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Optimal and efficient monetary policy rules in a forward-looking model
Journal of Macroeconomics 24 (2002) 41–49 www.elsevier.com/locate/econbase
Optimal and efficient monetary policy rules in a forward-looking model Alfred V. Guender
*
University of Canterbury, Christchurch, New Zealand Received 9 July 1999; accepted 30 January 2001
Abstract This paper shows that two monetary policy strategies––hybrid nominal income targeting and strict inflation targeting––are efficient strategies of monetary policy in the sense that they are special cases of the optimal monetary policy strategy. In the case of a hybrid nominal income targeting strategy, the policymaker chooses a unitary trade-off between real output and the rate of inflation, while under strict inflation targeting the policymaker attaches a zero weight on output in the optimal policy rule. Ó 2002 Elsevier Science Inc. All rights reserved. JEL classification: E5 Keywords: Forward-looking; Optimal and efficient policy; Inflation targeting; Hybrid nominal income targeting
1. Introduction Current discussions of issues in monetary policy share a number of characteristics. First, there has been a move toward a new framework where the rate of inflation––and not the price level––features prominently. The new approach emphasizes simplicity over complexity. It comprises a simple two-equation IS-Phillips Curve framework where real output and the rate of inflation enter as the two endogenous
*
Fax: +64-3-364-2635. E-mail address: [email protected] (A.V. Guender).
0164-0704/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 1 6 4 - 0 7 0 4 ( 0 2 ) 0 0 0 1 4 - 9
42
A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
variables. Practical considerations have led to the omission of the LM relation from the analysis. As most central banks use a short-term nominal interest rate to set the stance of monetary policy, the inclusion of money demand in the analysis will merely serve to determine the volume of the money supply that is consistent with the set interest rate. 1 Despite the adoption of an alternative framework for the analysis of monetary policy issues, there remains some disagreement among economists about the proper specification of the underlying structural relations. Of the competing specifications, two have attracted particular attention. One approach assumes that real output and the rate of inflation exhibit persistence, that is, that current real output and the rate of inflation are tied to their past behavior. In the literature this specification is referred to as ‘‘backward-looking’’. 2 In sharp contrast, the ‘‘forward-looking’’ specification adopts a rational expectations framework where current real output and the rate of inflation, respectively, depend on the next period’s expected level. 3 Another characteristic common to current and recent contributions to the literature is the renewed interest in the properties of monetary policy rules. In view of the widespread disagreement among economists about the proper specification of macroeconomic relationships, efforts have been made to examine the properties of simple, tractable rules across a wide variety of macroeconomic models. The idea here is to examine the robustness of candidate rules for inflation targets, price level targets, nominal income targets, exchange rate targets, and other rules such as the Taylor Rule. 4 For instance, Ball (1997) finds nominal income targeting to be inconsistent with the optimal policy rule in a simple backward-looking model. Indeed he labels nominal income targeting a disastrous strategy of monetary policy as it leads to instability in both the rate of inflation and the output gap. This result is challenged by McCallum (1997) who attributes Ball’s findings to the backward specification of the Phillips Curve. 5 The current paper takes the forward-looking model as the baseline model to derive the optimal monetary policy rule. The paper takes the view that optimal mon-
1 See, for instance, Taylor (1995), Ball (1997), McCallum (1997), Svensson (1997), Clarida et al. (1999) and McCallum and Nelson (1999) amongst others. 2 The papers by Taylor (1995), Ball (1997) and Svensson (1997, 1999) fall into this category. 3 Examples of this approach are McCallum (1997), McCallum and Nelson (1999), Clarida et al. (1999) and Woodford (1999). 4 A recent comprehensive contribution in this area is the papers presented at a symposium of the Sveriges Riksbank in 1998 and published in the Journal of Monetary Economics (1999). A treatment of monetary policy rules also appears in Taylor (1999). See also Bryant et al. (1993), Henderson and McKibbin (1993) and Taylor (1993) for earlier assessments of the empirical properties of various monetary policy rules. Analyzing solely the merits of nominal income targeting, Rudebusch (2000) finds that this strategy receives only modest empirical support from US data. 5 For a comparative analysis of various monetary policy rules in forward- and backward-looking models, the reader is referred to Guender (2000). He shows that the instability of nominal income targeting in the backward-looking specification disappears if the policymaker chooses to target a hybrid form of nominal income.
