Sophisticated Monetary Policies

Transcription

1 Federal Reserve Bank of Minneapolis Research Department Sta Report 419 January 2008 Sophisticated Monetary Policies Andrew Atkeson University of California, Los Angeles, Federal Reserve Bank of Minneapolis, and NBER V. V. Chari University of Minnesota and Federal Reserve Bank of Minneapolis Patrick J. Kehoe Federal Reserve Bank of Minneapolis, University of Minnesota, and NBER ABSTRACT The Ramsey approach to policy analysis nds the best competitive equilibrium given a set of available instruments but is silent about unique implementation, namely, designing policies so that the associated competitive equilibrium is unique. This silence is problematic in monetary policy environments, where many ways of specifying policy lead to indeterminacy. We show that sophisticated policies, which depend on the history of private actions and can di er on and o the equilibrium path, can uniquely implement any desired competitive equilibrium. This result is robust to imperfect information and runs counter to the claims made in a large literature which argues that monetary policy must adhere to the Taylor principle in order to avoid indeterminacy. We show that such adherence is unnecessary for either determinacy or e ciency. Atkeson, Chari, and Kehoe thank the National Science Foundation for nancial support and Kathleen Rolfe for excellent editorial assistance. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

2 The now-classic Ramsey (1927) approach to policy analysis under commitment speci- es the set of instruments available to policymakers and nds the best competitive equilibrium outcomes with those available instruments. This approach has been extended to situations with uncertainty, by Barro (1979) and Lucas and Stokey (1983), among others, by specifying the policy instruments as functions of exogenous events. 1 Unfortunately, the Ramsey approach does not tell policymakers how to achieve those best outcomes. Doing that requires an operational guide to policy. Such a guide would tell policymakers what to do for every history of private agent actions, policies, and exogenous events and would describe what the corresponding competitive equilibrium outcomes will be for every history. Here we provide an operational guide to policy by specifying policies after every history and describing continuation outcomes after every history. In building our guide we extend the language of Chari and Kehoe (1990) to an environment in which the policymaker has commitment. We allow policies both to depend on the history of past actions by private agents and to di er on and o the equilibrium path. We require that for all histories, including those with deviations by private agents, the continuation outcomes constitute a competitive equilibrium. We label such policies sophisticated policies and the resulting equilibrium a sophisticated equilibrium. If sophisticated policies lead to a unique equilibrium outcome which supports a desired outcome, then we say the policies uniquely implement the desired outcome. We use this framework to answer an important outstanding question in monetary economics: how to design policy to avoid indeterminacy and achieve unique implementation. It has been known, at least since the work of Sargent and Wallace (1975), that when interest rates are the policy instrument, many ways of specifying policy lead to indeterminate outcomes. Such indeterminacy is risky because it can lead to lead to bad outcomes including hyperin ation. Researchers are agreed that it is desirable to avoid indeterminacy. We illustrate our sophisticated policy approach in two standard monetary economies: a simple model with one-period price-setting and a model with Calvo (1983) (staggered) pricesetting. For both, we show that, under su cient conditions, any outcome of a competitive equilibrium can be uniquely implemented by appropriately constructed sophisticated policies. In particular, the Ramsey outcomes can be implemented uniquely.

3 The basic idea of the construction is the same in both models. The goal is to construct central bank policies that uniquely implement a desired competitive equilibrium. Our method is straightforward: Along the equilibrium path, choose the policies to be those given by the desired competitive equilibrium. Structure the policies o the equilibrium path, the reversion policies, to discourage deviations. Speci cally, if the average choice of private agents deviates from that in the desired equilibrium, then choose the reversion policies so that the optimal choice, or best response, of each individual agent is di erent from the average choice. When such reversion policies can be found, we say that the best responses are controllable. A su cient condition for controllability is that policies can be found so that after a deviation the continuation equilibrium is unique and varies with policy. The latter requirement typically holds, so if policies can be found under which the continuation equilibrium is unique (somewhere), then we have unique implementation (everywhere). This su cient condition suggests a simple way to state our message in a general way: uniqueness somewhere generates uniqueness everywhere. One concern with our construction is that it apparently relies on the idea that the central bank perfectly observes private agents actions and thus can detect any deviation. We show that this concern is unwarranted: our results are robust to imperfect information about private agents actions. Our main contribution here is to show how to construct an operational guide to policymaking which achieves unique implementation of desired outcomes. This guide can be applied by using the Ramsey approach to determine the best outcomes and then checking whether in that situation, best responses are controllable. If they are, then sophisticated policies can uniquely implement the Ramsey outcome. If best responses are not controllable, then the only option is to accept indeterminacy. Our work here is related to previous work that addresses problems of indeterminacy in monetary economies (Wallace 1981; Obstfeld and Rogo 1983; King 2000; Benhabib, Schmitt-Grohe, and Uribe 2001; Christiano and Rostagno 2001; and Svensson and Woodford 2005). The previous work pursues an approach di erent from ours (and from that in the microeconomic literature on implementation); we call it unsophisticated implementation. The idea of that approach is to specify policies and check only to see if the date 0 competitive 2

