case with a sole inflation target can be observed. The reason for choosing *0.5 yy OO is the fact that, as shown in Chapter III, Subsection 4.2.3, the coefficient values **0.5 y S OO yield a determinate rational-expectations equilibrium. 118 The responses of capital, the output gap and real wage determine the dynamics of real marginal cost, as evident from equation (3.33). Compared to the output gap dynamics, the path of real marginal cost reaches a more moderate peak negative value (0.4 times the shock), but does not adjust monotonically to its steady-state value. Instead, 4 quarters after the shock real marginal cost slightly overshoots the steady-state value; thereafter, it converges monotonically to the steady—state level. While the initial negative impulse to the real marginal cost is mainly a result of the sharp immediate decline of the output gap after the shock, the overshooting part is essentially explained by the slow and lasting response of capital and the real wage. A monetary policy unit shock has a quantitatively small and non-persistent impact on inflation, transmitted by the real marginal cost. The initial negative blip in inflation is less than 0.01 times the shock. In addition, until the effect subsides completely after 16 quarters, the path of inflation reveals overshooting dynamics before steady-state value is reached 16 quarters after the initial impulse. 125 Figure 4.12: Responses to a consumption preference unit shock under a passive rule with inflation- and output-targeting In conclusion, a consumption preference unit shock has quantitatively a relatively modest effect on the model variables. Its initial impact on consumption, the output gap and investment is alleviated by the increase in the real interest rate (caused by the nominal interest rate hike). The moderate responses of all other variables are determined by the magnitude of the deviations in the two latter variables in particular. 126 3.3. The case of inflation- and output-targeting with interest-rate smoothing The impulse responses in this subsection are calculated based on a passive Taylor rule with inflation and output coefficients both equal to 0.5130 and interestrate smoothing, whereby the lagged interest rate coefficient is equal to 0.8 as proposed by Casares and McCallum (2006). 3.3.1. Monetary policy unit shock Figure 4.13 reports impulse responses to a monetary policy unit shock (an upward blip of 1 percent in the shock term t i H in the policy rule (3.38) for the model specification with sticky prices and wages, endogenous capital and adjustment costs. The results under a monetary policy shock do not differ from these with a passive rule and inflation-targeting only. Negative spikes in investment, consumption and the output gap are observed immediately after the shock, followed by a gradual return to the steady-state level. Investment response is characterised by the sharpest fall (3 times as large as the shock) and gradual adjustment within 32 quarters. Compared to investment, consumption shows a more moderate response (0.5 times the shock) and a faster adjustment to its steady-state value. Since according to equation (3.26) the output gap is a weighted average of consumption and investment, its immediate response has an intermediate value approximately as large as the monetary policy shock. It takes both consumption and the output gap approximately 8 quarters until the effect of the monetary policy unit shock subsides completely. The paths of consumption, investment and output gap plotted on Figure 4.13 reveal the significance of the endogenous capital assumption for the model’s quantitative results. Under constant capital, the output gap response to a monetary policy unit shock would still be negative, but of a considerably smaller magnitude, as it would only incorporate the fall in consumption. The maximum negative response of capital (approximately 0.15 times the size of the shock) is significantly smaller than the initial fall in investment. Contrary to the immediate impact of the shock on investment, it takes 5 quarters until capital reaches its maximum negative response. Moreover, capital is also characterised by a longer-lasting adjustment to its steady-state level than investment- even after 60 quarters, capital remains under its steady-state level. In comparison to the relatively strong response of capital and the output gap the magnitude of the deviation of the marginal product of capital (-0.04 times the shock at its minimum) might at first seem worth further consideration. Algebraically this is determined by the relatively small value of the steady-state mar- 130 At the end of this subsection, the results under the values originally proposed by McCallum and Casares (2006) for (3.39) are shown. 