A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
43
etary policy ought to be framed in terms of the ultimate goal variables, real output and the rate of inflation, with the relative emphasis on the two goal variables in the optimal rule determined by the underlying preferences of the policymaker and the structural parameter in the Phillips Curve. In addition, this paper shows that two simple monetary policy rules, hybrid nominal income targeting and strict inflation targeting, are special cases of the optimal monetary policy rule. The paper is organized as follows. Section 2 presents the forward-looking model. The optimal monetary policy rule is derived in Section 3. Section 4 discusses two efficient monetary policy strategies. Section 5 concludes.
2. The forward-looking model The simple forward-looking model consists of the following two equations: yt ¼ brt þ Et ytþ1 þ mt ;
ð1aÞ
pt ¼ Et ptþ1 þ ayt þ ut ;
ð1bÞ
where yt is the output gap, rt is the real rate of interest, pt is the rate of inflation. Both ut and mt are random disturbances with distribution ð0; h2u Þ and ð0; r2u Þ, respectively. Both a and b are positive parameters. The model assumes that the policymaker has full control over the setting of the policy instrument, the real rate of interest, which is inversely related to the demand for real output. Eq. (1a) can be thought of as the counterpart to the IS relation in the standard IS-LM framework. There is one noteworthy difference between Eq. (1a) and the standard IS relation, however. While in standard IS-specifications current real output is positively related to that of the preceding period, in the forward-looking IS specification the current output gap responds to the expected output gap in the following period. 6 Eq. (1b) represents a forward-looking Phillips Curve where the output gap has a contemporaneous positive effect on the rate of inflation. 7 The link between the current rate of inflation and expected inflation next period is the direct result of optimal price-setting behavior by producers under imperfect competition. The property that current inflation depends on inflation next period distinguishes Eq. (1b) from other Phillips Curve specifications such as the backward-looking
6
There is some disagreement about what the variable yt represents. In McCallum and Nelson (1999) and McCallum (1999) it represents the level of real output. In contrast, Svensson (1997, 1999), Woodford (1999), Clarida et al. (1999) and Rudebusch (2000) employ the deviation of the level of real output from capacity output in both the IS and the Phillips Curve relation and term it the output gap. In this paper the latter convention is adhered to. 7 For an explicit derivation of Eq. (1b), the reader is referred to Roberts (1995) and the references cited there.
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A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
specification employed by Svensson (1999) and Ball (1997) or the New Classical specification employed by Woodford (1999) and others. 8 3. Optimal policy The policymaker sets a fixed nominal target for the sum of the ultimate goal variables: the real output gap and the rate of inflation. 9 The parameter h indicates the weight the policymaker attaches to the output gap relative to the rate of inflation in the policy rule. z ¼ ½hyt þ pt ¼ 0:
ð2Þ
As shown below, h P 0. Hence the optimal value of h determines the trade-off between real output and the rate of inflation. Inserting Eqs. (1a) and (1b) into (2) and solving for rt yields 1 1 ðEt ptþ1 þ ayt þ ut Þ þ ðEt ytþ1 þ mt Þ; ð3aÞ rt ¼ bh b or 1 1 ðpt Þ þ ðEt ytþ1 þ mt Þ: rt ¼ ð3bÞ bh b This reaction function illustrates that the policymaker will respond to pressure on real output arising on the demand side by systematically raising the real rate of interest by 1=b. The extent to which the real interest rate rises in response to an increase in observed inflation in contrast depends on the size of the policy parameter h. Substituting (3a) into the IS relation (Eq. (1a)) results in ðh þ aÞyt ¼ Et ptþ1 ut :
ð4Þ
This equation shows how real output behaves after the rule is imposed. Real output decreases in the wake of supply shocks and in response to expected future inflation. The extent to which output declines depends on the size of a, a structural parameter, and h, a policy parameter. The absence of mt from Eq. (4) implies that demand side disturbances have no effect on real output. This is a direct result of the policymaker’s ability to affect real output (and the rate of inflation) contemporaneously through
8 The forward-looking model differs from the backward-looking specifications employed by Ball (1997) and Svensson (1997, 1999) in two important respects. First, there are no control lags in the forwardlooking model. This implies that a change in monetary policy has contemporaneous effects on the output gap and the rate of inflation in the forward-looking model. Such contemporaneous effects are absent in the backward-looking model where a change in the real rate of interest affects the output gap with a oneperiod lag and the rate of inflation with a two-period lag. Second, while current inflation depends on expected future inflation in the forward-looking model, current inflation depends on past inflation in the backward-looking model. This property produces the intriguing result that the structural parameter on the current output gap in the backward-looking Phillips Curves is negatively related to the current rate of inflation. Woodford (1999) employs an expectations-augmented Phillips Curve where the output gap is sensitive to inflation surprises, that is, (pt Et1 pt ). 9 For convenience, the target value is set equal to zero.