4 equilibrium is unique. This approach to implementation has been criticized in the macroeconomic and the microeconomic literature. For example, in the macroeconomic literature, Kocherlakota and Phelan (1999), Buiter (2002), Ljungqvist and Sargent (2004), and Bassetto (2005) criticize this general idea in the context of the scal theory of the price level, and Cochrane (2007) criticizes it in the context of the literature on monetary policy rules. In the microeconomic literature, Jackson (2001) criticizes a similar approach to implementation. Unsophisticated implementation su ers from two problems. One is that the approach does not o er an operational guide to policy because it does not describe how the economy will behave after a deviation by private agents from the desired outcomes. The other problem is that, because the approach does not describe what will happen after a deviation, it leaves open the possibility that it achieves implementation via nonexistence. That is, this way of achieving implementation leaves open the possibility that if private agents deviate no continuation equilibrium exists. We agree with those who argue that implementation via nonexistence trivializes the implementation problem. To see why, consider the following policy: If private agents choose the desired outcome, then continue with the desired policy; if private agents deviate from the desired outcome, then forever after set government spending at a high level and taxes at zero. Clearly, under this policy, any deviation from the desired outcome leads to nonexistence of equilibrium, and hence, we trivially have implementation via nonexistence. Our approach, in contrast, insists that policies be speci ed so that a competitive equilibrium exists after any deviation. We achieve implementation in the traditional microeconomic sense, not by nonexistence, but by discouraging deviations. In our approach, policies are speci ed so that if an individual agent believes that other agents will deviate to some speci c action, that individual agent nds it optimal to choose a di erent action. Furthermore, not only do we ensure that the continuation equilibria always exist, our reversion policies have an important property: they are not extreme in any sense. They simply bring in ation back to the desired path and do not threaten the private economy with dire outcomes after deviations. Despite the shortcomings of the unsophisticated implementation approach, this liter- 3

5 ature has made two contributions that we nd useful. One is the idea of regime-switching. This idea dates back to at least Wallace (1981) and has been used by Obstfeld and Rogo (1983), Benhabib, Schmitt-Grohe, and Uribe (2001), and Christiano and Rostagno (2001). The basic idea in, say, Benhabib, Schmitt-Grohe, and Uribe (2001) is that if the economy embarks on an undesirable path, then the monetary and scal policy regime switches in such a way that the government s budget constraint is violated, and the undesirable path is not an equilibrium. We use regime-switching in some of our sophisticated monetary policies. We show that under su cient conditions, switching from an interest rate regime to a money regime after deviations can uniquely implement any desired outcome. The other valuable contribution of the literature on unsophisticated implementation is what Cochrane (2007) calls the King rule. This rule seeks to implement a desired equilibrium. It does so through an interest rate policy which makes the di erence between the interest rate and its desired equilibrium level a linear function of the di erence between in ation and its desired equilibrium level, with a coe cient greater than 1: This idea dates back to at least King (2000) and has been used by Svensson and Woodford (2005). As we show here, in our simple model, the King rule leads to indeterminacy and in the Calvo model there are some unbounded outcomes it cannot implement. Interestingly, however, in the Calvo model the King rule, when interpreted as a sophisticated policy can implement any bounded equilibrium. Under our de nition of equilibrium, in the periods following a deviation, equilibrium outcomes return to the desired outcome path. The policy in our equilibrium is set so that deviations are not optimal. In this sense, the King rule implements outcomes uniquely under our notion of equilibrium and ensures a well-de ned continuation equilibrium after deviations. Our work here is also related to another substantial literature that aims to nd monetary policy rules which eliminate indeterminacy. (See, for example, the work of McCallum 1981 and, more recently, Woodford 2003.) The recent literature argues that to achieve a unique outcome, interest rate rules should follow the Taylor principle: interest rates should rise more than one-for-one with rises in in ation rates. A related literature argues that the undesirable in ation experiences of the 1970s in the United States were due in large part to 4

6 the failure of monetary policy to obey the Taylor principle in that time period. (See, for example, the work of Clarida, Galí, and Gertler 2000.) The Taylor principle has been interpreted in a common, widely used way and in an alternative, less widely used way. The common interpretation, stressed by Taylor (1993) and Clarida, Galí, and Gertler (2000), among others, is that the Taylor principle refers to the comovements of interest rates and in ation rates along the equilibrium path. Here, these comovements are measured using the deviations of interest rates and in ation rates from some constants, thought of as the steady-state levels of interest rates and in ation rates. In the realized equilibrium path of the model, observed interest rates and in ation rates uctuate, so that deviations of these variables from their steady states are typically nonzero. Under this common interpretation of the Taylor principle, the stochastic processes for interest rates and in ation rates satisfy the principle if, on average, along the equilibrium path a rise in in ation rates is associated with a more than one-for-one rise in interest rates. We show here that the Taylor principle, interpreted in this way, is not necessary for determinacy. In particular, sophisticated policies can uniquely implement any desired competitive equilibrium, including those which violate the common interpretation of the Taylor principle. It follows that adherence to the Taylor principle is not needed for uniqueness. Moreover, our ndings imply that historical evidence that policy violated the Taylor principle in some periods does not necessarily imply that such policy was unwise. We also show that, under the common interpretation, following the Taylor principle is not necessary to support e cient outcomes. The less common interpretation of the Taylor principle is that it recommends that the monetary authority follow the King rule, which we have previously discussed. Here, we propose one way to eliminate indeterminacy under interest rate rules. For some other proposed resolutions to the indeterminacy issue, see the work of Bassetto (2002) and Adão, Correia, and Teles (2007). 1. A Simple Model with One-Period Price-Setting We begin by illustrating the basic idea of our construction of sophisticated policies using a model with one-period price-setting. The one-period nature of price-setting ensures 5

