Price level targeting versus inflation targeting in a forward looking model

Transcription

1 Price level targeting versus inflation targeting in a forward looking model David Vestin IIES, Stockholm University First draft: June 1999 This draft: January 2000 Preliminary - Comments Welcome Abstract This paper examines a price level target in a model with a forward-looking Calvo- Taylor Phillips curve. Contrary to conventional wisdom, it is found that price level targeting leads to a better inflation- output-gap variability tradeoff than inflation targeting, when the central bank acts under discretion. In some cases, price-level targeting results in the same equilibrium as inflation targeting under commitment. 1 Introduction The conventional wisdom emerging from the price level targeting discussion seems to be that price level targeting is bad because it generates unnecessary variability in the output-gap. Some recent papers have debated the relative merits of inflation targeting and price level targeting. Various results and indications that to some extent casts doubt on this conventional wisdom can be found in those papers. Svensson [15] found that price level targeting delivers a better outcome (lower variability of inflation) than inflation targeting when the central bank is unable to commit. This result holds for a Lucastype Phillips curve and requires some (realistic) output-gap persistence. Woodford [16] found that for an inflation targeting central bank, the optimal policy under commitment david.vestin@iies.su.se. I thank Lars E.O. Svensson for extensive comments on previous drafts. I have also benefited from discussions with, and comments from Henrik Jensen, Paul Klein, Stefan Laséen, Marianne Nessen and Per Pettersson and participants at seminars at the IIES, Stockholm University, Oslo University and the Swedish Central Bank. 1

2 is characterized by a significant degree of interest rate inertia. Woodford results suggest that given a central bank with no commitment, assigning a loss-function with the interest rate as an explicit argument induces the central bank mimic the commitment solution to some extent, since under the new loss-function there is an explicit reason to smooth interest rates. Jensen [6] finds that in some instances, nominal income growth targeting can dominate inflation targeting because of the same reason as in Woodford s paper, namely that it introduces the inertial behavior of interest rates that is a feature of the commitment solution. Importantly, both Clarida, Gali and Gertler [2] and Woodford [16] finds that the price level is stationary under commitment. This directs attention to the possibility that an explicit price level target might be preferable when the central bank acts under discretion (because the price level is stationary when there is a price level target). This paper compares price level targeting with inflation targeting under discretion, and finds that it is possible to improve the outcome of the discretionary inflation targeting case by assigning a price level target to the central bank. Thus, the question is the same as in Svensson [15], but posed in a model where forward looking behavior is emphasized. The emphasis on forward looking elements will turn out to play an important role when thinking about price level targeting. Clarida, Gali and Gertler [2] have recently stressed that in forward looking models, gains from commitment are possible also when the central bank aims at the natural rate of unemployment. This paper thus makes an attempt to see whether these benefits can bereachedevenwhennocommitmentdeviceexists. The main result is that price level targeting delivers a more favorable trade-off between inflation and output-gap variability than inflation targeting. With no (exogenous) persistence in the inflation process it is always possible to implement the commitment solution fully by assigning an appropriate price level targeting regime. The mechanism behind these results is the restraining effect of expectations. The private sector realizes that the central bank s incentive to offset shocks increases with a price level target, since the price level is persistent. Therefore, the central bank is helped by reduced expectations about future inflation when the economy is hit by an inflationary 2

3 shock. The paper proceeds as follows. Section 2 presents the model. Section 3 contains a summary of the optimal policy for the different regimes. Comparisons are made in section 4. Section five presents some conclusions. 2 The forward looking model The model has the following standard Phillips curve that relates inflation, π, to the output-gap, x, and expected future inflation, π t+1 t 1 π t = βπ t+1 t + κx t + u t, (1) where u t is an exogenous shock. Equation (1) is the central equation in what has become a work-horse model, dating back to Calvo [1], recently derived and extended by Rotemberg and Woodford [10] and thoroughly examined in Clarida, Gali and Gertler [2]. Inflation today is affected by two components. First, because of price rigidity, expected future inflation enters. Secondly, because of monopolistic competition, prices reflect the marginal cost conditions. In the model, this is captured by the inclusion of the output-gap, acting as a proxy for labor-market conditions that affect wages and thus the marginal cost. Finally, the cost push shock can be viewed as anything affecting real marginal cost working through channels other than the output-gap. This model can be contrasted with the model in Svensson [15], which is of Lucas-type. In that model it is the inflation surprise that affects the output-gap. In this model, it is the expected future inflation that drives the results. Thus, the model puts much emphasis on the forward looking elements of monetary policy. At this stage, I choose to abstract from transmission lags and, what the literature have labeled, endogenous persistence. Although these issues are an important part of practical monetary policy making, the main focus of the paper is to examine the effects of forward looking behavior. Gali and Gertler [5] finds empirical support for the relevance of forward looking behavior, whereas Fuhrer [4] questions its importance. 1 The notation t +1 t means conditional expectation of t +1given information at t. 3