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45
varying the setting of the policy instrument. Combining Eq. (4) with the evolution of the rate of inflation (Eq. (1b)), we obtain h ðEt ptþ1 þ ut Þ: ð5Þ pt ¼ hþa To solve the model, we pose putative solutions for the endogenous variables: yt ¼ s11 ut ; ð6Þ pt ¼ s21 ut : It therefore follows that Et ptþ1 ¼ 0; Et ytþ1 ¼ 0:
ð7Þ ð8Þ ð9Þ
Inserting (7) and (8) into (5) and matching coefficients yields h : ð10Þ s21 ¼ hþa Hence the solution for the rate of inflation is h pt ¼ ut : ð11Þ hþa Substituting Eqs. (8) and (11) into Eq. (1b) and solving for yt yields the expression for real output: 1 ut : yt ¼ ð12Þ hþa It follows then that 2 h VarðpÞt ¼ r2u ; ð13aÞ hþa
2 1 r2u : ð13bÞ hþa The objective of the policymaker is to minimize a loss function consisting of the variance of real output and the rate of inflation, respectively: 10; 11 Varðyt Þ ¼
10
This is the loss function employed by Ball (1997). Other specifications could include the variance of the policy instrument, V ðrt Þ, especially if it is believed that the policymaker desires to smooth movements in the policy instrument. However, this would necessitate choosing an arbitrary weight for V ðrt Þ. See Rudebusch and Svensson (1999) for examples of this type of loss function. Alternatively, Koenig (1995) suggests that the performance of policy rules be evaluated on the basis of whether a weighted average of the price level and real output equals a pre-announced target. He finds both price level and inflation targeting to be suboptimal because both strategies put a zero weight on real output. Woodford (1999) derives a loss function from a choice-theoretic utility maximizing framework. His loss function is essentially the same as the one employed in the current paper. 11 In the loss function the target value for real output is capacity (potential) output. For simplicity the target value for the rate of inflation is assumed to be zero.
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Fig. 1. The relationship between l and h for a ¼ 0:1, 0.25, 0.9.
Min Varðyt Þ þ lVarðpt Þ: h
ð14Þ
The solution to the above minimization problem is given by h ¼
1 P 0: la
ð15Þ
This setting represents the optimal choice for the policy parameter h. We notice that the optimum value for h is a function of l, the weight on the variance of inflation in the loss function, and a, the parameter on the output gap in the forward-looking Phillips curve. The relationship between h and l is illustrated for three different values of a in Fig. 1. An increase in the size of a shifts the curve downward, thus lowering h. The greater the effect of real output on the rate of inflation in the Phillips relation, the smaller the weight the policymaker attaches to real output in the optimal rule. This result is intuitively plausible as a larger value of a requires a smaller reduction in real output to engineer a desired decrease in the rate of inflation.
4. Efficient monetary policy strategies In this section we examine the properties of two strategies of monetary policy, both of which are special cases of the optimal monetary policy rule. The first strategy is geared towards attaining an announced hybrid nominal income target while the other strategy focuses solely on meeting a pre-specified inflation target. 4.1. A hybrid nominal income target The hybrid nominal income target consists of the sum of the deviation of real output from capacity and the rate of inflation: z ¼ yt þ pt . Again, let z ¼ 0 for simpli-
A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
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Table 1 The sensitivity of l to changes in the size of a under the hybrid nominal income targeting strategy l ¼ 10 l¼4 l ¼ 1:11
for a ¼ 0:1 for a ¼ 0:25 for a ¼ 0:9
city. 12 A monetary policy rule which targets the sum of the output gap and the rate of inflation constitutes an efficient form of monetary policy because the policymaker chooses a unitary trade-off between real output and inflation. Setting h ¼ 1 in Eqs. (11) and (12), we obtain yt ¼ pt ¼
1 ut ; 1þa
ð16Þ
1 ut : 1þa
ð17Þ
Notice the symmetric effect of the supply side disturbances on real output and the rate of inflation, respectively. The variances of real output and the rate of inflation under a hybrid nominal income target are then given by 13 V ðyt Þ
NIT
¼
1 ð1 þ aÞ
2
r2u ;
V ðpt Þ
NIT
¼
1 ð1 þ aÞ
2
r2u :
ð18Þ
Eq. (18) implies that in the special case where the policymaker places a weight of l ¼ 1=a on the variance of inflation in the loss function, hybrid nominal income targeting is consistent with the optimal rule. Table 1 gives an indication of the sensitivity of l to changes in a under the hybrid nominal income targeting strategy. 4.2. A strict inflation target Strict inflation targeting is an extreme, yet efficient form of the optimal policy rule. It occurs when l, the weight on the rate of inflation in the loss function,
12 Whether the hybrid nominal targeting strategy is actually operational is the subject of some controversy. The set-up implies that in a given time period the policymaker observes the current rate of inflation and the current output gap. Important issues regarding the availability of contemporaneous feedback data and the extent of measurement error are thus ignored. A study that addresses these concerns is by Croushore and Stark (1999). On the issue of output gap uncertainty in particular, see Orphanides (1998). In addition, the model assumes that the policymaker hits the target without fail. 13 Very different results emerge from an analysis of hybrid nominal income targeting in a backwardlooking model. There the variances of real output and the rate of inflation are not identical. Instead the effect of supply shocks is borne disproportionately by the variance of inflation. For more details, see Guender (2000).