7 that the dynamical system associated with the competitive equilibrium of this model is very simple, which lets us focus on the strategic aspects of sophisticated policies. With this model, we demonstrate that any desired outcome of a competitive equilibrium can be uniquely implemented by appropriate sophisticated policies. We also show that, in this model, regime-switching is necessary in order to reach the desired outcome uniquely and that policies satisfying the Taylor principle cannot uniquely implement desired outcomes. The model we analyze here is a modi ed version of the basic sticky price model with a New Classical Phillips curve (as in Woodford 2003, Chap. 3, Sec. 1.3). In order to make our results comparable to those in the literature, we here describe a simple, linearized version of the model. In Appendix A, we describe the general equilibrium version that, when linearized, produces the equilibrium conditions studied here. A. The Determinants of Output and In ation Consider a monetary economy populated by a large number of identical, in nitely lived consumers, a continuum of producers, and a central bank. Each producer uses labor to produce a di erentiated good on the unit interval. A fraction of producers i 2 [0; ) are exible price producers, and i 2 [; 1] are sticky price producers. In this economy, the timing within a period t is as follows. At the beginning of the period, sticky price producers set their prices, after which the central bank chooses its monetary policy by setting one of its instruments, either interest rates or the quantity of money. Two shocks t and t are then realized. We interpret the shock t as a ight to quality shock that a ects the attractiveness of government debt relative to private claims and the shock t as a velocity shock. At the end of the period, exible price producers set their prices, and consumers make their decisions. Now we develop necessary conditions for a competitive equilibrium in this economy and then, in the next subsection, formally de ne a competitive equilibrium. Here and throughout, we express all variables in log-deviation form. In particular, this way of expressing variables implies that none of our equations will have constant terms. Consumer behavior in this model is summarized by an intertemporal Euler equation 6

8 and a cash-in-advance constraint. We can write the linearized Euler equation as (1) y t = E t [y t+1 ] (i t E t [ t+1 ]) + t ; where y t is aggregate output, i t is the nominal interest rate, t (the ight to quality shock) is an i.i.d. mean zero shock, and t+1 = p t+1 p t is the in ation rate from time period t to t + 1, where p t is the aggregate price level. The parameter determines the intertemporal elasticity, and E t denotes the expectations of a representative consumer given that consumer s information in period t, which includes the shock t : The cash-in-advance constraint, when rst-di erenced, implies that the relationships among in ation t ; money growth t ; and output growth y t y t 1 are given by a quantity equation of the form (2) t = t (y t y t 1 ) + t ; where t (the velocity shock) is an i.i.d. mean zero shock. We turn now to producer behavior. The optimal price set by an individual exible price producer i satis es (3) p ft (i) = p t + y t ; where the parameter is the elasticity of the equilibrium real wage with respect to output (often referred to in the literature as Taylor s ). The optimal price set by a sticky price producer i satis es (4) p st (i) = E t 1 [p t + y t ] ; where E t 1 denotes expectations at the beginning of period t before the shocks t and t are realized. The aggregate price level p t is a linear combination of the prices p ft set by the exible price producers and the prices p st set by the sticky price producers and is given by (5) p t = Z 0 p ft (i) di + Z 1 p st (i) di: Using language from game theory, we can think of equations (3) and (4) as akin to the best responses of the exible and sticky price producers given their beliefs about the aggregate price level and aggregate output. 7

9 In this model, the exible price producers are strategically uninteresting. Their expectations about the future have no in uence on their decisions; their prices are set mechanically according to the static considerations re ected in (3). Thus, in all that follows, equation (3) will hold on and o the equilibrium path, and we can think of p ft (i) as being residually determined by (3) and substitute out for p ft (i). To do so, substitute (3) into (5) and solve for p t to get (6) p t = y t + 1 Z 1 p st (i) di; 1 where = =(1 ): We follow the literature and express the sticky price producers decisions in terms of in ation rates rather than price levels. To do so, let x t (i) = p st (i) p t 1, and rewrite (4) as (7) x t (i) = E t 1 [ t + y t ] : For convenience, we de ne (8) x t = 1 Z 1 x t (i) di 1 to be the average price set by the sticky price producers relative to the aggregate price level in period t 1; so that we can rewrite (7) as (9) x t = E t 1 [ t + y t ] : We can also rewrite (6) as (10) t = y t + x t : Note that in this model any competitive equilibrium gives rise to a New Classical Phillips curve: (11) t = y t + E t 1 [ t ]: To see this result, note that by taking expectations of (10) as of t 1 and substituting into (9), we obtain E t 1 [y t ] = 0; so that (11) follows from (9) and (10). Note that when it sets monetary policy, the central bank has to choose to operate under either a money regime or an interest rate regime. In the money regime, the central 8