4 Many empirical papers find that in order to fit the data, some persistence in the inflation process must be introduced. To avoid ignoring this issue completely, following Clarida, Gali and Gertler [2], I introduce exogenous persistence in the cost push shock captured by an AR(1) process. Thus, u t = ρu t 1 + ε t (2) ε t N 0, σ 2 ε. The Phillips curve is often coupled with an equation relating the interest rate to the output gap. However, unless interest rate smoothing is considered, this equation is redundant for solving the model in the sense that the problem is separable. First, a solution to the model is found treating the output gap as the control variable (instead of the interest rate). Then, the redundant equation is used to find the path of the nominal interest rate that is consistent with the optimal output gap path. If interest rate smoothing (captured by adding an interest rate term to the loss function) is considered, the problem is no longer separable. This is because then, the variability of the interest rate must be explicitly weighted against the variability of the output-gap and inflation, and the variability of the interest rate depends on the elasticity of the output-gap with respect to the interest rate. For the purpose of this paper, since interest rate smoothing is not considered, the Phillips curve gives a complete description of the dynamics that is of interest. The central bank behavior is assumed to be captured by minimizing min x t E t (1 β) X i=0 β i 2 L t+i (3) where L t will take different forms depending on whether inflation targeting or price level targeting is pursued. It is assumed that the loss-function of society takes the form L t = π 2 t + x 2 t To evaluate different policies, we will be interested in the average performance, measured by the unconditional expected value of the loss function. To simplify exposition of many of 4

5 the results in this paper, it is convenient to express the loss function in terms of variances of inflation and the output gap. Appendix G explains that when β 1 we get the following interpretation of the expected value of (3) E(L t )=Var(π t )+Var (x t ). In what follows, two different regimes for conducting monetary policy will be considered. One way to think about these regimes is in terms of delegation in the sense of Rogoff [9]. Society delegates a regime (defined in terms of a loss-function) to an independent central bank. Assuming that this delegation is enforceable, for example by finding a central banker with appropriate preferences or by conditioning re-election of the governor on performance evaluated against the assigned objectives, the implications of the different regimes are explored. Each regime will imply a different response to shocks which in turn will imply different time series properties for inflation and the output-gap. Given the interpretation of the loss-function discussed above, we will focus on the variance of inflation and the output-gap. The relative performance of the two regimes will be evaluated against the true social loss function. In particular we will assume the existence of a true, the relative weight on output-stabilization. The central bank is assumed to lack commitment, in the strict sense of not being able to credible announce future actions inconsistent with the assigned loss function. Nevertheless, the commitment solution for the social loss-function is calculated as a benchmark, toseehowclosethedifferent discretionary policies comes this optimal solution. To be able to evaluate the social loss-function, the next section calculates the implied variances of inflation and the output gap for each of the cases. 3 Solving the model The preferences of society regarding the trade-off short run trade-off between inflation and the output-gap is described by one of the following loss function. L t = π 2 t + x2 t (4) 5

6 Inthecaseofinflation targeting, the above loss function is assigned to the central bank. As an alternative, the central bank could be instructed to enforce a price level target. Formally, this would correspond to assigning the following loss function. L t = p 2 t + x 2 t (5) It is important to note the distinction between these two loss functions. The first one is corresponds to the true preferences, the second does not. Later in the paper, the different strategies for monetary policy will be evaluated against each other. This may be interpretedaswhatsvensson[13] referstoas flexible inflation (price level) targeting. It will be possible to express the optimal choice of the output gap (the control variable) and the evolution of the price level (the state variable) on a similar form in all three cases. x t = cp t 1 du t p t = ap t 1 + bu t where a, b, c and d will be determined by minimizing the respective loss-function defined by (4) and (5). Furthermore, appendix B shows that given a<1, the variance of inflation and the output-gap will take the form 2 Var (π t )=e 2 σ 2 u (6) Var (x t )=f 2 σ 2 u (7) e = 2b2 (1 ρ) (1 aρ)(1+a) f = b2 c 2 (1 + aρ)+d 2 (1 a 2 )(1 aρ)+2ρbcd (1 a 2 ) (1 a 2 )(1 aρ) The rest of this section examines the different cases. First, we consider the social benchmark, that is, inflation targeting under commitment. It is assumed that it is not possible to commit, and thus the next paragraphs examines inflation targeting and price level targeting under discretion. 2 This will be true for inflation targeting under commitment and for price-level targeting. For inflation targeting under discretion, a =1and then the variance calculation is trivial, as will be clear from the next section. 6

7 3.1 Social benchmark: Inflation targeting under commitment In the first best case, the central bank has complete credibility and is able to commit. Thus, it can credible announce any future path for the output gap, and thus affect the public sector expectations about future inflation with these statements. In this paper, it is assumed that this is not the environment the central bank faces in reality. That is why the next section deals with discretion. However, it is interesting to calculate the commitment case as a benchmark to use for evaluation of the regimes under discretion. Following Currie and Levine [3], Woodford [16] and the appendix in Clarida, Gali and Gertler [2], define the following Lagrangian. ( ) X β i min E t π 2 {x i } i=t 2 t+i + x 2 t+i + φt+i (π t+i κx t+i βπ t+i+1 u t+i ) i=0 As showed in appendix C, the optimal policy, represented by the optimal choice for the output gap (the control variable) is given by x t = c p t 1 d u t with c = (1 a β)(1 a ) κ d = 1 b [1 + β (1 ρ a )] κ a () = Ã r ³ ( (1 + β)+κ 2 ) 1 1 4β 2β 2 (1+β)+κ 2! (8) b = a 1 βa ρ This choice of the output gap gives an evolution of the price level given by p t = a p t 1 + b u t (9) 7