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A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
approaches infinity. With l ! 1, h ! 0 in the optimal policy rule. 14 The optimal policy rule thus reduces to z ¼ pt ¼ 0: Moreover, it follows that Et ptþ1 ¼ 0. Again the impact of shocks on the rate of inflation is completely neutralized by the appropriate change in the policy instrument. With pt ¼ Et ptþ1 ¼ 0, real output observes the following process: 15 1 yt ¼ ut : a
ð19Þ
The variance of real output is given by V ðyt Þ
SIT
¼
r2u : a2
ð20Þ
The smaller the sensitivity of the rate to inflation to the output gap in the Phillips Curve, the greater the variance of real output and hence the greater the losses under strict inflation targeting. 16
5. Conclusion This paper shows that two monetary policy strategies––hybrid nominal income targeting and strict inflation targeting––are efficient strategies in the sense that they are special cases of the optimal monetary policy strategy. Indeed, any monetary policy strategy that is centered on the ultimate goal variables––the output gap and the rate of inflation––satisfies this criterion. The critical issue is to determine the weight on real output in the optimal rule. This weight is shown to depend on the preferences of the policymaker and the structural parameter on the output gap in the Phillips Curve. With 0 < h < 1, the optimal monetary policy rule in the forward-looking model is consistent with what Svensson (1997) terms ‘‘flexible’’ inflation targeting in the backward-looking model.
Acknowledgements I am deeply indebted to Graeme Guthrie and Andreas Irmen for help with using computer software. In addition, I wish to thank Arthur Benavie, David Black, Rich-
14
Another extreme yet efficient form of monetary policy entails l approaching zero. In this case h goes toward infinity, and the policymaker will pay attention only to deviations of output from capacity. 15 Just like under the hybrid nominal income targeting strategy, the policymaker can offset completely any demand-side disturbances under strict inflation targeting. 16 It is worth noting that, in the backward-looking specification, the variances of both demand and supply disturbances appear in the loss function irrespective of whether the policymaker pursues a strict inflation target or a hybrid nominal income target. The difference in results is attributable to the existence of varying time lags in the effect of policy on the output gap and the rate of inflation. For further results see Guender (2000).
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ard Froyen, and Bennett McCallum for making helpful comments. The valuable suggestions by two referees and by the editor of this journal are gratefully acknowledged. All errors are the sole responsibility of the author.
References Ball, L., 1997. Efficient rules for monetary policy. NBER Working Paper 5952. Bryant, R., Hooper, P., Mann, C. (Eds.), 1993. Evaluating Policy Regimes: New Research in Empirical Macroeconomics. The Brookings Institution, Washington, DC. Clarida, R., Gali, J., Gertler, M., 1999. The science of monetary policy: a new Keynesian perspective. Journal of Economic Literature 27, 1661–1707. Croushore, D., Stark, T., 1999. A real-time data set for macroeconomists. Federal Reserve Bank of Philadelphia Working Paper no. 99-4. Guender, A.V., 2000. Alternative monetary policy rules and the specification of the Phillips Curve: a comparison of nominal income with strict inflation targeting, University of Canterbury, Mimeo. Henderson, D.W., McKibbin, W.J., 1993. A comparison of some basic monetary policy regimes for open economies. Carnegie-Rochester Conference Series on Public Policy 39, 221–317. Journal of Monetary Economics, 1999. 43 (3). Koenig, E.F., 1995. Optimal monetary policy in an economy with sticky wages. Federal Reserve Bank of Dallas, Economic Review, Second Quarter, pp. 24–31. McCallum, B.T., 1997. The alleged instability of nominal income targeting. Reserve Bank of New Zealand Discussion Paper. McCallum, B.T., 1999. Recent developments in monetary policy analysis: the roles of theory and evidence. NBER Working Paper 7088. McCallum, B.T., Nelson, E., 1999. An optimizing IS-LM specification for monetary policy and business cycle analysis. Journal of Money, Credit, and Banking 31, 296–316. Orphanides, A., 1998-03. Monetary policy rules based on real-time data. Finance and Economics Discussion Series, Federal Reserve Board. Roberts, J.M., 1995. New Keynesian economics and the Phillips Curve. Journal of Money, Credit, and Banking 27, 975–984. Rudebusch, G.D., 2000. Assessing nominal income rules for monetary policy with model and data uncertainty. European Central Bank Working Paper Series, No. 14. Rudebusch, G.D., Svensson, L., 1999. Policy rules for inflation targeting. In: Taylor, J.B. (Ed.), Monetary Policy Rules. University of Chicago Press, Chicago, IL, pp. 203–246. Svensson, L., 1997. Inflation forecast targeting: implementing and monitoring inflation targets. European Economic Review 46, 1111–1146. Svensson, L., 1999. Price level vs. inflation targeting: a free lunch? Journal of Money, Credit, and Banking 31, 277–295. Taylor, J., 1993. Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public Policy 39, 195–214. Taylor, J., 1995. The inflation/output trade-off revisited. In: Goals, Guidelines, and Constraints Facing Monetary Policymakers. Federal Reserve Bank of Boston, Boston. Taylor, J., 1999. Monetary Policy Rules. University of Chicago Press, Chicago, IL. Woodford, M., 1999. Interest and Prices. Princeton University., Mimeo.
