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The impact of learning and information dynamics on optimal policy Doyle, Matthew Stephen 2002

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THE IMPACT OF LEARNING AND INFORMATION DYNAMICS ON OPTIMAL POLICY by MATTHEW STEPHEN DOYLE B.A.(hons.), McMaster University, 1995 M.A., McMaster University, 1996 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 2001 ©Matthew Stephen Doyle, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract The goal of this dissertation is to analyze issues that arise when policy makers try to learn about the economy while their policies are affecting it. The dissertation takes the form of three essays. The first essay examines how optimal policy affects equiUbrium economic outcomes in environments in which agents are both imperfectly informed about the state of the economy and able to learn by observing the actions of others. This type of environment, in which there is social learning, has received growing attention, but to date there has been little examination of strategic policy making in such settings. In particular, the question of whether policy, in the absence of a commitment technology, can be designed to increase the speed of information revelation remains open. The essay builds on a real options model of investment and shows how this framework can be extended to derive time consistent policies and the related equilibrium outcomes in social learning environments. By comparing the equilibrium induced by a policy maker to both the laissez-faire outcome and the social optimum, it is shown that the policy maker is able to achieve the second best outcome and reduce delay to the efficient level even in the absence of commitment. The second essay raises the question of whether the fact that policy makers play a dual role, as both information gatherers and economic managers, can explain the flattening of the Phillips Curve relationship between inflation and real activity that has been observed in both Canada and the U.S over the 1990s. The paper models the central bank as both a provider of liquidity in a world where pre-set prices would otherwise cause potential gains from trade to go unrealized and a gatherer of information about real developments in the economy. The bank's information complements that of private agents so that, the central bank and private agents both wish to learn from the other. In equilibrium, this interaction gives rise to a Phillips curve relationship which both exhibits causality running from real activity to prices and justifies a feedback from prices to the setting of monetary instruments. The model implies that a decline in the slope of the Phillips curve may be a result of improvements in the manner in which central banks gather information about the economy. An investigation of the data for Canada and the U.S finds support for the model. The third essay attempts a more thorough empirical investigation of the issues raised in the previous chapter. The paper enriches the dynamic aspects of the model to further examine its properties, but focuses mainly on attempting to uncover whether the types of changes to the Phillips curve relationship which had been previously documented in Canada and the U.S have occurred in other OECD countries. The paper investigates this question using both single country and panel estimation and finds that the phenomenon of a declining slope in the Phillips curve relationship is prevalent in OECD countries throughout the 1980s and 1990s. Finally, the paper attempts to exploit the cross country data to provide more formal tests of the model's predictions regarding policy innovations and inflation targeting regimes. The results suggest that the model compares favourably to other potential explanations of the decline in the slope of the Phillips curve. ii Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Dedication vii 1 Overview and Summary 1 2 Informational Externalities, Strategic Delay and the Search for Opti-mal Policy 8 2.1 Introduction 8 2.2 Literature Review 11 2.3 An Investor's Problem 13 2.3.1 A One-Period Example 14 2.3.2 A Game of Timing 18 2.3.3 Equihbrium 24 2.4 The Policy Maker's Problem 26 2.4.1 Preferences and Information of the Policy Maker 27 2.4.2 The One-Step Property Revisited 30 2.4.3 Solving the Policy Maker's Problem 32 2.4.4 Equilibrium 40 2.5 Welfare Analysis 41 2.5.1 Social Preferences 42 2.5.2 The First Best Solution 43 2.6 Conclusion 45 2.7 Appendix 1 47 2.8 Appendix 2 47 2.9 Appendix 3 48 iii 3 What Happened to the Phillips Curve iri Canada and the U.S in the 1990s? 49 3.1 Introduction 49 3.2 Overview of the Output-Inflation Relationship in Canada and the U.S . 51 3.2.1 Basic Estimation and Results 52 3.2.2- The Changing Slope of the Phillips Curve 58 3.3 Why is there is a Phillips Curve and why might its slope change over time? 65 3.3.1 The Flattening Phillips Curve: Evidence of Optimal Policy or Downward Nominal Rigidities 76 3.3.2 The Flattening of the Philhps Curve and the Ball, Mankiw & Romer hypothesis 79 3.4 Conclusion 82 4 The Changing Nature of the Phillips Curve in OECD Economies 84 4.1 Introduction 84 4.2 A Model of the Phillips Curve 88 4.2.1 Other Properties of the Model 95 4.3 Some Stylized Facts 97 4.3.1 The Existence of Phillips Curve Relationships in the OECD . . . 97 4.3.2 The Changing Slope of the Phillips Curve: Rolling Regressions . 101 4.3.3 The Changing Slope of the Phillips Curve: Dummy Variables . . 102 4.3.4 The Variances of Output and Inflation 106 4.4 Further Tests of the Model 107 4.4.1 Monetary Contractions 107 4.4.2 Inflation Targeting 112 4.5 Conclusion 115 5 Conclusion 150 Bibliography 151 iv List of Tables 3.1 Basic Phillips Curve Estimates for Canada: 1961-1999 55 3.2 Basic Phillips Curve Estimates for the U.S.: 1961-1999 56 3.3 Phillips Curve Estimates for Canada: 1961-1999 57 3.4 Phillips Curve Estimates for the U.S.: 1961-1999 57 3.5 Rolling Sample Variances for Canada and the U.S: 1983-1999 75 4.1 Simple Phillips Curve Estimates for OECD Countries: 1961-1997 . . . 116 4.2 Alternate Estimates of Beta for OECD Countries: 1961-1997 118 4.3 Imposing the Unit Sum Constraint 120 4.4 Controlling for Energy Prices 122 4.5 Slope Estimates from Rolling Regressions 123 4.6 OLS Estimation with an Interacted End of Sample Dummy 124 4.7 SUR Estimation with an Interacted End of Sample Dummy 126 4.8 SUR Estimation with an Interacted End of Sample Dummy 128 4.9 SUR Estimation with an Interacted End of Sample Dummy: Unit Sum . 130 4.10 SUR Estimation with an Interacted End of Sample Dummy: Energy Pricesl32 4.11 SUR Estimation with a Monetary Contraction Dummy 133 4.12 SUR Estimation with Contraction Dummy, Separate estimates for end of sample, r = 1985 135 4.13 SUR Estimation with Contraction Dummy, Separate estimates for end of sample, r = 1981 137 4.14 SUR Estimation with Contraction Dummy, Separate estimates for end of sample, r = 1987 139 4.15 SUR Estimation with Contraction Dummy, Separate estimates for end of sample, Unit Sum 141 4.16 Inflation Targeters 143 4.17 Inflation Targeters (controlling for additional Lags of inflation) 144 List of Figures 2.1 Properties of Equilibrium 25 2.2 Optimal Delay is Similar to the Myopic Case 37 2.3 Optimal Delay Exceeds that of the Myopic Case 38 2.4 Optimal Delay is Less than that of the Myopic Case 39 3.1 Phillips Curve, Canada 1961-1999 53 3.2 Phillips Curve, U.S 1961-1999 54 3.3 Slope of Canadian Phillips Curve Over Time 59 3.4 Slope of U.S Phillips Curve Over Time 60 3.5 Canadian Phillips Curve Over Time, Alternate Specification 61 3.6 U.S Phillips Curve Over Time, Alternate Specification 62 3.7 Phillips Curve Over Time, Pooled Sample 63 3.8 Phillips Curve, 1985-1999, Pooled Sample 64 3.9 Output Gap Coefficient, Pooled Sample 77 3.10 Positive Gap Coefficient, Pooled Sample 78 3.11 Phillips Curve, Canada, 1980-1999 81 4.1 UK, Rolling Regressions, Base Case 145 4.2 France, Rolling Regressions, Base Case 146 4.3 Finland, Rolling Regressions, Base Case 147 4.4 Norway, Rolling Regressions, Base Case 148 4.5 Changes in the Variance of Inflation and the Output Gap 149 vi Dedication To Granny, Roscoe, and Jennifer (Mom). Chapter 1 Overview and Summary The process by which information about variables of economic importance is gener-ated and disseminated amongst agents has long been of interest to economists. Hayek (1945) writes that the distribution of information across agents in the economy is a cen-tral feature of a market economy, and crucial to our understanding of the functioning of such a system: The peculiar character of the problem of a rational economic order is determined precisely by the fact that the knowledge of the circumstances of which we must make use never exists in concentrated or integrated form, but solely as the dispersed bits of incomplete and frequently contradic-tory knowledge which all the separate individuals possess. The economic problem of society is thus not merely a problem of how to allocate 'given' resources...it is rather a problem of how to secure the best use of resources known to any members of society, for ends whose relative importance only those individuals know. Or, to put it briefly, it is a problem of the utiliza-tion of knowledge not given to anyone in its totality. Early Keynesian models of the macroeconomy generally avoided a serious treatment the role of information. However, attempts to create dynamic models out of the original, static Keynesian framework required some treatment of expectations and, therefore, information. Early efforts consisted of tacking on an expected inflation term to the original model, but the process by which these expectations were formed remained ad hoc. This approach led eventually to Lucas' (1976) famous critique which ushered in the "rational expectations revolution" foreshadowed by the work of Muth. The rational ex-pectations research agenda was a first step towards a reintroduction of a serious study of informational effects into macro. Early rational expectations models explicitly formalized the process by which agents receive information about the economy. With these models, macroeconomists became capable of understanding the role of information in a formal setting. However, the role of information in the standard rational expectations models bore 1 little resemblance to the economic order described by Hayek. Agents generally received signals about stochastic processes without the ability to affect the generation of these signals. Furthermore, the representative agent framework did not allow for a distribution of information amongst agents. Within the more conventional framework, Grossman &; Stiglitz (1980) argue that, when agents do not receive costless signals but instead have to pay for information, prices do not fully convey private information. The idea is that if prices are fully informative, there is no incentive for agents to pay for information. On the other hand, if no other agent pays for information (so that prices are uninformative) then every agent has an incentive to pay for private information. This leads to a breakdown of fully revealing equilibria. Vives (1993) derives a similar result examining the convergence of prices to a fully revealing equilibria over time. He shows that as prices become more informative, agents rely more on prices as a signal than on their own private information. Therefore, prices are slow to incorporate further private information. The result that prices incorporate agents' private information slowly occurs precisely because they transmit information successfully. Another literature, (see Townsend (1983), and Sargent (1991)) attempts to relax the representative agent assumption in an attempt to allow for a distribution of private in-formation. This literature ran into difficulties related to the infinite regress problem. Essentially, to form expectations about macro variables, agents need to form expectations about the behaviour of other agents. Since all agents have to do this, they must in turn form expectations about the expectations of others, and so on and so forth. When agents have to forecast the forecasts of the forecasts of others, the dimensionality of the state Of the economy (which includes expectations) becomes infinite. This infinite regress means that standard state space techniques can not be used to solve the models. .More recently, there has emerged a literature devoted to the consideration of informa-tion issues and their relation to macroeconomic issues. Due to the difficulties of dealing with informational issues in standard models, this literature departs from the stochas-tic dynamic general equilibrium paradigm and investigates the effects of information on macro variables (especially investment) in a variety of partial equilibrium and game the-oretic models. Examples of this literature include the papers on informational herding. Banerjee (1992) and Bikhchandani, Hirshleifer &; Welch (1992) model environments in which an exogenous sequence of agents make a decision on whether or not to invest in a risky project. Each agent receives an imperfect signal on the value of investing, and observes 2 the sequence of actions (but not signals) of previous agents. In such an environment, some agents have signals contrary to the information available through observing the history. Since agents must make a discrete choice, they must either follow their own information or the information embedded in the history of actions. In those cases where agents ignore their own signals and follow the public information, their private information is lost.1 This is an example of an informational externality: agents who invest early may choose to invest contrary to their private signals, not taking into account the fact that the information lost because of this choice has value to those who invest later. Both Banerjee (1992) and Bikhchandani, Hirshleifer & Welch (1992) show that beliefs converge on the wrong action with positive probability. Chamley & Gale (1994) extend the idea to allow for the endogenous timing of invest-ment decisions. In their model, agents choose if and when to invest by balancing the benefit of investing sooner (undiscounted profits) with the option value associated with waiting and making an investment decision in a future period based on more information. Their model also displays an informational externality, because investors do not take into account the informational value that their investment decision has to other agents when making their choices, and exhibits "investment" crashes which the authors compare to the cascades of Banerjee (1992) and Bikhchandani, Hirshleifer &: Welch (1992). Caplin &; Leahy (1994) provide a different informational explanation of crashes. They model an environment in which agents face costs to changing their behaviour and therefore their actions do not change to fully reflect any new information they may have. In such a situation, private information is not revealed until some agents become so pessimistic (or optimistic) that they are willing to pay the cost and change their behaviour. This releases their private information which causes other agents to update their own beliefs and may in turn cause more agents to change. This triggers a crash (surge) as many agents change their behaviour and a large amount of information is revealed in a short time span. Further examples of social learning are the papers by Rob (1991) and Ziera (1994). These papers examine the dynamics of investment in a situation where firms do not know the size of the market for the final good. Firms can only learn the capacity of demand to absorb further investment by investing more and observing the outcome. These papers show that firms enter the market gradually, and that the evolution of information combined with the possibility of costly overshooting creates a boom and bust cycle in investment. 1When agents all follow the public information in spite of signals to the contrary, it is called a herd or informational cascade. 3 Economists have long known that policy makers operate under uncertainty regarding both the state of the economy and the way it will react to any given policy. Early examples include Brainard (1967), who shows that under certain conditions it is optimal for a policy maker who is uncertain about how a policy target will react to changes in a policy instrument (instrument uncertainty) to take a cautious approach, and Poole (1970), who examines the choice of intermediate target when policy makers cannot continuously observe output. Blinder (1997) gives voice to the commonly held belief that uncertainty about the economy and its reaction to policy represents a major challenge for policy makers. The presence of informational issues and externalities as outlined in the social learning examples discussed above raises further questions about efficiency and the possibility for policy. To the extent that informational frictions represent a serious source of inefficiency in the economy, it is interesting to examine the role that policy makers might play in such environments. There has been some work on this subject. Rob (1991) examines policy in his environment and finds that a social planner who takes into account the informational value of investments as well as profits would want to increase the speed of entry into the market. Smith & S0renson (1997) examine optimal policy in the herding models of Banerjee (1992) and Bikhchandani, Hirshleifer &; Welch (1992) by observing the similarity of a policy maker's problem in these environments to the problem faced by an experimenter in the optimal experimentation literature. They show that a policy maker would subsidize early agents to give them incentives to take into account the value of the public value of their private information. In their paper, Chamley &: Gale (1994) show that the appropriate one period subsidy induces early investment and overcomes the externality. In their model, a large economy can be induced to reveal all the information in the first period, thus achieving the first best outcome. It is not clear that policy makers can always be effective when they must act in the face of uncertainty. Friedman (1967) argues that uncertainty about the effect of monetary policy on the economy, and the lags with which policy works, limits the effectiveness of policy makers. Caplin & Leahy (1996), discuss informational issues in the policy formation process itself. They argue that a policy maker who is uncertain about both the state of the economy itself and the economy's reaction to policy may be forced to search for the optimal policy. Agents who are aware of the search rule may choose not to react to policy innovations because they know that not doing so will result in more attractive policy 4 environments in the future. In this way, policy directed at reducing delay, for example, may be self defeating. This dissertation examines questions relating to policy making under uncertainty in both traditional models and social learning environments. In particular, it examines en-vironments in which both agents and policy makers face uncertainty. In such settings, the policy maker may learn by observing private agents, but private agents may simulta-neously be attempting to learn, not only from each other, but from the policy maker as well. Chapter 2 examines how optimal policy affects equilibrium economic outcomes in en-vironments in which agents are both imperfectly informed about the state of the economy and able to learn by observing the actions of others. This type of environment, in which there is social learning, has received growing attention, but to date there has been little examination of strategic policy making in such settings. In particular, the question of whether policy, in the absence of a commitment technology, can be designed to increase the speed of information revelation remains open. The essay builds on a real options model of investment and shows how this framework can be extended to derive time con-sistent policies and the related equiUbrium outcomes in social learning environments. By comparing the equilibrium induced by a policy maker to both the laissez-faire outcome and the social optimum, it is shown that the policy maker is able to achieve the second best outcome and reduce delay to the efficient level even in the absence of commitment. This result is quite striking. It is frequently the case in optimal policy games that the optimal policy in the case where the policy maker has access to a commitment technology is preferable to the case where the policy maker cannot commit (see Kydland and Prescott (1977) for a particularly well known example relating to the Phillips curve). Given the uncertainty and strategic possibilities in the environment studied here, one might expect that a policy maker might be particularly ineffective. Chapter 2, however, shows that this is not the case. Chapter 3 raises the question of whether the fact that policy makers play a dual role, as both information gatherers and economic managers, can explain both the existence of a Phillips Curve as well as the flattening of the Phillips Curve relationship between inflation and real activity that has been observed in both Canada and the U.S over the 1990s. This chapter begins by reviewing the empirical properties of the Phillips Curve in both Canada and the U.S over the last forty years. In particular, it documents the extent to which the slope of the Phillips Curve has declined in both countries over the nineties. The theoretical 5 part of the paper focuses on the nature of the Phillips Curve when monetary authorities are imperfectly informed about real developments in the economy but nevertheless try to set monetary policy optimally. The model explicitly recognizes two distinct activities performed by the central bank: on one hand, the central bank tries to provide sufficient liquidity to help private agents exploit gains from trade during periods in which prices are pre-set. On the other hand, the central bank also performs an information-gathering role as it continuously tries to infer the state of the economy. This dual role gives rise to a Phillips Curve relationship that both exhibits causality running from output to prices and justifies a feedback from prices to the setting of monetary instruments. Based on this model, it is argued that the observed flattening of the Phillips Curve may be the result of improvements in the manner in which central banks gather information regarding real forces affecting the economy, and that the flattening is not a reflection of a change in the output-inflation tradeoff faced by the central bank. Chapter 3 also attempts to distinguish this story of the Phillips curve, based on the interplay of information between policy makers and private agents, from other potential alternatives. Key features of the model include the prediction that a well informed policy maker reduces the variance of inflation without increasing the variance of output and the prediction that the Phillips curve observed during normal times is a reduced form object which differs from the short-run output inflation tradeoff which follows a disinflationary shock. These predictions are used to distinguish the model from alternative hypotheses. A preliminary examination of the data for Canada and the US suggests that the model performs well relative to alternate theories, though the data do not allow for a definitive test. Chapter 4 attempts a more thorough empirical investigation of the issues raised in the previous chapter. The paper enriches the dynamic aspects of the model to further examine its properties, but focuses mainly on attempting to uncover whether the types of changes to the Phillips curve relationship which had been previously documented in Canada and the U.S have occurred in other OECD countries. The paper investigates this question using both single country and panel estimation and finds that the phenomenon of a declining slope in the Phillips curve relationship is prevalent in OECD countries throughout the 1980s and 1990s. Finally, the paper attempts to exploit the cross country data to provide more formal tests of the model's predictions regarding policy innovations and inflation targeting regimes. The results suggest that the model compares favourably to other potential explanations of the decline in the slope of the Phillips curve, though 6 not without some problems. The remainder of the dissertation is structured as follows: Chapter 2 discusses the role of policy in a real options model of investment with informational externalities. Chapter 3 shows how informational asymmetries between private agents and policy makers might give rise to a Phillips curve, the slope of which will declines as policy makers become more well informed. Chapter 4 examines empirical evidence relevant to the model presented in Chapter 3. Chapter 5 offers concluding comments. 7 C h a p t e r 2 Informational Externalities, Strategic Delay and the Search for Optimal Policy 2.1 Introduction Policy makers often labour under uncertainty about both the state of the economy and how the economy will react to their policy initiatives. As a result, policy initiatives may play a dual role as policy makers try to both learn about and affect the economy at the same time, which can lead to rich interactions between policy makers and private agents, who may also be trying to learn about the economy. These interactions complicate the policy making process and may limit the ability of policy makers to exert a positive influence on the economy. In this paper I examine the effects of optimal policy on equilibrium outcomes in a social learning environment, in which all agents are imperfectly informed but learn by observing the actions of others. I focus on the possibility that the policy maker's attempts to learn about the economy through policy may exacerbate delay because private agents rationally believe that initiatives which fail to elicit the desired outcome will be followed by more attractive policies, as conjectured by Caplin & Leahy (1996). The paper builds on a real options model of investment and shows how this framework can be extended to derive optimal time consistent policies and the related equilibrium outcomes. It is shown, by comparing the equilibrium induced by the policy maker to both the laissez-faire outcome and the social optimum, that the policy maker is able to reduce delay in equilibrium and in fact achieves the first best outcome. 8 The real options theory of investment has been proposed as an explanation of a variety of phenomena including recessions (Bernanke (1983)), the time path of foreign investment in developing and transition economies (IMF (1996), Thimann &; Thum (1999)) and the decision of polluters to adopt abatement technologies (Berman & Bui (1999)). The idea is that firms undertaking irreversible investment in stochastic environments may postpone their investment decisions in hopes of obtaining more accurate information. If this infor-mation is generated endogenously by the actions of agents, as in Chamley k Gale (1994), this postponement represents inefficiency. It is not clear what role policy makers may play in such an environment. In particular, a policy maker wishing to influence the decisions of investors has to take into account the fact that policy initiatives also reveal information about the economy to both the private agents and the policy maker. Thus, active policy may actually increase delay because agents anticipate that policy measures which fail to stimulate investment will be followed by more attractive policies, as in Caplin &; Leahy (1996). The idea is that a policy maker who wishes to stimulate investment, through the use of an investment subsidy for example, but fears over stimulating wants to attempt a small subsidy and watch to see what happens to the economy. If the initial subsidy fails to induce sufficient investment, the policy maker infers that a more aggressive policy was required and increases the subsidy. Agents who understand this process therefore have an incentive to delay in order to take advantage of the higher subsidies that will occur in the future if investment remains low. This mechanism may represent a common and important limitation on the ability of policy makers to influence aggregate investment. Thimann k Thum (1999), for example, tell exactly this story in an attempt to explain the failure of policy makers, without access to a credible commitment technology, to stimulate foreign direct investment in transition economies. Berman &; Bui (1999) make a slightly different case, suggesting that polluters deciding on whether to adopt an abatement technology which may or may not be profitable may wait in the hopes that regulators in another part of the country will force other firms to adopt the technology first, thereby revealing its productivity. In both cases, the claim is that the actions of policy makers may be inducing the postponement of the very investment decisions they wish to encourage. In this paper, I undertake a formal investigation of this claim in an environment, based on the model of Chamley k Gale (1994), in which agents make a decision on whether or not to invest in a project the profitability of which depends on some unobserved state of nature. Agents choose the timing of any investment decision they make knowing that they 9 can learn about the state of nature by observing the actions of previous investors. Agents cannot internalize the informational value that their decision to invest has to others, so that there exists an informational externality in the investment process. As a result, all agents wish to adopt a wait and see strategy, which in turn results in inefficiently low investment. The model provides an appropriate framework in which to analyze the problems facing a policy maker who both learns about and affects the economy with policy. Since policy makers face uncertainty regarding the state of nature and the composition of agents in the economy, there will be uncertainty about the economy's reaction to policy. The timing of agents' actions is endogenized, which allows for endogenous delay in the response to policy initiatives. Since the paper asks about the kind of impact a policy maker is likely to have on the economy, it is important that there is a clear source of inefficiency in the equilibrium in the absence of a policy maker. This makes it possible to measure the performance of the policy maker, with and without commitment, against both the outcome which would obtain in the absence of a policy maker as well as against a clearly defined welfare benchmark. This is important if we wish to understand what policy may and may not achieve in such an economy. The paper shows that a policy maker operating in this environment who sets policy optimally unambiguously reduces delay in equilibrium. Because the initial equilibrium exhibits inefficient delay, the policy maker wishes to subsidize early investment. Agents know that the policy maker will likely also subsidize investment in the future and thus have an incentive to ignore the initial policy offering, in hopes of benefiting from a more attractive offer later. Recognizing this, the policy maker chooses a subsidy which com-pensates the agents not only for the informational value of early investment, which they would otherwise be unable to internalize, but also for any future subsidies which they forego by investing earlier in the game. Thus the impact of the strategic interaction be-tween the agents and policy maker manifests itself in an optimal set of subsidies which is more aggressive than would otherwise be the case, but this does not prevent the policy maker from reducing equilibrium delay. Given that the policy maker is able to take the strategic behaviour of agents into account and reduce equilibrium delay, the remaining questions concern whether or not the policy maker's ability to achieve efficiency gains would be enhanced by access to a commitment technology. Since the equihbrium outcome in the presence of the optimal 10 set of dynamic subsidies may still exhibit delay, in the sense that there are agents with projects with positive expected value who choose to postpone investment, it may be the case that the policy maker could benefit from access to a commitment technology. It is shown that this is not the case and that the optimal time consistent subsidies that the policy maker implements in the absence of commitment allow the policy maker to attain the first best outcome as an equilibrium. These results suggest that there is a role for policy makers in this type of environment, even in the absence of a credible commitment technology. The paper proceeds as follows: Section 2.2 reviews the relevant literature, Section 2.3.3 presents the benchmark model and discusses the problem of an investor both in the presence and absence of an investment subsidy. Section 2.4 analyzes the problem of a benevolent policy maker in this environment, and shows that such a policy maker reduces delay relative to the laissez-faire outcome. Section 2.5 shows that the equilibrium outcome achieved by the policy maker is a first best, and Section 2.6 discusses the results and offers concluding comments. 2.2 Literature Review This paper studies the problem of policy making under uncertainty allowing for strate-gic interactions between agents and the policy maker, while including elements of social learning. It is therefore related to the social learning literature, which studies environ-ments in which the actions of one agent endogenously generate information that is of value to others. The herd behaviour models of Banerjee (1992) and Bikhchandani, Hirshleifer and Welch (1992) study an environment in which agents acting in an exogenously given sequence may find it optimal to ignore their own information and follow the group. They show that in equilibrium, there is a positive probability that agents herd on the incorrect action, suggesting that there is socially insufficient experimentation among agents who act early in the sequence. Chamley & Gale (1994), on which the model economy of this paper is based, extends these models to allow for the endogenous sequencing of actions. This leads to strategic delay and an inefficiently slow path of investment as an equilibrium outcome. Caplin & Leahy (1994) also study the possibility that strategic inaction and informational externalities may lead to delay and crashes in real activity. Gonzalez (1999) investigates an alternate possibility in which the option value of investment depends on the information revealed by the actions of others. He argues that recoveries from recession 11 are slow because of the coordination problems that result from the social learning aspects of the investment process. Gradual changes in aggregate activity result in environments when information about the size of a product market, as in Rob (1991) and Ziera (1994), or the cost of production, as in Caplin & Leahy (1993), is generated by the actions of investors. In general, the presence of informational externalities implies inefficiency, which in turn suggests that there is a role for policy in social learning environments. Rob (1991) shows that the equilibrium rate of entry into a new industry is smaller than the social optimum in his model, and discusses the desirability of subsidizing early entrants in order to generate more information. Smith & S0renson (1997) appeal to the single agent optimal experimentation literature to characterize the optimal sequence of subsidies in the exoge-nous timing environments of rational herd behaviour. Vives (1997) studies the welfare properties of social learning models by allowing a social planner to impose decision rules on agents. These papers, however, do not focus on the strategic aspect of the interaction between private agents and the policy maker. Also related to this paper is the literature that studies the problems of policy making under uncertainty. Examples of early work in this area include Poole (1970), who examines the optimal choice of instruments under different sources of uncertainty, Brainard (1976), who suggests that when policy instruments themselves are uncertain, policy makers should act more cautiously, and Friedman (1968), who speculates on the potential for policy to do more harm than good in the presence of uncertainty, and advocated a non-activist rule as a solution to this problem. Recent work in this vein includes Bertocchi & Spagat (1993, 1997), and Wieland (2000), who analyze the potential for experimentation and learning on the part of policy makers, but do not emphasize the strategic nature of the interactions between policy makers and agents. The literature on policy games, which developed subsequent to the rational expec-tations revolution downplays the role of policy makers' uncertainty but incorporates the strategic behaviour of agents into the analysis. Well known papers in this vein include Kydland & Prescott (1977), Barro & Gordon (1983) and Rogoff (1995), which analyze issues of time consistency and inflationary bias in the context of monetary policy. Caplin & Leahy (1996) represents an attempt to incorporate the problems of policy making under uncertainty into a setting where agents understand the policy making process and behave strategically. Finally, the richness of the interactions between agents and the policy maker in this 12 framework, where both agents and policy makers can learn from one other, resembles the back and forth informational interactions which Blinder (1997) highlights as a potential source of inefficiency in the way monetary policy is conducted. 