8 Appendix F shows that lim 0 a () = 0 lim a () = 1. Thus, the price level is stationary except in the special case when limit when the central bank only cares about stabilizing the output-gap ( ). Consider a one-time positive shock to the inflation rate. Since c > 0, theoptimal policy requires the central bank to maintain the control variable (the output gap) below the steady state value (of zero) as long as the (log of the) price level remains above the steady state value (zero) even when no further shocks hit the economy. This is the gradual response that Woodford and Clarida et.al. found. As we shall see later, this will not be the case under discretionary inflation targeting, but it will turn out to be the case with a price level target. In the case of inflation targeting with commitment, the intuition is that with a gradual (credible) response, it is possible to affect expectations of future inflation through the forward-looking component of the Phillips curve and thereby reducing the amount of activism needed to stabilize inflation. Using a, b, c and d in (6) and (7) gives Var(π t )=(e ) 2 σ 2 u (10) Var(x t )=(f ) 2 σ 2 u (11) 3.2 Inflation targeting, discretion Next, let us turn to the discretionary setting. Since no credible promise can be made, and there is no endogenous state variable under inflation targeting discretion, the value functioncanbewrittenas V (u t ) = 1 E t min π 2 x t 2 t + x 2 t + βv (ut+1 ) = γ 0 + γ 1 u t + γ 2 2 u2 t where the minimization is subject to (1). The control variable x t will be a linear function of the exogenous variables. Also, the forward looking variable (being a linear function of 8

9 the exogenous variable and the control variable) will have the form (â =1, bc =0). π t = ˆbu t (12) Appendix D shows that x t = ˆdu t (13) ˆb = κ 2 + (1 βρ) ˆd = κ κ = ˆb κ 2 + (1 βρ) Thus, a positive shock to inflation will to some extent be offset by a negative output-gap. For later comparison, it is convenient to rewrite (12) in terms of the price level. >From (12) and (13) it is evident that p t = p t 1 + ˆbu t (14) Var (π t )=ˆb 2 σ 2 u (15) with 3.3 Price level targeting Var (x t )= ˆd 2 σ 2 u (16) σ 2 u = 1 1 ρ 2 σ2 ε In this section a price level target is considered. 34 The loss-function takes the form L t = 1 2 ³ p 2 t + x 2 t 3 It is possible to consider a trend in the price level by defining the loss function as the deviation of the price level from trend. This will not affect the variances of inflation and the output gap. 4 It can be shown that an inflation bias generated from an overambitious output-gap target (not present in this paper) can be removed with an appropriate selection of the (time-dependent) price level target. This case is discussed in Kiley [7]. 9

10 where is the weight delegated to the central bank together with the price level target. When society delegates the loss function, there is no reason why the relative weight on output stabilization must equal the true weight. Rogoff [9] showed that assigning a lower than society s true value (that is, a more conservative central banker) eliminated the inflation bias. Here, no bias is present, since we assumed that the output-gap target was consistent with the natural rate of unemployment. However, as will be clear from the results below, different values of will affect the trade-off between inflation and outputgap variability (whereas Rogoff s result was in terms of the level of inflation). Rewriting the Phillips curve (1) in terms of the price level yields p t p t 1 = κx t + β p t+1 t p t + ut To solve the model, note that in this case there are two state variables. As in the case of inflation targeting, the shock is one of them. However, the price level from the previous period also enters as a state variable. Intuitively, this is because actions affecting the price level will persist. In the case of inflationtargeting,anincreaseininflation today will not affect inflationtomorrow,whereas anincreaseinthepriceleveltodaywillaffect the price level tomorrow. This helps clarifying the difference between a price level target and targeting inflation at zero. In the latter case, a temporary deviation from the target will not affect future losses. In the price level targeting case, a temporary deviation from target will have to be countered with an offsetting deviation in the future. With two state variables, the loss-function will take the following form: 1 ³ V (p t 1,u t )=E t ½min p 2 t x t 2 + x ¾ 2 t + βv (p t,u t+1 ) u t = ρu t 1 + ε t Appendix E shows that the state variable we are interested in will also be a linear function of the state variables p t =ãp t 1 + bu t (17) where the coefficients are defined by the following equations: ω ã = κ 2 + ω 2 + β (1 ωã) 10

11 ³ i ω + βρ h b 1+βρ b + ω b b = κ 2 + ω 2 + β (1 ωã) (18) ω =1+β (1 ã) Note that ã is independent of ρ, the degree of persistence in the shock process. What will turn out to be important is that precisely in the same way as for the inflation targeting case under commitment, it is possible to show that lim 0 ã ³ lim ã ³ = 0 = 1. This exercise is done in appendix E. Notice that this means that the price level follows an AR(1) process and is stationary. Also, note that if there is no persistence in the residual process (i.e. ρ =0)thenã = b. The solution for the control variable x t is given by where x t = cp t 1 du t (1 ãβ)(1 ã) c = κ d = 1 b [1 + β (1 ρ ã)] κ Similar to the previous section, to find the variances of inflation and the output-gap, use ã, b, c and d in (27) and (28). Var(π t )=ẽ 2 σ 2 u (19) Var(x t )= f 2 σ 2 u (20) 4 Comparing results The main purpose of this paper is to examine the relative performance of an inflation target to a price level target. The essential insight is gained from comparing (17) with (9) and (14). 11