Optimal and efficient monetary policy rules in a forward-looking model Alfred V. Guender
*
University of Canterbury, Christchurch, New Zealand Received 9 July 1999; accepted 30 January 2001
Abstract This paper shows that two monetary policy strategies––hybrid nominal income targeting and strict inflation targeting––are efficient strategies of monetary policy in the sense that they are special cases of the optimal monetary policy strategy. In the case of a hybrid nominal income targeting strategy, the policymaker chooses a unitary trade-off between real output and the rate of inflation, while under strict inflation targeting the policymaker attaches a zero weight on output in the optimal policy rule. Ó 2002 Elsevier Science Inc. All rights reserved. JEL classification: E5 Keywords: Forward-looking; Optimal and efficient policy; Inflation targeting; Hybrid nominal income targeting
1. Introduction Current discussions of issues in monetary policy share a number of characteristics. First, there has been a move toward a new framework where the rate of inflation––and not the price level––features prominently. The new approach emphasizes simplicity over complexity. It comprises a simple two-equation IS-Phillips Curve framework where real output and the rate of inflation enter as the two endogenous
*
Fax: +64-3-364-2635. E-mail address: [email protected] (A.V. Guender).
0164-0704/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 1 6 4 - 0 7 0 4 ( 0 2 ) 0 0 0 1 4 - 9
42
A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
variables. Practical considerations have led to the omission of the LM relation from the analysis. As most central banks use a short-term nominal interest rate to set the stance of monetary policy, the inclusion of money demand in the analysis will merely serve to determine the volume of the money supply that is consistent with the set interest rate. 1 Despite the adoption of an alternative framework for the analysis of monetary policy issues, there remains some disagreement among economists about the proper specification of the underlying structural relations. Of the competing specifications, two have attracted particular attention. One approach assumes that real output and the rate of inflation exhibit persistence, that is, that current real output and the rate of inflation are tied to their past behavior. In the literature this specification is referred to as ‘‘backward-looking’’. 2 In sharp contrast, the ‘‘forward-looking’’ specification adopts a rational expectations framework where current real output and the rate of inflation, respectively, depend on the next period’s expected level. 3 Another characteristic common to current and recent contributions to the literature is the renewed interest in the properties of monetary policy rules. In view of the widespread disagreement among economists about the proper specification of macroeconomic relationships, efforts have been made to examine the properties of simple, tractable rules across a wide variety of macroeconomic models. The idea here is to examine the robustness of candidate rules for inflation targets, price level targets, nominal income targets, exchange rate targets, and other rules such as the Taylor Rule. 4 For instance, Ball (1997) finds nominal income targeting to be inconsistent with the optimal policy rule in a simple backward-looking model. Indeed he labels nominal income targeting a disastrous strategy of monetary policy as it leads to instability in both the rate of inflation and the output gap. This result is challenged by McCallum (1997) who attributes Ball’s findings to the backward specification of the Phillips Curve. 5 The current paper takes the forward-looking model as the baseline model to derive the optimal monetary policy rule. The paper takes the view that optimal mon-
1 See, for instance, Taylor (1995), Ball (1997), McCallum (1997), Svensson (1997), Clarida et al. (1999) and McCallum and Nelson (1999) amongst others. 2 The papers by Taylor (1995), Ball (1997) and Svensson (1997, 1999) fall into this category. 3 Examples of this approach are McCallum (1997), McCallum and Nelson (1999), Clarida et al. (1999) and Woodford (1999). 4 A recent comprehensive contribution in this area is the papers presented at a symposium of the Sveriges Riksbank in 1998 and published in the Journal of Monetary Economics (1999). A treatment of monetary policy rules also appears in Taylor (1999). See also Bryant et al. (1993), Henderson and McKibbin (1993) and Taylor (1993) for earlier assessments of the empirical properties of various monetary policy rules. Analyzing solely the merits of nominal income targeting, Rudebusch (2000) finds that this strategy receives only modest empirical support from US data. 5 For a comparative analysis of various monetary policy rules in forward- and backward-looking models, the reader is referred to Guender (2000). He shows that the instability of nominal income targeting in the backward-looking specification disappears if the policymaker chooses to target a hybrid form of nominal income.