2.3 A n Investor's Problem Since the objective of this paper is to analyze the problem of a policy maker in a social learning environment, it is useful to understand how such an economy operates in the absence of a policy maker. This section discusses the problem of an investor facing uncertainty, who can learn by observing the actions of others. It describes the equilibrium of such an economy in the absence of a policy maker, and shows how the behaviour of this equilibrium is influenced by the presence of (arbitrary) investment subsidies. These results are useful both as a first step in building up to the question of interest, and because the equilibrium in the absence of a policy maker provides a natural benchmark against which to measure the performance of the policy maker who will be introduced in subsequent sections. The economy is a version of Chamley &; Gale's (1994) model of information revelation and strategic delay, in which agents with an investment option are uncertain about a state of nature which affects the profitability of investment, but obtain information about this state by observing how many others have invested. Chamley & Gale assume that all agents are identical which results in a mixed strategy equilibrium, where the (symmetric) probability that an agent invests is such that all agents are indifferent between investing immediately and waiting for more information. In this paper I assume that agents possess heterogeneous costs of investing, which gives rise to a pure strategy equilibrium in which there is a unique cutoff cost such that all agents with a lower cost invest immediately while those with a higher cost choose to wait. That there is strategic delay in this form of the model can be seen from the fact that in equilibrium there are agents with costs low enough that they would expect to make a profit from investing immediately who nevertheless strictly prefer to wait. The assumption that agents are differentiated by cost has one important implication. Since information is generated by observing the investment decisions of other agents, more information is obtained by observing the investment decisions of a greater proportion of the agents in the economy. When agents are identical and the equilibrium is in mixed strategies, this means that a higher equilibrium probability of investment generates more 13 information. When agents are heterogeneous, a higher equilibrium cutoff generates more information. The difference is that in the latter case, more information can only be obtained by inducing agents with more costly projects to invest. This means that revealing all the information at the start of the game is not optimal, since this information can only be generated by inducing investment that is likely to be unprofitable. Therefore, a policy maker will wish to manage information flows in the economy on an ongoing basis, rather than by employing a one time only subsidy. This allows for the study of the kind of effect outlined by Caplin &: Leahy. To provide intuition on the mechanisms at work in the model, Section 2.3.1 analyzes the problem faced by an investor in a one-period version of the game. Section 2.3.2 proceeds to study the problem of an investor, in both the absence and presence of investment subsidies, in a dynamic game. The resulting equilibria are presented and discussed in Section 2.3. 2.3.1 A One-Period Example Consider a game in which there are N agents, n of whom will get an investment option. The number of players, TV, is given and known to all. The number of players with option, n is a random variable drawn from a known distribution g0(n). The realization of n is not observed. Al l agents are identical to begin the game. Therefore, conditional on n, each player receives an investment option with probability re/TV. A player who has an option uses Bayes' Theorem to derive posterior beliefs on n as follows: N g(n)=g0(n)-n[J29o(n')-n'}-1. (2.1) Tl'=0 Agents with investment options are indexed by i. Each agent with an investment option has a cost type C i , drawn independently from a known distribution F(c) (with the associated density function /(c)) on an interval Con the real number line, where c is the lower bound of C, and c is the upper bound. The realization of Cj is known only to player i. One can think of the process by which an agent gets an option and a type as follows: The number of options, n, is determined by a draw on g0(n). These n options are then randomly assigned amongst the N agents, where all agents have the same probability, (^), of receiving an option. Those agents with options are then assigned cost types via independent draws from F(c). 14 An investment generates revenue v(n), which depends on the unobserved state of the economy, n, is increasing in n, and is the same for all agents. The expected revenue for an agent with an investment option is: N V = ,£g(n)-v(n), (2.2) rz=0 and, since profits in state n are given by v(n) — Ci, the expected profit from investing for an agent with type Ci is V — c\. The policy maker has as an instrument an investment subsidy, s. The policy maker cannot observe either the state of the world, n, or agents' cost types and is therefore unable to either signal n to the agents or set different subsidies for agents with different costs. Section 2.4 discusses the policy maker's problem in detail, for now it is sufficient to note that agents face investment subsidies that are not conditioned on an individual's cost type. The existence of an investment subsidy requires a modification of agents' payoffs as follows. An agent who invests now receives both the revenue from the investment project and the subsidy, so that profits are v(n) + s — Cj. Using Equation (2 .2 ) , we can define expected profits for an agent with cost a in the presence of subsidy s as V + s — Ci. The Option Value of Waiting Consider the problem of an observer outside of the model described above who has an investment option and is allowed to observe the actions of the players in the model. Suppose further that an agent in the model who has an option, invests if and only if his cost type is an element of a given set C. That is, assume that there is some arbitrary subset of cost types such that an agent with a cost type in this set who has an investment project actually invests. How does the outside observer value his option to invest prior to his observing the actions of the other agents? Let k be the number of agents who invest given C. The expected revenue of the outside observer who decides to invest depends on k: V(k) = Y;g(n\k)-v(n). (2 .3) 71=0 Therefore his expected profit is V(k) — c, where c is his cost of investing. 15 Note that the outside observer is able to use the information revealed by the investment of the other agents to update his beliefs on n. Particularly, = ( 2 4 ) represents his beliefs over n, where, N Pr(fc) = Pr(k\n') • Pr{n). (2.5) n'=0 The probability of observing k investments given n and C follows a binomial distribu-tion, where the probability that an agent with a project invests is T = Pr(ci e C). That is, Pr(k\n) = Q - T ^ l - T ) n ~ k = b(k\n,C), (2.6) where, b(k\n,C) • g(n) I2%=0b(k\n',C)-g(n>) M k ) = ^ T . : v r ' , . . » - (2-7) If 0 < T < 1 and g(-) is non-degenerate, g(-\k) is increasing in k (in the sense of first order stochastic dominance).1 This implies that V(k) is increasing in k, so that, holding the set of cost types for which an agent with a project will invest fixed, if the outside observer observes a high realization of k he concludes that the expected revenue from an investment is high. This is because he is more likely to see a lot of investments being made, given C, when the number of agents with a project, ra, is high. Using this, we can define the outsider's valuation of the option to invest given that he will first observe the actions of the other agents as: W(c, C) = £ p(k)inax.{V(k) - c, 0}, (2.8) k where, TV p(k) = J£b(k\n,C)-g(n). (2.9) n=0 Note that in W(c, C) the c is the cost type of the outsider, while the C represents the set of cost types under which any given agent will invest. ^ h i s is basical ly L e m m a 1 from C h a m l e y & G a l e (1994). I reproduce the proof i n A p p e n d i x 1 for convenience. 16 Note that we can write the payoffs of an investor who must make his decision without observing the play of the game as: • V(k) - • c = X > ( * ) [ V ( * ) - c ] . (2.10) Therefore, the option value of waiting and observing the actions of others before investing equals W(c, C) — [V — c], which is always greater than or equal to zero, and strictly greater than zero if there are some states for which v(n) — c is negative. Essentially, the outsider is able to avoid investing in situations where it is likely that the state of the word is unfavourable, and this results in a higher payoff on average. How does this change if we allow for the presence of investment subsidies? Assume there exists some subsidy so, but that the outsider does not face SQ, but rather some other subsidy, si(k), which is conditional on the outcome of the one-period game. In such a situation, the outsider values his option to invest at: W(c,C,{Sl(k)}^0) = £ p ( * | C ) m a x { V ( * ) + Sl(k) - c, 0 } , (2.11) k which is the analogue to Equation ( 2 .8 ) . Note that an agent who does not observe others invests if and only if V + SQ — c = J2kP(k\C)[V(k)+so—c] > 0, so that, as before, the option value of observing the investment decisions of others is: W(c,C,{Sl}k)-V + s 0 - c . (2 .12) It is clear that the presence of investment subsidies conditional on the observed value of k can either increase or decrease the option value of waiting. Furthermore, there is no guarantee that this option value will be positive. It is always possible, for example, that the set of second period subsidies will be negative to such an extent that the outside observer never finds it profitable to invest. Information We have seen that an agent can benefit from observing whether others invest, so in this section we turn to the question of when an agent can be said to be more informed. Recall C, the set of cost types for which an agent with an investment project chooses to invest. In a precise sense, C measures the amount of information revealed by observing the number of agents who invest (both with and without subsidies). To see this, consider 17 the extreme cases where C = 0 and C = C and an intermediate case where C ^ 0 but C C C . When C = 0, then the probability that an agent with a project invests, is zero and k is trivially equal to zero for every possible value of n. That is, there is no chance that anyone invests and as a result the outsider learns nothing about n from the observation that no one in fact invests. If C = C on the other hand, then all players with projects invest with certainty and the outsider learns that n — k. In the final case, there is some probability that an agent with a project will invest and so the outsider learns something, but not everything, about n from observing k. Formally, consider two sets C and C, where T and T are the probabilities that Cj is an element of C and C respectively. Assume that C and C are such that T < T. Let X and X denote the number of agents who invest under C and C respectively. We can think of X as being generated as follows: let n agents be randomly distributed over C and let an agent invest with certainty if his cost type is in C. This is equivalent to a case where n agents randomize their investment decisions, choosing to invest with probability T. Take the k "successful" investors from this group and have them randomly invest with probability fi/F. The result is identical to the case where n agents randomly invest with probability T. Therefore, X equals X plus noise. Thus, by the criteria of Blackwell's "comparison of experiments", C is more informative than C whenever the probability that Cj is in C exceeds the probability that Ci is in C (i.e. when T < T). From Blackwell's theorem then, W(c,C) < W(c,C) whenever T < T, or W(c,C, < W(c,C, {si}k), for a given set of investment subsidies.2 Note that if players play cutoff rules of the form: I invest if my cost is less than some number c, then the set C is the interval [c, c] and higher values of the cutoff c correspond to more informative experiments. This is useful to know because it will turn out that agents play just such cutoff rules in equilibrium. 2.3.2 A Game of Timing We now turn to a multi-period version of the game described in the previous section. As before there are N players in the game, n of whom have investment options and choose the timing of any investment they decide to make. That is, agents decide whether to invest or not invest at any date t = 1,2,3, ..., oo, subject to the constraint that the decision to 2 See Cremer(1982) for a statement and proof of Blackwel l ' s theorem, and Grossman , K i h l s t r o m & M i r m a n (1977) for an appl ica t ion . 18 invest is irreversible and that a given agent can invest at most once during the play of the game. Agents discount the future at rate 5, so that in the absence of investment subsidies agent i receives a payoff of <J*-1(u(ra) — c») from investing at date t. The corresponding payoff in the presence of investment subsidies is d>t-1(i)(n) + st — Cj). Agents choose the timing of investment to maximize expected profits. At any period t of the game an agent who has yet to invest observes the sequence k* = {ko,ki,..., fa-i}, where kt is the number of agents who decide to invest in period £, and ko — 0. For now, it is assumed that any subsidies are exogenously given, and that agents know the path of subsidies that will obtain for every possible history of the game. In subsequent sections, in which the investment subsidies arise from optimizing behaviour on the part of the policy maker, the assumptions of rational expectations and common knowledge will imply that agents can infer the path of subsidies that will occur for any possible history. As the discussion of the previous section suggests, the information contained in an observation on kt depends on the probability that any given agent with an option invests, that is on the probability that a given agent has a cost in the set Ct, where Ct is the set of agents who invest at date t. The assumption of common knowledge implies that in equilibrium, agents will know the cost sets which generate investors, so that at date t they will use information on the past values of Ct and kt to interpret the expected value of investing. If we denote & = {Co,C\,... ,Ct-i}, where C = 0, then we can refer to the history3 of the game at date t as the pair {fc',C*}, and denote this by ht-If we define Ht as the set of possible histories at time t, and H = | J ~ 2 Ht, we can then regard S, the set of subsidies, as a mapping from H to the real number line. That is, S, a representative element of which is s(hi), relates the subsidy that would obtain for every possible history. We take <S as exogenously given for the time being. In this section we discuss the properties of a symmetric Perfect Bayesian Equilibrium of the game played by the agents, both for the special case of no subsidies as well as for the case where agents face an arbitrary, exogenous set of subsidies. Conditions on the set of subsidies under which the equilibria of these two games possess the same properties are stated and discussed. 3Strictly speaking the k's constitute the history vector and the equilibrium sets of investors, the Cs, are not observed, but agents are able to infer them through the assumption of common knowledge. I include the C's directly in h to make explicit the fact that agents use both the observed k's and their inferences on the C's when making their decisions. 19 Optimal Strategies The solution to the investors' problem when there are no subsidies takes a simple form. Agents play cutoff rules, whereby an agent invests if and only if his cost is below some critical value (which depends on the history at date t). Lemma 1 In the economy with no investment subsidies, agents play cutoff rules of the form: invest at t iff Q < c*(ht), ( 2 . 1 3 ) where Ci is the investor's type, andc*(ht) is some critical value depending on the observed history at date t. Proof See Appendix B. The intuition for Lemma 1 is that, since agents discount the future (and information is the same across agents with differing costs), those agents with lower costs have more to lose by waiting than agents with higher costs. Thus if it is optimal for an agent with a given cost c to invest at date t, it is optimal for all agents with lower cost to invest also.4 This result extends to the case with investment subsidies as long as the subsidies can not be conditioned on agents' cost types. Lemma 2 For an arbitrary set of subsidies conditioned on the history but not on an agent's cost type, agents play cutoff rules of the form: invest at t iff d < c*(ht,S), where ct is the investor's type andc*(ht, S) is some critical value depending on the observed history at date t, and the set of subsidies, S. Proof This follows directly from Lemma 1 because the government cannot condition its subsidies on the agents' types. Therefore, for any given set of subsidies, an agent with a lower cost type has more to lose from waiting than an agent with a higher cost type as shown previously. || Lemma 1 and Lemma 2 imply that, as long as subsidies are not conditioned on cost types, the equilibrium of the game will take the form of a cutoff cost value in each 4 T h i s is the same as the in tu i t ion which drives L e m m a 1 i n C a p l i n & Leahy(1996). 2 0 period, which determines the set of agents who are willing to invest in that period, and a corresponding realization of the number of investors, kt- We can therefore write ht = {h*-, c*}, where c* = {CQ, c*(ho),... c*(ht-i)}, where — c. It is also useful to note that ht+i = {ht,c*(ht),kt}. Let V(ht) be defined as the expected value of investing at period t given the history ht, so that undiscounted profits are V(ht) — c%, in the case with no investment subsidies, and V{ht) + s(ht) — Ci, in the presence of investment subsidies. Note that V(ht) is defined as J2nP(n\ht) • v(n), where p(n\ht) represents the agent's beliefs over n given the history of the game at date t. In order to be more precise about p(n\ht) it is useful to discuss how the history of the game maps into beliefs over ra. Because agents play cutoff rules in equilibrium it is possible to write Ftict), the probability that an agent has a cost type in the set of agents who will decide to invest this period, as:5 rt(ct) = f(c\c )dc = - T - ¥ ^ - r , (2.14) where T is interpreted by an agent in the game as the probability that any other agent with an option who has not yet invested invests this period. If we define K(ht) as the total number of investments that have been made at ht, that is K(ht) = X^ t=o then the probability of observing kt investment decisions at ht, given c*(ht), follows the following binomial distribution: p(kt\n,ht,c*t) = ^« - ~ 1^  [^(cj)]*. . [ ] L _ j r^ j j—ii rc fc , ) -* , - ! - b(kt\n-l,ht,c$). (2.15) This expression differs from the discussion of Section 2.3.1 for the obvious reason that the probability of observing any investment now depends on the number of agents who have already invested and also because we are now considering the case of how an agent in the game evaluates the probability of observing kt investment decisions, whereas Section 2.3.1 looked at the problem of how an agent outside the game would form beliefs. Since an agent in the game knows that he has an investment option, he knows that any investment he observes comes from ra — 1 other agents. As in Section 2.3.1, however, such an agent 5It is convenient to refer to the equihbrium cutoff, c*(ht) by the simpler notation c* when there is no confusion about the history. 21 will use Bayes' rule to update his beliefs, resulting in the following posterior assessment: b(kt-\\n - l,ht-i,c*t_x)-p(n\ht-i) p(n\ht) = £#=o 6(A*-i|n' - 1, ht-i, <4_j) • p(n'\ht-i) b(kt-i\n- l,fet,c*) • • • 6(fci|n - l,h0, c\) • gin) • (2-16) E £ U b(kt-i\n' - 1, ht,<*) • • • b(kx\n> - 1, ho, cJ) • g(n') Using this notation, we can characterize the expected payoff to an agent who invests at any history. In order to determine whether such an agent actually chooses to invest at ht, however we also need to compare the payoff to investing at ht to the expected payoff an agent receives by waiting at ht. Since the game is infinite, in principle this requires comparing the payoff to investing at ht to the payoff from waiting at ht and investing at every one of an infinite number of future dates. The next section provides a useful result which shows that in order to solve for the equilibrium cutoff we only need to find the cost type which would make a given agent exactly indifferent between investing at ht and waiting at ht and making a once and for all decision at date t + 1. The One-Step Property So far we have discovered that agents play cutoff rules in equilibrium and used this to calculate an agent's expected payoff from investing at a given history ht- Since, at any history, an agent essentially decides between investing immediately or waiting, we now turn to analyzing the payoff of an agent who waits at ht- First, consider the case where there are no investment subsidies and define W(d, c*(hi), ht) as the undiscounted payoff to an agent with cost type Ci who waits one period at ht before making an irrevocable decision when others play the cutoff c*(ht). Note that the first argument of W is the agent's own cost type, while the second refers to the set of other agents who invest. Suppose that at some ht, the equilibrium cutoff is c*(ht). The one-step property is said to hold if it is optimal for a player with type C j = c*(ht), who is indifferent between investing and waiting at ht, to make a once and for all decision at date t+1. Such a player will receive max{V(ht, c£, kt) — c£,0} at {ht,c%,kt} so that that player's undiscounted payoff from waiting is: Wicc^htlht) = Ylp(kt\ht,<t)rnsoi{V(ht,c!,kt)-ct,0}, (2.17) kt where p(kt\ht,c$) is the probability that kt people invest at ht given c\. Note that W(ci, ht, c%) corresponds closely to the expression W(c, C) of Section 2.3.1, and the analysis there implies that, for a given cj l j , a high c* is more informative than a lower c%. 22 Proposition 1 For any fixed but arbitrary symmetric Perfect Bayesian Equilibrium, the one-step property holds at any information set ht. Proof See Appendix C. The intuition for this result is as follows: An agent who is exactly indifferent between investing and waiting at some period t and decides to not to invest now has the lowest cost type of all remaining agents (by Lemma 1). If this agent does not find it optimal to invest at period t+1 then, again by Lemma 1 no one else will find it profitable to invest either. Therefore such an agent has no chance of learning by waiting at t +1. If he finds it unprofitable to invest given the information at t +1, then he must also find it unprofitable to invest at t + 2, since his information is the same. By induction, we can see that such an agent will never invest if he does not invest at t + 1. Therefore the One-Step Property holds. Note that whenever the one-step property holds, the game will end after a finite num-ber of periods. Since the only reason an agent ever chooses to delay is to gain further information, it must be the case that if he learns that the worst possible outcome has occurred, that is if no one invests, he will not invest. Therefore, investment only continues if at least one person invested in the previous period. Since there are a finite number of players, the game must have a finite number of periods. The intuition that an agent only waits in order to learn, only learns when other people invest, and can not expect to learn anything if he has the lowest remaining cost type does not hold true for the case with a policy maker. It is true, by Lemma 2, that such an agent indifferent between investing and waiting at ht, cannot expect to learn anything if he waits at t+1. However, it is not necessarily the case that such an agent has no incentive to wait at ht+i- If at £ + 1 the agent expects that a future subsidy will be sufficiently great that it compensates for the losses (due to discounting) of waiting, then he will wait. Furthermore, given that such an agent cannot hope to learn by waiting, an expectation of such a future subsidy is the only reason that the agent will wait at t + 1 and yet invest at some future date. Lemma 3 An agent indifferent between investing and waiting at ht finds it optimal to not make a once and for all decision att+1 if and only if he is expecting a sufficiently great subsidy at some date t + j where j > 0. Restriction 1 A set of subsidies, S, is said to satisfy Restriction 1 if it is the case that 23 given S, the One-Step Property holds. It is important to note, however, that we cannot derive the private sector equilibrium assuming the one-step property holds, then impose this equilibrium as a constraint on the policy maker's problem. The one-step property, if it holds, is an equilibrium outcome of the complete game. Therefore, we do not impose Restriction 1 on the equilibrium set of subsidies, rather we will show that the equilibrium set of subsidies, derived in the following section, satisfies this restriction. 2.3.3 Equilibrium We will now use the observations that in equilibrium agents will play cutoff rules and the one-step property will hold to solve for the private sector equilibrium. Using the one-step property, we can calculate the value of waiting at ht for an agent with type c\, when others play the cutoff rule c\. Recall W(ci, c*(ht), ht), the undiscounted value of waiting one period before making a once and for all decision at date t + 1. We can rewrite this as follows: N-\~K(ht) W(ci,c*(ht),ht) = YI P(h\ht,4)max{V(ht,c$,kt) - Ci,0} kt=o kt=o n=o v PKKt\nt,ct) J N-l-K(ht) N = YI rnax{J2p(n,kt\ht,c*t)[v(n)-Ci},0} kt=0 T1=0 N-\-K(ht) N = Y2 max{^6(A i |n- l , / i t ,Ct*)p(n|/ i t ) [«(n)-c i ] ,0}, (2.18) kt=0 n=Q where p(n\ht), b(kt\n — 1, ht, c£) and K(ht) are as previously defined. By the one-step property, the equilibrium c is that value which equates the value of investing at ht with the expected value of investing at t + 1. If the equilibrium value of c*(ht) is between tf._-y and c then it is that value of c which solves: V(ht)-c*=6W(^,c*t,ht). (2.19) The equilibrium is calculated by iterating on Equation (2.19). Lemma 1 and the 24 Figure 2.1. Properties of Equilibrium Ci V-6W(ci,c*t,ht) V-SW(ci,c*t,ht) continuity of W(c,c,ht) implies that the equation has a unique solution when c*(i%) is between c\_x and c. Figure 2.1 is helpful in understanding this relationship. By Lemma 1 we know that V — 5W(ci,Ct,ht) is decreasing in Cj (given an arbitrary set e£). Further, as c£ increases, the value to waiting increases, so that V — 8W(d,cZ,ht) shifts down as shown. The vertical distance V — W(c*,c*,ht) is the amount that a player with type c* would have to be compensated to be indifferent between waiting and investing. That is, it represents the subsidy that would be required to support c* as an equilibrium cutoff. The evolution of investment in the economy is as follows: At any information set in equilibrium one of three things will happen: i) beliefs will be sufficiently optimistic that agents of all cost types wish to invest, ii) beliefs will be sufficiently pessimistic that agents of all remaining cost types will choose to not invest, and iii) beliefs will be intermediate so that of the remaining cost types, some will wish to invest and others to wait and the equilibrium cutoff is determined by Equation (2.19). Note that there is delay in equihbrium. That is, some agents with projects whose expected value is positive at ht choose not to invest in period t, as can be seen from the fact that the value of waiting, W(ci,c*(ht),ht), is greater than or equal to zero, and strictly positive if the return to investment is negative for some states of the world which are possible at ht- It is also clear that the equihbrium cutoff is determined without consideration of the fact that a decision to invest reveals information that is of value to agents who have yet to make their decision. Individual agents balance the returns to investing immediately against the fact that they may learn if they wait, but are not able to internalize the informational value of their actions. In this sense, there is an informational externality which causes the set of agents who decide to invest in any period to be lower 25 than would be the case if agents could capture all of the returns to investing, which suggests that, at any given history, a policy maker will wish to encourage a larger set of agents to invest, and thereby reduce delay. We can apply the same logic to the case where agents face an arbitrary set of subsi-dies S. Assume that the set of subsidies «S, whose elements are denoted s(ht), satisfies Restriction 1. Then, the logic of the previous section applies, the one-step property holds, and the equilibrium cutoff is that value which solves: V(ht) + s(ht) - cst = 5W(ct,cst,ht,S+1) (2.20) where, S+1 = {8(ht, ct, kt)}^^-1. (2.21) Note that it is not necessarily the case that there is delay in the equihbrium with subsidies. If subsidies in early periods are high relative to those in subsequent periods, agents will have incentives to invest sooner in order to enjoy the higher subsidies and this may outweigh the desire to delay in order to learn. Alternately, if future subsidies are high relative to the current subsidy then a greater set of agents will desire to wait than would be the case were there no subsidies. In the next section we turn our attention to the problem of the policy maker who wishes to use such investment subsidies to reduce equilibrium delay. 2.4 The Policy Maker's Problem Thus far we have analyzed the outcome of the economy in the case without subsidies and seen that agents who have projects which are profitable in expected value given a history ht choose to wait at ht. Since there is an externality in the information generation process, we suspect that this delay is not efficient.6 This section examines the problem of a policy maker, facing the environment of Section 2.3.3, who wishes to maximize the (aggregate) expected welfare of the agents in the model; which in this case corresponds to pre-subsidy expected profits. The policy maker, therefore, internalizes the fact that agents who make investment decisions in the future use the information revealed by investment today. The policy maker will therefore wish to reduce delay, but it is not clear that this is possible, given that the policy maker has no extra information about the economy and 6 As yet it is not clear what level of delay, if any, should prevail in the economy. We will return to this question in Section 2.5 where we turn our attention to the social optimum. 26 that agents will understand and react to any systematic strategy that the policy maker wishes to employ in hopes of learning more. Note that the policy maker faces a non-trivial tradeoff concerning the amount of in-vestment he wishes to induce at any history. The distribution of cost types means that in order to get more information today, the government must induce agents with higher costs, and therefore less profitable investment opportunities, to invest. Thus the policy maker does not necessarily want to eliminate all delay in the model. In this section we compare the performance of a policy maker with no access to a commitment technology against the two benchmarks already established: i) the outcome of the economy in the absence of a policy maker, and ii) the outcome in the case of zero delay(where agents are myopic and therefore invest if and only if their projects are profitable in expected value). We will see that the policy maker always reduces delay relative to the outcome that would obtain in the absence of policy intervention. We will also see that the equihbrium outcome of the economy with the policy maker can exhibit either more or less delay than the myopic outcome. That is, it is possible that there remains delay, in the sense that agents with profitable projects choose to wait, even in the case when there is policy intervention. It is also possible that the policy maker induces some agents with unprofitable projects to invest in order to benefit from the information that this investment generates(which would involve 'negative delay' relative to the myopic, zero delay benchmark). Which of these outcomes obtains depends on the underlying parameters of the economy, and the way they affect the relative importance of the option value of delay and the informational value of early investment. 2.4.1 Preferences and Information of the Policy Maker The policy maker's objective is to maximize the aggregate, pre-subsidy, expected prof-its of the agents in the economy. The policy maker's instrument is a state contingent investment subsidy. We assume that the policy maker is able to finance this subsidy by levying non-distortionary, lump-sum taxes elsewhere in the economy. As a result, the policy maker neither benefits nor suffers from the use of subsidies. Therefore the policy maker only cares about the effects of subsidies on the path of investment. The policy maker observes neither the number of agents with an option, n, nor the realized cost type of any individual agent C\. Writing down the policy maker's objec-tive function therefore requires taking expectations over both n and Cj, which requires a discussion of the policy maker's information and beliefs. 27 The policy maker observes all public information in the economy at date t, which in this case consists of the number of agents who invest at every date r < t. Furthermore, common knowledge implies that the policy maker can infer the equilibrium cutoffs that generate these observations. Thus at any date t, the policy maker has the access to the same public information, ht, as the agents. For any given public information set ht, however, the policy maker generates different beliefs over n than an investor who observes the same history. This difference arises because an agent with an option has a piece of information that the policy maker does not possess; namely that he himself has an option. Thus, while an investor with an option who observes ht has beliefs given by Equation (2.16), a policy maker observing the same ht arrives at a somewhat different assessment over n, which is given by: pg(n\ht) = b(kt-^M^)---b{k^n,hQ,c\).g0{n) Yl„'=o Kh-i\ri, fh, c*) • • • b{k\\ri, ho, c|) • g0(ri) Furthermore, the policy maker, who does not observe the costs of individual investors, can use the known distribution of costs, /(c), to form beliefs over the expected cost of any agent who invests given the relevant cutoff values. In particular, given a value for the previous period's cutoff, cjLi, the cost of an agent who invests this period, weighted by the probability that any remaining agent has such a cost, is: i: where c\ is the current cutoff and F(c) is the cumulative distribution function associated with the density /(c). Using Equations (2.22) and (2.23), we can write the expected one period payoff for a policy maker at ht as a function of the period t cutoff, c*, as: J2 P9(n\ht)(n - K(ht)) r (v(n) - c) f® de (2.24) The integrated term is equal to the expected return per each remaining agent with an option if the state is n, given that the policy maker does not know the cost type of any such agent. This is multiplied by the number of agents who remain if the state is in fact n, and the policy maker takes expectations over n. The previous period's cutoff is treated as part of the state of the world, and the policy maker arriving at history ht chooses a subsidy to optimally influence c%. 