10 bank s policy instrument is money growth t ; it sets t, and the nominal interest rate i t is residually determined from (1) after the realization of the shock t : In the interest rate regime, the central bank s instrument is the interest rate; it sets i t, and money growth t is residually determined from (2) after the realization of the shock t : Of course, in both regimes, equations (1) and (2) hold. B. Competitive Equilibrium Now we de ne a notion of competitive equilibrium in the spirit of the work of Barro (1979) and Lucas and Stokey (1983). In this equilibrium, allocations, prices, and policies are all de ned as functions of the history of exogenous events, or shocks, s t = (s 0 ; : : : ; s t ), where s t = ( t ; t ): The actions of the sticky price producers, exible price producers, and consumers can be summarized by fx t (s t 1 ); t (s t ); y t (s t )g: In terms of the policies, for convenience, we let the regime choice as well as the policy choice within the regime be t (s t 1 ) = ( 1t (s t 1 ); 2t (s t 1 )), where the rst argument 1t (s t 1 ) 2 fm; Ig denotes the regime choice, either money (M) or the interest rate (I), and the second argument denotes the policy choice within the regime, either money growth t (s t 1 ) or the interest rate i t (s t 1 ): Let fa t (s t )g t0 = fx t (s t 1 ); t (s t 1 ); t (s t ); y t (s t )g t0 denote a collection of allocations, prices, and policies in this competitive equilibrium. Such a collection is a competitive equilibrium if it satis es (i) consumer optimality, namely, (1) and (2) for all s t ; (ii) optimality by sticky price producers, namely, (9) for all s t 1 ; and (iii) optimality by exible price producers, namely, (10) for all s t. We also de ne a continuation competitive equilibrium starting from any point in time. For example, consider the beginning of period t with state variables s t 1 and y t 1 : A collection of allocations, prices, and policies fa(s t 1 ; y t 1 )g rt = fx r (s r 1 js t 1 ; y t 1 ); r (s r 1 js t 1 ; y t 1 ); r (s r js t 1 ; y t 1 ); y r (s r js t 1 ; y t 1 )g rt is a continuation competitive equilibrium from (s t 1 ; y t 1 ) if it satis es the three conditions of competitive equilibrium above for all periods starting from (s t 1 ; y t 1 ): We de ne a continuation competitive equilibrium that starts at the end of period t given (s t 1 ; y t 1 ; x t ; t ; s t ) in a similar fashion. This de nition requires optimality by consumers and exible price producers from s t onward and optimality by sticky price producers from s t+1 onward. Obviously, a 9

11 continuation competitive equilibrium starting from the history h 1 is simply a competitive equilibrium. C. Sophisticated Equilibrium We now turn to what we call sophisticated equilibrium. The de nition of this concept is very similar to that for competitive equilibrium except that here we allow allocations, prices, and policies to be functions of more than just the history of exogenous events; they are also functions of the history of both aggregate private actions and central bank policies. For sophisticated equilibrium, we require as well that for every history, the continuation of allocations, prices, and policies for that history onward constitute a continuation competitive equilibrium. Setup and De nition Before we turn to our formal de nition, we note that our de nition of sophisticated equilibrium simply speci es policy rules that the central bank must follow; it does not require that the policy rules be optimal. We specify sophisticated policies in this way in order to show that our unique implementation result does not depend on the objectives of the central bank. We think of sophisticated policies as being speci ed at the beginning of period 0 and of the central bank as being committed to following them. We turn now to de ning the histories that private agents and the central bank confront when they make their decisions. The public events that occur in a period are, in chronological order, q t = (x t ; t ; s t ; y t ; t ). Letting h t denote the history of these events up to and including period t, we have that h t = (h t 1 ; q t ) for t 0. The history h 1 = y 1 is given. For notational convenience, we focus on perfect public equilibria in which the central bank s strategy (choice of regime and policy) is a function of only the public history. The public history faced by the sticky price producers at the beginning of period t when they set their prices is h t 1 : A strategy for the sticky price producers is a sequence of rules x = fx t (h t 1 )g for choosing prices for every possible public history. The public history faced by the central bank when it chooses its regime and sets either its money growth or interest rate policy is h gt = (h t 1 ; x t ): A strategy for the central bank f t (h gt )g is a sequence of rules for choosing the regime as well as the policy within the regime, 10

12 either t (h gt ) or i t (h gt ). Let g denote that strategy. At the end of period t, then, output and in ation are determined as functions of the relevant history h yt according to the rules y t (h yt ) and t (h yt ): We let y ={y t (h yt )} and ={ t (h yt )} denote the sequence of output and in ation rules. Notice that for any history, the strategies induce continuation outcomes in the natural recursive way. For example, starting at some history h t 1 ; these strategies induce outcomes fa r (s r jh t 1 ; )g as follows. The sticky price producer s decision in t is given by x t (s t 1 jh t 1 ; ) = x t (h t 1 ); where x t (h t 1 ) is obtained from x : The central bank s decision in t is given by t (s t 1 jh t 1 ; ) = t (h gt ); where h gt = (h t 1 ; x t (h t 1 )) and t (h gt ) is obtained from g : The consumer and exible price producer decisions in t are given by y t (s t jh t 1 ; ) = y t (h yt ) and t (s t jh t 1 ; ) = t (h yt ); where h yt = ((h t 1 ; x t (h t 1 ); t (h t 1 ; x t (h t 1 ))) and y t (h yt ) and t (h yt ) are obtained from y and : Continuing in a similar way, we can recursively de ne continuation outcomes for subsequent periods. We can likewise de ne continuation outcomes fa r (s r jh gt ; )g and fa r (s r jh yt ; )g following histories h gt and h yt ; respectively. We now use these strategies and continuation outcomes to de ne our notion of equilibrium. A sophisticated equilibrium given the policies here is a collection of strategies ( x ; g ) and allocation rules ( y ; ) such that (i) given any history h t 1 ; the continuation outcomes fa r (s r jh t 1 ; )g induced by constitute a continuation competitive equilibrium and (ii) given any history h yt ; so do the continuation outcomes fa r (s r jh yt ; )g. 2 Associated with each sophisticated equilibrium = ( g ; x ; y ; ) are the particular stochastic processes for outcomes that occur along the equilibrium path, which we call sophisticated outcomes. These outcomes are competitive equilibrium outcomes. A central feature of our de nition of sophisticated equilibrium is our requirement that for all histories, including deviation histories, the continuation outcomes constitute a competitive equilibrium. This requirement constitutes the most important di erence between our approach and that in the literature. Technically, one way of casting the literature s approach into our language of strategies and allocation rules is to consider the following notion of equilibrium. An unsophisticated equilibrium is a strategy for the central bank g and allocations, policies, and prices fa t (s t )g t0 = fx t (s t 1 ); t (s t 1 ); t (s t ); y t (s t )g t0 11