127 ginal product of capital (0.04) that enters equation (3.34) as a coefficient, implying a relatively weaker response of marginal product of capital to deviations in output and capital. The sluggish adjustment of the real wage is determined by the partial history-dependence of the variable as seen from (3.37) and by the assumption of sticky wages. The latter is modelled by the inclusion of a fixed probability W K (0.75 W K ) that nominal wages cannot be adjusted in the current quarter, which implies that on average wages are re-set once a year. The responses of capital, the output gap and real wage determine the dynamics of real marginal cost, as evident from equation (3.33). Compared to the output gap dynamics, the path of real marginal cost reaches a more moderate maximum negative value (0.5 times the shock), but does not adjust monotonically to its steady-state value. Instead, 4 quarters after the shock real marginal cost overshoots the steady-state value; thereafter, real marginal cost converges monotonically to the steady—state level. While the initial negative impulse to real marginal cost is mainly a result of the sharp immediate decline of the output gap after the shock, the overshooting part is essentially explained by the slow and lasting response of capital and the real wage. A monetary policy unit shock has a quantitatively small and non-persistent impact on inflation. As evident from equation (3.36), the monetary impulse is transmitted by real marginal cost. Under sticky prices, despite the strong response of real marginal cost to the shock, inflation remains hardly affected, as the value of the fixed probability that households cannot adjust their price 0.75 P K (i.e. prices adjusted once a year on average) implies that the real marginal cost coefficient (1 )(1 )(1 ) / (1 ) PP PFPPFPF E KKDKDDT   equals 0.02. Thus, the initial negative blip in inflation is less than 0.01 the shock. Until the effect subsides completely after 16 quarters, the path of inflation involves overshooting dynamics before steady-state value is reached. As far as the nominal interest rate is concerned, according to the policy rule (3.38) its deviation depends on the unit shock itself, the responses of the target variables (inflation differential and output gap) and the past deviation of the variable from its steady-state value. The monetary authority’s reaction to the unit shock involves an increase in the nominal interest rate half as large as the shock in the first quarter, followed by a gradual convergence to the steady-state level. The effect of the monetary impulse disappears completely after 8 quarters. The Fisher equation 1tt tt riE S   postulates that the dynamics of the real interest rate is approximated by the difference between the paths of inflation and the nominal interest rate. Due to the quantitatively weak response of inflation, the real interest rate response (a positive blip with a maximum value half as large as the shock, followed by a monotonic decrease) matches to a great extent the nominal interest rate dynamics. 128 129 Figure 4.13: Responses to a monetary policy unit shock under a passive rule with inflation- and output-targeting and interest-rate smoothing In conclusion, the analysis of the impact of a monetary policy unit shock in a model with sticky prices and wages, endogenous capital and adjustment costs shows that the only long-lasting deviation from the steady-state value concern the capital stock, which takes more than 60 quarters to converge. Consumption, the output gap, the real and the nominal interest rate, on the other hand, all return to their steady-state values within 8 quarters. These results basically confirm the long-run monetary neutrality of the model. 