28 Since we assume that the policy maker represents the interests of the agents in the economy, the policy maker has the same discount factor, 5, as the agents. We can use this and the one period payoff function to define the policy maker's objective function for the game as: N v9 = ] L > ( n W (v(n)-c)f(c)dc 7J=0 Jz N-l ( N fc1=0 U = 0 l - * ( C i J JV-A"(fci)-l k2=0 Y2pg(n\h2)(n-K(h2)) J Hn)-c)llf de n=0 ^ C 1 ! . ) + } ••• } } (2-25) Since we wish to compare the equilibrium with a policy maker to the outcome when there is no policy maker, it is more convenient to re-write both Vg and the equilibrium conditions of Section 2.3.3 in terms of the same beliefs. In particular, note that: p(n\ht) p(n, / a t - i , f c t - i , c * - i ) _ P(kt-i\n, ht-1,0^) • p(n,ht-i) p(ht-i, kt-i, c j _ i ) p{kt-i\ht-\,cl_l) b(kt-i\n- 1,/tt-i.c?,!) -p(n\ht-i) -p(At-i) p{kt-i\ht-\,c*t_x)-p(ht-i) b(kt-i\n-lM-i,<%-i)"-Kh\n-l,ho,<%) • 9Jn) p(kt-i\ht-i,c*t_l) • p(kt-2\ht-2,c*t_2) • • -p(ki\d{) (2.26) If we define: H(n\ht) = b(kt-i\n - l,ht-i,<$_i) • • • Kh\n - 1, ho, c\) • g(n), (2.27) then we can express an investor's assessment in terms of fi(n\ht) as: = -nr^i , \ ,^]htl —TZT^v (2-28) p(kt-i\ht-i,c*t_1)-(p(kt_2\ht-2,c$_2)---p(ki\c*1) This allows us to express the equilibrium conditions of Section 2.3.3 in terms of 29 fj,(n\ht), so that Equation (2.19) becomes: N N-l N Y, V.(n\ht)(v(n) - e£) = £ max{]T »(n\ht, h,<$)(v(n) - c*t), 0}. (2.29) n=0 kt=0 n=0 Furthermore, since: Kn\ht)[l - F(c*t)}[J2n,g0(n') • n'\ _ t _ K{ht)) ~ o(K t_i|n, c ) • • • b{ki\n, c x) • g0{n) (2.30) we can express the policy maker's beliefs as: n ( n \ h ) = Kn\ht)[l-F(ct)}[j:n>go(n')-n'} ( We can, therefore, re-write the policy maker's objective function Vg as follows: N Vg = £>(«) fCl(v(n)-c)f(c)dc E E Mnl^i) / . («(«) - c)f(c)dc fci=0 l n = 0 Jcl + 6 E E M * ) {v(n)-c)f(c)dc fc2=0 U = 0 ^ ( A i ) • • • } } (2.32) Note that we can, therefore, write the value to the policy maker of the continuation game at ht as: N c* N-K(ht)-1 Vg(ht) = max £ /i(n|^) / 1 («(n) - c)f(c)dc + 5 J2 Vg(ht+1). (2.33) 2.4.2 T h e One-Step P rope r t y Rev is i ted This section presents two results that are helpful for solving the policy maker's problem. The first shows that, for a given set of future subsidies, the policy maker can implement any desired cutoff. The second result shows that policy always wishes to implement a set 30 of subsidies that satisfy Restriction 1, so that the equilibrium cutoffs satisfy the one-step property. Together, these imply that the game with the policy maker will be finite and that we can solve for a self-enforcing equilibrium by applying backwards induction to the policy maker's problem. Furthermore, the result that the policy maker can implement any cutoff at date t via the appropriate subsidy implies that we can solve the policy maker's problem for the equilibrium cutoffs and allow the agents' equilibrium condition to determine the optimal subsidy residually. Lemma 4 For every c > cj?_i, given history ht and the continuation game beginning next period, there exists a subsidy which supports c as the equilibrium cutoff value at ht. Proof Take some arbitrary value c. Given ht and S (where S is some fixed set of finite subsidies), c implies some fixed set of continuation subsidies S+(ht,c). given c, ht, and «S+(/u,c) it is possible to calculate the value of the continuation game for an agent with type c. Denote this by W(c,c,ht,S+(ht,c)). Given W(c, c, ht,S+(ht,c)), an agent with type c will be exactly indifferent between investing and waiting at ht if: V(ht) -c + s = 8W(c, c, ht, S+(ht, c)). (2.34) Therefore, if we replace with s = 8W(c,c,ht,S+(ht,c)) — (V(ht) — c) the element of S denoted s(ht), c will be supported as the equilibrium cutoff. Since W(c,c,ht,S+(ht,c)), V{ht), and c are all real numbers, such a value of s always exists. || Lemma 4 implies that, given a set of finite continuation subsidies, the policy maker can choose a finite current subsidy so as to implement any desired cutoff as an equilibrium.7 The question remains as to what cutoffs the policy maker might choose to implement. Lemma 5 provides a first result along these lines. Lemma 5 The equilibrium set/sequence of subsidies is such that a player indifferent be-tween investing and waiting at ht finds it optimal to make a once and for all decision at t + l. That is, the equilibrium set of subsidies satisfies Restriction 1. Proof Suppose at (ht,c*t,kt) future subsidies {st+j}^0 are such that agents who are indifferent between investing and waiting at ht do not invest at (ht,cl,kt) but wish to invest at some future information set hT = (ht, c£, kt, 0, . . . , 0). 7It turns out that the model with a policy maker has a finite horizon, so that one can solve backwards for the equilibrium. Thus, Lemma 4 implies that the policy maker can target arbitrary sequences of cutoffs. 31 That is, at hT, the agent faces a sequence of subsidies {ST+J}JLQ and chooses to invest. Since the probability of investment is zero between (ht,Ct,kt) and hT, no information is revealed between (ht,c%,kt) and hT. This implies that the government's expected payoff at r is the same as the government's expected payoff at t + 1, but discounted. Since the policy maker induces the agent to invest at hr, the expected payoff to the policy maker from such an investment must be positive. Therefore, the expected payoff at (ht,c%,kt) must also be positive. The policy maker can induce the agent to invest at (ht,c%,kt) rather than at hr by setting st+j — sT+j for all j — {0,1,...}. Since the policy maker loses from discounting, it does so. || The implication of Lemma 5 is that in equilibrium, a player who is indifferent between investing and waiting at (ht, c£) finds it optimal to make a once and for all decision at That is, the one-step property holds in the model with a policy maker. Furthermore, notice that the only reason for the policy maker to desire that agents delay is that he hopes to generate more information which can be used by agents who delay. This implies that the contribution of the marginal cost type to the policy maker's objective function must be negative for some possible states of the world, which means that the policy maker will not wish to induce further investment should he observe the worst possible outcome.8 Therefore, the game involving the policy maker will be finite, with at most N periods, as investment will cease following any period in which there is not at least one investment. The finiteness of the game implies that we can solve the policy maker's problem by backwards induction. Furthermore, Lemma 4 implies that we can solve the policy maker's problem for the equihbrium cutoffs and then solve the agents' problem for the equilibrium subsidies. Since we know that these subsidies satisfy Restriction 1, the equilibrium subsidy is determined by Equation (2.20). 2.4.3 Solving the Policy Maker's Problem In this section we use the previous results to solve the policy maker's problem. We show that a policy maker always reduces delay relative to the equilibrium without a policy maker, but that there may be either more or less delay than the myopic benchmark. That is, for some values of the model's parameters the policy maker induces investment to the 8If there were any cost types in the previous period for which the value of investing was positive regardless of the information learned then the policy maker would have induced them to invest last period to avoid losses due to discounting. 32 point where agents whose projects are not profitable in expected value in order to generate information while in other cases some agents with profitable projects still choose to delay. As discussed in the previous section, the finiteness of the game allows us to solve the policy maker's problem via backwards induction, while Lemma 4 implies that we can solve the policy makers problem for the equihbrium cutoffs and solve the agents' for the equilibrium subsidies. Starting at T=N, the last possible period in which it is possible that the history is such that at least one agent invested so that investment still continues, we know that in equilibrium, V(kt+1) = 0. (2.35) This means that in period T=N, the policy maker wishes for the lone possible remaining agent to invest if and only if his project is positive in expected value. Since this is exactly what the agent would do if left alone, we can show that the optimal subsidy in period T=N is zero and that: N cs*(hr) = c*(hr) = ^p (n | / i r )« (n ) . 71=0 This makes sense intuitively. Since the possibility of learning generates the externality that creates inefficiency, in a situation where there is no possibility of learning, like the last period of a finite game, there will be no externality and therefore no inefficiency. Therefore, it is to be expected that the optimal policy in such a situation is to not intervene. For any T < N, the policy maker solves: N - c . N-K(ht)-\ maxJ>(n|M / ' (v(n) - c)f(c)dc +S £ Vg(ht+1), (2.36) c * n = 0 Jct-l kt=0 where Vg(ht+i) is defined as given by Equation (2.33). Since Vg(ht) is continuous in c* and bounded we know that there exists a solution to the policy maker's problem at each date t.9 There are three possibilities for a solution to Equation (2.36): i) it is optimal for all agents to invest immediately, so that cf * = c, ii) it is optimal that no more agents invest, in which case c£* = c£_i, or iii) it is optimal that agents with cost types less than some value cf* invest, where cf* € (c*_l:c). 9The function Vg(ht) is bounded by x = Yln=o ^ {nt\nt){v{N) — c*_a), which is the payoff which the policy maker receives if the best state of the world is realized and all remaining agents have the lowest possible remaining cost. 33 Consider case iii), where c£* e (c*_x,c). If this is the case then the optimal value of cf* must satisfy the following first order condition: f ] H(n\ht)(v(n) - c T ) / « * ) + g 9 E * ^ ( k + l ) k = < - = 0. (2.37) 71=0 * The first term in Equation (2.37) is the value to the policy maker of increasing the cutoff today, and thereby inducing more current investment. This term is positive if the cost type of the marginal investor in period t, c*, is such that that investor has a profitable project, in expected value, and is negative otherwise. The second term, 5(d^ktVg{htjr\)/ddt), represents the change in the value of the game beginning next period that obtains when the period t cutoff is increased a little. Since we know that higher values of the cutoff are more informative, it might seem to be the case that 5(dJ2kt Vg(ht+i) / dc*) should always be positive but this is not the case. There are two effects which determine whether this term is positive or negative: i) if the cutoff today is increased then some agents who would have been able to invest in the future with better information instead invest today, and ii) if the cutoff today is increased, those agents who still do not invest today will have more information on which to base their future investment decisions as they will have observed a more informative signal (a higher cf*). Basically, the term 5(dJ2kt Vg(ht+i) / ^ c*) captures the fact that there is both an option value of delaying investment (which suggests that a lower cf* is preferred) and an informational value of inducing current investment. The effects which make up S(d^2kt Vg(ht+i) / dc*) can be most easily iUustrated by considering the two period case. That is, consider the case where t = N — 1 and note that: dZkN Vg(hN) = £ l * ( r i \ h s ) ( v ( n ) ( 2 . 3 8 ) OCN-l kN-x n=0 +6 ^  ^2b'c(kN-i\n,hN^i,CN-i)fi(n\hN-i) / (v(n) - c)f(c)dc fcN_!n=0 JcN-l The first term represents the expected value of the profits that would be generated next period that will be lost if the cutoff is increased marginally. This equals the expected gain of an agent with c^_x who invests next period times the probability that there is such an agent. That is, it is the negative of the option value of waiting, since the summation over fcjv-i represents only those terms for which the future value is not equal to zero. The second term represents the value of the information to all agents who will not invest 34 this period of increasing the cutoff marginally. This is positive (by Blackwell's theorem), because higher values of cfL, are more informative. This is the informational effect that is taken into account by the policy maker but not the agents. We can decompose the effect of a change in c£* on J2kt Vg(ht+i) by isolating the infor-mational effect of changes in cf*. Define Vg(ht+Uc) = max^/ i (n|h t , fc t ,c£) / ' + (v(n)-c)f(c)dc-\-5 ^2 vg(ht+2), (2.39) where Vg(ht+2) is defined as before. We can see that c* affects Vg(ht+i,c) only through its effects on beliefs, that is through its informational value. Since higher values of c* are sufficient for lower values, by Blackwell's theorem, Vg(ht+i, c) is increasing in c*. Therefore, g S / ^ f r + i ) > 0 ( 2 4 0 ) Since c — e* implies Vg(ht, c) = Vg(ht), we can write: We can therefore express the first order condition of the policy maker's problem as: 0 = 5 > ( n | f c t ) ( « ( n ) - 0 / ( c n (2.42) Tl fct " t The first two terms sum to zero at the value of c, c*t, which solves the agents problem when there is no policy maker, and are negative when c is greater than this value. Since the third term is positive, it follows that any interior solution to the planner's problem involves cf* > c£, and therefore exhibits less delay. It remains to consider the two cases where the policy maker is not at an interior solution. First consider the case where the solution to the policy maker's problem at history t is cf* = c. If the policy maker induces agents of all types to invest then it is obvious that the equilibrium in the economy when there is no policy maker exhibits at least as much delay (because it is not possible for more than everyone to invest). 35 The remaining possibility is that the solution to the policy maker's problem involves cf* = c*_v That is, it is possible that the policy maker does not wish for any agents to invest, given the remaining cost types. In this case it is also true that the equihbrium outcome without a policy maker has solution c* = c*_j at history ht. To see this, assume that it is not the case. That is, assume that the solution to the policy makers problem is that cf* = c*_lt and that the solution without the policy maker involves c* > c*_x. Since c* > c*_lt it is the case that 5>("l^)(«(n) - c f ) / ( c D - * ££ / * (n | f c t + i ) ( u (n ) - cf*)f(cf*) > 0. (2.43) Since 5(dJ2kt v^iht+i, c = cf*) / dc*)\c*=c** > 0, it must be the case that the right hand side of Equation(2.42) is greater than zero. That is, the policy maker maximum utility can be increased by increasing cf*, which contradicts the assumption that cf* = c*_x is the solution to the policy maker's problem. Collecting the results for these three cases we can state: Theorem 1 The policy maker strictly reduces delay at any history ht relative to the equilibrium without a policy maker unless i) the solution to the policy maker's problem involves zero investment at ht, or ii) the equilibrium outcome with no policy maker involves all remaining cost types investing, in which cases the solutions to the problem both with and without the policy maker coincide. Theorem 1 allows us to compare the solution of the policy maker's problem to the equihbrium outcome in the absence of a policy maker, and we can see that a policy maker unambiguously reduces delay. We would also like to know whether the policy maker wants every agent whose project is profitable in expected value at a history ht to invest at ht-In particular, there are three possibilities: i) the policy maker wishes to have every agent whose project is profitable in expected value at a history ht invest at ht and for every agent whose project is unprofitable to wait, ii) the policy maker wishes to have some agents whose projects are unprofitable at ht to invest at ht in order to generate more information and iii) that policy maker wishes to have some agents whose projects are profitable at ht to delay in order to take advantage of information that will be generated while they wait. It turns out that the policy maker may wish to induce either more or less delay than the myopic benchmark at a history ht, depending on the parameters of the model. More precisely, it is not possible to sign 8(dJ2kt Vg(ht+i) / dc*) as either the option value effect 36 Figure 2.2. Optimal Delay is Similar to the Myopic Case or the informational effect can dominate, depending on the parameters of the model. To show that this is in fact the case, and to help generate some intuition about the type of effects that arise in this problem, we can compare the value function of a policy maker in a dynamic game to that of a myopic, one period policy maker for different parameter values. Figures 2.2-2.4 each plot two value functions, the payoff function of a policy maker in a three period game and the corresponding one period payoff function, for various values of c\. The optimal c\ is that which maximizes the relevant objective function, and by comparing the optimal period 1 cutoff in the three period game to the case of zero delay, we can investigate whether the policy maker wishes to induce more or less delay than would be the case in a one period game. As discussed previously, the cutoff which maximizes the one period objective function corresponds to the case where all agents with cost types profitable in expected value at ht invest at ht, and all others wait. In all three figures we assume that F(c) is uniform in the interval [0,1] and that the common discount factor, 5 equals 0.85. We then investigate the interaction between the option value of reducing the cutoff and the informational value of increasing the cutoff by changing our assumptions about go{n), the prior distribution of n, and v(n), the return to investing given n. 37 Figure 2.3. Optimal Delay Exceeds that of the Myopic Case 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 c1 Figure 2.2 is drawn under the assumption that go(n) = 0.25 for all n so that each value of n is equally likely a priori. We assume that v(n) is relatively flat and takes on the values {0.5,0.55,0.6,0.65}. In this case, the fact that the payoffs to investing do not depend strongly on the state of the world means that information plays little role in agents' decisions. The return to investing is similar enough across the various possible states of the world that, for most agents, investing will be either profitable in all states or not profitable in any state. Therefore, there is no strong incentive to deviate from the myopic outcome, and Figure 2 shows that, in fact, the optimum value of C\ in the dynamic game is similar to the one period solution. Figure 2.3 maintains the assumption that any realization of n is equally likely a pri-ori, but we assume that payoffs exhibit a greater dependence on this realization so that v(n) takes on the values {0,0.1,0.5,2}. Under these assumptions, the expected return to investing exceeds one so that the expected payoff from investing in period one is greater than zero for agents of all cost types (since c = 1) which means in a one period game it would be optimal for all agents with options to invest. However, agents with high costs stand to suffer substantial losses if they invest and the state of the world is poor. As a result, the option value of delaying such investment decisions is high so the policy maker wishes to implement a cutoff below the zero delay threshold. 38 Figure 2.4. Opt imal Delay is Less than that of the Myopic Case In some cases the option value of delaying investment does not dominate, and the informational value of inducing investment can be great enough that the optimal cut-off is greater than the zero delay benchmark. Figure 2.4 presents an example of this case, where we maintain our assumption on v(n), but now let go{n) take on the values {0,0.985,0.001,0.014}. Under these assumptions, the distribution of the state of the world is bimodal, and the payoffs associated with these modes are very different. It is no longer the case that agents expect profits regardless of cost type, and in fact the set of costs for which a project is expected to be profitable is quite small. Because the state will be either very good or very bad, the policy maker generates very valuable information by increasing the cutoff relative to the myopic case, since if an investment is observed in the first period all remaining agents know that the state of the world is almost certainly very profitable. Furthermore, since the myopic cutoff is so low there is not such a high cost associated with inducing bad investment. That is, the optimal cutoff cost is still relatively low, so that if an agent with such a cost invests and the state turns out to be bad, the loss is not great in contrast to the previous example where if the policy maker implements the myopic cutoff it is possible that an agent with a very high cost invests when the state is bad, which results in a large loss. Note that in all of these cases there is both an option value associated with delaying 39 investment and an informational value associated with increasing c\. Whether there is more or less delay than in the myopic case depends on which of these effects dominates, so that the zero delay benchmark does not provide the kind of bound to the solution to the policy maker's problem that the "laissez faire" equilibrium provides. 2.4.4 Equ i l i b r i um To find the equihbrium requires using backwards induction to solve the policy maker's problem for the optimal cutoffs. Since, by Lemmas 4 and 5, we know that these cutoffs will satisfy the one step property and that the equihbrium subsidies will, therefore, satisfy Restriction 1, we can use Equation (2.20) to solve for the equihbrium subsidies. That is, at any history ht, given that the pohcy maker wishes to implement the cutoff c*, and that all agents know that the set of subsidies in the continuation game satisfy Restriction 1, the value of waiting at ht is given by W(c, cf*,ht, «S+i), and the optimal subsidy solves: V(ht) + 5* - c f = 6W(cst*,c?,ht, S+1) (2.44) It appears logical that if the pohcy maker wishes to reduce delay in equihbrium, that the optimal subsidy should be declining in some sense. This is indeed the case. From equation (2.44) we can see: s*t = 5W(ct*,ct*,ht,S+1)-(V(ht)-ci*) = [5W(cf,cr,ht, S+i) - 8W(ct*,cr,ht)] + [8W(c?,cr,ht,) - (V(ht) - cf)] = J>(fetlfc, c T K + i + [SW(cr,crM) - (V(ht) - cf*)}, (2.45) fc where k represents those values of kt following which an investor with cost type c*s will decide to invest in period t+1. Thus, Y^kP(^t\ht,c*s) • represents the expected value of the period t + 1 subsidy to the marginal investor. Since 5W(cf, cf*,ht,) — (V(ht) — cf*) is greater than or equal to zero, the equilibrium subsidy in period t is greater than or equal to the expected value of the period t + 1 subsidy. Note that, except in the next to last period, the subsidy does not equal the informa-tional value of extra investment. It equals the informational value of extra investment plus the expected value of next period's subsidy.10 Essentially, the policy maker takes 1 0 To be precise, in addition to the informational value of investment, the policy maker may also have to take into account the fact that if in future periods he might induce agents with unprofitable projects to invest in order to generate information, this reduces the value of waiting. 40 into account the fact that agents know the policy making process when making decisions about the future and builds this into the optimal sequence of subsidies. Even though the agents understand that there will be future subsidies, and therefore may have an incentive to delay, the policy maker understands this and is able to take it into account when choosing the optimal subsidy. By setting a path of subsidies such that the early subsidies compensate agents for any future subsidies that they forego by investing early. The main result of this interaction is that the optimal subsidy at any given history is higher than would be the case were the agents ignorant of the fact that the policy maker will subsidize future investment. The equilibrium thus resembles that of Caplin & Leahy(1996) in that the policy maker must adopt more aggressive policies as a result of the agents' incentives to delay. However, unlike their model, it is not the case here that the optimal subsidy is increasing over time. The policy maker understands that an increasing path of subsidies will exacerbate delay, and since this is what the policy maker is trying to remove, he sets subsidies that reward early investment, and these subsidies are dynamically consistent. In this section, it was shown that a policy maker without commitment is able to set a set of subsidies to implement an equilibrium cutoff which reduces delay relative to the outcome that would obtain in the absence of the policy maker. Since this delay appears to result from an informational externality, we expect that reducing delay involves a gain in efficiency. However, we have also seen that the outcome induced by the policy maker may still involve some delay, and that it may also involve "negative delay" relative to the myopic, zero delay benchmark. Thus far, it is not clear whether this means that there remain efficiency gains which the policy maker is unable to exploit (due to the lack of a commitment technology, for example) or whether the policy maker achieves the first best. We examine this question in more detail next. 2.5 Welfare Analysis Thus far we have seen that a policy maker is able to reduce delay relative to the equilibrium of the benchmark economy by subsidizing early investors. Since delay in this model is a result of an informational externality, and since the optimal subsidy compen-sates agents for the informational value of their actions, it is natural to expect that the introduction of a policy maker increases efficiency, in contrast to the suggestion of Caplin & Leahy. 41 It remains to be shown to what extent the policy maker is able to induce efficiency gains in the economy. In particular, the observation that the pohcy maker is able to reduce equilibrium delay does not imply that the Caphn & Leahy effect is absent. It is possible, for example, that access to a commitment technology, or additional pohcy instruments, would enable the pohcy maker to improve upon the outcome of Section 2.4. In this section we show that this is not the case and that the pohcy maker is in fact able to achieve the first best outcome without any such commitment device. Therefore the three possible cases, that of the equihbrium with a policy maker without commitment, the equilibrium with a pohcy maker who has access to a commitment technology and the social optimum outcome all result in the same cutoffs. As our welfare benchmark we consider a world in which agents have access to a tech-nology which enables them to write binding contracts prior to the realization of agents' types. These contracts specify the action of an agent with cost type c at every possible history of the game. Since agents are a priori identical, the contract they will agree to is that which maximizes the expected profits of the representative agent. We refer to the set of cutoffs that arise from this Rawlsian process to be the first best, and judge the performance of the pohcy maker against this. The argument by which we show that the pohcy maker achieves this first best outcome through the use of a set of dynamic subsidies consists of two steps: i) we first derive the representative agent's objective function and show that it coincides with the preferences of the pohcy maker of Section 2.4, and ii) we show that the problem facing the Rawlsian representative agent is recursive, so that the solution to the policy makers problem studied in Section 2.4 is the same as the optimal Rawlsian contract. 2.5.1 Social Preferences We consider a situation in which all agents gather at the start of the game and commit to their future actions by means of a binding contract. These contracts specify the action of an agent contingent on his cost type for every possible observed history of the game. The fact that agents make this commitment prior to the realization of cost types means that they will take into account the fact that information revealed early on in the game is of value to agents whose cost types are such that they make their decisions later. Furthermore, at this stage all agents are identical and will therefore agree on the contract which maximizes the representative agent's expected payoff. The representative agent knows the probability distribution F(c) that wiU determine 42 his cost type, and forms expectations about the state of the world by observing the history of the game.11 His expected payoff for the game is then: Vr = E f f (n ) (C\v(n)-c)f{c)dc 71=0 J ± +5 £ { / . M«) - c)f(c)dc fci=0 171=0 Jcl JV-fci-1 f N * { h 2 ) +6 P(k2\ki)\Y2p(n\te) (v(n)-c)f(c)dc fc2=o U=o J<Z(i*i) + . . . } . . . } } (2.46) As in the previous section, we can re-write this in terms of /x(-) as: N re* Vr = Y2d(n) / \v{n)-c)f(c)dc 71=0 J ^ + S £ {£M(«,*I|C;) rihl\v(n)-c)f(C)dc fcl=0 171=0 + 5 E i Z > ( « > * 2 | f t i , c £ ) / (u(ra) - c)/(c)dc fc2=0 U = 0 •yc*(/i1) } • • • } } (2-47) + It is clear that the objective functions of our Rawlsian representative agent (Equation (2.47)) and the policy maker of Section 2.4 (Equation (2.32)) are identical. 2.5.2 The First Best Solution We have seen that the social welfare function coincides with the policy maker's objec-tive function. This by itself, however, does not guarantee that the solution to the policy maker's problem will be the first best. In this section we show that the problem of a "We could write the objective function more generally by integrating cost types over a set, as in general it is a set of cost types who choose to invest in each period. However, the logic of L e m m a 1 applies to the problem here, so that we know that the optimal contract will specify a set of cutoffs, so that the set of investing cost types is an interval. 43 planner wishing to maximize social welfare is recursive. As a result, the use of recursive methods allows us to solve for the optimum of the social welfare function. Since this is precisely the method used to solve the pohcy maker's problem, it follows that the pohcy maker achieves the first best outcome. Define as those terms of the social welfare function which apply after a given history hr. That is, = £ n(n\hT) I (v(n) - c)f(c)dc n = 0 J<-1 +5 YJ E ^ T ^ T ' C * ) / , (v(n) -c)f(c)dc + . . . } } (2.48) Let VT denote those terms pertaining to dates up to r, and V+T denote all terms dating from date r excepting those contained in V^. A strategy for the planner's problem is a set a = (c(fco), c(k\, ko), • • •) which specifies the cutoff cost for every possible history and for which Vr(a*) > VT(a) for all a. Let cr^ denote the subset of history dependent cutoffs which obtain after the first r periods if some history hr is observed. That is <ihT = (c(hT), c(hT, kT),...). An optimal strategy for the problem is a strategy, a*, for which Vr(a*) > Vr(a) for aU a. A strategy which is optimal for the continuation of the problem at a given history hT is a strategy, <7*^, for which ^(tr j j ) > V ^ C T / J - J . Using these definitions, we can define a strategy, a, which involves following the optimal plan, cr*, unless and until history hT is reached in which case the remainder of the problem is re-solved, thus resulting in the strategy a*^ being played subsequent to the observation of history hT. We wish to show that a is optimal for the original problem. That is, we wish to show that breaking the problem into pieces and solving the continuation problems sepa-rately results in the same solution as optimizing Vr with respect to ah of its arguments simultaneously. We proceed by assuming that this is not the case, so that Vr(a*) > VT(a), and looking for a contradiction. Note that we can decompose the payoffs under the two strategies as follows: Vr(a*) = Vt(<T*) + V+T(a*) + VhT{a*) (2.49) 44 and K ( C T ) = VT(a) + V+r(a) + Vhr(a). (2.50) Since VT(a*) and VT(a) are identical by construction, and V+T(a*) and V+T(a) are also identical by construction, for Vr(a*) > Vr(&) to hold it must be the case that V/^er*) > V/Ir(cr). This contradicts the fact that aT is optimal for the continuation problem V+T, which means that our assumption that a is not optimal for the whole problem is incorrect. Hence the problem can be solved recursively. The fact that the problem is recursive implies that we can solve the planner's problem by setting it up recursively. Since this is exactly the procedure we used to solve the policy maker's problem and since the objective functions are the same, it follows that the solutions of the two problems are identical. Therefore that the policy maker is able to achieve the first best outcome. 2.6 Conclusion This paper examined the effects of optimal policy in an environment in which both the policy maker and private agents learn about the economy through their actions, while interacting strategically with one another. Due to the central importance of uncertainty and strategic behaviour in this environment, it might be expected that the kind of effect identified by Caplin & Leahy (1996) would constrain the ability of the policy maker to improve the well being of the agents. Perhaps surprisingly, this turns out not to be the case. Even in the absence of a commitment technology, the policy maker is able to reduce equilibrium delay and achieve the socially optimal outcome. The policy maker reduces delay by subsidizing agents who invest early, taking into account the fact that agents are expecting subsidies in future periods as well. This recog-nition means that the optimal subsidy is declining over time, in expectation. The Caplin & Leahy intuition does not hold because the policy maker does not wish to increase the subsidy so much, following a period of very low investment, that agents are encouraged to delay. Furthermore, if the policy maker comes to believe that the state is bad, so that investment is unprofitable, the optimal policy is to allow agents to decide to not invest, rather than for the policy maker to increase the subsidy. The result that the policy maker achieves the first best without commitment is a strong outcome, and there are several factors of concern to policy makers that would alter this result. If, for example, the funds available to the policy maker came at some cost, the 45 policy would have an incentive to convince agents that future subsidies will be low in order to lower the cost of inducing current investment. Thus considering the cost of funds could introduce the elements of dynamic inconsistency into the policy maker's problem. This is in contrast to the speculation of Thimann Sz Thum (1998), who emphasize the role of the IMF in enhancing credibility by increasing the cost of funds made available to policy makers in developing and transition economies. A second issue concerns the possibility that the policy maker has private information, in contrast to the model in this paper where the policy maker influences the process by which the agents generate information, but has no further information to add. In such a situation, the fact that the policy maker's actions both influence the agents' decisions and signal the policy maker's information, may impede the policy maker's ability to manipulate the incentives facing private agents. Finally, positive payoff interdependency amongst agents, which is thought to play an important role in the experience of developing countries, would alter the policy maker's incentives. In particular, the possibility of pushing the economy into a better equilibrium might encourage the policy maker to respond to low investment by increasing the subsidy sufficiently aggressively so as to increase the incentives for delay on the part of the agents. Al l of these issues are of interest to policy makers operating in environments similar to that modelled in this paper, and might well impact the policy maker's ability to increase welfare. Future research might uncover the possibilities for optimal policy in environments where one or more of these factors complicates the policy maker's problem. 