12 Defined in ITC p t = a p t 1 + b u t (9) ITD p t =âp t 1 + ˆbu t (14) PTD p t =ãp t 1 + bu t (17) These equations define the optimal solution in terms of the price level for inflation targeting under commitment and discretion, and price level targeting under discretion. All other results such as variances of inflation and the output gap will be based on these equations. An implication of this is that if we can show that equations (9) and (17) are the same we know that the commitment solution can be implemented by assigning a price level target under discretion. To preview the results, this is almost what we will find. When there is no persistence (ρ =0)the commitment solution can be fully implemented with a price level target. That is, it is possible to find a under price level targeting such that a () =ã( ) and b () = b( ). In some of the experiments considered below, a numerical value for κ is needed. Roberts [8] estimates a version of the model above (eq. 9 p. 979): π t = π t+1 t + κx t + ² t and finds κ in the range of 0.25 to 0.36 depending on the measure of inflation expectations. On the basis of that result, I chose κ = No persistence (ρ =0) The main result of the paper is that price level targeting gives a better trade off between inflation- and output gap variability than inflation targeting. Later, this result will be proved for the case of persistence in the residual process. To gain understanding of this result, we start by first considering the special case of ρ =0. With this assumption, it is possible to find an analytical solution for the price level targeting case. Proposition 1 With no persistence in the residual process, the commitment solution can be implemented by assigning a price level target with a different. 12

13 With ρ =0, a = b and ã = b, so it is enough to prove that we can find a such that a () =ã( ). In fact, the preceding sections already provided the information needed to pursue this argument. To recapitulate: lim ã ( ) 0 = 0 lim ã( ) = 1 lim 0 a () = 0 lim a () = 1 Thus, since both the coefficient ã from the price level targeting case (eq. (??)) and the counterpart from the commitment case (eq. (8)) is limited by the interval [0, 1) we know that for a given value of, implying a fixed value for a () it is always possible to find a value of that sets ã( ) =a (). That is, it is always possible to implement the commitment solution for an inflation target by assigning a price level target with a different (namely ). 4.2 Persistence With persistence in the residual process, we have two conditions that must be satisfied in order to implement the inflation targeting commitment solution with a price level target: ã( ) = a () b( ) = b () Figure 1 and 2 gives the a and b coefficient values for the different cases, for different values of. Examiningthesefigures reveals that it is not possible to perfectly replicate the commitment solution with a price level target. This does not mean that an inflation target is preferred, it only suggest that full commitment through a price level target is not available when ρ > 0. Tofind out whether the price level target dominates the inflation target we will find the policy frontiers for the two cases. To do this, both the variance of inflation and the output-gap must be recovered under the two regimes. 13

14 The a coefficient, κ= , ρ=0.5 1 ITD ITC PTD Figure 1: 2 The b coefficient, κ= , ρ= ITD 1.4 ITC PTD Figure 2: 14

15 Var(π) 5 Variance of inflation, κ= , ρ= ITD ITC PTD Variance results Figure 3: To summarize, the variance of inflation under inflation targeting and price level targeting is given by (15), (10) and (19). Similarly, the output gap variances are given by (16), (11) and (20) respectively. Var (π t ) Var (x t ) ITD ˆb 2 σ 2 u ˆd 2 σ 2 u ITC (e ) 2 σ 2 u (f ) 2 σ 2 u PTD ẽ 2 σ 2 u f 2 σ 2 u In Svensson [15], comparing the two cases is more clear-cut since there is no difference in output gap variability. In the forward looking case, this is not true. Both output gap and the inflation variability will differ under the two regimes and thus it is hard to judge the result by just inspecting the equations. To interpret previous findings in the literature, examine the following variance plots. >From these figures, it is tempting to make the conclusion that price level targeting generates higher output-gap variability that does inflation targeting. This conclusion is reached by fixing and vertically examining figure 4. This leads Kiley [7] to conclude that 15

16 Var(y) 4 Variance of the output-gap, κ= , ρ= ITC PTD 0.5 ITD Figure 4: a price level target is worse than an inflation target, since it generates higher variability of the output-gap (he compares to Svensson [14] who finds that (given some conditions) a price level target gives the same variability of the output-gap as does an inflation target, but a lower variability of inflation and thus concludes that a price level target is preferable). However, the same experiment in figure 3 reveals that the variance of inflation is lower with a price level target that with an inflation target. It thus seems inconclusive which is the better. This paper suggests is that it is more instructive to read the figure horizontally. A given variance of inflation resulting from a particular value of can always be implemented by assigning a different value under a price level target. It is not obvious that both the variance of inflation and the output-gap under price level targeting can be made smaller compared to inflation targeting (both under discretion) simultaneously by inspecting the figures. To evaluate this, the next section plots efficiency frontiers. 4.4 Efficiency frontiers An illustrative way of describing the implications of the two regimes is by plotting the frontiers in the two dimensional space of inflation- and output-gap variance. A frontier 16