3.3.2. Technology unit shock Figure 4.14 reports impulse responses to a technology unit shock (an upward blip of 1 percent in the shock term t A H in the AR(1) process 1t tAt A AA U H   that enters the real marginal cost condition (3.33) for the model specification with sticky prices and wages, endogenous capital and adjustment costs. Variable deviations, caused by a technology unit shock, are generally of a much smaller magnitude that those occurring as a result of a monetary policy unit shock and in each case of a lower value than the shock itself. Another distinction in the case of a technology shock is the slower adjustment to the steadystate values of nearly all variables (except real marginal cost and the nominal and real interest rate). The strongest deviations are reported for real marginal cost, real wage and investment. The adjustment paths of near all variables resemble the ones under a passive interest-rate rule and a sole inflation target. Compared to the active specification with interest-rate smoothing, the impulse responses of consumption, investment and the output gap are of a smaller magnitude. Through the moderate nominal interest rate cuts, passive policy generates a rise in the real interest rate that acts in a stabilising manner. The initial transmission of the shock occurs through the immediate effect on real marginal cost. A large negative spike of almost 0.9 times the shock is registered immediately after the shock, followed by a gradual return to the steadystate level. This negative initial impulse is then transmitted to inflation as evi- 130 dent from (3.36). The assumption of sticky prices determines the negative deviation of 0.09 times the shock of inflation from steady state. As a result of the increased inflation differential, the nominal interest rate is reduced in accordance with the policy rule (3.38), but the response is quantitatively smaller. The initial positive blip in the real interest rate is explained by the initial decrease in expected inflation in (3.36) that in the first 10 quarters quantitatively exceeds the negative effect on the nominal interest rate. Thus, in accordance with the Fisher equation 1tt tt riE S   , the real interest rate reports a positive deviation from steady state. Investment, consumption, the output gap, capital and the marginal product of capital all reveal an initial positive response to the technology shock. The strongest deviations in consumption, the output gap and the marginal product of capital all occur within the first 8 quarters, while it takes longer until the maximum impact of the shock on capital unfolds. The technology shock is channelled to consumption and investment through its impact on the real interest rate; the considerably differing magnitudes of the maximum responses of investment (0.18 times the shock) and consumption (0.05 times the shock) are again observed. The immediate response of the output gap reaches a maximum value of about 0.08 times the size of the technology shock. The maximum positive response of capital is significantly smaller than the strongest increase in investment. In addition, the deviations in investment are channelled with a time lag to the dynamics of capital. Once again, capital is also characterised by a longer-lasting adjustment to its steady-state level. In comparison to the response of capital and the output gap the magnitude of the deviation of the marginal product of capital (about 0.002 times the shock in both directions) is again quite modest, due to the inclusion of steady-state unit and marginal adjustment costs. Both the sluggish adjustment of marginal product of capital over 60 quarters and the undershooting path registered after the twentieth quarter following the shock can be traced back to the different adjustment dynamics of capital and output gap discussed above. 