13 such that fa t (s t )g t0 is a period 0 competitive equilibrium and the policies induced by g from fa t (s t )g t0 coincide with f t (s t 1 )g t0. In our view an unsophisticated equilibrium is not an operational guide to policy. While an unsophisticated equilibrium does tell policymakers what to do for every history, it does not specify what will happen under their policies for every history, in particular, for deviation histories. Achieving implementation using the notion of unsophisticated equilibrium is, in general, trivial. One way of achieving implementation is via nonexistence: simply specify policies so that no competitive equilibrium exists following deviation histories. We nd this way of achieving implementation unpalatable. Equilibrium with Sophisticated Policies We now show that in the simple sticky price model, any competitive equilibrium in which the central bank uses the interest rate as its instrument can be uniquely implemented with sophisticated policies. (Later we show that when the central bank uses money as its instrument, unique implementation is trivial.) The basic idea behind our sophisticated policy construction is that the central bank starts by picking any desired competitive equilibrium allocations and sets its interest rate policy on the equilibrium path consistent with them. The central bank then constructs its policy o the equilibrium path so that if an individual agent believes that all other agents will deviate to some speci c action, that individual agent nds it optimal to choose a di erent action. To set up our construction of sophisticated policies, recall that in our economy the only strategically interesting agents are the sticky price producers. Rewriting (9) using our language of strategies, their choices must satisfy a key property, that (12) x t (h t 1 ) = E [ t (h yt ) + y t (h yt )] ; where h yt = ((h t 1 ; x t (h t 1 ); t (h t 1 ; x t (h t 1 )); s t ): Notice that x t (h t 1 ) shows up on both sides of equation (12), so we require that the optimal choice x t (h t 1 ) satisfy a xed point property. To get some intuition for this property, suppose that each sticky price producer believes that all other sticky price producers will choose some value, say, ^x t : This choice, together with the central bank s strategy and the in ation and output rules, induces the 12

14 outcomes t (^h yt ) and y t (^h yt ); where ^h yt = (h t 1 ; ^x t ; t (h t 1 ; ^x t ); s t ): The xed point property requires that in order for ^x t to be part of an equilibrium, each sticky price producer s best response must coincide with ^x t. Our implementation result relies on constructing policies so that the xed point property is satis ed at only the desired allocations. To show this formally, we begin by proving a lemma which says that the model has a key controllability property: after a deviation, the central bank can revert to a money regime and choose money growth rates so as to make it not optimal for any individual price-setter to cooperate with the deviation. Lemma 1. (Controllability of Best Responses with One-Period Price-Setting) For any history (h t 1 ; ^x t ); if the central bank chooses the money regime, then there exists a choice for money growth t such that (13) ^x t 6= E h t (^h yt ) + y t (^h yt ) i ; where h yt = (h t 1 ; ^x t ; M; t ): Proof. Substituting (2) into (10), we have that if the central bank chooses the money regime with money growth t ; then output y t and in ation t are uniquely determined and given by (14) (15) y t = t + t + y t 1 ^x t 1 + t = y t + ^x t : Hence, E h t (^h yt ) + y t (^h yt ) i = ( t + y t 1 ^x t ) + ^x t : Clearly, then, any choice of t 6= ^x t y t 1 will ensure that (13) holds. Q:E:D: We use this lemma to establish that sophisticated policies can uniquely implement any desired competitive equilibrium. To do so, we consider sophisticated policies with one-period reversion to money. Under these policies, following a deviation, the central bank switches to a money regime for one period. In particular, after a deviation, the central bank switches to a level of the money supply which generates the same expected in ation as in the original 13

15 equilibrium. (Of course, we could have chosen many other values that also would discourage deviations, but we found this value to be the most intuitive. 3 ) More precisely, x a desired competitive equilibrium outcome path (x t (s t 1 ); t (s t ); y t (s t )) together with central bank policies i t (s t 1 ). Consider the following trigger-type policy: If sticky price producers choose x t in period t to coincide with the desired outcomes x t (s t 1 ), then let central bank policy in t be i t (s t ^x t 6= x t (s t 1 ): If not, and these producers deviate to some 1 ); then for that period t; let the central bank switch to a money regime with money growth set so that (16) t = ^x t y t h x t (s t 1 ) ^x t ) i : Note that t 6= ^x t y t 1 : With such a money growth rate, expected in ation is the same in the reversion phase as it would have been in the desired outcome. From Lemma 1, such a choice of ^x t cannot be part of an equilibrium. We have established the following proposition. Proposition 1. Unique Implementation with Sophisticated Policies. Any competitive equilibrium outcome in which the central bank uses interest rates as its instrument can be implemented as a unique equilibrium with sophisticated policies with one-period reversion to money. A simple way to describe our unique implementation result is that uniqueness of best responses under some regime guarantees unique implementation of any desired outcome. Note that if the variance of the money shock t is large, then all of the outcomes under the money regime are undesirable. Nevertheless, the money regime is useful as an o -equilibrium commitment that helps support desirable outcomes along the equilibrium path under interest rate regimes. As a technical aside, note from the proof of Proposition 1 that we do not need uniqueness for all money growth policies. Rather, all we need is to be able to nd for every deviation a central bank policy that induces a best response correspondence which does not include the deviation. So far we have focused on uniquely implementing competitive outcomes when the central bank uses interest rates as its instrument in the desired equilibrium. Our controllability lemma implies that implementing competitive outcomes is trivial when the central 14