17 Variance of outputgap 12 Policy frontier, κ= , ρ= ITD 2 ITC PTD Variance of inflation Figure 5: plots all combinations of output gap variance and inflation variance that are attainable for different values of the preference parameter. Since there is a tension between these variances, there will always be a trade-off of increased inflation variability in order to reduce output gap variability. Technically, the frontiers are constructed by fixing κ and then plotting inflation variance and output-gap variance for different values of. With β 1, the slope of the efficiency frontier is equal to. Proposition 2 Price-level targeting gives a better inflation- output-gap variance trade-off than inflation targeting. Figure 5 reveals that price level targeting dominates inflation targeting (when the central bank acts under discretion) since the frontiers never cross. Thus, in the absence of commitment it is preferable to use a price level target. Note that the variance frontiers for inflation targeting and price level targeting almost coincide. If persistence is increased up to almost one, there will be a more pronounced difference in the two cases, but the price level target will sill dominate the inflation target. 5 From an economic point of view, a price 5 For very high values of ρ, there will be a discrepancy between the commitment case and the price level targeting case. However, the price level target will still dominate the inflation target under discretion. 17

18 level target adds credibility in the following sense: Under an inflation target, a temporary increase in inflation is disregarded in the next period. With a price level target, this is no longer true. Instead, a temporary increase in inflation must be countered sooner or later by a reduction in inflation below target. This has been used as an argument against using price level targets, and the claim is that it would increase volatility of inflation. However, with a forward looking agents, the anticipated reduction that must take place in the future reduces inflationary expectations and thus helps the central bank to fight inflation. 4.5 Interpreting When Rogoff found that assigning a less than the one found in the social welfare function, the interpretation was that a more conservative central banker should be appointed. Therefore, it is tempting to draw the conclusion that finding > means that a less conservative central banker should be appointed. However, this is premature. The reason is that and have different interpretations as can be seen from comparing the two loss-functions, for convenience reproduced in terms of variances. E[L t ]=Var(π t )+Var (y t ) E[L t ]=Var(p t )+ Var (y t ) In the first loss-function, canbeinterpretedastherelativeweightplacedonthevariability of the output-gap compared to the variability of inflation. Inthecaseofprice level targeting, measures the relative weight placed on output-gap variability compared to the variability of the price level. Since the two weights have different benchmarks, not much is gained from comparing their absolute values. However, since the inflation rate is tightly linked to the price level, it is possible to interpret the relative size of the two weights. The simplest case is when ρ =0. The price level follows p t =ãp t 1 + bu t 18

19 With no persistence in the residual process, u t evaluating the variance expressions (26) and (27) result in Var (p t )= b 2 1 ã 2 σ2 u Thus,bytakingtheratioweget or Var (π t )= 2b2 1+ã σ2 u Var (π t ) Var (p t ) = 2(1 ã2 ) 1+ã Var (p t )= In order to get a comparable weight (i.e. 1 2(1 ã) Var (π t) to have the same normalization in both loss-functions) the following equation must be satisfied: ³ ³ =2 1 ã ³ ³ That is, a > 2 1 ã imply that the central bank should be more conservative. 5 Conclusions The main result of the paper is that in a forward looking model used by several authors, a price level target dominates an inflation target even when preferences are concerned with the variability of inflation and the output-gap. The result can be interpreted in line with Rogoff s classic result that by assigning a loss-function different from society s (in Rogoff s case, a more conservative central banker in the sense that banker < society ), a better outcome can occur. In this case there is a two-dimensional assignment. First, the inflation target is replaced by a price level target. Second, a different value of is assigned. In previous literature there has been a misinterpretation of the effects of a price level target. There, it is recognized that for a given value of, a price level target generates more variability of the output-gap than does an inflation target. But, it is also 19

20 recognized, inflation variability is lower under the price level target. Conventional wisdom explained this result by claiming that in the price level targeting case, a positive shock must later be countered by a monetary tightening, which will induce more volatility of the output-gap than in the inflation targeting case where bygones are treated as bygones. The point of this paper is that this comparison is not the most interesting. By instead examining the policy frontiers, it is clear that the price level target dominates the inflation target since it is always possible to implement a better outcome by assigning a different in the price level targeting case. With no persistence, the commitment solution of the inflation targeting case can be implemented. With persistence, this is not true. However, it is still always the case (in the model examined!) that a price level target generates a better outcome than the inflation target, and is almost as good as the commitment solution. With price level targeting, the private sector expects the central bank to counter an above average inflation (normalized to zero in this paper) with a below average inflation somewhere in the future. In other words, a positive shock to inflation reduces the expected future inflation and thus lowers the amount of intervention the central bank must engage in. 20