46 2.7 Appendix 1 The claim is trivially true if C equals 0 or C or if the distribution g(n) is degenerate, so suppose that g(n) is non-degenerate and 0 c C c C . If n and m are in the support of g(-), then for any n > fc, p(n|fe) = b(k;n,C)g(n) g(n + m,\k) b(k;n + m,C)g(n + m) (n + m — fc)! • n! • g(n) (n - fc)! • (n + m)! • (1 - "^(C))mc7(n + m)' which is decreasing in fc. Compare <7(-|fc) and g(-\k + 1). For n = 1,..., fc — 1, g(n\k) = g(n\k + 1) = 0, and for n = fc, g(n\k) > 0 = g(n\k + 1). For any n > fc, if g(n\k) < g(n\k + 1), then g(n + m\k) < g{n\k) < 1 g(n + m, |fc + 1) #(n, |fc + 1) Since we know that for some n that g(n\k) > g(n\k + 1), the two density functions have the single crossing property: g(n\k + 1) < g(n\k) if n < no and, #(n|fc+l) < g(n\k) if n > no-This implies first order stochastic dominance, which implies that the claim is correct. 2.8 Appendix 2 Consider the first period. If it is not optimal for an agent to invest for any c then c*(ho) = c. If an agent would strictly prefer investment for some c, let C denote the set of ah types c for which the agent strictly prefers investing to waiting. Since C C [c,c], C has a least upper bound c. Consider an agent with type c < c. If such an agent invests in period one, he receives c — c more than an agent with c. If he waits, he receives c — c more than an agent with c, but this is discounted. Since all agents have the same information, the agent with c < c has less incentive to wait than the agent with c (that is, gets the same information and loses more because of discounting). Therefore, all agents with c < c strictly prefer investment to waiting in the first period, so c*(ho) = c. By induction, this applies to all subsequent periods. 47 2.9 Appendix 3 If c*(ht,kt) > c*(ht) then it is (clearly) optimal for the player with type c*(ht) to invest at (ht, kt) (by Lemma 1). Therefore, it is only necessary to consider the case where c*(ht,kt) = c*(ht) (explain why c*(ht, kt) < c*(ht) is not relevant). i) If V(ht, kt) — c(ht) > 0 and c*(ht,kt) = c*(ht), then an agent with c*(ht) must be planning to invest at some future information set h'. Let h' — (ht, kt, 0 . . . , 0) be the first information set after (ht,kt) at which c* > c*(ht). Since the probability of investment is zero between (ht, kt) and h' (because, by Lemma 1 if the agent with the lowest remaining cost (c*(ht)) doesn't find it optimal to invest then no-one does), the information is the same at both information sets so that V(h') —c*(ht) — V(ht,kt)—c*(ht) > 0. Discounting makes the agent better off if he invests at (ht,kt), meaning that c*(ht,kt) > c*(ht). ii) If V(ht,kt) - c*(ht) < 0 and c*(ht,kt) = c*(ht), then at h' = (fh,kt,0..., 0) no new information has been revealed (again this is an implication of Lemma 1) and V(h') — c*(ht) < 0. Therefore it is not optimal to invest at any such information set h', so that it is never optimal to invest following (ht,h). iii) If V(ht, kt) — c*(ht) — 0 and c*(ht,kt) = c*(ht), then at h' = (ht,kt,0,... ,0) no new information has been revealed and V(h') — c*(ht) = 0. Since v(n) ^ c*(ht)) for all n (that is, I have been assuming a non-degenerate distribution on n without saying so explicitly), there is some uncertainty about V(h') — c*(ht) (that is, it could be positive or negative). If c*(h') > c*(ht), then a player expects a payoff of zero now, but there is some uncertainty about it. Furthermore, because c*(h') > c*(ht) if the agent waits he can learns about the uncertainty and thereby increase his expected payoff, so a player with c*(ht) will strictly prefer to wait at h'. This contradicts c*(h') > c*(ht) and we have that c*(h') = c*(ht) for all h' (the logic in this part is the basically the idea that you have a positive option value when your payoff is negative for some realizations of n). 48 Chapter 3 What Happened to the Phillips Curve in Canada and the U.S in the 1990s? 3.1 Introduction There are at least two broad classes of interpretation to consider when examining observations on a country's output-inflation relationship (a country's Phillips curve). On one hand, there is the traditional interpretation, which emphasizes how such a relationship mainly reflects a country's wage and price setting institutions. In this case, the Phillips curve is viewed primarily as a structural object in the sense that the slope of the Phillips curve is governed foremost by the institutional aspects of the wage setting mechanism-and hence is an object that constrains monetary policy. On the other hand, there is the view that the Phillips curve is essentially a reduced form relationship, which mainly reflects rather than constrains the behavior of monetary authorities. In this paper we will argue that this second view helps explain recently observed changes in the Phillips curve. We begin the paper by reviewing the changing nature of the PhiUips curve relationship in both Canada and the U.S from 1961-1999. We define the PhiUips curve as the statistical relationship between the change in inflation and the deviation of output from trend and, based on this definition, show that in both Canada and the U.S the slope of the Phillips curve has become much smaller over the last twenty years, with a sharp reduction observed in the nineties. This observation raises two related issues: (1) what explains the decline in slope and (2) what does this decline imply for the proper conduct of monetary pohcy. Our goal is to provide new insight on these issues by presenting an explanation of the 49 observed flattening of the Phillips curve based on the notion that, since the seventies, central banks have continuously increased their awareness and understanding of the real forces that determine aggregate output. Hence, we believe that the current observation of a nearly horizontal Phillips curve may best be interpreted as sign of well executed, neutral stance, monetary policy. Our explanation of the flattening of the Phillips curve is presented in a simple model that recognizes the role of both price rigidities and real disturbances in explaining macroe-conomic fluctuations. In effect, our model extends the monopolisticalfy competitive model of Blanchard & Kiyotaki (1987) in a manner which allows for real disturbances (as in the real business cycle literature) and for imperfect information.1 However, in contrast to much of the macroeconomic literature with imperfect information (for example Lucas (1972) and Barro &; Gordon (1983)), the information asymmetry we emphasize is such that the central bank is imperfectly informed regarding real developments in the economy and hence the central bank is constantly trying to infer the state of the economy while simultaneous affecting it.2 We believe that this type of informational limitation is preva-lent in all central banks and is important for understanding both the conduct of monetary policy and the co-movement between output and inflation. Our story most closely resembles that of Sargent (1999) and Cooley and Quadrini (1999). In these accounts of the Phillips curve in the U.S. and the U.K. respectively, the authors attempt to explain changes in the Phillips curve relationship as a result of the central bank's attempts to learn about the economy. Our story differs in that the central bank in our model is trying to learn about shocks affecting the real economy and this learning gives rise to a Phillips curve in equilibrium. In previous accounts the Phillips curve exists as a structural object, and the central bank is trying to learn about its slope. Within this simple model, we derive the properties of the output-inflation relationship under the assumption that monetary policy is conducted optimally, subject to the central bank's hmited information. We show how a statistical Phillips curve can arise in this environment, with the causality running from real developments to nominal outcomes. Moreover, we show how the central bank will use observations of output and inflation to readjust the path of its monetary instruments. 1In the terminology of King & Goodfriend (1997), our model is a small scale "new neoclassical synthesis" model. 2In this respect, our model capture some of the elements present in Caplin & Leahy (1996) regarding the Interaction between the central bank and private agents when the central bank is uninformed about the state of the economy. 50 We derive two main results from the model. Our first result is to show how, as the central bank becomes more aware and sensitive to real developments in the economy, the slope of the Phillips curve will tend to approach zero. The intuition for this result is rather straightforward. The objective of monetary policy should be to simultaneously support the well functioning of the economy and maintain price stability. However, in the absence of complete information on the state of the economy, the central bank cannot achieve this perfectly. The interaction between private agents and the central bank, both of whom are trying to learn from the other, gives rise to a Phillips curve relationship. As the central bank learns to perform its information gathering role more adequately, the positively sloped Phillips curve gradually disappears. We will argue that this mechanism helps understand the observed flattening of the Phillips curve over the last twenty years, as the central banks in both the U.S and Canada first became aware of the importance of real shocks in the seventies, and then learned to identify and react to them more appropriately throughout the eighties and nineties. The second result we wish to highlight is that a flattening of the Phillips curve does not mean that the short run output-inflation tradeoff faced by the central bank has changed. In effect, we show why the Phillips curve and the output-inflation tradeoff should be considered as two distinct objects, and why a flattening of the statistical Phillips curve can arise without there being a change in the relevant output-inflation tradeoff faced by the central bank. The remaining sections of the paper are structured as follows. In Section 3.2, we doc-ument the changing nature of the Phillips curve for both the U.S and Canada over the period 1961-1999. In Section 3.3, we present our model of the Phillips curve. In particular, we derive the properties of the output-inflation relationship under the assumption that monetary authorities are imperfectly informed about the state of the economy but never-theless try to conduct monetary policy optimally. We go on to compare the plausibility of our explanation of the flattening of the Phillips curve to one based on nominal wage rigidities. Finally, in Section 3.4 we offer concluding comments. 3.2 Overview of the Output-Inflation Relationship in Canada and the U.S In this section we review the evidence related to the existence of a positively sloped Phillips curve for both the U.S and Canada over the period 1961 to 1999. In particu-51 lar, we present evidence to suggest that the PhiUips curve relationship is robust to various specifications and is roughly similar in Canada and the U.S.3 We also present evidence sug-gesting that the relationship between inflation and output has changed in recent decades. In particular, we show that the PhiUips curve has flattened over the past 20 years. We find that the reduction in slope, which has occurred in both the U.S and Canada, is quite substantive. 3.2.1 Basic Estimation and Results In its simplest form, the PhiUips curve can be expressed as a relationship between inflation, lags of inflation, and the deviation of output from its trend level (referred to as the output gap). In the absence of clear theoretical guidance on the appropriate measure of prices, the Phillips curve literature uses various measures, from broad ones like the GDP deflator to measures which try to capture the notion of core inflation. In our basehne estimations, we use the percentage change in the GDP deflator as our measure of inflation.4 Measuring the output gap raises further issues. The literature arrives at output gap series by employing a variety of techniques, including HP filters, structural VARs, struc-tural macroeconomic models and simple time trends, to infer the trend level of output. We explored several alternatives and choose as our baseline measure the output gap se-ries created by applying an HP filter to the natural logarithm of real GDP. 5 Since we recognize that the PhiUips curve can be expressed as a relationship between inflation and unemployment, we also explored the nature of the inflation-unemployment PhiUips curve to provide a check on our results.6 As a starting point, we estimate the foUowing very simple Phillips curve, using annual data on inflation and the output gap: Ant = a + p • GAPt + e t . In Figures 3.1 and 3.1 we plot this relationship, along with the associated regression line, for Canada and the U.S from 1961 to 1999. The slope of the estimated PhiUips curve 3Fiflion and Leonard(1997) present linear Phillips curve estimates for Canada which resemble our own. 4We also used the CPI as an alternate measure of prices in order to check the robustness of our results. 5We have verified that the results are robust to various values of A, the smoothing parameter of the HP filter. The results presented in the paper set A to 1600, which with annual data implies that we are unlikely to be over-smoothing. 6We take our data for Canada from Cansim, and our U.S data from Basic Economics (formerly Citibase). 52 Figure 3.1. Phillips Curve, Canada 1961-1999 • • • a aa 0 o o • D ° o 0 a O^y - ^ p • u Slope=0.214 J 2 ^ " o D • D • • o D • D I I • 1 1 -6 -4 -2 0 2 4 6 Output Gap (HP-Filtered) 53 Figure 3.2. Phillips Curve, U.S 1961-1999 4 3 H 2 H -2 H -3 H -6 -4 -2 0 2 4 6 Output Gap (HP-Filtered) 54 Table 3.1. Basic Phillips Curve Estimates for Canada: 1961-1999 Avrt A7Tt Constant (Std. Error) -0.0360 (0.2259) -0.0690 (0.2231) 0.0021 (0.2219) HP-GAP t HP-GAP t _ i T -GAP t _ i 0.2141 (0.0828) 0.1890 (0.0814) 0.2599 (0.0998) for the U.S is 0.256, suggesting -that a positive output gap of 1 percent is associated with an increase in inflation of around one quarter of one percent on average. The Canadian estimate of 0.214 is similar to that of the U.S. In both countries, we reject the hypothesis that the slope of the PhiUips curve is zero at conventional levels. To aUow for the possibUity that inflation responds to real developments with some delay, in column 2 of Tables 3.1 and 3.2, we aUow for lagged values of the output gap to enter as the right-hand side variable. This specification will be particularly relevant when discussing our theoretical model. As can be seen in the tables, our estimated Philhps curve relationship is not strongly affected by the choice of the lag of the output gap rather than its contemporaneous value as a regressor. In order to illustrate the robustness of these results, we consider a variety of alternate PhiUips curve specifications. As mentioned above, one specification issue concerns our measure of the output gap. Since we derive our output gap series by decomposing the level of output into trend and gap components using an HP filter, we wish to repeat our analysis using alternate detrending methods. In column 3 of Tables 3.1 and 3.2, we report the results of estimating our simple PhiUips curve equation using a cubic time trend to create the output gap measure. Our point estimates of the slope differ depending on the choice of gap measure as iUustrated in the Tables, but the differences are not very large. We also wish to check the robustness of these results when we aUow for a freer speci-fication of the inflation process and when we control for supply side factors. In short, we estimated several variants of the following equation : nt = const + a(L)nt-i + b(L)(GAPt) + cXt + e t , 55 Table 3.2. Basic Phillips Curve Estimates for the U.S.: 1961-1999 A7Tt ATTf A7Tt Constant (Std. Error) -0.0014 (0.1567) -0.0531 (0.1567) 0.0133 (0.1576) HP-GAP t HP-GAPt_! T -GAP t _ i 0.2560 (0.0650) 0.2404 (0.0619) 0.2719 (0.0714) where 7rt is inflation in period t, G A P t is a measure of the output gap, and X t is a vector of supply side variables. We present a set of such results in Tables 3 and 4. As can be seen in Tables 3.3 and 3.4, allowing for lags of the change in inflation as re-gressors can have noticeable effects on our estimated slope coefficients. Comparing column 1 of Table 3.4 to our base results from Table 3.2 shows that the respecification has the effect of increasing the coefficient on the output gap for the U.S. The same respecification, however, has almost no effect on the Canadian estimate. While the addition of lags of the change in inflation as regressors affects our slope estimates, in no case does the re-specification overturn our initial results that there is a positive and statistically significant co-movement between output and changes in inflation over the period 1961-1999. Respecifying the problem in terms of inflation rather than the change in inflation, as shown in columns 3 and 4 of Tables 3.3 and 3.4, allows for a freer specification of the inflation process. We find that this specification of the inflation process also affects on our slope estimates. In general this results in a higher estimated coefficient on the output gap term, where the estimate tends to increase with the number of lags of inflation included. Finally, the inclusion of supply side variables appears to have moderate effects on our slope estimates. Columns two and four report results where inflation in relative energy prices is included as a regressor.7 We find that the inclusion of energy prices has a small to moderate effect on our estimates of the coefficient on the gap variable, and that this effect differs in size and sign depending on the specification and country. To summarize, we find that the data since 1960 strongly supports the existence of a 7 W e define inf la t ion i n relative energy prices as the percentage change i n the ra t io of the C P I for energy to the a l l i tems C P I . 56 Table 3.3. Phillips Curve Estimates for Canada: 1961-1999 A 7 T t Ant Constant -0.0787 (0.2310) -0.0184 (0.2319) 1.2991 (0.4042) 1.7040 (0.5207) HP-GAP^- ! 0.1935 (0.0922) 0.2301 (0.0947) 0.3646 (0.0889) 0.3799 (0.0909) A7Tt_l 0.0975 (0.1725) 0.0743 (0.1718) - -TTt-1 - -0.8332 (0.1596) -0.1218 (0.1445) 0.7391 (0.1739) -0.1280 (0.1471) 7T t ENERGY TTt-i ENERGY --0.0142 (0.0615) -0.0886 (0.0590) -0.0665 (0.0570) 0.0427 (0.0623) Table 3.4. Phillips Curve Estimates for the U.S.: 1961-1999 A 7 T t Ant Constant -0.0559 (0.1556) -0.0299 (0.1474) 0.4776 (0.3095) 1.2633 (0.2969) HP-GAP t _ i 0.3161 (0.0771) 0.2724 (0.0779) 0.3054 (0.0742) 0.2734 (0.0604) A7r t_! -0.1449 (0.1637) -0.1726 (0.1792) - -7Ti -2 - -0.8042 (0.1593) 0.0694 (0.1618) 0.4578 (0.1595) 0.2429 (0.1397) nt ENERGY 7 r t _ i ENERGY -0.0719 (0.0300) -0.0072 (0.0328) -0.0980 (0.0239) 0.0723 (0.0305) 57 positively sloped Phillips curve in both Canada and the U.S, and that this observation is robust to alternate specifications. Our estimates of the slope of the Phillips curve vary mostly between 0.2 and 0.3. In all cases, the estimated slope of the Phillips curve is positive and significantly different from zero at conventional levels. 3.2.2 The Changing Slope of the Phillips Curve Having reviewed the case for the existence of a positive co-movement between inflation and output in both Canada and the U.S from 1961 to 1999, we now turn our attention examining whether the Phillips curve relationship may have changed over time. As we will show, the slope of the Phillips curve in both Canada and the U.S has declined markedly from its peak in the late 1970's. To examine the slope of the Phillips curve relationship over time, we employ a series of rolling regressions on a 15-year moving window of data. That is, for each year in our sample, starting in 1978, we estimate the Phillips curve for the most recent 15 years. For example, the estimates for 1983 are derived from observations over the period 1969-1983. Figures 3.3 and 3.4 present results from running the change in inflation on the lag of the output gap. We use this as our baseline specification as it is easily tied in to the theoretical results presented in subsequent sections. Note that the estimated slope of the Phillips curve peaks around 1982 in both countries. The slope then declines throughout the 1980's and 1990's. In the U.S the slope begins to fall around 1988, and declines smoothly through to the end of the sample. This decline does not occur until 1992 in Canada leading to a much sharper decline in the 1990's. By the end of the 1990's, the slope of the Phillips curve is not significantly different from zero in either Canada or the U.S. As described in the previous section, we performed a variety of robustness checks of our baseline specification. We find that the pattern of a flattening of the Phillips curve in the 1980's in both Canada and the U.S is robust across different specifications. The results presented in Figures 3.3 and 3.4 also seem to suggest that the slope of the Phillips curve may also have been quite low prior to the the mid 1970's. This implication, however, is not robust to the choice of estimation framework. Figures 3.5 and 3.6 present one example where we include as an additional regressor inflation in relative energy prices. As can be seen from the figure, the pattern of a declining slope in the 1980's and 1990's remains essentially unchanged. Note, however, that our estimate of the Phillips curve's slope in Canada at the end of the 1970's almost triple 58 59 Figure 3.4. Slope of U.S Phi l l ips Curve Over Time 15 Year Rolling Regression of Change in Inflation on Lag of Output Gap i i I I I I i i I I i I I — i — i — i — i — i — i — i — i — i — 1978 1981 1984 1987 1990 1993 1996 1999 60 Figure 3.5. Canadian Phillips Curve Over Time, Alternate Specification Change in Inflation on Lag of Output Gap and Inflation in Relative Energy Prices 0.7 - , 0.6 - \ . — - — \ / » i . > 0.5 -CO 0.0 r - ^ : -0.1 --°- 2 I I I I I I I I I I I I I I I I 1 1 1 1 1 1 1978 1981 1984 1987 1990 1993 1996 1999 under the new specification (relative to the estimate in Figure 3.3). As a result, we do not believe that low slope coefficient observed in Figure 3.3 in the 1970 is a robust feature of the data.8 Since we are attempting to examine changes in the Phillips curve relationship over time, we also ran a series of weighted rolling regressions in which we imposed declining weights on more distant years. This procedure reduces the chance that one or two observations might unduly influence the profile of our estimates. We found that this approaches yields similar results to that presented in Figures 3.3 to 3.6. That is, the slope of the Phillips curve appears to decline substantially over the 1980's and 1990's. The point estimates presented in Figures 3.3 to 3.6 are not very precise, as can be seen from the size of the standard error bands. Since the imprecision of our estimates 8 T h e ou t l y ing observation of 1975 (which was a year characterized by large movements i n c o m m o d i t y prices) m a y expla in w h y our estimates for the 1970's are sensitive to alternate specifications. 61 62 Figure 3.7. Phillips Curve Over Time, Pooled Sample Rolling Regression of Change in Inflation on Lag of Output Gap 0.7 - • 0.6 -0.5 H O CO 0.0 --0.1 -1978 1981 1984 1987 1990 1993 1996 1999 is a function of the size of our moving sample, we face a tradeoff: we can increase the precision of our estimates only by including more distant years in our sample, in which case the composition of our sample tends to change much more slowly. Given this choice, we prefer to present estimates, which may be imprecise, but more fully capture any possible changes in the Phillips curve relationship. We believe that the magnitude of the change in the point estimates is economically important enough to warrant interest, even if the statistical significance can be questioned. As a further check on the robustness of our results, we pool our U.S and Canadian data to increase the number of observations in each sample. The slope of the Phillips curve estimated on the full sample is 0.2239, which is in the same range as our previous estimates. As before, we find our estimate is robust to a variety of alternate specifications. Figure 3.7 reports the results of a series of rolling regressions each on 15 years of pooled data and using the baseline specification.9 We find the slope of the pooled U.S and Canadian 9 T h a t is we regress the change i n inflat ion on the lag of the output gap. 63 Figure 3.8. Phillips Curve, 1985-1999, Pooled Sample Phillips curve exhibits the same profile as in the individual samples. That is, the slope of the Phillips curve peaks in the early 1980's, and declines thereafter. As in previous cases, the decline in slope from its peak to its 1999 level is substantial. To illustrate the flatness of the Phillips curve since the mid-1980's, Figure 3.8 plots the relationship between the change in inflation and the output gap for the pooled sample from 1985 to 1999. The estimated slope for this, sample is 0.1108 with a standard error of 0.0683. This is substantially lower than any of our full sample estimates, and is not significantly different from zero at conventional levels. As can be seen from the figure, one outlier drives much of this slope: the Canadian observation for 1992. If we include a dummy variable to control for this observation, we find the slope of the Phillips curve for the U.S and Canada since 1985 to be merely 0.0212. The evidence presented above leads us to believe that the Phillips curve relationship has changed significantly in recent decades. Particularly, we find that the Phillips curve has flattened substantially in both the U.S and Canada. Furthermore, at least in Canada, 64 this flattening has occurred mainly in the 1990's. 3.3 Why is there is a Phillips Curve and why might its slope change over time? In this'section we explore the theoretical nature of the output-inflation relationship. Our goal is to illustrate the mechanism by which optimal monetary policy can give rise to a Phillips curve and how, in such a case, the slope of the Phillips curve relates to the fundamentals of the economy. In particular, we want to highlight the link between the slope of the Phillips curve and the degree to which monetary authorities are imperfectly informed about the state of the economy. We present this issue by building upon a commonly used monopohstic competition macro model (see, for example, Blanchard & Kiyotaki (1987)) which we specify to allow for both real and nominal disturbances to affect output. We consider an environment in which one final good, Yt, is produced using a set of N intermediate goods, Xn, where i = 1,..., N. The intermediate goods are produced by monopolistically competitive firms, which must preset prices at the beginning of each pe-riod, before the demand for intermediate goods is determined. The final good is produced by competitive firms according to the CRS production function given by equation (3.1). Yt = (tx$)&N<^ (3.1) Each intermediate goods firm has access to a production technology given by (3.2). Xit = At^Ll (3.2) where, Lu is the quantity of labour employed in firm i and At is the productivity index. We assume that the productivity index, At, is common to all intermediate goods, and that the log of At follows the stationary stochastic process given by (3.3).10 oo oo °* = JZ^fr-j, V>0 = 1, £ $ < 0 0 (3-3) j=0 i = l where, et is assumed to be a normally distributed mean zero random variable with variance cr£, and the ipiS are assumed to be positive. This last restriction is meant to capture the notion that deviations of technology from trend are positively autocorrelated. 10In all that follows, we use lower case letters to denote the logarithm of a variable. 65 In order to keep the presentation of the model as simple as possible, we do not ex-plicitly include a trend in the process for At. Nonetheless, we think it is best to interpret the variables of the model as deviations from a trend induced by growth in At. Fur-thermore, our assumption of a common technology process across intermediate goods is clearly restrictive but is justifiable on the grounds that we are interested only in aggregate fluctuations. The representative household in this economy has preferences defined over consump-tion, labour supply and real balances, as given by (3.4). We assume that the household's utility is linear in labour so as to generate a constant real wage. Hence, the model can alternatively be interpreted as a model with an exogenously fixed real wages. U(CU ^,Lt)= C f ^ 1 6 ~ (3-4) The household's budget constraint is given by (3.5), where Pt is the price of the final good, Wt is the nominal wage rate, Mt is money demanded and Mt is the money balances distributed by the central bank at the beginning of each period. PtCt + Mt = WtLt + Mt (3.5) In order to solve for the private sector's equilibrium behavior, we start by examining the household's decision problem. The representative household takes prices as given and chooses consumption, labour and money balances so as to maximize utility. The first order conditions associated with this maximization imply that money demanded satisfies equation (3.6), and that labour is supplied elastically at the real wage given by (3.7). Mt = PtCv^Q- (3.6) ^ = (1 - 9)l-09e<f> (3.7) Pt Final good producers also take prices as given and maximize profits by choosing the amount of intermediate inputs to use. This gives rise to a demand for intermediate goods given by (3.8), where Pa is the price of the i'th intermediate good. X ^ = ( ^ ) ^ (3-8) The problem facing an intermediate good firm is more complicated given that the prices of intermediate goods must be set before the realizations of either At or Mt- The firm's 66 objective is therefore to set Pn to maximize expected profits conditional on the information set £lt-i, which contains all information dated t — 1 or earlier, including realizations of past values of e. A n intermediate good producer's problem can therefore be expressed as follows: max E[PitXit - WtLa/Ot-i] s.t. (3.2), (3.3), (3.7), (3.8) Using the market clearing conditions for both the goods market and the money market, and imposing symmetry on the behavior of intermediate goods producers, one can easily derive Equations (3.9) and (3.10) which describe the behavior of the aggregate price level and aggregate output. 1 1 In these two equations, constant terms have been dropped. oo Pt = E[mt/nt-i] - £ fat-i (3.9) oo yt = mt-Pt = (mt - E[mt/Sk-i\) + £ ipi€t-i (3.10) i=l Equations (3.9) and (3.10) represent the equihbrium behavior of private agents, for arbitrary processes of money supplied. Note that both prices and output depend on real and monetary forces. In particular, the aggregate price level depends on real shocks and expected money, while aggregate output depends on real shocks and unexpected money. It is important to note that the et's in Equations (3.9) and (3.10) can be interpreted very broadly as reflecting any real shocks that affect the potential gains from trade, as opposed to the narrow technology shock representation. The model thus far is a typical preset prices macro model and generates a structure common to models of this type. The novel aspect of our analysis concerns the nature of the interaction between the private sector and the central bank. We now introduce the objectives and the constraints facing the central bank, and highlight the key elements of our model. r 1 1In order to derive equation (3.9) from the intermediate good firm's problem, it is easiest to first use (3.2) and (3.7) to eliminate Wt and Lit from the firm's objective function. Then, using the market clearing conditions Ct = Yt and Mt = Mt in combination with (3.6) and (3.8), the demand facing the firm can be written as simply a function of Pu,Pt and Mt. Finally, by imposing that Pt = Pit in the first order condition associated with the firm's optimal choice of Pit, and taking logs one obtains to Equation (3.9). 67 We assume that the objective of the central bank is to minimize deviations of output and prices from target levels y\ and p*, as given by (3 .11) . co £ (?E[(yt - y*t)2 + * • ( ? * - P*t)2m (3 .11) i=0 In (3 .11) , $ is the weight the central banker places on deviations of inflation from its target relative to output deviations. With respect to the target for output, we assume that it is the level of output that would arise in the competitive equilibrium in the absence of any price rigidities or informational imperfection, that is, y* = I ^ o ^ 6 * - * (note that we have again dropped the constant term).12 This choice of output target may be controversial. In particular, it may be the case that the central bank would like to attempt to use monetary policy to overcome the other source of inefficiency in the economy: the presence of market power on the part of the monopolistically competitive firms. We believe, however, that central bankers do not think that monetary policy is an appropriate tool for overcoming this kind of inefficiency. Thus we believe that the assumption that the central bank targets the "flexible-price' equilibrium output is a fair description of how central banker's view their role as monetary policy makers, and is, therefore, the most reasonable assumption for the model at hand. With respect to the target for the price level, we assume that it is driven by an exogenously given inflation target 7ft, such that p* — pt-i + Ttt- For our purposes, the process for the inflation target can be thought as being either stochastic or deterministic; the key simplifying assumption being that it is exogenous. In order to allow for the possibility that the inflation target be stochastic, we denote the agents expectation of target inflation as of time t — 1 by t-i^t- By assuming that the inflation target follows a known exogenous process, we are obviously sidestepping important issues related to the signaling of inflation targets. The key assumptions of our model relate to the timing of moves and the information available to the central bank and private agents when making decisions. The assumptions are chosen to capture the notion that, in the short run because of sticky prices, the central bank has the important but difficult task of helping private agents achieve gains from trade by providing the right amount of liquidity to the system. In effect, we model the central 1 2 By assuming that the central bank's objective is to attaint the competitive equilibrium outcome, we are ehminating a standard channel which gives rise to time consistency problems (and inflationary bias). 