A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
43
etary policy ought to be framed in terms of the ultimate goal variables, real output and the rate of inflation, with the relative emphasis on the two goal variables in the optimal rule determined by the underlying preferences of the policymaker and the structural parameter in the Phillips Curve. In addition, this paper shows that two simple monetary policy rules, hybrid nominal income targeting and strict inflation targeting, are special cases of the optimal monetary policy rule. The paper is organized as follows. Section 2 presents the forward-looking model. The optimal monetary policy rule is derived in Section 3. Section 4 discusses two efficient monetary policy strategies. Section 5 concludes.
2. The forward-looking model The simple forward-looking model consists of the following two equations: yt ¼ brt þ Et ytþ1 þ mt ;
ð1aÞ
pt ¼ Et ptþ1 þ ayt þ ut ;
ð1bÞ
where yt is the output gap, rt is the real rate of interest, pt is the rate of inflation. Both ut and mt are random disturbances with distribution ð0; h2u Þ and ð0; r2u Þ, respectively. Both a and b are positive parameters. The model assumes that the policymaker has full control over the setting of the policy instrument, the real rate of interest, which is inversely related to the demand for real output. Eq. (1a) can be thought of as the counterpart to the IS relation in the standard IS-LM framework. There is one noteworthy difference between Eq. (1a) and the standard IS relation, however. While in standard IS-specifications current real output is positively related to that of the preceding period, in the forward-looking IS specification the current output gap responds to the expected output gap in the following period. 6 Eq. (1b) represents a forward-looking Phillips Curve where the output gap has a contemporaneous positive effect on the rate of inflation. 7 The link between the current rate of inflation and expected inflation next period is the direct result of optimal price-setting behavior by producers under imperfect competition. The property that current inflation depends on inflation next period distinguishes Eq. (1b) from other Phillips Curve specifications such as the backward-looking
6
There is some disagreement about what the variable yt represents. In McCallum and Nelson (1999) and McCallum (1999) it represents the level of real output. In contrast, Svensson (1997, 1999), Woodford (1999), Clarida et al. (1999) and Rudebusch (2000) employ the deviation of the level of real output from capacity output in both the IS and the Phillips Curve relation and term it the output gap. In this paper the latter convention is adhered to. 7 For an explicit derivation of Eq. (1b), the reader is referred to Roberts (1995) and the references cited there.