21 References [1] G. Calvo. Staggered prices in a utility-maximizing framework. Journal of Monetary Economics, 12:383 98, [2] Richard Clarida, Jordi Gali, and Mark Gertler. The science of monetary policy. Journal of Economic Literature, page?, Forthcoming [3] D. Currie and P. Levine. Rules, Reputation and Macroeconomic Policy Coordination. Cambridge University Press, [4] Jeffrey C. Fuhrer. The (Un)Importance of forward-looking behaviour in price specifications. Journal of Money, Credit and Banking, 29(3): , [5] Jordi Galí and Mark Gertler. Inflation dynamics: A structural econometric analysis. Journal of Monetary Economics, forthcoming. [6] Henrik Jensen. Targeting nominal income growth or inflation? mimeo, University of Copenhagen, June [7] Michael T. Kiley. Monetary policy under neoclassical and new-keynesian phillips curves, with an application to price level and inflation targeting, May [8] John M. Roberts. New keynesian economics and the phillips curve. Journal of Money, Credit, and Banking, 27(4): , [9] Kenneth Rogoff. The optimal degree of commitment to an intermediate monetary target. Quarterly Journal of Economics, 100: , [10] Julio Rotemberg and Michael Woodford. An optimization-based econometric framework for the evaluation of monetary policy. NBER Macroeconomics Annual, [11] Glenn Rudebusch and Lars E.O. Svensson. Policy rules for inflation targeting. In Monetary Policy Rules, pages NBER, [12] Frank Smets. What horizon for price stability? mimeo, European Central Bank, August

22 [13] Lars E.O. Svensson. Inflation forecast targeting: Implementing and monitoring inflation targets. European Economic Review, 41(6): , [14] Lars E.O. Svensson. Price stability as a target for monetary policy: Defining and maintaining price stability. Mimeo, Stockholm University, IIES, March [15] Lars E.O. Svensson. Price level targeting vs. inflation targeting. Journal of Money, Credit and Banking, 1999, forthcoming. [16] Michael Woodford. Optimal monetary policy inertia. NBER Working Paper no. 7261, August

23 A Finding c and d To get the c and d coefficients when the price level follows p t = ap t 1 + bu t use the stationary version of (42) in (43) and simplify to get x t = 1 aβ 1 βb (1 ρ) (bp t 1 + bu t p t 1 ) u t κ κ (1 aβ)(1 a) 1 b [1 + β (1 ρ a)] = p t 1 u t κ κ = cp t 1 du t B Variance calculations This appendix calculates the variance for inflation and the output-gap given a model implying an evolution of the price level of the form p t = ap t 1 + bu t (21) y t = cp t 1 + du t (22) Subtracting p t 1 from (21) gives π t = (1 a) p t 1 + bu t Var (π t )=(1 a) 2 Var (p t 1 )+b 2 Var (u t ) 2(1 a) bcov (p t 1,u t ) (23) Var (p t )=a 2 Var (p t 1 )+b 2 Var (u t )+2abCov (p t 1,u t ) (24) Cov (p t 1,u t ) = Cov(ap t 2 + bu t 1, ρu t 1 + ε t ) = aρcov (p t 2,u t 1 )+bρvar (u t 1 ) = bρ 1 aρ σ2 u (25) 23

24 Using that the price level is stationary and substituting (25) into (24) yields 1 a 2 Var (p t ) = b 2 σ 2 u + 2ab2 ρ 1 aρ σ2 u Var (p t ) = b 2 (1 + aρ) (1 a 2 )(1 aρ) σ2 u (26) Using (26) and (25) in (23) gives (1 a 2 ) b 2 (1 + aρ) Var (π t ) = + b 2 2(1 a) b2 ρ σ 2 (1 a 2 u )(1 aρ) 1 aρ = 1 a 1+aρ 1 aρ 1+a + 1 aρ 1 a 2ρ b 2 σ 2 u 2(1 ρ) = (1 aρ)(1+a) b2 σ 2 u (27) To calculate the variance of the output-gap, it is convenient to use the form (22) and note that this implies Var (x t )=c 2 Var (p t 1 )+d 2 Var (u t )+2cdCov (p t 1,u t ) Substituting (26) and (25) into the above equation gives Var (x t ) = c 2 b 2 (1 + aρ) (1 a 2 )(1 aρ) σ2 u + d2 σ 2 bρ u +2cd 1 aρ σ2 u = b2 c 2 (1 + aρ)+d 2 (1 a 2 )(1 aρ)+2ρbcd (1 a 2 ) σ 2 (1 a 2 u (28) )(1 aρ) C Inflation targeting, commitment Following Currie and Levine [3], Woodford [16] and the appendix in Clarida, Gali and Gertler [2], define the Lagrangian ( ) X β i min E t π 2 {x i } i=t 2 t + x 2 t + φt+i (π t+i κx t+i βπ t+i+1 u t+i ) i=0 We start by taking the first order conditions with respect to inflation: π t+i = β i 1 φ t+i 1 β + β i π t+i + φ t+i =0 π t+i = φ t+i 1 φ t+i, i>0 (29) π t = φ t (that is, i =0) 24

25 Next,wetakethefirst order conditions with respect to x t+i. x t+i = x t+i φ t+i κ =0 φ t+i = κ x t+i, i 0 (30) Combining the first order conditions by substituting (30) into (29) gives π t+i = κ (x t+i x t+i 1 ) (31) Substituting (31) into the constraint finally gives a second order stochastic difference equation for x t : with a = κ (x t x t 1 ) = κx t β xt+1 t x t + ut κ x t = ax t 1 + aβx t+1 t κa u t. Rewrite the last equation as (1+β)+κ 2 x t+1 t 1 aβ x t + 1 β x t 1 = κ β u t Factorizing the right hand side lead polynomial by solving Denoting the stable root by δ, wehave h 2 1 aβ h + 1 β =0 δ = 1 p 1 4βa 2 2αβ and, since h 1 h 2 = 1 we have the unstable root h β 2 = 1. Thus, we can rewrite our equation βδ as µ (1 δl) 1 1 βδ L x t+1 = κ β u t Solving this finally gives (1 δl) x t = κδ x t = δx t δβρ u t κδ (1 δβρ) u t (32)