68 bank as having both an informational disadvantage and a timing advantage relative to the private sector. The central bank's disadvantage is that it does not directly observe the et's, and therefore must infer their values from past developments in the economy. Its advantage is that it has some information on the current state of the economy, which it can use during the period over which prices are preset. In effect, we assume that the central bank receives a signal, St, from its research department each period. This signal is an unbiased indicator of real developments in the economy as captured by equation (3.12), where fit is a normally distributed mean zero 2 random variable with variance cr2. We denote by T 2 the noise to signal ratio - | . 8t = €t + fit (3.12) The timing of moves is as follows. At the beginning of a period, intermediate good firms set prices and the central bank simultaneously decides on the money supply. How-ever, since private agents and the bank are differentially informed, the information used to determine these elements differs. Private agents know all past developments in the econ-omy but do not know the realization of et that is to arise during the period. In contrast, the central bank has past information only on output and prices (not the e's), but has the advantage of observing St- We will denote the information set of the central bank at the beginning of time t by Clt = {st, St~i,... ,pt-i, • • •, Vt-i • • •}, and the information set of the private agents as Qt-i = • • •, st-i, • • • ,Pt-i • • •, yt-i •••}• Our justification for giving the central bank an informational advantage through St captures the notion that the central bank has a timing advantage over the private sector. Since the private sector has pre-set prices, the central bank has more flexibility within a period to react to current shocks but is, nevertheless, imperfectly informed regarding the right way to react. The problem facing the central bank is to choose a monetary policy rule so as to minimize (3.11) subject to its informational restrictions and the optimizing behavior of the private economy, given by equations (3.9) and (3.10). This problem is more intricate than standard optimal policy problems since the information sets of the private agents and the central bank are neither identical nor subsets of each other. In fact, our setup is similar to a simultaneous move game in which both sides have private information. As discussed in Townsend (1983), this can give rise to infinite regress problems. In this case, however, we have kept the problem simple enough to allow for an explicit solution. The policy rule which solves the central banks problem is given by equation (3.15), 69 with the implied equilibrium solution for inflation (71*) and output given by equations (3.13) and (3.14) respectively. n =t-i n + ih(Qst-i - et-i), e = (3.13) yt = G-St + ^2ipiet-i (3.14) mt =Pt-l +t-l TTt ~ -J-{n-l -t-2 7T*_i) + G£V iS t - t + £ ViCt-i (3.15) i=0 i=3 In order to gain intuition about equations (3.13)-(3.15), it is helpful to first recognize that the term 0 • St is the central bank's best estimate of the current supply shock et-Since the central bank's objective is to accommodate real shocks while maintaining price stability (around target), it adjusts the money supply so as to reflect its best guess of the current supply shock. Given that prices are fixed, an expansion of the money supply is first reflected in output, as desired, and not in prices. That is, the central bank uses the money supply to allow the real economy to react to its signal on the current supply shock, thus partially overcoming the nominal rigidities inherent in the economy. In the following period, the private sector becomes informed about the realization of last period's supply shock and adjusts prices accordingly. Note that inflation only deviates from the target level of inflation to the extent that the central bank's estimate of the real shock in the previous period was mistaken. In effect, by adjusting prices in response to the central bank's error, the private sector actually reveals to the central bank the extent of its past error. The reason that private agents react to past mistakes is that they foresee that the central bank will continue to accommodate the effects of a perceived shock until it becomes aware that it has made an error. Hence, the profit maximizing price setting rule is to increase prices in response to past excessive expansion on the part of the central bank. Correspondingly, once the central bank recognizes that it has made an error, by ob-serving a deviation of inflation from its target, it readjusts the money supply. This can be seen from equation (3.15), where the past deviation of inflation from the target level enters negatively in the money supply rule. Although monetary authorities never directly observe the e's, within two periods they are able to perfectly infer their values from ob-serving developments in the economy. This explains why the money supply rule can be 70 written as a function of lagged values of the e's. 1 3 We now turn our attention to the implications of the above model for the nature of the Phillips curve. For now let us define the Phillips curve, as we did in Section 3.2, as a purely statistical object. In particular, let the slope of the Phillips curve be the slope of the relationship between the change in inflation and the deviation of output from trend. Since our model is in terms of deviations from trend, the theoretical analogue to this slope is the covariance between the change in inflation and output, divided by the variance of output. The analytical expression for this slope is reported in equation (3.16) and is denoted by /?. ffi 9g_ ( 3 1 6 ) var{yt) ( 5 X i ipf X 1 + T ) d r Note that our model suggests that we focus on the relationship between the change in inflation and the lagged deviation of output from trend, since it is only after one period that prices in the model can react to demand disturbances. Recall from Section 3.2, that in the data, such a distinction (at the annual level14) does not make much difference. If we enriched the dynamics of the model to allow for an autoregressive component to the output gap, this distinction would not matter in the theory either. 1 5 The first thing to note from equation (3.16) is that the model generates a statistical Phillips curve; that is, even though monetary policy is set optimally, the economy never-theless exhibits a systematic positive co-movement between inflation growth and output. Moreover, this co-movement actually represents causality running from money to output and then to inflation, as is usually thought to be the case in discussions of the Phillips 1 3Using the money supply rule (3.15) to calculate expected and unexpected money, it is rather simple to verify that equations (3.13) and (3.14) are consistent with private agents optimal behavior given by Equations (3.9) and (3.10). It is only slightly more difficult to verify that the money supply rule given by (3.15) is in effect optimal. To see this, note that for both prices and output the deviation from target is simply the difference between the central bank's guess of et, @ • st, and it realization. Hence, since this difference is minimized by setting 0 = n2_£a* , this confirms the optimality of the policy. 1 4With quarterly data, we generally found the lagged output gap to be a better predictor of inflation than the contemporaneous output gap. 1 5One of the limitations of the current model is that, because we have not included any state variables, there is no endogenous propagation mechanism. This explains why money expansions only affect output for one period. If we included adjustment costs, such as a convex cost of changing labour, monetary shocks would have persistent effects and hence the distinction between the co-variance of Airt with either yt or yt-i would be, as in the data, rather minor. Given the small returns and added complexity associated with adding such elements, we do not pursue this generalization here. 71 curve. Essentially, the bank responds to its signal by changing the money supply. As prices are fixed in the very short run, this has an immediate effect on output in the economy. In the subsequent period, firms observe the true value of the shock and set new prices responding to both the new information they have received about last period's shock, and any error made by the pohcy maker in the last period. The second aspect to note is that the slope of the Phillips curve is strictly increasing in r 2 ( the noise to signal ratio for St). In other words, equation (3.16) implies that when the central bank becomes more aware of real developments in the economy (perhaps by expending greater effort to gather information about these developments and thereby reducing r 2), it will make fewer errors conducting monetary policy and this will lead to a flatter Phillips curve. This is the first result we want to highlight from this model: a flat Phillips curve may be a reflection of a well run monetary policy. In particular, if a 2 were to go to zero, monetary authorities would make no errors and the statistical Phillips curve would become perfectly horizontal. The reason is that, in such a case, monetary authorities would be able to stabilize prices while allowing the economy to respond efficiently to real forces. In contrast, the Phillips curve would tend to be more steeply sloped in an environment with substantial variations in real shocks or a poorly informed central bank. Before discussing the potential relevance of equation (3.16) for explaining the changing nature of the Phillips curve, it is interesting to note the difference between the statistical Phillips curve implied by this model and the short-run output-inflation tradeoff faced by the central bank. In particular, even in a situation where the slope of the statistical Phillips curve is almost zero, this model does not imply that the central bank should perceive the short run trade-off between inflation and output to be close to zero. In effect, such a tradeoff could still be quite large. To see this, we can use Equations (3.13) and (3.14) to derive the short-run relationship between inflation, target inflation, output and supply shocks. This relationship is given by Equation (3.17). CO 7T* =i_l 7Tt* + tplVt-1 - £ V^ t - i , (3-17) i = l The term xfriyt-i in Equation (3.17) represents the effect on inflation induced by the central bank stimulating (or contracting) output in a one time deviation from its optimal monetary pohcy. This equation nicely captures the type of short-run output-inflation 72 tradeoff often used to discuss the short run effect of monetary shocks.16 The distinction in this model between the statistical Phillips curve and the short run output-inflation tradeoff reflects the difference between the effect of a systematic pohcy rule and the effects of monetary shocks conditional on agents believing that the policy rule is being followed. In particular, the statistical PhiUips curve tends to become horizontal precisely when monetary authorities do not try to exploit the short-run tradeoff and instead try to correlate output with the real shocks. This result is reminiscent of that derived in Lucas (1972,1973), but there is an important difference. In the Lucas model, when the statistical PhiUips curve is horizontal, the output-inflation tradeoff is zero. Here, this does not arise since private agents are not confused between real and monetary shocks. If the central bank decides to arbitrarily stimulate (or contract) the economy, the agents recognize this and respond by adjusting prices. This property of the model is, we beheve, quite interesting since it can potentially explain why strong monetary contractions are often associated with faster declines in prices than would be predicted by the statistical PhiUips curve. Now that we have described the functioning of the model, let us return to our question of interest: what insight does this model provide towards explaining the flattening of the statistical Philhps curve in Canada and the U.S over the last 15 years? The answer suggested by the model is that the decline in slope may have arisen because monetary authorities have learned to better identify and properly respond to real developments in the economy thereby allowing such real developments to take place without large price effects. In other words, the flattening of the PhiUips curve may be a reflection of improvements in the manner in which monetary pohcy is executed. In the rest of the paper we will present both empirical and anecdotal evidence in support of this view and compare its merits to alternative explanations. Our first argument in favour of this view is entirely anecdotal as it reflects the change in macroeconomic thinking throughout the last 25 years and in relation to the conduct of monetary policy. Prior to the seventies, the importance of real shocks on the macroe-conomy was perceived to be rather minimal. The substantial fluctuations in oil prices changed this view and lead central banks to rethink the way they conducted monetary pohcy. The focus of macroeconomic research also changed over this period. In particular, the arrival of real business cycle theory showed that a weU functioning economy might 1 6 T h e only major difference between E q u a t i o n (3.17) and the more s tandard s t ruc tura l Ph i lUps curve is that the relevant t e rm for expected inflat ion is the agents' expectat ion of the central bank 's inf lat ion target as opposed to agents' expectat ion of ac tua l inf la t ion. 73 optimally fluctuate around its steady state growth path, and the rational expectations literature questioned the potential for monetary policy to have systematically large effects on the real economy. Correspondingly, it appears reasonable to think that central banks (at least in Canada and the U.S) responded to these changes by focusing more effort on identifying the underlying real forces in the economy and learning how to respond to them. In the context of the model, we believe that such a process would correspond to a reduction in r 2, since ^ captures the degree to which central banks are informed about changes in the fundamentals of the economy. As central banks focused more attention on understanding economic fundamentals throughout the eighties and nineties, and came to believe that market forces were appropriate for the short run determination of economic activity, the quality of their economic indicators (captured by st) likely improved and the degree to which central banks acted upon these signals (captured by 0 in the money rule (3.15)) likely increased. These developments are exactly of the type that our model suggests would lead to a flattening of the Phillips curve.17 In order to examine the plausibility of the idea that improvements in the manner in which monetary policy is conducted could be the cause behind the observed flattening of the Phillips curve, it is useful to consider alternative explanations. One such potential explanation, often heard in the press, is that, over the eighties and nineties, monetary authorities began to disregard their role in controlling output fluctuations and conducted monetary policy with the sole aim of stabilizing prices. According to this view, greater price stability is achieved only at the cost of greater output instability. To help evaluate this view, Table 5 reports a series for the variance for the output gap and for the change in inflation for both the U.S and Canada since the early eighties (the period over which we observe the decline in the slope of the Phillips curve). The variance reported for each year is calculated using the observations on the previous 15 years. As can be seen from the Table, in both the U.S and Canada, the variance of inflation growth has decreased quite substantially over the last fifteen years. In contrast, the variance of output for Canada has remained about the same while that for the U.S appears to have declined. In particular, the variance of output in Canada was about 7.5 throughout much of the eighties and is approximately at the same level by the end of the nineties. The main inference we draw from Table 3.5 is that the variance of output does not appear to have increased during the period in which the slope of the Phillips curve flat-1 7In is interesting to note from Figures 3.3 and 3.4 that the period in which the statistical Phillips curve appears steepest is in the late seventies and early eighties, which is generally thought as being a period of high variation in real shocks and substantial confusion. 74 Table 3.5. Rolling Sample Variances for Canada and the U.S: 1983-1999 Canada U.S Year Var(A7rt) Var(Gapt-i) Var(A7r() Var(Gapt_i) 1983 3.61 5.15 2.82 7.34 1984 4.04 7.40 2.79 8.07 1985 4.04 8.01 2.79 7.02 1986 3.65 7.62 2.82 6.71 1987 3.65 7.56 2.86 7.08 1988 3.24 7.56 2.72 6.86 1989 2.04 7.62 2.04 5.90 1990 2.04 7.62 1.99 6.45 1991 1.59 7.62 1.35 6.00 1992 2.19 8.41 1.37 6.15 1993 2.28 9.92 1.37 6.05 1994 2.28 10.76 1.37 6.35 1995 2.07 9.55 1.23 6.05 1996 1.88 9.06 1.12 5.20 1997 1.88 8.64 0.52 5.24 1998 1.34 8.06 0.31 3.57 1999 1.04 7.24 0.32 2.46 75 tened. While such an observation is not inconsistent with our proposed explanation, it is somewhat at odds with the view that greater price stability was achieved at the cost of greater output variability. 3.3.1 The Flattening Phillips Curve: Evidence of Optimal Policy or Downward Nominal Rigidities One possible explanation for the observed flattening of the Phillips curve, as suggested by Akerlof, Dickens & Perry (1996) and Fortin(1997) among others, is downward nominal wage rigidity. The reasoning is as follows: when inflation is very low, the unwillingness of workers to accept nominal wage reductions prevents real wages from adjusting in response to excess supply in the labour market. In this case, if prices are a fixed markup on wages, then prices will not fall in response to a negative output gap. This causes the Phillips curve to be flatter in periods of lower inflation. The proponents of this explanation claim that this is the relevant difference between the experience of the nineties relative to the early eighties. In this section we attempt to differentiate between the downward nominal wage rigidity theory and our proposed explanation to the flattening of the Phillips curve which relies on improved monetary policy. One implication of the downward nominal rigidity explanation, not shared by our explanation, is the prediction that the flattening of the Phillips curve should be associated with an increase in its degree of non-linearity. In particular, the downward nominal rigidity hypothesis suggests that as inflation decreases, it is mainly the segment of the Phillips curve that relates to negative values of the output gap that should flatten (because downward nominal wage rigidities are not relevant when the labour market is tight). In order to explore this hypothesis empirically, we estimated several variants of the type of non-linear Phillips curve given by Equation (3.18) and examined how the coefficients changed over time.1 8 TTt+i = fa + Tr, + fa (GAPt) + 02PosGapt + et+1 (3.18) In Equation (3.18), the variable PosGapt-i takes the value of zero if the output gap is negative and is equal to the value of the output gap if the latter is positive.19 Figures 3.9 1 8 D u p a s q u i e r and Ricketts(1997) provide a good entrance po in t to the l i terature on es t imat ing non-linear P h i l l i p s curves for Canada . 1 9 O u r approach is to est imate a P h i l l i p s curve which has a k i n k at a zero output gap. W e 76 F i g u r e 3.9. O u t p u t G a p C o e f f i c i e n t , P o o l e d S a m p l e o ui CO a c O T3 C CO 9= CD O u a. ce O 3 Q . 3 o 0.84 0.72 0.60 - \ 0.48 H 0.36 0.24 H 0.12 - \ 0.00 -0.12 Change in Inflation on Lag of Gap and Lag of Positive Gap "i i i i i i i i i i i i i i i r 1983 1985 1987 1989 1991 1993 1995 1997 1999 and 3.10 report respectively values for /?i and 02 associated with successively estimating Equation (3.18) based on pooled U.S and Canadian data over periods of 15 years. We present the results for the pooled estimates since they are the most precise. However, it should be noted that we also estimated this equation for each country individually and for several different specifications, and obtained results similar to that represented in Figures 3.9 and 13.10. As can be seen in Figure 3.9, the value of 0\ decreased substantially over the late eighties and nineties. Since this coefficient represents the slope of the Phillips curve for negative values of the output gap, its decline is consistent with the hypothesis downward nominal rigidities may have caused the Phillips curve to flatten. However our estimates of 02, as shown in Figure 3.10, suggest that the degree of non-linearity of the Phillips curve has not increased over this period, an observation which is inconsistent with the nominal adopt this simple approach to evaluate the presence of a non-linearity even though the nominal wage rigidity hypothesis does not precisely predict a kink at a zero output gap. 77 wage rigidity hypothesis. In effect, our estimates of @2 suggest that the Phillips curve remained linear throughout the period, whereas an increase in would be expected if downward nominal wage rigidity was the cause of the flattening of the Phillips curve. Our evidence against the downward nominal rigidity hypothesis can be inferred visually from the simple scatter plot presented in Figure 3.8. Since the mid 1980's, the PhiUips curve has been very flat over the range of both positive and negative output gaps. In fact, the only evidence of non-linearity relates to the out-lying observation of Canada in 1992. However, for this observation the output gap was negative and large, and inflation fell substantially. Hence, we take this evidence as contradicting the downward nominal rigidity hypothesis as an explanation to the observed flattening of the PhiUips curve. 78 3.3.2 The Flattening of the Phillips Curve and the Ball, Mankiw &c Romer hypothesis A second potential explanation to the flattening of the Phillips curve is the one pro-posed by Ball, Mankiw and Romer (1988) based on menu costs. This theory suggests that in a period of low trend inflation, firms do not find themselves on the boundary of the set of acceptable prices (that is, the S,s boundary of acceptable prices defined by the size of the menu cost) very often. Therefore, firms do not change their individual prices as frequently when trend inflation is low as when it is high. This greater sluggishness in individual prices increases the degree of overall nominal rigidity in the economy and therefore leads to a flatter Phillips curve. Since the trend level of inflation has fallen over the past twenty years, the menu costs hypothesis predicts that the PhiUips curve should have become flatter over this period, which is exactly what we observe in the data. The menu costs explanation and our model, however, have important differences re-garding the effects of monetary surprises, which arise due to their respective implications for the short-run output-inflation tradeoff and the statistical PhiUips curve. In the menu cost explanation, when inflation is low the PhiUips curve is flat. Since there is no distinc-tion in this story between the statistical Phillips curve and the short-run output-inflation tradeoff, such a flattening imphes that the output-inflation tradeoff has increased. In con-trast, while our model predicts that the statistical PhiUips curve (whose slope is given by Equation (3.16)) becomes flatter when policy makers monitor the economy properly, this does not imply that the short-run output-inflation tradeoff changes. In effect, the relevant output-inflation tradeoff -that is, the tradeoff induced by a deviation from the perceived pohcy rule- is governed by the parameter ipi in Equation (3.17) which is independent of the trend level of inflation. In short, the difference between the two models is that the menu costs story imphes that the effect of a monetary shock varies inversely with the trend level of inflation, while our model predicts that the effect of a monetary shock on inflation is independent of the level of trend inflation. This difference indicates how the two models can be distinguished empiricaUy. In effect, one can differentiate the two models by examining whether the co-movement of inflation and the output gap foUowing a monetary shock differs in periods of high relative to low trend inflation.20 2 0 L i k e the menu cost theory, the hypothesis of downward n o m i n a l wage r ig id i ty does not i m p l y a d i s t inc t ion between the short run output- inf la t ion tradeoff and the s ta t i s t ica l P h i l l i p s curve, therefore, the evidence presented i n th is section also relates to that poten t ia l explanat ion . 79 The major limitation of this strategy involves data. In order to compare these two competing theories, we need to observe monetary shocks in periods of both high and low trend inflation. Since monetary shocks are infrequent, we find ourselves confronted with the problem of having few observations. Nevertheless, using the Bank of Canada's Annual Reports as our source, we can identify two important disinflationary shocks in Canada between 1980 and 1999: the first occurred in 1982-83 and the second in 1991-92. In particular, the 1980 and 1981 Annual Reports suggest that the Bank of Canada was troubled by the high inflation of the late 1970's, but unable to act because of the need to respond to changes in U.S interest rates and large capital flows out of the country. In 1982, then Governor Gerald Bouey wrote that "the Canadian economy has shown strong resistance to becoming less inflationary", and noted that "inflation must sooner or later be fought". A year later, he reflected on the "strong monetary medicine" that had been required to "beat the fever of inflation" that gripped the Canadian economy in the late 1970's. We count the experience of 1982-83 as a disinflationary shock, since the Bank of Canada appeared to be focused on reducing what it regarded as an unacceptably high rate of inflation rather than responding to real developments in the economy. The reports from 1984 through to 1990 portray a Bank of Canada on guard against a renewal of inflation but not actively seeking to reduce the trend rate. In 1991, in response to the increased inflation of the late 1980's, the Bank of Canada jointly with the Government of Canada announced a set of inflation targets that were to take effect starting in 1992. The targets essentially mandated a reduction in inflation, which was then around 5 percent, into a target band of 2 to 4 percent. Governor Crow, in the 1991 annual report, wrote that the purpose of the inflation targets was "to provide Canadians with a clear affirmation that price stability remains the goal of monetary policy". Two years later, Governor Thiessen reflected that "a key purpose in establishing (the inflation targets) was to indicate as clearly as possible not a path for sustaining inflation, but a path for reducing inflation". We regard the experience of 1991-92 as a second disinflationary shock. Subsequent to 1992, the Bank of Canada reduced the target band in 1994 and 1995, but since this had been announced in 1991 it is not clear that we would wish to count it as a monetary shock. After 1995, the Bank chose to maintain the target band at its 1995 level through 1998 and later through 2001. Therefore, we conclude that there have been two disinflationary shocks in Canada since 1980. The important difference between the two being that the 1982-83 shock occurred 80 Figure 3.11. Phillips Curve, Canada, 1980-1999 Regression includes dummy variables for 1983 and 1992 -2 0 2 Output Gap (HP-Filtered) during a period of relatively high inflation, while the 1991-92 shock occurred while the trend rate of inflation was much lower. In principle, these two episodes provide an excellent opportunity to test the different theories. In order to make this comparison, Figure 3.11 plots the change in inflation against the output gap for Canada for the sample 1980 to 1999. The surprising and noticeable aspect is that the observations for 1983 and 1992 lie almost exactly on top of one another.21 We see this as providing some, albeit hmited, support for the view that the short-run output-inflation tradeoff did not change as inflation decreased. We also find it informative to contrast the inferred size of the output-inflation tradeoff 2 1While we do not include 1994 as a shock, it is interesting to note that it does lie along the same line as the two stronger shocks. 81 under the two views. Under the assumptions that our model is correct and that the 1983 and 1992 points are representative of the short-run output-inflation tradeoff, Figure 3.11 imphes that the cost of reducing inflation by one percent is a negative output gap of approximately 1.3 percent (the slope imphed by the 1983 and 1992 observations). If, on the other hand, the menu costs theory is correct then the slope of the Phillips curve is the proper estimate of the output-inflation tradeoff. In this case, using the final estimate of the slope of the PhiUips curve from the rolling regressions for Canada (which is around 0.1) as the measure of the tradeoff, the negative output gap induced by a one percent reduction in inflation would be on the order of ten percent. Clearly, the two interpretations differ by orders of magnitude and hence suggest the need to provide further evidence to differentiate the two views more convincingly. 3.4 Conclusion Our answer to the title of the paper "What has happened to the PhiUips curve over the 1990 in Canada?" is both empirical and theoretical. From a statistical point of view, we have shown that the slope of the Phillips curve in Canada has decreased substantially over the period. We also document that the same phenomenon is observed in the U.S. Since we are interested in interpreting these observations for policy discussion, we have used a prototypical macro model to attempt to understand why the slope of the Phillips curve may have changed over time and what implication this may have for the output-inflation tradeoff faced by the central bank. In particular, we have shown why a change in monetary policy that incorporates a better understanding of the real side of the economy wiU lead to a flatter PhiUips curve. The reason we believe that the conduct of monetary pohcy may have changed in this direction is that, after the oil shocks of the seventies, central banks appear to have devoted more effort towards tracking the real forces affecting aggregate output and have likely incorporated the improved knowledge into their behaviour.22 The second insight drawn from the model is that a flatter PhiUips curve does not necessarily 2 2 One possibility we have not addressed in the paper is that monetary authorities may have learned how to properly unlink real and nominal developments in the economy over the last twenty years, which would explain the flattening of the Phillips curve, but that this simple unlinking is not necessarily the best pohcy to follow. In effect, if there are other imperfections in the market besides those associated with preset prices, it may be socially optimal for monetary authorities to favor lower output variability than that which would arise by letting market forces work freely. We believe that this possibility is very relevant and should be examined in future research. 82 imply a change in the output-inflation tradeoff faced by the central bank. In effect, we showed why the PhiUips curve can become flatter while the relevant output-inflation tradeoff remains constant. Based on several pieces of evidence, we have argued that our model provides a reason-able framework for interpreting recent observations on the PhiUips curve. As we explained in the paper, the main imphcation of this view for pohcy is that the best guess of the po-tential costs associated with a disinflation undertaken today is that inferred from the disinflationary episodes of the early eighties and nineties. In other words, we believe that the evidence on inflation and output over the last twenty years supports the view that the costs associated with reducing inflation have hkely neither increased nor decreased over the last twenty years even if the statistical PhiUips curve appears to have flattened. 83 Chapter 4 The Changing Nature of the Phillips Curve in OECD Economies 4.1 Introduction This paper presents a model in which a Phillips curve relationship results from bilateral information flows between private agents and monetary authorities and examines the empirical support for the model in a data set of OECD countries for the period 1960-1997. The model is an extension of the simple pre-set prices macro model of Chapter 3, in which agents and the monetary authority are differently informed about real developments. In equilibrium the policy maker and private agents each learn from the other, and this bilateral learning gives rise to a Phillips curve relationship between inflation and real activity. The model predicts that: i) any increase in the policy maker's ability to identify and react to real shocks will be associated with a corresponding decline in the slope of the Phillips curve, ii) such a change will be associated with a reduced variance of inflation but the variance of output will remain unchanged, iii) the co-movement of inflation and output will differ in response to a monetar^jijmpvation depending on whether such an innovation is the policy maker's attempt to respond to real developments or an attempt by the policy maker to lower the trend inflation rate, and iv) deviations of inflation from target should not be serially correlated. The paper examines the empirical plausibility of the model by estimating these aspects of the Phillips curve relationship about which the model makes predictions. 