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A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
specification employed by Svensson (1999) and Ball (1997) or the New Classical specification employed by Woodford (1999) and others. 8 3. Optimal policy The policymaker sets a fixed nominal target for the sum of the ultimate goal variables: the real output gap and the rate of inflation. 9 The parameter h indicates the weight the policymaker attaches to the output gap relative to the rate of inflation in the policy rule. z ¼ ½hyt þ pt ¼ 0:
ð2Þ
As shown below, h P 0. Hence the optimal value of h determines the trade-off between real output and the rate of inflation. Inserting Eqs. (1a) and (1b) into (2) and solving for rt yields 1 1 ðEt ptþ1 þ ayt þ ut Þ þ ðEt ytþ1 þ mt Þ; ð3aÞ rt ¼ bh b or 1 1 ðpt Þ þ ðEt ytþ1 þ mt Þ: rt ¼ ð3bÞ bh b This reaction function illustrates that the policymaker will respond to pressure on real output arising on the demand side by systematically raising the real rate of interest by 1=b. The extent to which the real interest rate rises in response to an increase in observed inflation in contrast depends on the size of the policy parameter h. Substituting (3a) into the IS relation (Eq. (1a)) results in ðh þ aÞyt ¼ Et ptþ1 ut :
ð4Þ
This equation shows how real output behaves after the rule is imposed. Real output decreases in the wake of supply shocks and in response to expected future inflation. The extent to which output declines depends on the size of a, a structural parameter, and h, a policy parameter. The absence of mt from Eq. (4) implies that demand side disturbances have no effect on real output. This is a direct result of the policymaker’s ability to affect real output (and the rate of inflation) contemporaneously through
8 The forward-looking model differs from the backward-looking specifications employed by Ball (1997) and Svensson (1997, 1999) in two important respects. First, there are no control lags in the forwardlooking model. This implies that a change in monetary policy has contemporaneous effects on the output gap and the rate of inflation in the forward-looking model. Such contemporaneous effects are absent in the backward-looking model where a change in the real rate of interest affects the output gap with a oneperiod lag and the rate of inflation with a two-period lag. Second, while current inflation depends on expected future inflation in the forward-looking model, current inflation depends on past inflation in the backward-looking model. This property produces the intriguing result that the structural parameter on the current output gap in the backward-looking Phillips Curves is negatively related to the current rate of inflation. Woodford (1999) employs an expectations-augmented Phillips Curve where the output gap is sensitive to inflation surprises, that is, (pt Et1 pt ). 9 For convenience, the target value is set equal to zero.
A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
45
varying the setting of the policy instrument. Combining Eq. (4) with the evolution of the rate of inflation (Eq. (1b)), we obtain h ðEt ptþ1 þ ut Þ: ð5Þ pt ¼ hþa To solve the model, we pose putative solutions for the endogenous variables: yt ¼ s11 ut ; ð6Þ pt ¼ s21 ut : It therefore follows that Et ptþ1 ¼ 0; Et ytþ1 ¼ 0:
ð7Þ ð8Þ ð9Þ
Inserting (7) and (8) into (5) and matching coefficients yields h : ð10Þ s21 ¼ hþa Hence the solution for the rate of inflation is h pt ¼ ut : ð11Þ hþa Substituting Eqs. (8) and (11) into Eq. (1b) and solving for yt yields the expression for real output: 1 ut : yt ¼ ð12Þ hþa It follows then that 2 h VarðpÞt ¼ r2u ; ð13aÞ hþa
2 1 r2u : ð13bÞ hþa The objective of the policymaker is to minimize a loss function consisting of the variance of real output and the rate of inflation, respectively: 10; 11 Varðyt Þ ¼
10
This is the loss function employed by Ball (1997). Other specifications could include the variance of the policy instrument, V ðrt Þ, especially if it is believed that the policymaker desires to smooth movements in the policy instrument. However, this would necessitate choosing an arbitrary weight for V ðrt Þ. See Rudebusch and Svensson (1999) for examples of this type of loss function. Alternatively, Koenig (1995) suggests that the performance of policy rules be evaluated on the basis of whether a weighted average of the price level and real output equals a pre-announced target. He finds both price level and inflation targeting to be suboptimal because both strategies put a zero weight on real output. Woodford (1999) derives a loss function from a choice-theoretic utility maximizing framework. His loss function is essentially the same as the one employed in the current paper. 11 In the loss function the target value for real output is capacity (potential) output. For simplicity the target value for the rate of inflation is assumed to be zero.
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A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
Fig. 1. The relationship between l and h for a ¼ 0:1, 0.25, 0.9.
Min Varðyt Þ þ lVarðpt Þ: h
ð14Þ
The solution to the above minimization problem is given by h ¼
1 P 0: la
ð15Þ
This setting represents the optimal choice for the policy parameter h. We notice that the optimum value for h is a function of l, the weight on the variance of inflation in the loss function, and a, the parameter on the output gap in the forward-looking Phillips curve. The relationship between h and l is illustrated for three different values of a in Fig. 1. An increase in the size of a shifts the curve downward, thus lowering h. The greater the effect of real output on the rate of inflation in the Phillips relation, the smaller the weight the policymaker attaches to real output in the optimal rule. This result is intuitively plausible as a larger value of a requires a smaller reduction in real output to engineer a desired decrease in the rate of inflation.