16 bank uses money as its instrument in the desired equilibrium. Clearly, we can use a simple generalization of Proposition 1 to uniquely implement a competitive equilibrium in which the central bank uses interest rates in some periods and money in others: in periods of an interest rate regime, use sophisticated policies with reversion to money, while in periods of a money regime, make the money growth independent of the decisions of private agents. Necessity of Regime-Switching for Unique Implementation We now show that in this model regime-switching is necessary for unique implementation. To do so, we consider a common way of modeling what are called restricted policies. The restriction is to require policies to be the same on and o the equilibrium path. Such policies are typically assumed to be linear functions of private agents actions. Here we show that any interest rate policies that are linear functions of actions and shocks that the central bank has observed lead to a continuum of equilibria. Hence, such policies cannot uniquely implement any desired outcome. In this model, therefore, in order to uniquely implement any desired outcome, after deviations, the central bank must switch from an interest rate regime to a money regime. Consider a class of restricted policies of the form 1X 1X 1X (17) i t = { t + xs x t s + ys y t s + s t s ; s=0 s=1 s=1 where the intercept term { t can depend on the history of stochastic events. Notice that policies of this kind are linear feedback rules on variables in the central bank s history. We then can establish the following result. Proposition 2. Indeterminacy of Equilibrium under Restricted Policies. Competitive equilibria with interest rate rules of the feedback form (17) have outcomes of the form (18) x t+1 = i t + c t ; t = x t + (1 + c) t ; and y t = (1 + c) t : For every feedback rule, the economy has a continuum of competitive equilibria indexed by x and the parameter c: Proof. In order to verify that the outcomes which satisfy (18) are part of an equilibrium, we need to check that they satisfy (1), (9), and (10). That they satisfy (9) follows by 15

17 taking expectations of the second and third equations in (18). Substituting for x t+1 from (18) and i t from (17) into (1), we obtain that y t = (1 + c) t, as required by (18). Inspecting the expressions for t and y t in (18) shows that they satisfy (10). Q:E:D: Proposition 2 shows that if the central bank follows an interest rate regime in all periods for all histories, then the economy has a continuum of competitive equilibria. In this sense, unique implementation requires regime-switching. The intuitive idea behind the multiplicity of equilibria associated with the initial condition x 0 is that interest rate rules induce nominal indeterminacy and do not pin down the initial price level. The intuitive idea behind the multiplicity of stochastic equilibria associated with c 6= 0 is that interest rate rules pin down only expected in ation and do not completely pin down the state-by-state realizations of either nominal or real variables. Next we show that the class of linear feedback rules in (17) includes a popular speci- cation of the Taylor rule of the form (19) i t = { t + E t 1 t + be t 1 y t : When the parameter > 1, such policies are said to satisfy the Taylor principle: the central bank should raise its interest rate more than one-for-one with increases in in ation. When < 1, such policies are said to violate that principle. Of course, the Taylor rule is not a well-de ned function of histories until we ll in how expectations are formed. To do so, we show that in any history h t 1, (20) E [y t jh t 1 ] = 0 and x t = E [ t jh t 1 ]. To see these results, take expectations of (10) as of h t 1 and substitute into (9). We then obtain E [y t jh t 1 ] = 0; and using this result in (10), we obtain x t = E [ t jh t 1 ]. Also, if history h t 1 leads to an interest rate regime, we have that (21) E [ t+1 jh t 1 ] = i t (h gt ); where h gt = (h t 1 ; x t (h t 1 )): To see this result, take expectations of the Euler equation (1) with respect to h t 1 to get that (22) E [y t jh t 1 ] = E [y t+1 jh t 1 ] (i t (h t 1 ) E [ t+1 jh t 1 ] ): 16

18 Using the law of iterated expectations gives that E [y t+1 jh t 1 ] = 0: From (22), we then have (21). From these results, policies of the Taylor rule form (19) can be written as (23) i t = { t + x t : Corollary. (Indeterminacy with Taylor Rules) Consider Taylor rules of the form (23) with { t = 0. This economy has a continuum of equilibria regardless of the value of the parameter : If 1, the economy has a continuum of unbounded equilibria indexed by c and x 0 as well as a unique bounded equilibrium with c = 0 and x t = 0 for all t. If < 1, then the economy has a continuum of bounded equilibria indexed by c and x 0. Proof. From Proposition 2 we know the equilibria are of the form given in (18). The proof of this corollary follows from substituting from (23) into the rst equation in (18) to obtain a di erence equation in expected in ation: (24) x t+1 = x t + c t : Clearly, if > 1; either x t is unbounded or x t = c = 0 for all t: If < 1; clearly the economy has a continuum of bounded equilibra. Q:E:D: In the literature, the policy rule (23) is said to satisfy the Taylor Principle if > 1 and adherence to this principle is thought to be necessary and su cient for uniqueness. Proposition 1 shows that adherence to this principle is not necessary for unique implementation and Proposition 2 shows that adherence to this principle is not su cient either. One justi cation for adherence to the Taylor Principle advanced in the literature is that, if we restrict attention to bounded equilibria, the economy has a unique equilibrium. Here, we argue that equilibria in which in ation is unbounded cannot be dismissed on logical grounds. Equilibria in which in ation is unbounded are perfectly reasonable in this model because an in ation explosion is associated with a money supply explosion. To see this association, suppose that policy is described by a Taylor rule of the form (23) with { = 0 and > 1, and for simplicity, that the economy has no stochastic shocks so 17