84 It is argued that there are a set of phenomena relating to the Phillips curve that are broadly applicable across the OECD and that the model presented in the paper is consistent with these observations. The paper begins by documenting the existence of a robust Phillips curve relationship across OECD countries over the period 1960-1997. The paper then proceeds to give evidence supporting the claim that the slope of the Phillips curve has declined in many OECD countries over the past fifteen or twenty years. The paper compares estimates of the co-movement of output and inflation during episodes of monetary contraction with the Phillips curve relationship prevailing in more normal periods. The results suggest that the output-inflation tradeoff during these episodes differs from the co-movements observed during more normal periods. Finally, the paper examines various correlations predicted by the model which appear to be counterfactual under explicit inflation targeting regimes, in order to examine the behaviour of deviations of inflation from target rather than the raw rate of inflation, as suggested by the theory. The evidence is that analyzing the more appropriate empirical counterparts to the theoretical variables helps to reconcile some but not all of the seemingly counterfactual predictions of the model with the empirical record. A large number of papers examine the Phillips curve from an empirical perspective. Ex-amples of recent work include Roberts (1995), Fuhrer (1997) and Gordon (1997), amongst others, argue that there is clear evidence of a Phillips curve relationship in U.S data. There is also a fair amount of evidence on the existence of a Canadian Phillips curve (see Fillion and Leonard (1997) and Dupasquier and Ricketts (1997) for examples). Other papers find evidence of a Phillips curve relationship in other countries. Gruen, Pagan and Thompson (1999) is one example of the literature which finds empirical support for the Phillips curve in Australian data. Haldane and Quah (1999) argue that observations of U.K wage and unemployment data post 1980 exhibit the negative correlation of Phillips' original sample. Cross country studies include Turner and Seghezza (1999), who find evidence of a Phillips curve in a number of OECD countries. Gali, Gertler and Lopez-Salido (2001) estimate a New Keynesian Phillips curve (using MC as excess demand measure) for the EU, and find some evidence of a relationship between real activity and inflation. The existing literature contains several models which give rise to a Phillips curve corre-lation between inflation and real activity through a variety of different mechanisms. Lucas (1973) derives a Phillips curve from a model in which agents are confused about whether price changes that they observe are the;result of real shocks (changes in relative demand) or nominal shocks (changes to the money supply) and therefore change their production 85 decisions in response to shocks which may be partly nominal. New Keynesian models, starting with the staggered wage contracts model of Taylor (1980), present nominal rigidi-ties as a potential cause of linkages between real and nominal variables, though Taylor's model highlights the tradeoff between the variance of output and the variance of inflation. Subsequent New Keynesian sticky price models, in the spirit of Blanchard and Kiyotaki (1988) and Rotemberg (1987), imply a Phillips curve style relationship between inflation and an appropriate measure of excess demand.1 Mankiw and Reis (2001) attempt to ad-dress some of the shortcomings of the New Keynesian Phillips, and derive a Phillips curve in a model in which the stickiness is moved from prices to information, so that any agent updates the information on which he or she bases pricing decisions in any given period with probability less than one. Taking a different approach, Cooley and Quadrini (1999) examine the PhiUips curve in a limited participation model of money in which the labour market exhibits elements of search and matching behaviour. The model in this paper differs from those discussed above in that the PhiUips curve in those papers is essentiaUy a structural relationship embedded in the preferences, tech-nology and information of the private economy. The PhiUips curve is thus a relationship which policy maker's take as given, though, as in Lucas (1973), this is not necessarily a relationship which pohcy makers can exploit advantageously. In this paper, the Phillips curve is a reduced form relationship which arises as a result of the interaction between the monetary authority and private agents in the economy. Thus the Phillips curve does not represent a structural component of the economy, which may or may not be amenable to exploitation on the part of policy makers. Rather, it reflects interactions in the economy which include the actions of policy makers. One interesting feature of a model in which the PhiUips curve results from the inter-action between policy makers and private agents is that aspects of the PhiUips curve, the slope for example, wUl depend on the actions, information and attitudes of the monetary authority. This is of interest as many authors (including, for example Sargent (1999) and Haldane and Quah (1999)) argue that the past two decades have seen a shift in pohcy makers' attitudes towards and information about aspects of the PhiUips curve relation-ship. This paper uses the idea that pohcy makers have become more informed about real developments in the economy and how they might appropriately use monetary pohcy to offset or accommodate these disturbance, and shows that this can cause a decrease in the 1See Clarida, Gali and Gertler (2000) and Goodfriend and King (2000) for useful discussion of these models 86 slope of the observed Phillips curve. Existing work which finds evidence of such a change for the U.S includes Lown and Rich (1997), Hogan (1998), Brayton, Roberts and Williams (1999)). Similarly, there is some evidence that in recent years, particularly the 1990s, the correlation between output and inflation has declined significantly in Canada (see Chapter 3 for some discussion of this point). There is also scattered evidence to suggest that such a change has occurred elsewhere. Haldane and Quah (1999) argue that while the period from 1958-1979 saw a very steep Phillips curve, the years since have witnessed substantial a flattening in slope, while Nishizaki and Watanabe (2000) argue that such a flattening has also occurred in Japan. In this paper I present evidence that the flattening of the Phillips curve in recent years is a fairly widespread phenomenon which has occurred (or is occurring) in a number of OECD countries. The fact that this decline has occurred across a variety of countries with a variety of different institutions suggests that there may be a common cause. This is at least consistent with the idea that central bankers in the industrial world have become more adept at performing their functions over the past twenty years may be the driving force behind the changing nature of the Phillips curve.2 A second interesting feature of the model is that the Phillips curve which arises, being a reduced form object reflecting in part the conduct of monetary policy, does not reflect the short run tradeoff between output and inflation that would be associated with an exogenous monetary contraction. Thus, a distinguishing feature of the model is that it predicts that the economy will react differently to a monetary contraction designed to lower the target rate of inflation without regard to real conditions than one which attempts to stabilize the inflation rate around the monetary authority's target in a situation where the economy is thought to be overheating. In other words, a flattening of the observed statistical Phillips curve, perhaps due to a more informed policy maker, does not imply a flattening of the output-inflation tradeoff associated with a disinflationary monetary contraction. In this paper I attempt to test the notion that the relationship between output and inflation differs in situations in which monetary conditions are tightened for the purpose of lowering the trend inflation rate. The appropriate contractions are identified using both 2Given the decline in the average-rate of inflation observed in most OECD countries over the past twenty years, this decline could also be consistent with both theories of downward nominal wage rigidities and theories involving menu costs. See section 4.4 for attempts to distinguish between these competing hypotheses. 87 mechanical means and by examining the historical record in selected countries. The results suggest that there is be a difference between the co-movements of output and inflation during episodes of monetary contraction as opposed to normal times. There is also some evidence that the slope of the short-run output-inflation tradeoff that obtains in the face of monetary contraction has remained stable over the same period during which the slope of the statistical Phillips curve has been observed to decline. The mechanism by which the model gives rise to a Phillips curve is the learning of agents and policy makers each from the other. In particular, it turns out that inflation responds to those errors made by pohcy makers which become incorporated in to output. This, at least in the simplest version of the model, imphes that deviations of inflation from trend ought not to be correlated with lagged deviations from target. This is because the policy maker's error, which affects current inflation, is orthogonal to the state of the economy in the previous period. Therefore, the model imphes that serial correlation in inflation is driven by serial correlation in the pohcy maker's target rate of inflation. To test these propositions, the paper examines the experience of three early inflation targeting countries, Canada, New Zealand and the United Kingdom, each of which moved to an exphcit inflation targeting regime in the early nineties. Using the announced targets as measures of the central bank's actual target rate of inflation it is possible to recover data on the deviations of inflation target from target. Using this data I test predictions of the model, such as the lack of serial correlation in inflation deviations from target, on inflation targeting countries. The results support some, but not all of the model's predictions. The paper proceeds as follows: Section 4.2 presents the model and discusses the various predictions of the model that will be examined empirically in subsequent sections. Section 4.3 presents evidence on the existence of a Phillips curve across the countries of the OECD and documents how this relationship appears to have change in many countries in recent years. Section 4.4 apphes the predictions of the model regarding monetary contractions and examines the evidence that can be brought to bear on the other predictions of the model. Section 4.5 offers concluding comments. 4.2 A Model of the Phillips Curve This section explores the theoretical nature of the output-inflation relationship using the model of Chapter 3. Essentially the model is a standard sticky prices macro model which shows how the interaction between private agents and policy makers, each with 88 imperfect information, might give rise to a Phillips curve relationship between output and inflation. This section analyzes the robustness of various features of the model The environment is one in which one final good, Yj, is produced using a set of N intermediate goods, Xu, where i = 1,... ,7V. The intermediate goods are produced by monopolistically competitive firms, which must set prices at the beginning of each period, before the demand for intermediate goods is determined. The final good is produced by competitive firms according to the CRS production function given by equation 4.1. yt = (Exs)<±> t 4 - 1 ) i = l The representative household in this economy has preferences denned over consump-tion, labour supply and real balances, as given by (4.2). We assume that the household's utility is linear in labour so as to generate a constant real wage. Hence, the model can alternatively be interpreted as a model with an exogenously fixed real wages. U(Cu^,Lt)=C°^~e -<pLt (4.2) The household's budget constraint is given by (4.3), where Pt is the price of the final good, Wt is the nominal wage rate, Mt is money demanded and Mt is the money balances distributed by the central bank at the beginning of each period. PtCt + Mt = WtLt + Mt (4.3) The household's problem and the final goods firm's problem are the identical to the problems analyzed in Chapter 3. As there, the household's problem results in the following first order conditions: Mt = ^ - P t • Ct, (4.4) ^ = (1 - ef-OQU = 9. (4.5) Pt The final goods producer's problem yields the demand for intermediate goods: Xit = (^)^Yt, (4.6) where Pu is the price of the ith intermediate good. Each intermediate goods firm has access to a production technology given by (4.7). XU = 41_7) • LI • YT-i (4-7) 89 where, Lu is the quantity of labour employed in firm i and At is the productivity index. This differs from the corresponding production function in in the previous paper in that it allows the firm's production to depend on the aggregate level of activity in the economy. This allows for the internal dynamics of the model to display propagation independently of the serial correlation of the technology shocks, albeit in a limited and arbitrary fashion. One could think of a positive value of cr as representing something like learning by doing, so that if output was high in a previous period, firms learned a lot and are more productive now. We assume that the productivity index, At, is common to all intermediate goods, and that the log of At follows the stationary stochastic process given by (4.8).3 oo oo at = Yl^jCt-j, Vt> = l , £ V » i < o o (4-8) where, et is assumed to be a normally distributed mean zero random variable with variance ae, and the ipiS are assumed to be positive. This last restriction is meant to capture the notion that deviations of technology from trend are positively autocorrelated. The problem facing an intermediate good firm is more complicated given that the prices of intermediate goods must be set before the realizations of either At or Mt. The firm's objective is therefore to set Pu to maximize expected profits conditional on the information set Qt-i, which contains all information dated t — 1 or earlier, including reahzations of past values of e. An intermediate good producer's problem can therefore be expressed as follows: max E[PitXit - WtLu/Sk-i] Pit s.t. (4.7), (4.8), (4.5), (4.6) Using the market clearing conditions for both the goods market and the money market, and imposing symmetry on the behavior of intermediate goods producers, one can easily derive Equations (4.9) and (4.10) which describe the behavior of the aggregate price level and aggregate output. In these two equations, constant terms have been dropped. oo Pt = E[mt/nt-i] - £ ipiet-i - cyt-i (4.9) i=l 3In all that follows, lower case letters are used to denote the logarithm of a variable. 90 oo yt=mt-pt = (mt - E[mt/Clt-\\) + £ ^ t - i + ayt-u (4-10) i = l where a — Equations (4.9) and (4.10) represent the equilibrium behavior of private agents, for arbitrary processes of money supplied. Note that both prices and output depend on real and monetary forces. In particular, the aggregate price level depends on real shocks and expected money, while aggregate output depends on real shocks and unexpected money. It is important to note that the et's in Equations (4.9) and (4.10) can be interpreted very broadly as reflecting any real shocks that affect the potential gains from trade, as opposed to the narrow technology shock representation. The novel aspect of the analysis concerns the nature of the interaction between the private sector and the central bank. We assume that the objective of the central bank is to minimize deviations of output and prices from target levels y* and p*, as given by (4.11). oo J2 FE[(vt - y*tf + * • (pt - p*t)2\nt} (4.11) i=0 In (4.11), $ is the weight the central banker places on deviations of inflation from its target relative to output deviations. With respect to the target for output, we assume that it is the level of output that would arise in the competitive equilibrium in the absence of any price rigidities or informational imperfection, that is, y\ = YliZo ipiet-i + cn/t_i (note that we have again dropped the constant term). With respect to the target for the price level, we assume that it is driven by an exogenously given inflation target 7ft, such thatp* = pt-i-|-7ft. For our purposes, the process for the inflation target can be thought as being either stochastic or deterministic; the key simplifying assumption being that it is exogenous. In order to allow for the possibility that the inflation target be stochastic, we denote the agents expectation of target inflation as of time t — 1 by t_i7ft. The key assumptions of the model relate to the timing of moves and the information available to the central bank and private agents when making decisions. The assumptions are chosen to capture the notion that, in the short run because of sticky prices, the central bank has the important but difficult task of helping private agents achieve gains from trade by providing the right amount of liquidity to the system. In effect, the central bank has both an informational disadvantage and a timing advantage relative to the private sector. The central bank's disadvantage is that it does not directly observe the et's, and 91 therefore must infer their values from past developments in the economy. Its advantage is that it has some information on the current state of the economy, which it can use during the period over which prices are preset. In effect, we assume that the central bank receives a signal, st, from its research department each period. This signal is an unbiased indicator of real developments in the economy as captured by equation (4.12), where \xt is a normally distributed mean zero 2 random variable with variance cr2. We denote by r 2 the noise to signal ratio The timing of moves is as follows. At the beginning of a period, intermediate good firms set prices and the central bank simultaneously decides on the money supply. However, since private agents and the bank are differentially informed, the information used to determine these elements different. Private agents know all past developments in the economy but do not know the realization of et that is to arise during the period. In contrast, the central bank has past information only on output and prices (not the e's), but has the advantage of observing St- We will denote the information set of the central bank at the beginning of time t by £lt = {st, s t - \ , . . . ,Pt-i, • • •, yt-i • • •}, and the information set of the private agents as Ut-i = {Q-i , • • •, s t - i , • • • ,Pt-i • • •, Vt-i • • •}• Our justification for giving the central bank an informational advantage through s t captures the notion that the central bank has a timing advantage over the private sector. Since the private sector has pre-set prices, the central bank has more flexibility within a period to react to current shocks but is, nevertheless, imperfectly informed regarding the right way to react. The problem facing the central bank is to choose a monetary pohcy rule so as to minimize (4.11) subject to its informational restrictions and the optimizing behavior of the private economy, given by equations (4.9) and (4.10). The pohcy rule which solves the central banks problem is given by equation (4.15), with the implied equihbrium solution for inflation (7rt) and output given by equations (4.13) and (4.14) respectively. St = et + fM (4.12) 7Tt = t _ i irt + ipi(Qst-i - et-i), 0 = (4.13) oo yt = 6 • st + £ ipiet-i + ayt-i (4.14) 92 CO mt=P*t +vyt-i + <3>st + ipi&st-i + £ ipi€t-i (4.15) i=2 In order to gain intuition about equations (4.13)-(4.15), it is helpful to first recognize that the term 0 • St is the central bank's best estimate of the current supply shock et. Since the central bank's objective is to accommodate real shocks while maintaining price stability (around target), it adjusts the money supply so as to reflect its best guess of the current supply shock. Given that prices are fixed, an expansion of the money supply is first reflected in output, as desired, and not in prices. That is, the central bank uses the money supply to allow the real economy to react to its signal on the current supply shock, thus partially overcoming the nominal rigidities inherent in the economy. In the following period, the private sector becomes informed about the realization of last period's supply shock and adjusts prices accordingly. Note that inflation only deviates from the target level of inflation to the extent that the central bank's estimate of the real shock in the previous period was mistaken. In effect, by adjusting prices in response to the central bank's error, the private sector actually reveals to the central bank the extent of its past error. The reason that private agents react to past mistakes is that they foresee that the central bank will continue to accommodate the effects of a perceived shock until it becomes aware that it has made an error. Hence, the profit maximizing price setting rule is to increase prices in response to past excessive expansion on the part of the central bank. Correspondingly, once the central bank recognizes that it has made an error, by ob-serving a deviation of inflation from its target, it readjusts the money supply. Although monetary authorities never directly observe the e's, within two periods they are able to perfectly infer their values from observing developments in the economy. This explains why the money supply rule can be written as a function of lagged values of the e's. We now turn our attention to the implications of the above model for the nature of the PhiUips curve. For now let us define the Philhps curve as a purely statistical object. In particular, let the slope of the PhiUips curve be the slope of the relationship between the change in inflation and the deviation of output from trend. Since our model is in terms of deviations from trend, the theoretical analogue to this slope is the covariance between the change in inflation and output, divided by the variance of output. The analytical expression for this slope is reported in equation (4.16) and is denoted by /3: 93 var(yt) (1 + 2 E t=i Wi) + (E£ i W+ + " ^ " (4.16) where, * = 2 E ~ i E ^ + i V ' i - iV^ . 4 The first thing to note from equation (4.16) is that the model generates a statistical Phillips curve; that is, even though monetary policy is set optimally, the economy never-theless exhibit a systematic positive co-movement between inflation growth and output. Moreover, this co-movement actually represents causality running from money to output and then to inflation, as is usually thought to be the case in discussions of the Phillips curve. The second aspect to note is that, under reasonable conditions, the slope of the Phillips curve is strictly increasing in r 2 ( the noise to signal ratio for s t). In other words, equa-tion (4.16) implies that when the central bank becomes more aware of real developments in the economy (perhaps by expending greater effort to gather information about these developments and thereby reducing r 2 ) , it will make fewer errors conducting monetary policy and this will lead to a natter Phillips curve. This is the first result we want to highlight from this model: a flat Phillips curve may be a reflection of a well run monetary policy. In particular, if cr2 were to go to zero, monetary authorities would make no errors and the statistical Phillips curve would become perfectly horizontal. The reason is that, in such a case, monetary authorities would be able to stabilize prices while allowing the economy to respond efficiently to real forces. In contrast, the Philhps curve would tend to be more steeply sloped in an environment with substantial variations in real shocks or a poorly informed central bank. Before discussing the potential relevance of equation (4.16) for explaining the changing nature of the Phillips curve, it is interesting to note the difference between the statistical Phillips curve implied by this model and the short-run output-inflation tradeoff faced by the central bank. In particular, even in a situation where the slope of the statistical Phillips curve is almost zero, this model does not imply that the central bank should perceive the short run trade-off between inflation and output to be close to zero. In effect, such a tradeoff could still be quite large. To see this, we can use Equations (4.13) and (4.14) to derive the short-run relationship between inflation, target inflation, output and 4 The slope of the Philhps curves is always positive. A sufficient condition for it to decline as the bank becomes more informed is a > 0. A necessary and sufficient condition is 1+2 YliLi 0* + £ . ~ i # + *>o . 94 supply shocks. This relationship is given by Equation (4.17). oo n =t-i *t + ipiyt-i - Y2 i>iCt-i, (4.17) i = l The term ip\yt-\ in Equation (4.17) represents the effect on inflation induced by the central bank stimulating (or contracting) output in a one time deviation from its optimal monetary policy. This equation nicely captures the type of short-run output-inflation tradeoff often used to discuss the short run effect of monetary shocks.5 The distinction in this model between the statistical Phillips curve and the short run output-inflation tradeoff reflects the difference between the effect of a systematic policy rule and the effects of monetary shocks conditional on agents believing that the policy rule is being followed. In particular, the statistical Phillips curve tends to become horizontal precisely when monetary authorities do not try to exploit the short-run tradeoff and instead try to correlate output with the real shocks. This result is reminiscent of that derived in Lucas (1972, 1973), but there is an important difference. In the Lucas model, when the statistical Phillips curve is horizontal, the output-inflation tradeoff is zero. Here, this does not arise since private agents are not confused between real and monetary shocks. If the central bank decides to arbitrarily stimulate (or contract) the economy, the agent recognize this and respond by adjusting prices. This property of the model is, we believe, quite interesting since it can potentially explain why strong monetary contractions are often associated with faster declines in prices than would be predicted by the statistical Phillips curve. 4.2.1 Other Properties of the Model While the previous section outlined the main properties of the model, it is worth while discussing some of the other aspects of the model and their robustness. The discussion above focused entirely on the simple correlation between the change in inflation and the lag of the output gap. In the simplest version of the model (where (7 = 0), this is the only moment for which a positively sloped Phillips curve relationship exists. If, however, the model is extended to allow for more internal dynamics (as is the case when a > 0)6 then the Phillips curve relationship is robust to various lags of the 6The only major difference between Equation (4.17) and the more standard structural Phillips curve is that the relevant term for expected inflation is the agents' expectation of the central bank's inflation target as opposed to agents' expectation of actual inflation. 6This is clearly an unappealing way to model internal propagation in the model, but it is 95 output gap. Equation (4.18) presents the correlation analogous to equation (4.16) using the contemporaneous output gap rather than the lag. = = cov(Ant,yt) = frMVs + tyi - V>I)T2 . . var(yt) (1 + 2 E £ i ^ i ) + ( E £ i $ + + r2) 1 " ' It is clear that, subject to the usual conditions relating to the denominator, the J3 is positive if a is sufficiently large that (V^  + o^i — tpi) is greater than zero. If this is the case, then the J3 is also strictly decreasing in r 2. Thus result that the model gives a positively sloped Phillips curve, where the slope is declining in the policy maker's signal to noise ratio, applies whether one uses the contemporaneous or lagged output gap. The inclusion of simple propagation, however, does not alter some of the other cor-relations which arise in the model. In particular, it is the case that cov(itt+\, yt) = 0, cov(itt,yt) < 0, and cov(-Kt+i,irt) = 0, where ir here is properly interpreted as a deviation from target. The intuition behind the first and third of these covariances is simply that future deviations of inflation from target reflect the central bank's errors, which must be orthogonal to any variables observable at time t (or else the bank would use the infor-mation contained in these variables to update its forecasts of et). The second covariance comes from the fact that current inflation adjusts to the error the bank made in the pre-vious period and if the bank's error was to under predict output last period then output is, on average, high today. These correlations, despite the fact that they are in terms of deviations from target, are somewhat disturbing given that Phillips curve type relationships can be found in inflation-output gap space, rather than just change in inflation-output gap space and that inflation is known to be persistent.7 The first and third of these correlations, however, are artifacts of the simplified structure of the economy and not robust, to extending the degree of uncertainty facing the central banker for example. To see this, suppose that the central banker does not perfectly observe last period's price level (a reasonable assumption given the lags involved in data gathering). In this case, the bank cannot perfectly infer the value of et_i, but can only use information in pt-i to infer the value of et-2- In such a case the equilibrium outcomes of the economy become:8 simple and suffices for the purpose of illustrating the robustness of the Phillips curve result to various lags of the output gap. 7See Fuhrer and Moore (1995) for evidence relating to the US experience. 8The policy rule for money is particularly messy and thus unreported. 96 *t =t-i K + 4>i(0st-i - €t-l) + (1P2 + aipi)(0sT-2 - e t _ 2 ) (4.19) yt = 0st + J2 ^ 6 * - * + &yt-i (4-2°) This yields the usual Phillips curve in change in inflation-output gap space that has the usual positive slope which declines as the bank becomes more informed. However, in this case there is also a positive correlation between inflation and its lags, as given by equation (4.21).9 c o r r ( , u , t - 1 ) ^ ^ ± ^ ± l ^ > 0 , if fc + o ^ X ) . (4.21) This version of the model also allows for the possibility that corr(irt+i,yt) > 0, but since corr(TTt,yt) < 0, this only occurs if V2 + vipi < 0. Thus the conditions for corr(irt+i,yt) > 0 are inconsistent with the requirement that inflation be positively seri-ally correlated. As mentioned above, the negative correlation between 717 and yt is robust to this extension of the model. In a later section I will consider whether the fact that the model predicts this negative correlation in deviations of inflation from target rather than raw rates of inflation can reconcile these predictions with the data. 4.3 Some Stylized Facts In this section I present evidence of a direct relationship between inflation and real ac-tivity in a number of OECD countries. In particular, I show that for a number of countries, the PhiUips curve exists and is robust to various possible specifications. Furthermore, I show that the evidence supports the hypothesis that the slope of the PhiUips curve has declined in the last two decades in a number of these countries. This decline in slope is important in magnitude and widespread across OECD economies. 4.3.1 The Existence of Phillips Curve Relationships in the OECD The Phillips curve is an observed relationship between inflation, its lags and some measure of real activity. In order to estimate a PhiUips curve it is necessary to decide on some measure of the price level and a measure of real activity. In the absence of compelling 9 I t might be wor th not ic ing that this correlat ion is invariant to the pohcy maker 's s ignal to noise ra t io . 97 theoretical guidance, it is not immediately apparent which of the many possible measures ought to be employed. The usual recourse in the empirical PhiUips curve literature is to run the estimation with a variety of different measures and check for robustness of result. That is the approach employed here. As a baseline, I employ the percentage change in the all items Consumer Price index as a measure of inflation. To provide a check for robustness I also use the percentage change in the implicit GDP deflator as an alternate series. The issue of how to measure real activity is more problematic. In general the PhiUips curve literature takes the approach that the relationship to be estimated is between in-flation and some measure of excess demand rather than, say, total output. EssentiaUy there are two standard approaches to estimating a Phillips curve. The first is to estimate a relationship in inflation-unemployment space, where some measure of the cyclical com-ponent of unemployment is taken to be related to excess demand. This is problematic as it requires an estimate of structural unemployment (see Gordon (1997), Blanchard & Katz (1997)). An alternate is to attempt to detrend real output and estimate a PhiUips curve in inflation-output gap space, where the deviation of output from trend is taken to measure excess demand. This approach encounters problems of its own as it is not im-mediately apparent how one ought to go about separating output into trend and cyclical components. The literature employs a variety of techniques (HP filters, structural VARs, structural macroeconomic models, simple time trends) to arrive at trend series for output. In this paper I use detrended output as the measure of excess demand, and arrive at the trend by using an HP filter. I replicate the results using an output gap arrived at by detrending output using a cubic time trend as a check of robustness, but do not consider other possible measures of excess demand. The estimation covers 21 OECD countries: Austraha, Austria, Belgium, Canada, Denmark, Finland, France, Germany10, Greece, Ireland, Italy, Japan, Luxembourg, the Netherlands, New Zealand, Norway, Spain, Sweden, Switzerland, the United Kingdom and the United States.11 The data are annual and cover the period from 1960-1997, and are taken from the OECD's national accounts (real GDP, implicit GDP deflator) and the IFS database (CPI). This section presents the results of both country by country regressions as well as the results of regressions using Seemingly Unrelated Regressions to estimate the OECD 1 0 G e r m a n da t a covers the ter r i tory of the former West Germany. " E x c l u d e d are M e x i c o , Iceland and Turkey because of hyper-inflat ions. A l s o excluded are the formerly communis t economies of Eas te rn Europe , because the da ta does not go back very far. 98 countries as a system, so as to control for the possibility of correlated contemporaneous residuals. In all cases, the Phillips curves estimated are some variant of one of the following equations: fc / irt = a + (3 • Gap + £ & • itt-i + £ 7i • %t-j + et, (4.22) i=0 i=o (4.23) i = 0 j = 0 where nt represents the rate of inflation at time t, A714 represents the change in inflation at time t, Gap represents either the contemporaneous or lagged value of the output gap, and xt represents the value of some other control variables, such as energy price shocks for example, at date t. Table 4.1 presents the results of estimating a simple Phillips curve equation of the form: for the 21 OECD countries using the method of Seemingly Unrelated Regressions. The results show a statistically significant Phillips curve for 19 of the 21 countries (the excep-tions are France and Luxembourg), with two thirds of the point estimates of the slope between 0.25 to 0.45. A test of the hypothesis that the vector of slope coefficients equals zero rejects the null hypothesis at any conventional level of significance. The results presented in table 4.1 are robust to a number of possible variations, in-cluding using the GDP deflator as an alternate measure of prices, using a cubic time trend rather than an HP-filter to derive the output gap, using the contemporaneous rather than the lagged output gap, allowing for further lags of inflation to enter as right hand side variables, and estimating country by country Phillips curves rather than estimating as a system. Table 4.2 presents the estimates of beta, the coefficient on the output gap, for a number of these specifications. Column one of table 4.2 presents the estimated beta from estimating Equation 4.24 using the implicit GDP deflator as the measure of prices instead of the CPI. As in the base case, most of the estimated coefficients, 20 out of 21 with Luxembourg being the exception, are statistically significant and 13 of 21 are between .25 and .45. Column two presents the estimates using the CPI again, but using a cubic time trend to derive the output gap. Again 20 out of 21 of the estimated coefficients are statistically significant, njt =ctj+l3j- Gapjtt-i + 5jiTjtt-i + j = 1, • • •, 21 (4.24) 99 with New Zealand the only exception. Less than half of the estimated coefficients he between .25 and .45, with a greater number lying between 0.2 and 0.25 than in previous cases. The specification for column three involves adding one extra lag of inflation to the right hand side of Equation 4.24. In this case all of the estimated coefficients are statistically significant and 13 of 21 he between 0.25 and 0.45. Column four uses the CPI and the contemporaneous value of the HP-filtered output gap an again implements Equation 4.24. In this case, there are a number of countries, five, for which the estimated coefficient is not statistically significant and for sixteen of the countries the point estimate of the coefficient for this specification is lower than in the base case, though 10 of the 21 coefficients are between 0.25 and 0.45. The most notable exception to this is Italy, for which the point estimate increases by more than 100 percent. Column five shows the results of estimating Equation 4.24 using OLS rather than SUR. In this case, six of the estimated slope coefficients are not statistically significant. However, the results using OLS are essentially the same as those obtained using the SUR approach. Table 4.3 shows the effects of estimating the Phillips curve relationship using the change in inflation, rather than the level, as the dependent variable. In this case the estimated slope of the PhiUips curve is statisticaUy significant for every country except three: the Netherlands, New Zealand and Switzerland. Note that for Switzerland in particular, the unit sum constrain on lags of inflation does not seem to be a reasonable restriction (see the coefficient on lagged inflation in table one). A final robustness check involves including lags of other controlling variables into Equation 4.24. Common choices in the literature are energy prices and import prices (I don't have data on import prices so I only look at energy prices). Table 4.4 reports the results of a regression of the level of inflation on the lag of the gap, lagged inflation and and a lag of the percentage change of the relative price of oil. That is, nt = a + 0 • Gap + 8 • nt-i + 7 • xt-\ + et, (4.25) where Xt = (pt — pt-i)/Pt-i and pt = p^/CPIt. As in previous cases, the estimated slope coefficient is statisticaUy significant for most of the sixteen countries (with the exception of New Zealand). Furthermore, with the exception of France, the estimates are quite close to those estimated using the baseline specification. In summary, the results presented in Tables 4.1 through 4.4, as well as other unreported specifications, suggest that there are fairly robust Phillips curve relationships in many OECD countries. Of the countries examined, only Luxembourg and New Zealand reject 100 the hypothesis of a positive relationship between inflation and real activity with any regularity across various possible specifications. 