4. Efficient monetary policy strategies In this section we examine the properties of two strategies of monetary policy, both of which are special cases of the optimal monetary policy rule. The first strategy is geared towards attaining an announced hybrid nominal income target while the other strategy focuses solely on meeting a pre-specified inflation target. 4.1. A hybrid nominal income target The hybrid nominal income target consists of the sum of the deviation of real output from capacity and the rate of inflation: z ¼ yt þ pt . Again, let z ¼ 0 for simpli-
A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
47
Table 1 The sensitivity of l to changes in the size of a under the hybrid nominal income targeting strategy l ¼ 10 l¼4 l ¼ 1:11
for a ¼ 0:1 for a ¼ 0:25 for a ¼ 0:9
city. 12 A monetary policy rule which targets the sum of the output gap and the rate of inflation constitutes an efficient form of monetary policy because the policymaker chooses a unitary trade-off between real output and inflation. Setting h ¼ 1 in Eqs. (11) and (12), we obtain yt ¼ pt ¼
1 ut ; 1þa
ð16Þ
1 ut : 1þa
ð17Þ
Notice the symmetric effect of the supply side disturbances on real output and the rate of inflation, respectively. The variances of real output and the rate of inflation under a hybrid nominal income target are then given by 13 V ðyt Þ
NIT
¼
1 ð1 þ aÞ
2
r2u ;
V ðpt Þ
NIT
¼
1 ð1 þ aÞ
2
r2u :
ð18Þ
Eq. (18) implies that in the special case where the policymaker places a weight of l ¼ 1=a on the variance of inflation in the loss function, hybrid nominal income targeting is consistent with the optimal rule. Table 1 gives an indication of the sensitivity of l to changes in a under the hybrid nominal income targeting strategy. 4.2. A strict inflation target Strict inflation targeting is an extreme, yet efficient form of the optimal policy rule. It occurs when l, the weight on the rate of inflation in the loss function,
12 Whether the hybrid nominal targeting strategy is actually operational is the subject of some controversy. The set-up implies that in a given time period the policymaker observes the current rate of inflation and the current output gap. Important issues regarding the availability of contemporaneous feedback data and the extent of measurement error are thus ignored. A study that addresses these concerns is by Croushore and Stark (1999). On the issue of output gap uncertainty in particular, see Orphanides (1998). In addition, the model assumes that the policymaker hits the target without fail. 13 Very different results emerge from an analysis of hybrid nominal income targeting in a backwardlooking model. There the variances of real output and the rate of inflation are not identical. Instead the effect of supply shocks is borne disproportionately by the variance of inflation. For more details, see Guender (2000).
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A.V. Guender / Journal of Macroeconomics 24 (2002) 41–49
approaches infinity. With l ! 1, h ! 0 in the optimal policy rule. 14 The optimal policy rule thus reduces to z ¼ pt ¼ 0: Moreover, it follows that Et ptþ1 ¼ 0. Again the impact of shocks on the rate of inflation is completely neutralized by the appropriate change in the policy instrument. With pt ¼ Et ptþ1 ¼ 0, real output observes the following process: 15 1 yt ¼ ut : a
ð19Þ
The variance of real output is given by V ðyt Þ
SIT
¼
r2u : a2
ð20Þ
The smaller the sensitivity of the rate to inflation to the output gap in the Phillips Curve, the greater the variance of real output and hence the greater the losses under strict inflation targeting. 16
5. Conclusion This paper shows that two monetary policy strategies––hybrid nominal income targeting and strict inflation targeting––are efficient strategies in the sense that they are special cases of the optimal monetary policy strategy. Indeed, any monetary policy strategy that is centered on the ultimate goal variables––the output gap and the rate of inflation––satisfies this criterion. The critical issue is to determine the weight on real output in the optimal rule. This weight is shown to depend on the preferences of the policymaker and the structural parameter on the output gap in the Phillips Curve. With 0 < h < 1, the optimal monetary policy rule in the forward-looking model is consistent with what Svensson (1997) terms ‘‘flexible’’ inflation targeting in the backward-looking model.
Acknowledgements I am deeply indebted to Graeme Guthrie and Andreas Irmen for help with using computer software. In addition, I wish to thank Arthur Benavie, David Black, Rich-
14
Another extreme yet efficient form of monetary policy entails l approaching zero. In this case h goes toward infinity, and the policymaker will pay attention only to deviations of output from capacity. 15 Just like under the hybrid nominal income targeting strategy, the policymaker can offset completely any demand-side disturbances under strict inflation targeting. 16 It is worth noting that, in the backward-looking specification, the variances of both demand and supply disturbances appear in the loss function irrespective of whether the policymaker pursues a strict inflation target or a hybrid nominal income target. The difference in results is attributable to the existence of varying time lags in the effect of policy on the output gap and the rate of inflation. For further results see Guender (2000).
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49
ard Froyen, and Bennett McCallum for making helpful comments. The valuable suggestions by two referees and by the editor of this journal are gratefully acknowledged. All errors are the sole responsibility of the author.
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