19 that t = t = 0 for all t: Using (20), then, we know that y t = 0 for all t, and hence, from (2) the growth of the money supply is given by (25) t = x t = t x 0 : Thus, in these equilibria, in ation explodes because money growth explodes. Each equilibrium is indexed by a di erent initial value of the endogenous variable x 0 : This endogenous variable depends solely on expectations of future policy and is not pinned down by any initial condition or transversality condition. The idea that the central bank s printing of money at an ever-increasing rate leads to a hyperin ation is at the core of most monetary models. In these equilibria, in ation does not arise from the speculative reasons analyzed by Obstfeld and Rogo (1983), but from the conventional money-printing reasons analyzed by Cagan (1956). In this sense, our model predicts, for perfectly standard and sensible reasons, that if the central bank follows a Taylor rule which satis es the Taylor principle, then the economy can su er from any one of a continuum of very undesirable paths for in ation. (Cochrane 2007 makes a similar point for a exible price model.) We now show that rules of the form speci ed in (17) include rules of the form discussed by King (2000) and Svensson and Woodford (2005): (26) i t = i t + (E t 1 t E t 1 t ); where i t and t can depend on the history of stochastic events. The idea behind a King rule of the form (26) is that i t and t are the interest rates and in ation rates associated with a competitive equilibrium which the central bank wants to implement uniquely. From (20), we know that the King rule can be written in the form (27) i t = i t + (x t x t ): Clearly, such a rule is of the linear feedback rule form (17). Note that the King rule is a Taylor rule with the additional requirement that i t and t are competitive outcomes. That the King rule is neither necessary nor su cient for unique implementation follows from Propositions 1 and 2. 18

20 In Proposition 2, we focused on linear competitive equilibria that can be described as time-invariant linear functions of the shocks: Clearly, there are other competitive equilibria in which the coe cients of the allocation rules depend on period t as well as the history of the shocks. There are also competitive equilibria in which the allocations depend on exogenous sunspots. Our theorems about supporting competitive equilibrium outcomes with sophisticated policy rules apply to all of these equilibria too. Extension to Interest-Elastic Money Demand So far, to keep the exposition simple, we have assumed a cash-in-advance setup in which money demand is interest-inelastic. This feature of the model implies that if a money regime is adopted in some period t, then the equilibrium outcomes in that period are uniquely determined by the money growth rate in that period. This uniqueness under a money regime is what allows us to switch to a one-period money regime to support any desired competitive equilibrium. Now we consider economies with interest-elastic money demand. We argue that under appropriate conditions, our unique implementation result extends to such economies. For such economies, consider sophisticated policies that specify an interest rate regime along the equilibrium path and an in nite reversion to money after a deviation. Such policies can uniquely implement any desired outcome if, again, best responses are controllable. A su cient condition for such controllability is that competitive equilibria are unique under a suitably chosen money regime. Here, as with inelastic money demand, the uniqueness under a money regime is what allows us to use reversion to money to support any desired competitive equilibrium. A sizable literature has analyzed the uniqueness of competitive equilibrium under money growth policies with interest-elastic money demand. Obstfeld and Rogo (1983) and Woodford (1994) provide su cient conditions for this uniqueness. For example, Obstfeld and Rogo (1983) consider a money-in-the-utility function model with preferences of the form u(c) + v(m), where m is real money balances, and show that a su cient condition for uniqueness under a money regime is for lim m!0 mv0 (m) > 0: Obstfeld and Rogo focus attention on exible price models, but their results can be 19

21 readily extended to our simple sticky price model. Indeed, their su cient conditions apply unchanged to a deterministic version of that model because our model without shocks is e ectively identical to a exible price model. Hence, under appropriate su cient conditions, our unique implementation result extends to environments with interest-elastic money demand. 2. A Model with Staggered Price-Setting We turn now to a version of our simple model with staggered price-setting, often referred to as the New Keynesian model. Our main point here is to show that the primary result of the simple model, that sophisticated policies can uniquely implement any equilibrium allocation, carries through to this widely used setting. To make this point in the simplest way, we abstract from aggregate uncertainty. We show that, along the lines of the argument developed above, sophisticated policies can uniquely implement any desired outcome under an interest rate regime with an in nite reversion to money after a deviation. We also study a question relevant to the recent use of interest rates as the main instrument of monetary policy: Can sophisticated policies uniquely implement desired outcomes using interest rate regimes both on and o the equilibrium path? We have seen that in the simple model, they cannot; now we show that here, under su cient conditions, they can. Finally, as we did with the simple sticky price model, here we investigate the implications of our analysis for the King rule and the Taylor principle. A. Setup We begin by setting up the model with staggered price-setting. The model has no aggregate uncertainty, and in it, prices are set in a staggered fashion as in the work of Calvo (1983). At the beginning of each period, a fraction 1 of producers are randomly chosen and allowed to reset their prices. After that, the central bank makes its decisions, and then, nally, consumers make theirs. This economy has no exible price producers. The nonlinear version of this economy is described in Appendix A. The linearized equations in this model are similar to those in the simple model. The Euler equation (1) and the money growth equation (2) are unchanged except that here they have no shocks, t ; t. The price set by a producer that is permitted to reset its price is given 20