4.3.2 The Changing Slope of the Phillips Curve: Rolling Regressions In this section I provide evidence in support of the hypothesis that the slope of the PhiUips curve in many OECD countries appears to have declined over the past two decades. This decline may have been gradual or sudden, depending on the country, but the evidence suggests that the PhiUips curve relationship observed today differs significantly from that of the late 1970s. The estimating framework in this section is based on that of Equation 4.24 of the previous section. I run a series of country by country roUing regressions, whereby I estimate a PhiUips curve for each country on a 15 year moving window of data, so that I obtain an estimate of the slope of the Phillips curve in each year, based on the most recent 15 years of data, and track how this slope changes over time for each country. That is, the slope coefficient estimated in 1981, for example, is estimated on data from 1967-1981. Figures 4.1 to 4.4 plot the results of these regressions for the UK, France, Finland, and Norway for the basehne specification. These countries are chosen to represent the four basic patterns found in the data: 1 2 i) gradual decline throughout (Denmark, Greece, Lux-embourg, Netherlands, Sweden, Spain, UK), ii) low at start, increases and then dechnes, either graduaUy or sharply, towards the end of the sample (Austria, Canada, France, Germany, Italy, US), iii) stable at start with a fairly sharp decline (Australia, Belgium, Finland, Ireland, Switzerland), and iv) no evidence of a decline (New Zealand, Norway). Note that the point estimate of the slope of the PhiUips curve dechnes quite substantiaUy from its peak to its end of sample in Figures 4.1 to 4.3. The experiences of the earlier part of the sample are less dramatic and also less robust to the choice of country and specification. The main common feature is the decline in slope usuaUy beginning in the mid 1980's and continuing throughout the 1990s. Table 4.5 reports the results from the basehne set of rolhng regressions for all of the countries over the latter portion of the sample. For most countries, the point estimate of the slope of the Phillips curve is high for the early periods, starts to decline with the estimates of the late 1980s and early 1990s, and continues to decline throughout the 1990s. Partial exceptions are Ireland, in which no decline in slope is observed until the late 1990s, and Italy, for which the slope estimates are low to begin with, peak in the late 1980s and 12I do not present the US or C a n a d i a n figures as these were analyzed i n C h a p t e r 3. 101 early 1990s, and return to their previous level in the late 1990s. As mentioned previously, New Zealand and Norway do not exhibit a decline in slope. While Table 4.5 does not present standard error bands, it is generally the case, as may be inferred visually from Figures 4.1 to 4.4, that the precision of the estimates is not great in relation to the magnitude of the observed declines in slope for most countries. However, despite the statistical imprecision, the change in slope is economically important in many countries. For example, in Finland, at the start of the sample an increase in the output gap of one percent (say from 2% to 3%), holding all else would be associated with an increase in inflation of nearly one full percent. By the end of the 1990s, there would be essentially no change in inflation associated with a similar change in the output gap. Alternate specifications, including more lags of the variables as well as experimenting with the alternate measures of inflation and the output gap, including energy prices in the estimation and imposing the unit sum constraint on lags of inflation produce similar results. In every case, the slope estimates decline for most countries, and this decline is often large enough to be of economic importance, though the estimates are not precise enough to establish the statistical significance of the decline. While most countries exhibit a decline, for every specification there are one or two countries which do not, with Norway and New Zealand being the primary repeat offenders. This suggests that the story of a widespread decline in the slope of the Philhps curve across the OECD is not without merit. 4.3.3 The Changing Slope of the Phillips Curve: Dummy Variables To further explore the possibility of a decrease in the slope of the Phillips curve across OECD countries, I employ the use of dummy variables in the context of the OLS and SUR regressions of Section 4.3.1. The general technique is to divide the sample using a year dummy. This dummy can then be interacted with the output gap to give an estimate of the difference in slopes in the two parts of the sample (in particular, it represents the difference between 0, the estimated slope for the first part of the sample, and the slope for the second part of the sample). Thus all Phillips curves estimated in this section are some variant of one of the following equations: k i n = eto + <*i • Dt + 0o • Gapt-i + 0i • A - i • Gapt-i + £ Si • irt-i + £ 7i • %t-j + e t, (4.26) i=0 j=0 102 fc I Airt = a0 + cn • Dt + 0 • Gapt-i + (3X • Bt-\ • Gapt-1 + £ & • &*t-i + £ 7i • zt-t + et, (4-27) where Dt = {1 if t > T, 0 otherwise }, and r is the date at which the sample is divided. Table 4.6 presents the results of running a version of Equation 4.26 in which inflation is run on the constant terms, the lag gap terms and one lag of inflation, and r is 1985. Column five is the term representing the output gap interacted with the dummy variable. This coefficient represents the difference between the slope of the Phillips curve from 1985-1997 with the slope from 1961-1984. Subtracting this number from the usual output gap term, given in column two, gives the point estimate of the slope of the Phillips curve from 1985-1997. For most countries (New Zealand and Switzerland are the exceptions) the point estimates suggest that the slope of the Phillips curve has declined. For a number of countries this decline is both economically important and statistically significant (one rejects the null hypothesis that the estimated coefficient is zero at a 5% level of significant for Finland and the UK and at a 10% level for Denmark, France and Japan). For many other countries (Australia, Austria, Belgium, Canada, Germany, Greece, Ireland, Italy, Luxembourg, Spain and the U.S) the decline is statistically imprecise but economically important, representing a change of at least a third, and often more, in the observed output-inflation relationship. For Spain, for example, the point estimates suggest that prior to 1985 the slope of the PhiUips curve was 0.626 and feU post-1985 to 0.113, so that whereas an decrease in the output gap of one percent (from -2% to -3%, say) was associated with an decrease in inflation of 0.626 percent, while after 1985, the same output gap would be associated with an decrease in inflation of only 0.113 percent. Table 4.7 presents the results for the same regression, using SUR rather than OLS as the regression technique. Again the pattern of a lower slope in the second portion of the sample appears for many countries. In this case, however, the decline is both economicaUy important and statisticaUy significant for most countries. One rejects the null hypothesis that /?i equals zero, that is that there is no difference in the slope of the PhiUips curve in the two periods, against the alternate hypothesis that /?i does not equal zero at a 5% level of significance for Australia, Austria, Belgium, Canada, Denmark, Finland, Germany, Ireland, the UK and the US, and at a 10% level of significance for France, Japan, Luxembourg, and Spain. Of the remaining countries, the dechne in slope is economically important (greater than one third of the slope of the Phillips curve is 103 my ad hoc benchmark) in Greece, Italy and Sweden. The estimate of Pi is positive and economically large for New Zealand, Norway and Switzerland and positive but small for the Netherlands. It is not statistically different from zero in any of these cases. Table 4.8 presents a robustness check on the results of Table 4.7 by adopting various alternate values for r, the choice of break point. The results reinforce the results of Table 4.7. As before there are four countries (the Netherlands, New Zealand, Norway and Switzerland, for which the hypothesis of a decline in slope does not fit, regardless of which year is chosen for dividing the sample.13 The other seventeen countries, however, display evidence of a decline in the slope of the Phillips curve, almost regardless of the choice of break year. The decline in slope is almost universally large relative to the estimate of po- Furthermore, for sixteen of these seventeen countries, there is a breakpoint for which the null hypothesis of no change in the slope is rejected against the alternative at a significance level of 5%.14 For many countries, (Australia, Belgium, Denmark, Finland, Ireland, the UK and the US), the data reject the null hypothesis at a 5% significance level regardless of the choice of r. The results of Table 4.8 suggest that there is evidence of a decline in slope in the latter half of the sample, and that this evidence is fairly robust to exactly where the sample is split. The results of Table 4.8 are also robust to the inclusion of further lags of inflation in the specification, choice of the GDP deflator rather than the CPI as the measure of prices as well as the choice of a cubic time trend filtered output gap rather than an HP-filtered gap. Tables 4.9 and 4.10 present examples of other robustness checks. Table 4.9 imposes the unit sum constraint on lags of inflation and regresses the change of inflation on the lag of the output gap and the various dummies. That is, it estimates the following equation: A7Tt = O J 0 + ai • Dt + p • Gapt-i + Pi • Dt-i • Gap t-i + et. (4.28) As in the previous instance, most of the point estimates shown in Table 4.9 are negative and economically large. Several countries reject the null hypothesis of no change in the 1 3 I n the case of N e w Zealand, the point est imate becomes negative, and large relative to the estimate of /3b, i f the sample is sp l i t i n 1989. Fur ther tests show that b o t h for N e w Zealand and the Nether lands , one obtains a negative point estimate of f3i i f one chooses a later year, such as 1991 or 1993, to d iv ide the sample. T h i s suggests that is is possible that has been a recent decline i n slope i n N e w Zealand's P h i l l i p s curve relat ionship. However, i n no case is the est imated coefficient s ta t i s t ica l ly significant, and there are too few observations to make inferences w i t h much confidence. 1 4 T h e exception is Italy, which rejects the n u l l at 10% when r is set to 1981, but otherwise does not reject the nu l l . 104 slope of the PhiUips curve. The null hypothesis that Pi equals zero is rejected at the 5% level of significance for Austraha, Denmark, Finland, Ireland, Sweden, the UK and the US, and at the 10% level of significance for Canada and Luxembourg. Unhke the previous case, the point estimates of Pi for New Zealand and Norway are both negative, and large relative to the estimate of Po, but not statistically significant. The point estimates for the Netherlands and Switzerland, however, remain positive, and the point estimate for Italy is also positive. A check on the robustness of the results to the choice of r — 1985 analogous to that reported in Table 4.8 finds that for ten countries (Austraha, Belgium, Denmark, Finland, Ireland, Japan, Luxembourg, Sweden, the U K and the US) there exits at least one choice (and often several) of r between 1981 and 1989 for which the estimate of Pi is negative, economicaUy meaningful and statistically significant at the five percent level and four others for which Pi is negative, economically meaningful and statisticaUy significant at the ten percent level of significance. Only the Netherlands and Switzerland exhibit consistently positive estimates of pi.15 The only positive and statisticaUy significant estimates are for the Netherlands when r is 1981 or 1983. OveraU, the results of the estimation imposing the unit sum constraint also suggests that there has been a decline in the slope of the PhiUips curve in recent years, though the evidence is somewhat weaker than was the case when the coefficient on the lags of inflation were not restricted. Table 4.10 presents an example of the results when controlhng for changes in relative energy prices. Here, the equation estimated is: 7rt = a 0 + cxi • Dt + Po • Gapt-i + Pi • Dt-i • Gapt_i + 5 • TTt-\ + 7 • ^ t - i + £t, (4.29) where Xt-i is as previously defined. The estimation is run on a sample of sixteen countries (for which I have energy price data). For four countries, the Netherlands, New Zealand, Norway and Switzerland, the point estimate of Pi is positive. For the remaining countries, the estimate of Pi is negative, and economicaUy meaningful. It is statisticaUy significant at the 5% level for six countries (Denmark, Finland, France, Ireland, The UK and the US) and at the 10% level for Australia, Austria and Japan. Checking robustness to the choice of r reveals that there is a choice of r for which the estimate is negative, 1 5The estimate for Italy is positive if r equals 1983 or 1985, and the estimate for Austria is positive if r is 1983 and 1989. As before, the estimate for Canada is positive if the sample is split in the early 1980s. In this case it takes a positive value if r equals 1981 or 1983. 105 economically important and statistically significant (at the 5% level in 10 countries (Aus-tralia, Denmark, Finland, France, Ireland, Italy, Japan, Sweden, the UK and the US) and at the 10% level for Austria). Estimates for the Netherlands and New Zealand become negative, though not statistically significant, if r is set to 1987 or 1989, while the estimates for Norway and Switzerland remain positive essentially across all choices of r. The only other positive point estimate is for Canada when r is set to 1981. In sum, the results in this section suggest that there has been a decline in the slope of the Phillips curve across OECD countries sometime over the past two decades. Of the countries analyzed, all but four repeatedly reject the hypothesis that the slope of the Phillips curve has remained the same (or increased). Of these four countries (the Netherlands, New Zealand, Norway and Switzerland) two show at least some evidence of a decline when the unit sum constraint is imposed on the lags of inflation, and both the Netherlands and New Zealand return point estimates which would indicate a decline if one contrasts very recent years with the rest of the sample. 4.3.4 The Variances of Output and Inflation This section presents evidence pertaining to the relationship between the variance of output gaps and the variance of inflation over time. In particular, it examines whether or not the data for OECD countries support the hypothesis that there is a tradeoff between the variance of output and of inflation. The method of analysis is straightforward. I compute sample variances for inflation and the output gap for different periods and examine whether or not decreases (increases) in the variance of inflation are associated with increases (decreases) in the variance of the output gap. For consistency with the previous section, the data are divided into two subsections where the break point, r, takes the value 1985. The variances of inflation and the output gap are calculated for both sub-samples, and the change in these variances is calculated by subtracting the variances of the earlier sub-sample from the variances of the later sub-sample. Figure 4.5 presents a scatter plot of the change in the variance of the output gap against the change in the variance of inflation. The first thing to note from Figure 4.5 is that the variance of inflation declines in almost all countries, with New Zealand as the lone exception (the average change is -14.206). On the other hand, the variance of the output gap increases on average (by 0.647).16 Thus there is, in averages, an increase in 1 6 M u c h of the increase can be a t t r ibuted to the ou t l y ing observation of F i n l a n d . If F i n l a n d is 106 the variance of output associated with the decline in the variance in inflation. A look a Figure 4.5, however, suggests that this is as far as it goes. In particular, it is clear from the figure that there is no strong inverse relationship between the magnitude of the change in the variance of inflation and the change in the variance of inflation. In fact, for several countries both the variance of inflation and the variance of the output gap decline. 4.4 Further Tests of the Model Having described a number of stylized facts about the Phillips curve in OECD countries in the previous section, this section attempts to further test the predictions of the model with an eye to contrasting these predictions to those of other potential theories of the Philhps curve. The first such tests involve the predictions of the model regarding the slope of the Phillips curve in the presence of disinflationary policy contractions. In this case, the key element is the fact that the model distinguishes between the reduced form Phillips curve as a statistical object and the short run output-inflation tradeoff facing a policy maker who wishes to reduce the trend rate of inflation. The tests involve testing for a difference between the slope of the statistical Phillips curve during disinflationary contractions as opposed to more normal periods under a variety of specifications. Finally, the paper attempts to use the fact that during explicit inflation targeting regimes it is possible to calculate the deviations of inflation from target. The model of Section 2 gives a Phillips curve which is in deviations of inflation from target and the output gap. This sections examines whether the predictions of the model, which include the fact that inflation deviations from target ought not to be autocorrelated and ought not to be able to predict the current output gap, can be reconciled with the experience of five recent inflation targeting regimes. 4.4.1 Monetary Contractions The model of Section 4.2 predicts the the comovement between inflation and output ought to differ during a period of monetary contraction, in which the goal of the monetary authority is to reduce the trend or target level of inflation, from the comovement observed during "normal" episodes during which the policy maker is attempting to use monetary policy of manage real developments. Note that this is not the case in models in which some form of Phillips curve is a structural object that the policy maker takes as a given excluded, the average change in the variance of inflation becomes negative. 107 constraint on behaviour. In that case the statistical Phillips curve and the short run output-inflation tradeoff should be the same at any point in time, and while structural factors may cause the slope of the statistical Phillips curve to change over time, these ought to change the slope of the short run output inflation tradeoff in the same manner. This section presents evidence relating to the prediction of the model that the statistical Phillips curve that one observes in the data does not represent the short run output-inflation tradeoff that obtains following an intentional monetary contraction using a variety of specifications. The main goal is to test whether the slope of the Phillips curve is different in episodes of disinflationary policy contractions. To test the predictions of the model in the presence of monetary contractions requires that one has available data on monetary contractions. There is a large literature in macro attempting to extract monetary shock series from the data. One popular method involves the use of structural VAR's in which identifying restrictions are imposed (frequently these involve causal orderings on the impact matrix or long run restrictions). An alternate approach, sometimes referred to as the narrative approach and best exemplified in the widely used dummy variable of Romer and Romer (1989), is to examine the historical record, for example by looking at meetings of the minutes of the pohcy making body, and attempting to infer the motives of pohcy makers when they made their decisions. This paper employs an approach initiated by Gordon (1994). The approach uses essentially mechanical methods to supplement a loose reading of the historical record and consists of identifying large declines in the rate of inflation (1.5%) as monetary shocks. A disinflation episode is any period that starts at an inflation peak and ends at a trough with the inflation rate at least 1.5% lower than at the peak. The dummies obtained via this procedure coincide with the historical record of policy contractions as embodies in the OECD's Economic Outlook series (at least for the G-7 countries, which are covered in reasonable detail). The dummies also coincide fairly closely with the dummies presented by Romer and Romer (1989) and what I know of the Canadian experience.17 This is not an ideal approach but it is difficult to access the necessary information to create proper Romer and Romer type dummies for the whole OECD. Structural VAR's can be problematic too, especially in this case where it is not clear how to distinguish between a monetary innovation arising due to pohcy maker error and a monetary innovation due to a change in the inflation target. 1 7 B a l l (1994) c la ims that they also coincide closely w i t h the dummies presented i n Fernandez (1992). I have not had the oppor tun i ty to verify this c l a im. 108 Alternately, the Gordon dummies are properly identified to the extent that disinfla-tionary contractions are the cause and the only cause of 1.5% declines in inflation, and that these contractions are unexpected be agents and do not represent anticipated changes in the target rate of inflation (which the model predicts ought not to affect output at all). If nothing else, the dummies clearly identify periods where the inflation rate is falhng-so that any model of the Phillips curve ought to be able to explain why the Phillips curve relationship ought to differ in such periods. This section employs the SUR technique to examine the properties of the Phillips curve under disinflationary contractions versus more normal episodes. The lag of the monetary contraction dummies are interacted with the lagged output gap to obtain an estimate of the difference between the slope of the Phillips curve during the initial periods of monetary contractions and the usual slope. These tests are particularly interesting because leading alternate explanations of the decline in the slope of the Phillips curve are df the more structural variety. In particular, the menu costs idea of Ball, Mankiw and Romer (1988) and the downward nominal wage rigidity hypothesis of Akerlof, Dickens and Perry (1996) are both consistent with the observation that the slope of the Phillips curve has declined over the past twenty years.18 However, neither of these theories make a distinction between the statistical Phillips curve and the short-run output-inflation tradeoff. Thus, they predict that if the slope of the Phillips curve has declined then the slope of the output-inflation facing policy makers ought to have changed in the same way. I attempt to distinguish between these two competing ideas by estimating the following equation: 7rt = a0 + po • Gapt-i + Pi • DtGapt-i +p2-Slt-i- Gapt-i + P3 • S2t-i • Gapt-i + 5 • irt-i + et, (4.30) where SI takes on the value one if there is a monetary contraction and the year is less than some value r and zero otherwise, while S2 takes on the value one if there is a monetary contraction and the year is greater than or equal to r and zero otherwise, and Dt takes on the value one if the year is greater than or equal to tau and there is no contraction. Thus we can interpret p\ as the estimate of the difference between the slope of the Phillips curve during normal episodes at the start of the sample (given by Po) and the slope during contractionary episodes in the first portion of the sample. Similarly, Ps is the estimate of 1 8Both of these models suggest that the slope of the Phillips curve should fall as the average rate of inflation falls. Since this is exactly what has happened in OECD countries over the past twenty years, these model predict the observed decline in the slope of the Phillips curve. 109 the difference between the slope of the Phillips curve during normal episodes throughout the sample and the slope during contractionary episodes in the second portion of the sample. 0i is the estimate of the difference between the slope in normal times at the start of the sample and the slope in normal episodes at the end of the sample. The model of section 4.2 suggests that the slope of the Phillips curve has fallen in recent years (that 0\ is negative) but that the response of the economy to monetary contractions ought not to have changed over the same period (that is, that 02 equals 0^). A more structural explanation of the PhiUips curve that allows for a decline in slope would suggest that the response of the economy to monetary shocks ought to have changed in the same way. In particular, the slope during monetary contractions ought to be the same as the slope in other periods (so that 02 equals zero and 0s equals 0\). Given a decline in the slope of the PhiUips curve (so that 0\ is less than /3b), these two restrictions ought to be mutually exclusive. Table 4.12 presents the results from estimating Equation 4.30 when r is set to equal 1985. Of the 20 countries in the sample (Belgium is excluded due to the fact that there are no identified contractionary episodes post-1985), the estimated slope of the PhiUips curve increases in four countries (as in section 4.3, these are the Netherlands, New Zealand, Norway, and Switzerland). Of the other sixteen countries, there is a statistically significant decline in slope in 8 of these, and in four others (Greece, Italy, Japan, and Sweden) there is a decline which is economicaUy meaningful although not statistically significant. These results are similar to the results obtained in the previous section, although the estimated decline in slope is statistically significant in fewer countries. The nuU hypothesis, suggested by the model of section 4.2, that 02 equals 03 is rejected in six of twenty countries. It is worth noting that this nuU hypothesis is rejected only three times in the sixteen countries for which a decline in slope is observed in the point estimate of 0\.19 This suggests that there is some evidence that the slope of the Phillips curve during monetary contractions has not changed over the past twenty years, at least in a significant subset of countries.20 Given the small number of episodes of monetary 1 9The fact that the same countries tend to reject both the hypothesis of a falling slop and the restriction that the response to monetary contractions has not changed suggests that there is a small set of countries whose experiences cannot be explained by the model. 2 0It is worth noting that of those six countries for which the hypothesis that equals p\ is rejected, in three cases (Canada, Greece and the Netherlands) the estimate increases over the second part of the sample. A structural interpretation of the Phillips curve would suggest that the slope of the output inflation tradeoff ought to have fallen along with the slope of the statistical Phillips curve. 110 contraction, there is some cause for concern that the failure to reject the null hypothesis may be due to the fact that 02 and 03 are estimated imprecisely. Indeed, a joint test of the null hypothesis that the vector of /Vs equals the vector of /Vs rejects the null hypothesis at any conventional levels of significance. However, a joint test of the null hypothesis that the 02's equal the /Vs for those countries which do not reject the initial hypothesis that the their own estimated 02 equals 03 does not reject the null. This suggests that though there may be a significant number of countries for which the slope of the short-run output-inflation tradeoff has not changed over recent years. The more structural hypothesis that the slope of the Phillips curve equals the slope of the short run output-inflation tradeoff, that 02 is zero and 03 equals 0i, is rejected in fifteen of twenty counties (and 11 of the sixteen for which 0\ is negative). Of the five cases for which this hypothesis is not rejected, in two instances, Germany and Sweden, the point estimates suggest that the slope of the short-run output-inflation tradeoff became steeper over the latter part of the sample in which the Phillips curve otherwise flattened. This suggests that there is some evidence in support of the idea of a distinction between the short-run output-inflation tradeoff and the statistical Phillips curve over the idea that these two objects are the same. Tables 4.13 and 4.14 investigate the robustness of these results to the choice of r equal to 1985. Table 4.13 presents the results when r is set to 1981. In this case there is a statistically significant decline in the slope of the Phillips curve (0\ is negative) in 13 of 21 countries as well as an economically large decline in the slope in Italy. As usual, the slope appears to increase in four countries: the Netherlands, New Zealand, Norway and Switzerland. The hypothesis that 02 equals 03 is rejected in 8 of 21 countries, and 5 of the 17 countries for which a decline in slope is observed. The structural hypothesis, that 02 is zero and 03 equals 0i, is rejected in 18 of 21 countries, and 14 of the 17 for which a decline in slope is observed. These numbers reinforce the conclusions and interpretation of the previous results. The choice of r equal to 1987 also makes little difference to the results, as can be seen from Table 4.14. There is a statistically significant decline in the slope of the Phillips curve (0\ is negative) in 8 of 19 countries, and the slope increases only in New Zealand, Norway, and Switzerland (there is no contraction post 1986 for the Netherlands, so I exclude that country from the estimation). The hypothesis that 02 equals 03 is rejected in 6 of 19 cases (and in four of the 16 countries for which a decline in slope is observed). The hypothesis that 02 is zero and 03 equals 0\, is rejected in 14 of 19 cases (and 11 of the 16 countries 111 for which a decline in slope is observed). The results of Tables 20 and 21 reinforce the conclusions and interpretation of the previous results. Table 4.15 presents results which check the robustness of the previous results to the imposition of the unit sum constraint on lags of inflation. Specifically, it presents the results of estimating the equation: Airt = ao+Po-Gapt-i+pi-DtGapt-1+p2-Slt-i-Gapt-i+p3-S2t-1-Gapt-i+et, (4.31) where, as before, S i takes on the value one if there is a monetary contraction and the year is less than some value r and zero otherwise, while S2 takes on the value one if there is a monetary contraction and the year is greater than or equal to r and zero otherwise, Dt takes on the value one if the year is greater than or equal to tau and there is no contraction, and tau equals 1985. As in previous cases, there is a statistically significant decline in the slope of the Phillips curve in the second half of the sample for a number of countries, 7 in this case, and an economicaUy large decline which is not statisticaUy significant in several more (five in this case). As usual, the estimate of P\ is positive for the Netherlands, New Zealand, Norway, and Switzerland. The hypothesis that P2 equals ps is rejected in 7 of 20 countries and in four of the 16 countries for which a decline in slope is observed. The hypothesis that P2 is zero and Ps equals Pi is rejected in 17 of 20 countries (and 13 of the 16 for which a dechne in slope is observed). The results of this section provide some support for the hypothesis that there is a statistical PhiUips curve distinct from the short-run output-inflation tradeoff. Neither hypothesis fits the data extremely weU, the hypothesis that there is an output-inflation tradeoff which does not change over time and is distinct from the statistical Phillips curve relationship observed in normal time appears to be a reasonable description of the experience of some countries. On the other hand, the notion that the short-run output-inflation tradeoff and the statistical Philhps curve are the same finds support in a noticeably smaller set of countries. 4.4.2 Inf lat ion Target ing This section examines the recent experiences of three exphcit inflation targeting coun-tries (Canada, New Zealand and the United Kingdom), and attempts to investigates whether some of the other correlations discussed in section 4.2 can be observed in the data. Inflation targets were introduced first in New Zealand in 1990, and subsequently 112 in Canada in 1991 and the United Kingdom in 1992.21 Since the inflation targets are published, it is possible to derive deviations of inflation from target, in which the model of section 4.2 is cast, for these countries over this period. The aim is to discover whether the fact that the model expresses correlations in terms of inflation deviations from target can reconcile some of the moments reported in section 4.3.2 with the data. I use quarterly data on real GDP and the CPI to examine the output-inflation rela-tionship in these countries in recent years. The output gap is obtained by applying an HP-filter to the quarterly GDP series from the first quarter of 1970 to the fourth quarter of 1997.22 Data on inflation targets is obtained from the web sites of the respective central banks. The deviation of inflation from target is the difference between the target and the annualized value of the quarterly inflation rate.23 Since each of these countries adopted explicit inflation targets at different dates, the sample differs for each of the regressions. The Canadian regressions use data from the first quarter of 1992, the regressions from New Zealand start with the first quarter of 1991 and those for the UK begin with the fourth quarter of 1992. Al l of the regressions use data up to and including the fourth quarter of 1997. The main moments of interest here are the correlation of current deviations of inflation from target and the current output gap, which the model predicts should be negative, and the serial correlation of inflation deviations from target, which can be positive (negative) if the correlation between future deviations of inflation from target and the current output gap is negative (positive). Table 4.16 presents some results. The first row presents the results of running current inflation deviations from target on the current output gap and one lag of the deviation of inflation from target. The results show that a positive correlation, as all three point estimates are positive, and only the estimate for Canada is statistically close to zero. These results are robust to including further lags of inflation on the right hand side (I have tried as many as six) or to excluding inflation as a right hand side variable altogether. Thus it does not appear that looking at deviations of inflation from target reconciles the model's 2 1For a detailed discussion on the early years of inflation targeting, see Bernanke, Laubach, Mishkin and Posen (1999). 