26 By substituting (32) into (31), inflation can be recovered as π t = δπ t 1 + δ 1 δβρ (u t u t 1 ) Or, summarizing the maximized policy in terms of the price level as p t = a p t 1 + b u t (33) b = a 1 βa ρ a = r ³ ( (1 + β)+κ 2 ) 1 4β 2β 2 (1+β)+κ 2 (34) D Inflation targeting, discretion Since there is no credible promise can be made, and there is no endogenous state variable, the value function in the case of inflationtargetingcanbewrittenas 1 V (u t ) = E t min π 2 x t 2 t + x 2 t + βv (ut+1 ) = γ 0 + γ 1 u t + γ 2 2 u2 t where the minimization is subject to (1). The control variable x t will be a linear function of the exogenous variables. x t = ˆdu t (35) Also, the forward looking variable (being a linear function of the exogenous variable and the control variable) will have the form Using the above in () gives π t = ˆbu t (36) π t = κx t + u t + βe t u t+1 ³ = κx t + 1+βρˆb u t (37) 26

27 Solving the minimization problem results in the following first order condition: µ π t 0 = E t π t + x t x t = π t κ + x t or, x t = κ π t (38) Substituting (38) into (37) gives ³ π t = κ κ π t + ³ 1+βˆbρ π t = Comparing (36) and (39) it is clear that or, solving for ˆb ³ 1+βˆbρ u t + κ 2 u t (39) ˆb = ˆb = ³ 1+βρˆb + κ 2 κ 2 + (1 βρ) Substituting this into the first order condition gives x t = ˆdu t where ˆd = κ κ 2 + (1 βρ). 27

28 E Price-level targeting, discretion Value function: 6 µ 1 V t (p t 1,u t ) = E t µmin p 2 y t 2 t + x 2 t + βvt+1 (p t,u t+1 ) (40) = γ 0,t + γ 1,t p t γ 2,tp 2 t 1 +γ 3,t p t 1 u t + γ 4,t u t γ 5,tu 2 t u t = ρu t 1 + ε t where the minimization in the above problem is subject to (1). The guessed value function will only be used when taking conditional expectations at t of the derivative with respect to p t,thatis: E t V (p t,u t+1 ) = γ p 1,t+1 + γ 2,t p t+1 + γ 3,t+1 E t u t+1 t so without loss γ 0,t, γ 4,t and γ 5,t can be set to zero. Note that if there is no persistence in the residual process (ρ =0) γ 3,t can also be set to zero. Finally, γ 1,t will only concern a drift in the price level. If x, π and p t are all set to zero, γ 1,t will be of no interest and can also be set to zero. It is possible to show that the inflation bias resulting from x > 0 can be eliminated with a price level target with a drift. Thus, the guess of loss function can be written V t (p t 1,u t )=γ 1,t p t γ 2,tp 2 t 1 + γ 3,t p t 1 u t (41) The state variable will follow a linear path (the quadratic loss function ensures that the policy instrument x t is a linear function of the state variables) p t+1 = a t+1 p t + b t+1 u t+1 p t+1 t = a t+1 p t (42) 6 In the optimization, there is no difference in considering the price level or the output-gap as the control variable, since it is just two ways of substituting equation (??) into the loss function. However, for consistency and as a preparation for future extensions when this property will not hold, the outputgap will be considered as the control variable and all derivatives of the output-gap with respect to the price level will be replaced by the derivative of the price level with respect to the output-gap. 28

29 where the coefficients are left to determine. Rewrite (1) using π t = p t p t 1 : x t = 1 κ (1 + β) p t β κ p t+1 t 1 κ (p t 1 + u t ) Inserting (42) into the above equation gives or, p t = x t = 1 κ [1 + β (1 a t+1)] p t 1 κ p t 1 1+βρb t+1 u t (43) κ κ 1+β (1 a t+1 ) x 1 t + 1+β (1 a t+1 ) p t βρb t+1 1+β (1 a t+1 ) u t (44) p t κ = x t 1+β (1 a t+1 ) V t+1 (p t,u t+1 ) = γ p 2,t+1 p t + γ 3,t+1 u t+1 t Solving the minimization (40) results in the following first order conditions: p t 0 = E t p t + x t + β V t+1 (p t,u t+1 ) p t x t p t x t κ = E t 1+β (1 a t+1 ) p t + x t + β V t+1 (p t,u t+1 ) p t p t x t = p t + [1 + β (1 a t+1)] x t + βγ κ 2,t+1 p t + βγ 3,t+1 E t u t+1 >From (2) follows that E t (u t+1 ) = ρu t. simplifying gives Inserting this and (43) into the above and p t = (1 + β (1 a t+1 )) κ 2 + (1 + β (1 a t+1 )) 2 + βκ 2 γ 2,t+1 p t 1 + (1 + β (1 a t+1 )) (1 + βρb t+1 ) βρκ 2 γ 3,t+1 κ 2 + (1 + β (1 a t+1 )) 2 + βκ 2 γ 2,t+1 u t (45) In order to solve the above equation, γ 2,t+1 and γ 3,t+1 must be identified. This can be done by differentiating (41) with respect to V p,t (p t 1,u t )=γ 2,t p t + γ 3,t u t (46) 29