131 Figure 4.14: Responses to a technology unit shock under a passive rule with inflation- and output-targeting and interest-rate smoothing 132 Finally, the impulse response of the real wage is worth some consideration. As a result of the labour-augmenting technology shock, the variable reaches a quantitatively significant131 maximum positive deviation from steady state of 0.4 times the shock after 10 quarters and converges monotonically to its steady-state value thereafter. The response is initially triggered by the inflation differential and then sustained by the dynamics of inflation expectations and the historydependence on own past realisations of the real wage. In conclusion, the analysis of the impact of a technology unit shock in a model with sticky prices and wages, endogenous capital and adjustment costs shows that the long-lasting deviations from steady-state values concern the marginal product of capital, inflation and the real wage that take more than 60 quarters to converge. Compared to the impact of a monetary policy shock convergence to the steady-state values generally appears to be a longer-lasting process. For most variables except the real marginal cost, inflation and the real wage the magnitude of the observed deviations is relatively smaller than in the case of a monetary policy unit shock. The reason for the milder impact of a technology unit shock is it induces nominal interest rate cuts as a reaction to the increased inflation differential that quantitatively exceed the latter and thus generate a rise in the real interest rate as well. Through the real-interest-rate channel the positive impulse to investment, consumption and the output gap remains moderate and the variables report positive deviations significantly smaller than the size of the shock. 3.3.3. Consumption preference unit shock Figure 4.15 reports impulse responses to a consumption preference unit shock (an upward blip of 1 percent in the shock term t X H in the AR(1) process 1t tt XX XUX H   that enters the consumption equation (3.25) for the model specification with sticky prices and wages, endogenous capital and adjustment costs. Variable deviations, caused by a consumption preference shock, are of a very small magnitude and in each case of a much lower value than the shock itself. Another distinction in the case of a preference shock is that the adjustment to the steady-state values for all variables occurs much faster than in the case of a technology or a monetary policy shock. Within approximately 16 quarters after the initial impulse all variables apart from capital and the real wage have returned to their steady-state values. The strongest deviations as a result of the shock are reported for consumption, the output gap and real marginal cost. The initial transmission of the shock occurs through the immediate effect on consumption as in (3.25). A positive blip of 0.2 the shock is observed immediately after the shock, followed by a fast return to the steady-state level within the first 4 quarters. Through its effect on consumption, the preference shock is 131 The magnitude of the impulse response of real wage is significant especially compared to the magnitude of the responses of the other model variables. 133 channelled to the output gap (see equation (3.26) which also reports a sharp rise by 0.1 percent, followed by a rapid convergence to steady state. As a result of the positive output gap, the nominal interest rate is increased in accordance with the policy rule (3.38), but the response is of a smaller magnitude (0.005 times the shock at its peak). After the first quarter, the nominal interest rate is gradually decreased; the path of the real interest rate mimics that of the nominal interest rate with an initial positive blip of slightly under 0.005. The rise in the real interest rate releases a positive response of investment, followed by an adjustment within 12 quarters. The impulse to capital is also extremely modest. The deviations of capital and the output gap from steady state determine the responses of marginal product of capital and real marginal cost respectively. Due to the combined impact of the output gap and the capital fluctuations, the maximum response of marginal product of capital (0.004 times the shock) is stronger than that of capital. After the initial positive blip, the marginal product of capital converges to steady state within 6 quarters. The response in real marginal cost shows a similar path – an initial positive spike of 0.06 times the shock, followed by a gradual convergence to steady state within the first 6 quarters. The positive initial effect on real marginal cost is transmitted to inflation as evident from (3.36), whereby the maximum impact on inflation occurs immediately after the shock and is quantitatively small (an increase of 0.002 