22Quarterly data is only available from the second quarter of 1982 in the case of New Zealand. 2 3 The discussion in this section relies on the assumption that the announced target represents the bank's true target. This may not be the case. In Canada, for example, the inflation rate has been below target, although within the acceptable bands, without prompting a response by the Bank of Canada. This has led some to suggest that the stated target may not represent the Bank's real target. 113 prediction of a negative correlation between current inflation and the current output gap with the data. The second row of Table 4.16 presents the results of regressing inflation deviations on the lagged output gap and lagged inflation deviations. The first thing to note is the fact that there is no strong positive serial correlation in inflation. For both Canada and New Zealand the estimates are positive but statistically close to zero, while for the UK the estimate is negative and statistically significant at the 10% level of significance. The correlation between inflation and lagged output is positive in all three cases. It is statistically insignificant for Canada but significant for both the UK and New Zealand. Recall that the baseline model predicts that inflation deviations from target ought to be serially uncorrelated and that the correlation between inflation deviations and the lagged gap (or future inflation deviations and the current gap) ought to be zero, which is essentially what we observe for Canada. Also, the model with extended uncertainty, discussed in section 4.3.2, suggests a positive correlation between inflation deviations ought to be associated with a negative correlation between current and past inflation deviations, which is observed for the UK. Table 4.17 investigates this further by allowing for further lags of inflation deviations from target to enter on the right hand side. The results suggest that adding further lagged deviations does not change the basic result that inflation deviations from target do not exhibit strong, positive serial correlation. For all three countries the point estimates are generally small and not significantly different from zero. In Canada and New Zealand, the sum of the coefficients on lagged inflation is positive, but in neither case is it possible to reject the null hypothesis that the vector of coefficients is equal to zero. For the UK, the estimated vector of coefficients on lagged inflation is jointly statistically significantly different from zero, but the sum of these coefficients is negative, suggesting the lack of any strong positive serial correlation. The coefficient on the output gap is not statistically significantly different for zero for Canada or for the UK, but is for New Zealand. Thus the results of Table 4.17 reconfirm the previous evidence that suggests that for the Canada and the UK the consideration of inflation deviations from target reconciles the model results about the serial correlation of inflation deviations and the correlation of future inflation deviations and the current output gap. For New Zealand, the consideration of inflation deviations from target explains why the model might reasonably predict low or no serial correlation in inflation deviations, but the positive correlation of future inflation deviations and the current output gap is not explained. 114 The results of this section go part of the way in explaining how some of the seemingly counterfactual predictions of the model are consistent with the data when one has access to the variables predicted by the theory. In particular, the prediction that inflation deviations from target need not be strongly positively correlated appears to be borne out by the data. 2 4 However, the model's prediction that current inflation deviations from target should be negatively correlated with the output gap remains a puzzle that does not appear to fit with the observations. 4.5 Conclusion The evidence presented in this paper shows that there appears to have been a decline in the slope of the Phillips curve in recent decades across a wide variety of countries. The paper presents a model which shows how such a decline may be the result of the fact that pohcy makers have become better informed about real developments in the economy in recent years. The paper presents some evidence that argues in favour of this kind of explanation, in which the Phillips curve emerges as an equilibrium, over a more structural interpretation Phillips curve and the decline in its slope. The model, however, has some difficulty in explaining the recent experiences of three exphcit inflation targeting regimes. 2 4 T h u s the mode l w o u l d suggest tha t observed serial correla t ion i n inflat ion rates might be a t t r ibu ted to a serial correlat ion i n inf lat ion targets. 115 Table 4.1. Simple Phillips Curve Estimates for OECD Countries: 1961-1997 Country Constant HP-Gap t - 1 TTt-1 Australia 2.017 (0.485) 0.517 (0.103) 0.674 (0.058) Austria 2.480 (0.345) 0.430 (0.069) 0.399 (0.069) Belgium 1.780 (0.323) 0.336 (0.075) 0.614 (0.050) Canada 1.598 (0.361) 0.375 (0.079) 0.688 (0.056) Denmark 1.771 (0.483) 0.290 (0.093) 0.716 (0.058) Finland 3.342 (0.599) 0.262 (0.069) 0.505 (0.065) France 1.060 (0.355) 0.181 (0.101) 0.815 (0.041) Germany 1.637 (0.259) 0.349 (0.045) 0.500 (0.062) Greece 2.030 (0.763) 1.192 (0.158) 0.841 (0.043) Ireland 2.930 (0.611) 0.285 (0.090) 0.623 (0.053) Italy 2.344 (0.577) 0.269 (0.115) 0.719 (0.045) Japan 1.739 (0.528) 0.362 (0.088) 0.617 (0.057) Luxembourg 1.430 (0.333) 0.157 (0.401) 0.666 (0.056) 116 Table 4.1. cont... Country Constant HP-Gap* - 1 Netherlands 1.652 (0.409) 0.329 (0.113) 0.632 (0.073) New Zealand 3.004 (0.786) 0.228 (0.109) 0.617 (0.076) Norway 3.687 (0.574) 0.420 (0.114) 0.397 (0.074) Spain 2.000 (0.657) 0.402 (0.112) 0.782 (0.055) Sweden 3.104 (0.617) 0.449 (0.121) 0.507 (0.078) Switzerland 3.180 (0.406) 0.568 (0.074) 0.120 (0.093) UK 3.167 (0.644) 0.715 (0.115) 0.570 (0.060) US 1.712 (0.331) 0.445 (0.067) 0.653 (0.049) 117 'able 4.2. Alternate Estimates of Beta for OECD Countries: 1961-1997 Country Spec. 1 Spec. 2 Spec. 3 Spec. 4 Spec. 5 (Deflator) (T-Gapt-i) ( T * - 2 ) (Ga P i ) (OLS) Australia 0.380 0.338 0.463 0.277 0.598 (0.051) (0.089) (0.108) (0.095) (0.188) Austria 0.469 0.245 0.471 0.274 0.451 (0.070) (0.053) (0.074) (0.085) (0.139) Belgium 0.450 0.218 0.264 0.199 0.502 (0.090) (0.057) (0.081) (0.077) (0.146) Canada 0.271 0.366 0.379 0.426 0.446 (0.062) (0.071) (0.089) (0.077) (0.124) Denmark 0.288 0.234 0.393 0.120 0.505 (0.069) (0.088) (0.101) (0.073) (0.170) Finland 0.128 0.267 0.254 0.122 0.303 (0.062) (0.064) (0.069) (0.085) (0.128) France 0.346 0.187 0.237 0.194 0.502 (0.093) (0.082) (0.101) (0.101) (0.193) Germany 0.353 0.287 0.328 0.296 0.361 (0.075) (0.041) (0.051) (0.054) (0.084) Greece 0.773 0.782 1.514 0.904 1.299 (0.190) (0.122) (0.166) (0.164) (0.295) Ireland 0.366 0.178 0.307 0.191 0.323 (0.170) (0.061) (0.088) (0.083) (0.228) Italy 0.315 0.328 0.352 0.791 0.444 (0.105) (0.105) (0.158) (0.100) (0.266) Japan 0.272 0.214 0.383 0.239 0.589 (0.077) (0.058) (0.090) (0.070) (0.177) Luxembourg 0.178 0.082 0.089 0.049 0.229 (0.129) (0.030) (0.040) (0.050) (0.081) 118 Table 4.2. cont... Country Spec. 1 Spec. 2 Spec. 3 Spec. 4 Spec. 5 (Deflator) (T-Gap-t - 1) ( T * - 2 ) (Gap*) (OLS) Netherlands 0.346 0.220 0.460 0.261 0.332 (0.112) (0.073) (0.122) (0.097) (0.183) New Zealand 0.458 0.197 0.294 0.175 0.236 (0.118) (0.103) (0.110) (0.115) (0.170) Norway 0.675 0.335 0.351 0.285 0.497 (0.139) (0.079) (0.131) (0.114) (0.242) Spain 0.351 0.308 0.455 0.418 0.329 (0.103) (0.082) (0.115) (0.112) (0.182) Sweden 0.271 0.317 0.447 0.359 0.539 (0.135) (0.102) (0.124) (0.127) (0.310) Switzerland .411 0.331 0.444 0.434 0.495 (0.076) (0.053) (0.072) (0.049) (0.119) UK 0.746 0.539 0.739 0.465 0.769 (0.100) (0.099) (0.128) (0.116) (0.220) US 0.202 0.390 0.365 0.490 0.574 (0.058) (0.064) (0.093) (0.069) (0.129) 119 Table 4.3. Imposing the Unit Sum Constraint Country Constant HP-Gapt - 1 Australia -0.031 (0.341) 0.429 (0.110) Austria -0.060 (0.219) 0.192 (0.079) Belgium 0.016 (0.258) 0.282 (0.074) Canada 0.031 (0.250) 0.290 (0.079) Denmark -0.034 (0.325) 0.335 (0.099) Finland 0.005 (0.448) 0.200 (0.079) France -0.040 (0.266) 0.222 (0.102) Germany -0.026 (0.191) 0.224 (0.050) Greece 0.116 (0.610) 1.100 (0.160) Ireland -0.008 (0.477) 0.155 (0.085) Italy -0.045 (0.448) 0.330 (0.118) Japan -0.127 (0.512) 0.278 (0.090) Luxembourg 0.013 (0.247) 0.184 (0.039) 120 Tal ale 4.3. cont... Country Constant HP-Gapt - 1 Netherlands 0.036 (0.276) 0.136 (0.102) New Zealand -0.019 (0.543) 0.202 (0.120) Norway 0.017 (0.392) 0.366 (0.121) Spain -0.057 (0.425) 0.456 (0.125) Sweden -0.053 (0.399) 0.438 (0.128) Switzerland -0.050 (0.289) 0.100 (0.068) UK 0.011 (0.512) 0.609 (0.133) US 0.050 (0.262) 0.353 (0.070) 121 Table 4.4. Controlling for Energy Prices Country Constant HP-Gapt - 1 Energy Prices Australia 2.320 0.652 0.648 0.158 (0.488) (0.142) (0.058) (0.082) Austria 2.299 0.383 0.443 0.002 (0.395) (0.076) (0.080) (0.032) Canada 2.007 0.478 0.590 0.104 (0.409) (0.095) (0.071) (0.056) Denmark 1.769 0.315 0.695 0.017 (0.569) (0.116) (0.080) (0.027) Finland 3.564 0.258 0.466 • 0.099 (0.635) (0.075) (0.071) (0.042) France 0.784 0.448 0.849 -0.061 (0.404) (0.114) (0.055) (0.037) Ireland 3.563 0.304 0.535 0.173 (0.600) (0.092) (0.051) (0.035) Italy 2.249 0.387 0.730 -0.020 (0.636) (0.143) (0.053) (0.034) Japan 1.515 0.428 0.613 -0.127 (0.544) (0.095) (0.062) (0.053) Netherlands 1.300 0.358 0.708 -0.018 (0.465) (0.134) (0.088) (0.034) New Zealand 3.181 0.212 0.596 0.060 (0.875) (0.135) (0.088) (0.082) Norway 3.282 0.376 0.467 -0.057 (0.642) (0.140) (0.086) (0.122) Sweden 2.934 0.509 0.539 -0.101 (0.688) (0.141) (0.089) (0.089) Switzerland 3.099 0.557 0.132 0.021 (0.464) (0.085) (0.111) (0.016) UK 4.666 0.812 0.412 0.259 (0.730) (0.135) (0.073) (0.066) US 2.176 0.568 0.564 0.082 (0.416) (0.100) (0.069) (0.092) 122 Table 4.5. Slope Estimates from Rolling Regressions 1981 1983 1985 1987 1989 1991 1993 1995 1997 Australia 1.009 0.612 0.753 0.919 0.382 0.238 0.352 0.374 0.287 Austria 0.501 0.526 0.699 0.738 0.571 0.532 0.472 0.370 0.332 Belgium 0.943 0.809 0.832 0.950 0.667 0.381 0.316 0.302 0.210 Canada 0.350 0.795 0.775 0.685 0.496 0.413 0.419 0.398 0.297 Denmark 1.078 0.868 0.813 0.602 0.199 0.531 0.374 0.101 0.123 Finland 0.752 0.828 0.711 0.691 0.324 -0.031 -0.015 0.069 0.092 France 0.811 0.845 1.046 1.250 0.882 0.555 0.484 0.229 0.138 Germany 0.424 0.495 0.510 0.512 0.457 0.330 0.369 0.330 0.313 Greece 2.004 1.837 1.685 1.451 1.339 1.132 0.848 0.658 0.614 Ireland 0.731 0.510 0.698 0.720 0.710 0.787 0.921 0.061 -0.110 Italy 0.286 0.133 0.490 1.131 0.582 0.726 0.592 0.238 0.269 Japan 0.807 0.973 0.973 1.607 0.479 0.343 0.100 0.099 0.117 Lux. 0.415 0.273 0.307 0.383 0.306 0.212 0.276 0.122 0.086 Net. 0.491 0.538 0.576 0.413 0.361 0.368 0.387 0.281 0.191 N.Z. 0.185 0.205 0.262 0.230 0.125 0.098 0.241 0.546 0.770 Norway 0.422 0.670 0.677 0.478 0.559 0.649 0.672 0.562 0.389 Spain 0.984 0.761 0.765 0.816 0.636 0.265 0.065 0.122 0.132 Sweden 0.630 0.427 0.344 0.577 0.419 0.417 0.198 0.351 0.278 Swi. 0.797 0.790 0.820 0.569 0.321 0.299 0.210 0.413 0.430 UK 2.013 1.486 1.439 1.413 0.922 0.542 0.594 0.353 0.425 US 0.813 0.936 0.983 0.838 0.630 0.516 0.548 0.235 0.199 123 Table 4.6. OLS Estimation with an Interacted End of Sample Dummy Country Constant Constant (Dummy) HP-Gap t_i Inflationt_i H P - G a p ^ (Dummy) Austraha 1.424 (0.717) -0.918 (0.705) 0.826 (0.224) 0.832 (0.084) -0.548 (0.380) Austria 3.108 (0.601) -1.522 (0.418) 0.630 (0.147) 0.373 (0.116) -0.317 (0.225) Belgium 2.236 (0.554) -1.465 (0.515) 0.683 (0.182) 0.619 (0.088) -0.435 (0.274) Canada 1.791 (0.541) -1.184 (0.513) 0.579 (0.152) 0.739 (0.077) -0.236 (0.231) Denmark 3.007 (0.992) -2.227 (0.801) 0.818 (0.201) 0.655 (0.116) -0.653 (0.323) Finland 3.696 (0.979) -2.612 (0.900) 0.655 (0.174) 0.597 (0.103) -0.519 (0.219) France 1.930 (0.658) -1.598 (0.608) 0.804 (0.262) 0.755 (0.079) -0.642 (0.359) Germany 2.072 (0.458) -0.925 (0.362) 0.464 (0.101) 0.467 (0.106) -0.158 (0.145) Greece 2.257 (1.091) -0.074 (1.357) 1.393 (0.314) 0.817 (0.079) -0.619 (0.763) Ireland 2.814 (1.113) -2.547 (1.194) 0.540 (0.291) 0.726 (0.097) -0.646 (0.445) Italy 1.984 (1.011) -1.616 (0.990) 0.498 (0.312) 0.828 (0.085) -0.241 (0.584) Japan 3.402 (1.000) -2.736 (1.113) 0.794 (0.181) 0.478 (0.124) -0.651 (0.363) Luxembourg 1.732 (0.538) -1.351 (0.521) 0.269 (0.087) 0.711 (0.089) -0.114 (0.171) 124 Table 4 .6. cont... Country Constant Constant (Dummy) HP-Gap t_i Inflationt_i HP-Gap t_i (Dummy) Netherlands 3.317 (0.805) -2.251 (0.696) 0.491 (0.186) 0.437 (0.131) -0.103 (0.364) New Zealand 2.473 (1.189) -1.293 (1.159) 0.162 (0.206) 0.742 (0.112) 0.257 (0.392) Norway 3.043 (1.019) -1.546 (0.831) 0.505 (0.324) 0.595 (0.129) -0.028 (0.457) Spain 3.252 (1.060) -2.223 (0.909) 0.626 (0.247) 0.727 (0.086) -0.503 (0.352) Sweden 2.832 (0.972) -1.638 (0.808) 0.510 (0.299) 0.642 (0.121) -0.045 (0.425) Switzerland 4.450 (0.620) -2.067 (0.451) 0.670 (0.106) -0.028 (0.134) 0.130 (0.186) UK 2.910 (0.986) -1.756 (1.000) 1.354 (0.296) 0.705 (0.092) -1.011 (0.403) US 1.863 (0.527) -1.001 (0.504) 0.676 (0.135) 0.703 (0.080) -0.365 (0.309) 125 Table 4.7. SUR Estimation with an Interacted End of Sample Dummy Country Constant Constant (Dummy) HP-Gap*.! 7 T t - l HP-Gapt_i (Dummy) Australia 2.034 (0.574) -1.075 (0.644) 0.792 (0.132) 0.743 (0.062) -0.547 (0.241) Austria 3.822 (0.378) -1.833 (0.351) 0.551 (0.074) 0.232 (0.066) -0.257 (0.126) Belgium 2.727 (0.370) -1.757 (0.442) 0.488 (0.085) 0.537 (0.049) -0.329 (0.144) Canada 2.445 (0.434) -1.440 (0.462) 0.535 (0.100) 0.629 (0.058) -0.359 (0.159) Denmark 3.376 (0.641) -2.429 (0.627) 0.531 (0.109) 0.606 (0.069) -0.392 (0.176) Finland 4.473 (0.712) -3.028 (0.776) 0.515 (0.099) 0.502 (0.068) -0.366 (0.129) France 2.173 (0.457) -1.790 (0.513) 0.423 (0.147) 0.730 (0.049) -0.386 (0.210) Germany 2.406 (0.303) -1.077 (0.310) 0.489 (0.058) 0.382 (0.063) -0.232 (0.085) Greece 1.929 (0.891) 0.342 (1.212) 1.279 (0.193) 0.852 (0.056) -0.628 (0.474) Ireland 4.384 (0.751) -3.449 (0.945) 0.638 (0.100) 0.567 (0.054) -0.645 (0.152) Italy 3.355 (0.695) -2.180 (0.882) 0.428 (0.136) 0.691 (0.047) -0.307 (0.275) Japan 2.961 (0.698) -2.393 (0.908) 0.616 (0.104) 0.543 (0.072) -0.420 (0.221) Luxembourg 2.253 (0.381) -1.591 (0.454) 0.244 (0.046) 0.610 (0.054) -0.152 (0.079) 126 Table 4.7. cont... Country Constant Constant (Dummy) HP-Gap t_i HP-Gap t_i (Dummy) Netherlands 4.033 (0.549) -2.700 (0.553) 0.459 (0.110) 0.312 (0.083) 0.070 (0.227) New Zealand 3.617 (0.957) -1.613 (1.062) 0.082 (0.139) 0.610 (0.083) 0.385 (0.282) Norway 4.829 (0.713) -2.223 (0.734) 0.319 (0.149) 0.342 (0.082) 0.347 (0.234) Spain 3.839 (0.747) -2.455 (0.797) 0.624 (0.137) 0.675 (0.054) -0.390 (0.219) Sweden 4.042 (0.695) -1.927 (0.731) 0.340 (0.149) 0.472 (0.079) -0.332 (0.228) Switzerland 4.886 (0.419) 0.119 (0.384) 0.720 (0.066) -0.129 (0.084) 0.163 (0.119) UK 3.717 (0.719) -2.133 (0.884) 1.344 (0.152) 0.614 (0.057) -1.033 (0.213) US 2.160 (0.368) -1.057 (0.452) 0.603 (0.066) 0.647 (0.046) -0.596 (0.164) 127 Table 4.8. SUR Estimation with an Interacted End of Sample Dummy T 1981 1983 1987 1989 Country Gapt_i Gapt_i (Dum.) Gapf_! Gap t_i (Dum.) Gapt_i Gapt_i (Dum.) Gap t_i Gap t_i (Dum.) Australia 1.113 -0.954 1.088 -0.899 0.853 -0.613 0.737 -0.501 (0.146) (0.199) (0.141) (0.196) (0.122) (0.222) (0.118) (0.229) Austria 0.540 -0.196 0.503 -0.219 0.603 -0.255 0.614 -0.143 (0.076) (0.123) (0.082) (0.138) (0.072) (0.133) (0.073) (0.157) Belgium 0.611 -0.546 0.492 -0.378 0.534 -0.402 0.537 -0.535 (0.103) (0.140) (0.096) (0.149) (0.082) (0.149) (0.085) (0.191) Canada 0.272 0.033 0.439 -0.166 0.501 -0.295 0.454 -0.238 (0.181) (0.206) (0.189) (0.219) (0.095) (0.157) (0.089) (0.154) Denmark 0.796 -0.783 0.527 -0.403 0.545 -0.550 0.422 -0.479 (0.109) (0.139) (0.111) (0.155) (0.094) (0.162) (0.092) (0.215) Finland 0.492 -0.412 0.503 -0.387 0.515 -0.362 0.489 -0.368 (0.093) (0.119) (0.097) (0.125) (0.100) (0.131) (0.100) (0.132) France 0.355 -0.342 0.327 -0.291 0.539 -0.568 0.502 -0.534 (0.151) (0.205) (0.153) (0.213) (0.135) (0.219) (0.132) (0.247) Germany 0.454 -0.143 0.473 -0.219 0.496 -0.246 0.491 -0.200 (0.064) (0.083) (0.059) (0.079) (0.058) (0.088) (0.057) (0.094) Greece 1.482 -0.839 1.477 -1.00 1.341 -0.713 1.276 -0.437 (0.203) (0.392) (0.204) (0.440) (0.154) (0.422) (0.168) (0.559) Ireland 0.663 -0.556 0.668 -0.744 0.687 -0.639 0.562 -0.667 (0.109) (0.148) (0.103) (0.146) (0.107) (0.171) (0.094) (0.174) Italy 0.498 -0.501 0.313 -0.233 0.509 -0.401 0.492 -0.445 (0.160) (0.257) (0.143) (0.263) (0.120) (0.289) (0.122) (0.314) Japan 0.569 -0.383 0.606 -0.447 0.612 -0.349 0.584 -0.375 (0.100) (0.199) (0.102) (0.204) (0.097) (0.225) (0.096) (0.253) Lux. 0.311 -0.238 0.258 -0.175 0.283 -0.291 0.278 -0.308 (0.055) (0.083) (0.050) (0.078) (0.042) (0.084) (0.042) (0.106) 128 Table 4.8. cont... r 1981 1983 1987 1989 Country Gap t_i Gap t _! (Dum.) Gap t_i Gapt_i (Dum.) Gapt-i Gap t_i (Dum.) Gap t_i Gapt_i (Dum.) Net. 0.207 0.136 0.238 0.031 0.448 0.185 0.407 0.151 (0.139) (0.199) (0.134) (0.539) (0.114) (0.272) (0.127) (0.335) N.Z. 0.045 0.217 0.073 0.347 0.181 0.095 0.196 -0.097 (0.140) (0.247) (0.129) (0.262) (0.127) (0.285) (0.112) (0.284) Norway -0.084 0.579 0.193 0.240 0.293 0.300 0.153 0.185 (0.228) (0.268) (0.187) (0.242) (0.130) (0.226) (0.124) (0.290) Spain 0.723 -0.530 0.698 -0.561 0.620 -0.255 0.596 -0.210 (0.189) (0.262) (0.165) (0.240) (0.124) (0.228) (0.121) (0.264) Sweden 0.828 -0.503 0.833 -0.575 0.611 -0.242 0.701 -0.389 (0.184) (0.236) (0.170) (0.228) (0.139) (0.218) (0.140) (0.229) Swi. 0.616 0.069 0.680 -0.054 0.730 0.267 0.674 0.271 (0.078) (0.124) (0.074) (0.121) (0.059) (0.112) (0.065) (0.133) UK 1.890 -1.542 1.557 -1.147 1.263 -0.793 1.154 -0.800 (0.186) (0.225) (0.172) (0.218) (0.153) (0.222) (0.148) (0.232) US 0.742 -0.631 0.783 -0.629 0.581 -0.540 0.568 -0.600 (0.074) (0.117) (0.072) (0.115) (0.067) (0.167) (0.063) (0.170) 129 Table 4.9. SUR Estimation with an Interacted End of Sample Dummy: Unit Sum Country Constant Constant (Dummy) HP-Gap t_i HP-Gap f_i (Dummy) Australia 0.244 (0.403) -0.529 (0.670) 0.784 (0.139) -0.774 (0.248) Austria 0.058 (0.273) -0.330 (0.456) 0.208 (0.092) -0.012 (0.176) Belgium 0.160 (0.319) -0.440 (0.532) 0.406 (0.091) -0.137 (0.148) Canada 0.212 (0.305) -0.438 (0.508) 0.405 (0.101) -0.299 (0.164) Denmark 0.205 (0.361) -0.511 (0.601) 0.711 (0.112) -0.760 (0.184) Finland 0.288 (0.523) -0.738 (0.870) 0.447 (0.121) -0.418 (0.157) France 0.171 (0.323) -0.625 (0.538) 0.347 (0.147) -0.153 (0.214) Germany -0.033 (0.236) -0.007 (0.392) 0.323 (0.068) -0.181 (0.112) Greece 0.450 (0.752) -1.129 (1.263) 1.236 (0.177) -0.495 (0.463) Ireland 0.087 (0.588) -0.702 (0.993) 0.406 (0.121) -0.461 (0.192) Italy 0.301 (0.552) -0.949 (0.918) 0.327 (0.139) 0.081 (0.308) Japan -0.167 (0.614) 0.103 (0.102) 0.419 (0.100) -0.231 (0.230) Luxembourg 0.242 (0.301) -0.610 (0.501) 0.289 (0.046) -0.145 (0.077) 130 Table 4.9. cont... Country Constant Constant (Dummy) HP-Gap t_i HP-Gap t_i (Dummy) Netherlands 0.087 (0.345) -0.091 (0.575) 0.066 (0.113) 0.350 (0.260) New Zealand 0.202 (0.677) -0.591 (0.127) 0.140 (0.151) -0.123 (0.292) Norway 0.185 (0.483) -0.453 (0.803) 0.655 (0.154) -0.333 (0.235) Spain 0.230 (0.523) -0.869 (0.868) 0.545 (0.165) -0.183 (0.262) Sweden 0.329 (0.486) -0.960 (0.810) 0.790 (0.166) -0.539 (0.244) Switzerland 0.046 (0.360) -0.339 (0.602) 0.103 (0.073) 0.237 (0.199) UK 0.292 (0.589) -0.457 (0.979) 1.308 (0.187) -1.191 (0.266) US 0.206 (0.319) -0.357 (0.531) 0.456 (0.071) -0.463 (0.188) 131 Table 4.10. SUR Estimation with an Interacted End of Sample Dummy: Energy Prices Country Constant Constant (Dummy) HP-Gapt-! 7 T t - l Energy Pricest_i HP-Gap^ i (Dummy) Australia 2.273 (0.585) -1.168 (0.629) 0.938 (0.175) 0.728 (0.063) 0.162 (0.083) -0.424 (0.258) Austria 3.642 (0.434) -1.860 (0.363) 0.542 (0.086) 0.283 (0.079) -0.026 (0.029) -0.248 (0.134) Canada 2.839 (0.504) -1.597 (0.479) 0.566 (0.120) 0.552 (0.076) 0.074 (0.058) -0.206 (0.183) Denmark 3.120 (0.856) -2.195 (0.695) 0.614 (0.138) 0.617 (0.103) 0.017 (0.025) -0.464 (0.223) Finland 5.239 (0.789) -3.409 (0.810) 0.522 (0.109) 0.415 (0.075) 0.057 (0.040) -0.323 (0.142) France 1.444 (0.545) -1.430 (0.496) 0.830 (0.166) 0.815 (0.065) -0.073 (0.038) -0.678 (0.225) Ireland 5.295 (0.745) -3.850 (0.916) 0.505 (0.110) 0.473 (0.054) 0.143 (0.032) -0.535 (0.164) Italy 3.380 (0.776) -2.248 (0.897) 0.473 (0.161) 0.394 (0.055) -0.039 (0.033) -0.210 (0.295) Japan 2.396 (0.725) -2.190 (0.920) 0.568 (0.109) 0.599 (0.076) -0.132 (0.054) -0.394 (0.224) Netherlands 3.606 (0.679) -2.512 (0.612) 0.368 (0.139) 0.400 (0.107) -0.026 (0.032) 0.089 (0.285) New Zealand 4.375 (1.108) -2.022 (1.118) 0.077 (0.164) 0.537 (0.099) 0.064 (0.085) 0.490 (0.314) Norway 4.391 (0.830) -2.071 (0.766) 0.286 (0.202) 0.419 (0.099) -0.139 (0.118) 0.126 (0.299) Sweden 3.747 (0.805) -1.793 (0.759) 0.730 (0.181) 0.513 (0.093) -0.435 (0.085) -0.370 (0.264) Switzerland 5.063 (0.489) -2.278 (0.401) 0.748 (0.076) -0.174 (0.101) 0.015 (0.013) 0.185 (0.135) UK 4.864 (0.832) -1.882 (0.914) 1.101 (0.196) 0.475 (0075) 0.180 (0.072) -0.567 (0.267) US 2.500 (0.447) -1.238 (0.486) 0.636 (0.103) 0.59 (0.066) -0.020 (0.090) -0.659 (0.214) 132 Table 4.11. SUR Estimation with a Monetary Contraction Dummy Country Constant HP-Gap t _! HP-Gap t _! (Dummy) Australia 2.211 (0.472) 0.635 (0.121) 0.640 (0.057) -0.489 (0.206) Austria 2.461 (0.336) 0.392 (0.071) 0.403 (0.066) 0.227 (0.136) Belgium 1.659 (0.345) 0.292 (0.082) 0.654 (0.060) 1.701 (0.802) Canada 2.032 (0.357) 0.268 (0.081) 0.608 (0.056) 1.029 (0.281) Denmark 2.033 (0.486) 0.337 (0.093) 0.672 (0.059) -0.860 (0.717) Finland 3.413 (0.610) 0.202 (0.081) 0.489 (0.066) 0.238 (0.162) France 1.301 (0.358) 0.313 (0.113) 0.786 (0.045) -0.481 (0.261) Germany 1.657 (0.262) 0.402 (0.047) 0.512 (0.066) -0.286 (0.104) Greece 2.659 (0.752) 1.095 (0.162) 0.789 (0.042) 0.637 (0.404) Ireland 3.065 (0.593) 0.370 (0.078) 0.615 (0.051) -0.618 (0.305) Italy 2.632 (0.585) 0.426 (0.145) 0.701 (0.051) -0.620 (0.335) Japan 1.876 (0.493) 0.623 (0.099) 0.590 (0.055) -0.584 (0.159) Luxembourg 1.642 (0.315) 0.189 (0.039) 0.612 (0.054) -0.314 (0.109) 133 Tab: e 4.11. cont.. Country Constant HP-Gap t_i TTt-l HP-Gap t_i (Dummy) Netherlands 1.694 (0.406) 0.381 (0.115) 0.618 (0.072) -0.349 (0.389) New Zealand 2.988 (0.798) 0.188 (0.127) 0.620 (0.079) -0.161 (0.308) Norway 3.550 (0.570) 0.496 (0.124) 0.420 (0.073) -0.116 (0.356) Spain 1.684 (0.657) 0.496 (0.131) 0.830 (0.058) -0.769 (0.335) Sweden 3.148 (0.608) 0.608 (0.148) 0.491 (0.076) -0.284 (0.238) Switzerland 2.947 (0.408) 0.612 (0.076) 0.207 (0.097) -0.408 (0.142) UK 3.120 (0.661) 0.381 (0.134) 0.575 (0.065) -0.117 (0.377) US 1.722 (0.330) 0.436 (0.069) 0.651 (0.050) -0.023 (0.302) Joint Test that vector equals zero X 2 (21) = 106.1742 P-Value = 0.0000 134 Table 4.12. SUR Estimation with Contraction Dummy, Separate estimates for end of sample, r = 1985 Country Const. Gapt_i 7 T t - l Gapt_i Post-85 Shock Pre-85 Shock Post-85 Pre = Post Shock = non-shock Australia 1.609 (0.458) 1.123 (0.150) 0.736 (0.056) -0.838 (0.261) -0.993 (0.228) -1.270 (0.397) p=0.50 p=0.00 Austria 2.553 (0.340) 0.373 (0.083) 0.380 (0.067) -0.048 (0.148) 0.327 (0.175) 0.080 (0.228) p=0.36 p=0.14 Belgium • Canada 1.633 (0.332) 0.509 (0.096) 0.728 (0.055) -0.566 (0.159) -0.629 (0.522) 0.790 (0.311) p=0.02 p=0.00 Denmark 1.436 (0.433) 0.717 (0.117) 0.779 (0.053) -0.609 (0.186) -1.210 (0.635) 5.186 (4.332) p=0.14 p=0.07 Finland 3.584 (0.575) 0.267 (0.119) 0.485 (0.063) -0.073 (0.156) 0.684 (0.246) -0.276 (0.230) p=0.00 p=0.02 France 0.979 (0.378) 0.630 (0.174) 0.835 (0.051) -0.501 (0.249) -1.180 (0.463) -0.439 (0.469) p=0.23 p=0.04 Germany 1.717 (0.277) 0.432 (0.069) 0.490 (0.072) -0.081 (0.113) -0.214 (0.176) -0.342 (0.148) p=0.54 p=0.12 Greece 2.181 (0.764) 1.274 (0.203) 0.827 (0.045) -0.475 (0.468) 0.326 (0.451) 6.674 (2.585) p=0.01 p=0.03 Ireland 2.692 (0.591) 0.689 (0.106) 0.645 (0.052) -0.547 (0.158) -0.952 (0.344) -1.299 (0.918) p=0.72 p=0.00 Italy 2.288 (0.605) 0.575 (0.184) 0.739 (0.054) -0.374 (0.341) -0.994 (0.415) -0.130 (0.653) p=0.23 p=0.05 Japan 1.814 (0.477) 0.861 (0.112) 0.603 (0.056) -0.242 (0.250) -0.635 (0.191) -0.964 (0.295) p=0.30 p=0.00 Lux. 1.584 (0.326) 0.196 (0.054) 0.625 (0.058) -0.021 (0.103) -0.455 (0.165) -0.485 (0.206) p=0.91 p=0.00 135 Table 4.12. cont... Country Const. Gapt_i 7Tt-l Gap t_i Shock Shock Pre = Shock = Post-85 Pre-85 Post-85 Post non-shock Net. 1.982 0.268 0.555 0.268 -1.625 0.141 p=0.04 p=0.08 (0.403) (0.129) (0.073) (0.296) (0.729) (0.494) N.Z. 2.806 -0.088 0.691 0.770 0.600 -1.087 p=0.01 p=0.00 (0.768) (0.144) (0.081) (0.273) (0.327) (0.577) Norway 3.243 0.249 0.495 0.705 0.996 -0.265 p=0.10 p=0.03 (0.587) (0.170) (0.081) (0.264) (0.572) (0.498) Spain 1.773 0.733 0.820 -0.440 -1.257 -0.932 p=0.58 p=0.00 (0.640) (0.168) (0.058) (0.249) (0.410) (0.481) Sweden 3.047 0.663 0.509 -0.263 -0.373 -0.261 p=0.80 p=0.66 (0.636) (0.193) (0.084) (0.310) (0.408) (0.291) Swi. 3.049 0.530 0.170 0.227 -0.370 0.019 p=0.34 p=0.08 (0.430) (0.087) (0.105) (0.169) (0.168) (0.396) UK 2.822 1.211 0.638 -0.992 0.621 -1.107 p=0.03 p=0.60 (0.615) (0.194) (0.063) (0.272) (0.632) (0.509) US 1.676 0.634 0.667 -0.713 -0.123 -0.522 p=0.53 p=0.89 (0.323) (0.077) (0.049) (0.199) (0.392) (0.515) Test that pre shock equals post shock column 5: x2 (20) = 50.568 P-value: 0.00 136 Table 4.13. SUR Estimation with Contraction Dummy, Separate estimates for end of sample, r = 1981 Country Const. Gapt_i TTt-l Gapt_i Shock Shock Pre = Shock = Post-81 Pre-81 Post-81 Post non-shock Austraha 2.188 1.043 0.642 -0.600 0.263 -1.149 p=0.00 p=0.06 (0.446) (0.145) (0.053) (0.228) (0.395) (0.223) Austria 2.487 0.386 0.394 -0.046 0.370 0.093 p-0.27 p=0.07 (0.326) (0.082) (0.063) (0.134) (0.171) (0.204) Belgium 1.537 0.525 0.662 -0.424 1.738 -1.547 p=0.02 p-0.09 (0.344) (0.122) (0.061) (0.152) (1.009) (1.185) Canada 1.926 0.490 0.632 -0.353 -0.151 0.754 p=0.15 p=0.00 (0.358) (0.171) (0.058) (0.196) (0.578) (0.313) Denmark 1.280 0.847 0.773 -0.713 -1.377 -1.058 p=0.82 p=0.00 (0.409) (0.122) (0.049) (0.156) (0.700) (1.201) Finland 3.423 0.239 0.506 -0.076 0.613 -0.319 p=0.00 p=0.01 (0.576) (0.119) (0.063) (0.153) (0.239) (0.200) France 0.984 0.588 0.832 -0.443 -1.197 -0.678 p=0.33 p=0.02 (0.362) (0.180) (0.047) (0.250) (0.441) (0.384) Germany 1.770 0.412 0.475 -0.030 -0.214 -0.324 p=0.60 p=0.09 (0.273) (0.078) (0.070) (0.106) (0.180) (0.151) Greece 2.838 1.505 0.748 -1.229 0.055 0.773 p=0.48 p=0.04 (0.731) (0.206) (0.043) (0.396) (0.547) (0.874) Ireland 2.576 0.748 0.685 -0.514 -2.543 -1.014 p=0.06 p=0.00 (0.590) (0.111) (0.052) (0.154) (0.764) (0.343) Italy 2.212 0.709 0.752 -0.445 -0.796 -1.259 p=0.42 p=0.06 (0.590) (0.180) (0.052) (0.303) (0.504) (0.419) Japan 1.668 0.925 0.618 -0.585 -0.733 -1.158 p=0.18 p=0.00 (0.486) (0.118) (0.059) (0.232) (0.195) (0.295) Lux. 1.679 0.263 0.591 -0.145 -1.438 -0.418 p=0.39 p=0.04 (0.315) (0.056) (0.057) (0.087) (1.191) (0.125) 137 Table 4.13. cont... Country Const. Gapt_i Gap t_i Shock Shock Pre = Shock = Post-81 Pre-81 Post-81 Post non-shock Net. 2.114 0.244 0.540 0.173 -2.278 0.215 p=0.02 p=0.01 (0.384) (0.152) (0.068) (0.215) (0.732) (0.429) N.Z. 3.262 -0.159 0.598 0.605 0.780 -0.294 p=0.03 p=0.01 (0.780) (0.138) (0.077) (0.256) (0.390) (0.354) Norway 3.110 -0.258 0.541 0.964 1.821 -0.191 p=0.01 p=0.00 (0.558) (0.243) (0.078) (0.282) (0.614) (0.497) Spain 2.207 0.864 0.755 -0.640 -1.310 -0.882 p=0.45 p=0.01 (0.636) (0.205) (0.061) (0.278) (0.426) (0.446) Sweden 2.898 0.801 0.525 -0.490 -0.200 -0.620 p=0.40 p=0.82 (0.602) (0.219) (0.076) (0.293) (0.534) (0.288) Swi. 3.007 0.551 0.183 0.089 -0.440 0.042 p=0.15 p=0.02 (0.408) (0.086) (0.098) (0.142) (0.155) (0.318) UK 2.510 1.989 0.611 -1.576 -0.570 -1.787 p=0.06 p=0.53 (0.568) (0.229) (0.056) (0.272) (0.585) (0.436) US 1.608 0.765 0.655 -0.644 -0.095 -1.033 p=0.12 p=0.66 (0.313) (0.089) (0.050) (0.139) (0.407) (0.444) Test that pre shock equals post shock: x2 (21) = 73.239 P-value: 0.00 138 Table 4.14. SUR Estimation with Contraction Dummy, Separate estimates for end of sample, r = 1987 Country Const. Gapf-x Gapt-i Post-81 Shock Pre-81 Shock Post-81 Pre = Post Shock = non-shock Australia 1.309 (0.460) 1.117 (0.159) 0.779 (0.056) -1.002 (0.275) -0.927 (0.249) -1.197 (0.424) p=0.55 p=0.00 Austria 2.457 (0.350) 0.419 (0.091) 0.404 (0.071) -0.118 (0.167) 0.193 (0.174) 0.051 (0.373) p=0.72 p=0.48 Belgium • Canada 1.774 (0.338) 0.490 (0.095) 0.705 (0.057) -0.530 (0.169) -0.615 (0.553) 0.844 (0.326) p=0.02 p=0.00 Denmark 1.590 (0.457) 0.696 (0.110) 0.749 (0.056) -0.570 (0.196) -1.354 (0.655) 5.564 (4.698) p=0.14 p=0.05 Finland 3.503 (0.586) 0.279 (0.122) 0.496 (0.066) -0.087 (0.161) 0.664 (0.255) -0.269 (0.236) p=0.00 p=0.02 France 1.013 (0.367) 0.682 (0.154) 0.842 (0.051) -0.648 (0.262) -1.322 (0.475) -0.557 (0.462) p=0.22 p=0.02 Germany 1.691 (0.277) 0.442 (0.068) 0.499 (0.073) -0.142 (0.115) -0.201 (0.174) -0.354 (0.149) p=0.47 p=0.22 Greece 2.312 (0.765) 1.249 (0.194) 0.820 (0.044) -0.086 (0.492) 0.306 (0.443) 5.232 (2.742) p=0.07 p=0.11 Ireland 2.670 (0.616) 0.674 (0.123) 0.644 (0.057) -0.562 (0.187) -0.804 (0.395) -1.424 (1.073) p=0.59 p=0.00 Italy 2.121 (0.600) 0.590 (0.173) 0.759 (0.055) -0.320 (0.388) -0.878 (0.418) 0.104 (0.645) p=0.17 p=0.09 Japan 1.847 (0.473) 0.876 (0.109) 0.603 (0.056) -0.221 (0.252) -0.647 (0.175) -1.029 (0.318) p=0.25 p=0.00 Lux. 1.633 (0.311) 0.273 (0.049) 0.620 (0.056) -0.199 (0.112) -0.552 (0.153) -0.597 (0.197) p=0.85 p=0.00 139 Table 4.14. cont... Country Const. Gapt_i Gapt_i Post-87 Shock Pre-87 Shock Post-87 Pre = Post Shock = non-shock Net. • • N.Z. 2.677 (0.792) 0.105 (0.148) 0.690 (0.080) 0.233 (0.310) 0.314 (0.332) -1.490 (0.743) p=0.02 p=0.05 Norway 3.073 (0.618) 0.380 (0.175) 0.530 (0.086) 0.461 (0.296) 1.114 (0.621) -0.403 (0.538) p=0.07 p=0.06 Spain 2.028 (0.662) 0.643 (0.152) 0.798 (0.060) -0.330 (0.278) -1.039 (0.430) -0.773 (0.494) p-0.67 p=0.04 Sweden 3.085 (0.652) 0.725 (0.199) 0.503 (0.087) -0.276 (0.324) -0.368 (0.428) -0.357 (0.305) p=0.98 p=0.57 Swi. 3.074 (0.422) 0.546 (0.084) 0.162 (0.102) 0.234 (0.169) -0.407 (0.163) 0.009 (0.384) p=0.30 p=0.04 UK 2.954 (0.618) 0.979 (0.192) 0.644 (0.065) -0.617 (0.283) 1.052 (0.567) -1.306 (0.609) p-0.00 p=0.10 US 1.699 (0.325) 0.603 (0.078) 0.665 (0.050) -0.579 (0.209) 0.050 (0.397) -0.686 (0.545) p=0.28 p=0.98 Test that pre shock equals post shock: x 2 (19) = 46.070 P-value: 0.00 140 Table 4.15. SUR Estimation with Contraction Dummy, Separate estimates for end of sample, Unit Sum Country Const. Gapt-i Gapt_i Post-81 Shock Pre-81 Shock Post-81 Pre = Post Shock = non-shock Australia -0.045 (0.315) 1.088 (0.157) -0.971 (0.268) -0.962 (0.238) -1.319 (0.403) p=0.40 p=0.00 Austria -0.070 (0.218) 0.213 (0.092) -0.099 (0.182) 0.025 (0.207) 0.326 (0.268) p=0.33 p=0.28 Belgium • • Canada 0.365 (0.224) 0.496 (0.106) -0.588 (0.175) -1.820 (0.526) 0.833 (0.351) p=0.00 p=0.00 Denmark 0.029 (0.286) 0.855 (0.121) -0.819 (0.198) -0.910 (0.677) 1.578 (4.725) p=0.60 p=0.35 Finland 0.136 (0.422) 0.242 (0.137) -0.176 (0.178) 0.837 (0.278) -0.409 (0.252) p=0.00 p=0.01 France 0.035 (0.243) 0.657 (0.184) -0.459 (0.260) -1.455 (0.440) -0.554 (0.501) p=0.14 p=0.00 Germany 0.064 (0.182) 0.289 (0.071) -0.014 (0.127) -0.570 (0.170) -0.310 (0.155) p=0.23 p=0.00 Greece 0.103 (0.596) 1.291 (0.210) -0.468 (0.482) 0.362 (0.466) 7.940 (2.666) p=0.00 p=0.01 Ireland 0.109 (0.453) 0.478 (0.114) -0.297 (0.174) -1.427 (0.358) -0.595 (0.986) p=0.43 p=0.00 Italy 0.131 (0.418) 0.620 (0.186) -0.058 (0.361) -1.103 (0.384) -0.323 (0.686) p=0.28 p=0.02 Japan -0.136 (0.465) 0.812 (0.137) -0.340 (0.300) -0.615 (0.233) -0.853 (0.352) p=0.53 p-0.01 Lux. 0.033 (0.241) 0.210 (0.056) -0.015 (0.108) -0.192 (0.157) -0.499 (0.211) p=0.24 p=0.05 141 Table 4.15. cont... Country Const. Gap t_i Gapt_i Shock Shock Pre = Shock = Post-81 Pre-81 Post-81 Post non-shock Net. 0.025 0.079 0.397 -1.863 0.008 p=0.04 p=0.04 (0.275) (0.134) (0.325) (0.761) (0.537) N.Z. 0.494 0.008 0.497 0.470 -2.049 p=0.00 p=0.00 (0.493) (0.140) (0.264) (0.321) (0.542) Norway 0.282 0.541 0.074 1.979 -1.307 p=0.00 p=0.00 (0.368) (0.159) (0.247) (0.532) (0.444) Spain 0.110 0.682 -0.285 -1.421 -0.560 p=0.19 p=0.00 (0.386) (0.185) (0.272) (0.460) (0.539) Sweden 0.008 0.511 -0.254 0.261 -0.306 p=0.20 p=0.81 (0.402) (0.207) (0.332) (0.417) (0.316) Swi. 0.111 0.158 0.191 -0.740 -0.772 p=0.95 p=0.08 (0.264) (0.087) (0.210) (0.196) (0.493) UK 0.275 1.058 -1.115 1.738 -1.277 p=0.00 p=0.03 (0.458) (0.209) (0.295) (0.668) (0.561) US 0.117 0.563 -0.812 0.528 -0.612 p=0.12 p=0.00 (0.254) (0.082) (0.223) (0.425) (0.588) Test that pre shock equals post shock: x2 (20) =114.396 P-value: 0.00 142 Table 4.16. Inflation Targeters Canada Constant Gapt Gap t_i -0.734 (0.371) 0.216 (0.265) 0.277 (0.209) -0.699 (0.361) 0.358 (0.265) 0.228 (0.208) New Zealand Constant Gapt Gapt-i 0.433 (0.292) 0.323 (0.363) 0.829 (0.182) 0.661 (0.269) 0.105 (0.148) 0.109 (0.219) United Kingdom Constant Gapt Gap t _! 7 T t - l 0.354 (0.527) 0.777 (0.479) -0.394 (0.218) 0.483 (0.523) 0.844 (0.427) -0.399 (0.210) 143 Table 4.17. Inflation Targeters (controlling for additional Lags of inflation) Canada NewZealand UK Constant -0.643 (0.433) 0.557 (0.374) 0.460 (0.363) Gapt-i 0.445 (0.361) 0.553 (0.303) 0.504 (0.431) 0.334 (0.244) 0.235 (0.271) 0.031 (0.247) -0.145 (0.243) 0.104 (0.208) -0.303 (0.186) 0.137 (0.234) -0.018 (0.208) -0.278 (0.127) 7T t _4 -0.167 (0.150) -0.252 (0.192) 0.456 (0.123) 0.054 (0.150) 0.233 (0.214) -0.337 (0.175) 0.094 (0.138) -0.244 (0.127) 0.062 (0.213) Sum on Inf. 0.307 0.058 -0.369 F-test: Lags of Inf p=0.745 p=0.186 p=0.001 144 Figure 4.1: U K , Rolling Regressions, Base Case Std. Error 145 Figure 4.2: France, Rolling Regressions, Base Case 146 Figure 4.3: Finland, Rolling Regressions, Base Case 147 Figure 4.4: Norway, Rolling Regressions, Base Case 148 gure 4.5: Changes in the Variance of Inflation and the Output Gap a a a — a > 0 O) c a o 20 15 10 5 • » -65 -55 -45 • -35 -25 -15 -5H -10 Change in Var(pi) 149 Chapter 5 Conclusion In this dissertation I have examined issues that arise when both agents and policy makers face uncertainty so that, while the policy maker may learn by observing private agents, private agents may simultaneously be attempting to learn from observing each other and the policy maker. I have argued that these issues represent important challenges to policy makers. The first essay of the dissertation showed that, contrary to what one might expect, it is possible for policy makers to play an effective role despite uncertainty and the possibility that agents may react strategically to policy initiatives. In fact, in the model presented there, the policy maker is able to eliminate all inefficiency in the economy. In the second essay it was shown that bilateral learning between policy makers and private agents can give rise to a Phillips curve relationship between output and inflation. It was also shown that the slope of such a relationship changes with the degree of information possessed by the policy maker in a fashion which matches the observed changed in Canada and the US in recent years. Finally, the third essay undertook an empirical investigation of these issues. In partic-ular, it showed that the model presented in the second essay can explain the experiences of many OECD countries. An effort to distinguish the story based on informational flows from other theories met with some success. 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