30 and comparing this to the equivalent expression obtained using the envelope theorem on (40) µ V p,t (p t 1, u t ) = E t x t 1 κ = E t µ κ {[1 + β (1 a t+1)] p 2 t p t 1 (1 + βρb t+1 ) u t } = κ 2 {[1 + β (1 a t+1)] (b t u t + a t p t 1 ) p t 1 (1 + βρb t+1 ) u t } = κ (1 [1 + β (1 a t+1)] a 2 t ) p t 1 + (47) κ {(1 + βρb t+1) [1 + β (1 a 2 t+1 )] b t } u t (48) comparing (46) and (47) it is clear that γ 2,t = κ 2 {1 [1 + β (1 a t+1)] a t } (49) γ 3,t = κ 2 {(1 + βρb t+1) [1 + β (1 a t+1 )] b t } (50) Using (49) and (50) in (45) gives p t = [1 + β (1 a t+1 )] κ 2 + [1 + β (1 a t+1 )] 2 + β{1 [1 + β (1 a t+2 )] a t+1 } p t 1 + [1 + β (1 a t+1 )] + βρ{b t+1 (1 + βρb t+2 )+[1+β(1 a t+2 )] b t+1 } κ 2 + [1 + β (1 a t+1 )] 2 + β{1 [1 + β (1 a t+2 )] a t+1 } u t Finally, comparing this to (42), the following equations must hold: a t = [1 + β (1 a t+1 )] κ 2 + [1 + β (1 a t+1 )] 2 + β{1 [1 + β (1 a t+2 )] a t+1 } (51) b t = [1 + β (1 a t+1)] + βρ{b t+1 (1 + βρb t+2 )+[1+β (1 a t+2 )] b t+1 } κ 2 + [1 + β (1 a t+1 )] 2 + β{1 [1 + β (1 a t+2 )] a t+1 } (52) 30

31 F Limit calculations F.1 Inflation targeting, commitment We have lim a () = lim = à r ³ ( (1 + β)+κ 2 ) 1 1 4β ³ (1 + β)+ κ2 1 = lim à r ³ (1 + β) 1 1 4β = 1. 2β 2β s µ 1 4β 2β 1 (1+β)! 2 2 (1+β)+κ 2 1 (1+β)+ κ2! 2 Next, the lower limit lim 0 a () = lim 0 1 = lim 0 1 = lim e 0 wherewehaveusedl Hospital srule. à r ³ ( (1 + β)+κ 2 ) 1 1 4β r ³ 1 4β 2β (1+β)+κ 2 2β 2 (1+β)+κ 2 p 1 4βe 2, with e = 2βe (1 4βe 2 ) 1 2 ( 8βe) = lim e 0 2β = 0 2 (1+β)+κ 2 (1 + β)+κ 2! F.2 Price-level targeting, discretion lim ã ³ 0 lim ã ³ 0 ω = lim 0 κ 2 + ω 2 + β (1 ωã) = 0 31

32 ω lim ã = lim κ 2 + ω 2 + β (1 ωã) ω lim ã = lim κ 2 + ω2 + β (1 ωã) ω lim ã = ³ ω 2 + β 1 ω lim ã ³ >From the last line we see that lim ã =1is a solution (remember that ω =1+ β (1 ã). G The loss function This appendix deals with explaining how the loss function can be rewritten in terms of variances. This section formalizes the argument in Rudebusch and Svensson ([11]). The loss function is given by X (1 β) β i L t+i. i=0 The central bank is minimizing this conditional upon the information available at time t. However, we are interested in how the policy preforms on average. This is represented by taking the unconditional expected value of the loss function. That is " # X X E (1 β) β i L t+i = (1 β) β i E[L t+i ] i=0 = E[L t ] The central bank is minimizing # X X E t "(1 β) β i L t+i. =(1 β) β i E t [L t+i ] i=0 i=0 i=0 In the case of discretion and no exogenous persistence, it is not possible to affect the conditional expected future value. Thus, L t is a constant, and all future terms are equal 32

33 to the unconditional expected value of L t+i.thisgives = (1 β) L t +(1 β) X β i E[L t+i ] i=1 = (1 β)(l t E(L t )) + (1 β) X β i E[L t ] i=0 = (1 β)(l t E(L t )) + (1 β) E(L t) 1 β = (1 β)(l t E(L t )) + E (L t ) In the limit when β 1, thefirst term disappears. In the case of commitment (and/or residual persistence), the story is slightly more complicated. Then, it is possible to affect future expected values of the loss function today. However, in the limit when β 1, wegetthesameresult. Thisisbecausewhen i becomes big enough, the expected level of the variables in the loss function will return to their steady state values. Thus, the conditional expected value will converge towards the unconditional expected value. Given that β 1, the weight of the values in the beginning of the sum, that are out of steady state, will eventually be dominated by the vast number of terms that has converged into the steady state. 33