times the shock). The impulse disappears in less than 4 quarters. Finally, as in (3.37) the real wage response is driven by the positive inflation differential and then sustained by the dynamics of inflation expectations and the history-dependence on own past realisations of the real wage. Unlike inflation, the real wage takes about 20 quarters to reach its highest negative deviation of 0.005 times the shock. This negative deviation persists for more than 60 quarters after the initial impulse. In conclusion, a consumption preference unit shock has quantitatively a relatively modest effect on the model variables. Its initial impact on consumption, the output gap and investment is alleviated by the increase in the real interest rate. The moderate responses of all other variables are determined by the magnitude of the deviations in the two latter variables in particular. 134 m m l 1 11 tt inv k kt G GG    . (3.35’) Equations (3.33) and (3.34) provide the algebraic terms to be substituted for n t rmc n and in (3.32’) and (3.36): t mpk l    1  tt tt PF rmc k y PF D (3.33) ZD    l l ( tt mpk mpk y k ) t. (3.34) Next, l t y can be substituted out according to equations (3.26) and (3.35’)135: l l m l 1 1 inv tct t inv t y ck Z ZZ GG   §·   ¨¸ ©¹ k G . (3.26’) From (3.41) 1tt tt riE S   and after rewriting (3.25) as  1t tt rEc c V     t, (3.25’) the nominal interest rate can be substituted in the policy rule equation (3.38) or (3.39). After substituting l t y out according (3.26’), the interest-rate rule relation becomes136: m l m  l ** * * 11 11 1yc yinv yinv tt t t t t t t Ec c k k E S OZ OZ OZ G O 1 SS VVG VG VV  §·       ¨¸ ©¹ . (A3.12) Finally, equations (3.25’), (3.33), (3.34), (3.35’), (3.26’) and (3.41) can be combined with (3.32’), (3.36), (3.37) and (A3.12) to yield the system’s reduced forms: m      m 2 1 * 416 17 3175 121 t tinv y inv kkbbb bb bbbb N GZO Z   ªº ;; ;  ¬¼ 1      ** 4 3 17 5 6 6 17 1 7 3 17 5 11111 t t c yy kb b bb b b b bb b b b bb b c NN * y ZG GO O O ;ªº ªº ;  ;     ¬¼ ¬¼    l m 1 *5 17 2 17 5 59 111 ttt b bb b bb b bb NN N S GG G OS ZZ EE E  §· ;; ;    ªº ¨¸ ¬¼ ©¹ (3.32’’) m   1 36 13 11t tt inv tt t c kkc bb b Eb 2 3t b Z Z SS Z E GG  §·     ¨¸ ©¹  E (3.36’)    m   1 36 92 3 11 3 1 t tt inv ttt c t E kk bb bb b b ZZZ EE G ZZ GG     §·  ¨¸ ©¹  c (3.37’) 135 Equation (3.26’) dies not include the term l g t g Z as in the calibration .0 g Z 136 Here * S O and * y O denote the effective values assigned to inflation and the output gap in the monetary rule. For the Taylor rule (3.38) *1 SS OO  and y * y OO . For the policy specification (3.39) *(1 )(1 ) i SS OOO   and y *(1 ) yi OOO  . For the purpose of the subsequent determinacy analysis no interest-rate smoothing is assumed, i.e. . The possibility of pursuing interest-rate smoothing will be further considered when the implications of shocks are examined. 0 i O 141   m   l 11 ** ** 2 73 7 3 36 6 11 1 1 1 1 tt ttt t yy y Ec c k k b bb b b bb b S t OOGOOSZ GV VG V E  §· §· ªº ªº           ¨¸ ¨¸ ¬¼ ¬¼ ©¹ ©¹  VE (A3.12’) where 1 *10 1 PF PF rmc bmpk D D §·  ¨¸  ©¹ ! 2 (1 )(1 )(1 ) (1 ) P PP PPFPF bF E KKD KDDT   ,2 01b 3 (1 )(1 ) (1 ) P FPP PPFPF b DEKK E KDDT   , for 01 32 01bb PF DE  4 1 1 b G , for 4 01b 0 G ! 5*0bmpkrmc !  61 inv b ZG ,6 01b 70 c b Z V ! 81 PF P F b D D ,8 01b 911b E  ! 1* ** 1 11 PF inv PF rmc Nmpk D Z GD §· ; ! ¨¸  ©¹ 0. For convenience, the system (3.32’’), (3.36’), (3.37’) and (A3.12’) can be rewritten as: m m  l  1 21 123456 t ttt tt t kkkaaaacaa Z SZ     (A3.13) m   11 7 8 9 10 11 t tt tt t kk cEaaaaa t Z SS        (A3.14)  m   11 12 13 14 15 16 t ttt t Ekkaaacaa 1tt ZZZ    (A3.15) m  l 11 17 18 19 20 21 t tttt t Ec k k caaaaa S t Z    (A3.16) where     * 1416 17 3175 121 inv y inv abbb bb bbbb N GZO Z ªº ;; ;  ¬¼  1  * 2431756 617 111 y abbbbbb bbb N GO ªº ;  ;  ¬¼   * 31 11ab N S GOE §· ;  ¨¸ ©¹ 7 b   ** 4173175 11 c yy abbbbbb N ZG OO ;ªº  ¬¼   52175 1abbbbb N59 b G E ;  ªº ¬¼ 142 5 6 b aN G E ; 3 7 invb a Z G 6 83 1 b ab G §  ¨ ©¹ · ¸ 9 1 a E 10 3c ab Z 2 11 b a E 3 12 invb a Z G  36 13 bb a G G  14 3c ab Z 92 15 bb a E  16 1 a E  * 17 7 3 11 y ab O GV §·   ¨¸ ©¹ b  * 18 3 6 6 1 y abb GO VG ªº  ¬¼ b * 19 11 a S O VE §·  ¨¸ ©¹  * 20 7 3 1y abb O ªº   ¬¼ 2 21 b a VE . 143 References Abel, A. 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