22 by the analog of (4), which is " 1 # X (28) p st (i) = (1 ) () r t (y r + p r ) ; r=t where is the discount factor. Here, again, Taylor s is the elasticity of the equilibrium real wage with respect to output: Letting p st denote the average price set by producers that are permitted to reset their prices in period t; we can recursively rewrite this equation as (29) p st (i) = (1 ) [y t + p t ] + p st+1 ; together with a type of transversality condition lim T!1 () T p st (i) = 0: The aggregate price level can then be written as (30) p t = p t 1 + (1 )p st : To make our analysis parallel to the literature, we again translate the decisions of the sticky price producers from price levels to in ation rates. Letting x t (i) = p st (i) p t 1 and letting x t denote the average of x t (i); with some manipulation, we can rewrite (29) as (31) x t = (1 )y t + t + x t+1 : We can also rewrite (30) as (32) t = (1 )x t : The transversality condition above can be rewritten in terms of in ation rates as lim T!1 () T x t (i) = 0: Using (32), this restriction is equivalent to (33) lim T!1 ()T t = 0: In addition to these conditions, we now argue that in this staggered price-setting model a competitive equilibrium must satisfy two boundedness conditions. Such conditions are controversial in the literature. Standard analyses of New Keynesian models impose strict boundedness conditions: in any reasonable equilibrium, both output and in ation must be bounded both above and below. Cochrane (2007) has forcefully criticized this practice, arguing that any boundedness conditions must have a solid economic rationale. 21

23 Here we provide rationales for two such conditions. 4 Since the economy has a nite amount of labor, it can produce only a nite amount of output and it makes sense to require that output y t be bounded above, so that (34) y t y for some y. Since the nominal interest rate must be nonnegative.it makes sense to require that in ation be bounded below, so that (35) for some : We think of the boundedness conditions (34) and (35) as being minimal. These bounds allow for outcomes in which y t ; (the log of) output, falls without bound (so that the level of output converges to zero). The bounds also allow for outcomes in which in ation rates explode upward without limit. For completeness, we provide below conditions under which our unique implementation result holds with stricter and weaker boundedness conditions as well. Here, then, a collection of allocations, prices, and policies a t = fx t ; t ; t ; y t g is a competitive equilibrium if it satis es (i) consumer optimality, namely, the deterministic versions of (1) and (2); (ii) sticky price producer optimality, (31) (33); and (iii) the boundedness conditions, (34) and (35). Note that any allocations that satisfy (31) (33) also satisfy the New Keynesian Phillips curve: (36) t = y t + t+1 ; where now = (1 )(1 )=: To see this result, use (32) to substitute for x t and x t+1 in (31) and collect terms. Here, as in the simple sticky price model, we also de ne continuation competitive equilibria. For example, consider the beginning of period t with a state variable y t 1 : A collection of allocations fa r (y t 1 )g rt = fx r (y t 1 ); r (y t 1 ); r (y t 1 ); y r (y t 1 )g rt is a continuation competitive equilibrium with y t 1 given if it satis es the three conditions of competitive equilibrium above in all periods r t: This de nition requires optimality by sticky price producers and consumers from t onward. A continuation competitive equilibrium that starts at 22

24 the end of period t given (y t 1 ; x t ; t ) is de ned similarly. This de nition requires optimality by consumers from t onward and optimality by sticky price producers only from t+1 onward. B. Sophisticated Policies The de nition of a sophisticated equilibrium in the staggered price-setting model parallels that in the simple sticky price model. The elements needed for that de nition are basically the same. The public events that occur in a period are, in chronological order, q t = (x t ; t ; y t ; t ). We let h t 1 denote the history of these events up until the beginning of period t: A strategy for the sticky price producers is a sequence of rules x = fx t (h t 1 )g. The public history faced by the central bank is h gt = (h t 1 ; x t ); and its strategy, f t (h gt )g: The public history faced by consumers in period t is h yt = (h t 1 ; x t ; t ): We let y ={y t (h yt )} and ={ t (h yt )} denote the sequences of output and in ation rules. Strategies and allocation rules induce continuation outcomes in the obvious recursive fashion. We write these continuation outcomes as fa r (h t 1 ; )g rt and fa r (h yt ; )g rt. Formally, then, a sophisticated equilibrium given the policies here is a collection of strategies ( x ; g ) and allocation rules ( y ; ) such that (i) given any history h t 1 ; the continuation outcomes fa r (h t 1 ; )g rt induced by constitute a continuation competitive equilibrium and (ii) given any history h yt ; so do the continuation outcomes fa r (h yt ; )g rt. We now show that here any competitive equilibrium can be uniquely implemented with sophisticated policies. The basic idea behind our construction is that the central bank starts by picking any competitive equilibrium allocations and sets its policy on the equilibrium path consistent with this equilibrium. The central bank then constructs its policy o the equilibrium path so that any deviations from these allocations would never be a best response for any individual price-setter. In so doing, the constructed sophisticated policies support the chosen allocations as the unique equilibrium allocations. In this model, as in the simple sticky price model, the choices of the sticky price producers must satisfy a key xed point property, that (37) x t (h t 1 ) = (1 )y t (h yt ) + t (h yt ) + x t+1 (h t ); where h yt = (h t 1 ; x t (h t 1 ); t (h t 1 ; x t (h t 1 ))); and h t = (h yt ; t (h yt ); y t (h yt )): Here, as in the simple sticky price model, x t (h t 1 ) shows up on both sides of the xed point equation on 23