We propose a simply yet flexible framework for the analysis of optimal monetary policy
rules that produces the type of non-linear responses derived in the literature as
special cases. Perhaps more importantly, our framework suggests a richer set of nonlinear
responses than have been considered yet and thus may prompt further work in
this area

Boinet, V

Martin, C

English

Brunel University Research Archive

  1 Optimal non-linear monetary policy rules  Virginie Boinet   and  Christopher Martin    Brunel Business School Brunel University Uxbridge Middlesex UB8 3PH  July 2006  Abstract We propose a simply yet flexible framework for the analysis of optimal monetary policy rules that produces the type of non-linear responses derived in the literature as special cases.  Perhaps more importantly, our framework suggests a richer set of non-linear responses than have been considered yet and thus may prompt further work in this area.   1) Introduction Recent models of optimal monetary policy rules have allowed for non-linearity in the response of interest rates to inflation and output.  In some of these models, non-linearity arises because the standard assumption of quadratic preferences has been amended to allow for asymmetry (eg Cukierman and Gerlach, 2003) or for zone-like preferences that might match the features of some inflation targeting regimes (Orphanides and Wieland, 2000). In other models, non-linearity arises because the aggregate supply or Phillips curve is non-linear (eg Nobay and Peel, 2000).  In this paper, we propose a simple yet flexible framework that produces the type of non-linear responses derived in the literature as special cases.  Perhaps more importantly,   2our framework suggests a richer set of non-linear responses than have been considered yet and thus may prompt further work in this area.  The paper is structured as follows. The canonical linear Taylor rule model is briefly derived in section 2) as a benchmark.  Section 3) considers a general model of policymakers’ preferences and shows how this generalises and extends models proposed in the literature.  Section 4) does the same for the case of a non-linear Phillips curve.  Section 5) considers the interactions between non-quadratic preferences and a non-linear Phillips curve, showing how this implies a richer set of non-linear models than considered thus far in the literature.  Section 6) concludes.  2) The Taylor rule In this section we outline the derivation of the Taylor (1993) policy rule, as a foundation for the extensions in subsequent sections.  We assume the economy comprises conventional aggregate demand and aggregate supply or Phillips curves.  The aggregate demand curve is   (1)   1 1( )dt t t t t t ty i E E yρ π ε+ += − − + +    where y  is the output gap, i  is the nominal interest rate, π is the inflation rate, ε dt is an i.i.d demand shock and ρ  is a positive coefficient.   The Phillips curve is   (2)  1st t t t ty Eπ γ θ π ε+= + +   where stε is an i.i.d supply shock and γ  and θ  are positive coefficients.  We assume that policymakers choose the nominal interest rate at the beginning of period t on the basis of information available at the end of period (t-1).  Their optimisation problem is   (3)  { } 10Min  tjt t jijE Lδ∞− +=∑     subject to (1) and (2), where δ  is the discount factor and   3 (4)  ( )2* 2 * 21 ( )2 2 2t t t tL y i iλ µπ π= − + + −   is a conventional per-period quadratic loss function where *π  is the inflation target or desired inflation rate, *i  is the equilibrium or desired nominal interest rate and λ  and µ  are positive coefficients.   Following the existing literature (eg Clarida et al, 1999) by solving this optimisation problem under discretion, the optimal monetary policy rule can be expressed as  (5)  ( )* *1 1t t t t ti i E y Eρλ ργ π πµ µ− −= + + −   This is the celebrated Taylor rule, with the familiar constant proportional response of interest rates to inflation and output.    3) Optimal monetary policy with non-quadratic preferences We generalise the quadratic per-period loss function in (4) by assuming  (6) ( ) ( )* * * 22 21 11 ( )2 2 2yt y t yyt y tt ty ye e yL i iβπ βππβα π π α βππ πα π π αλ µβ α β α−− − −− −= + + −   where πβ  and yβ  are integers and πα  and αy are real numbers.  The optimal monetary policy rule then becomes  (7)  ( )* *1 1( ;  , ) ( ;  , )t t t y y t ti i E f y E f π πρλ ργβ α π π β αµ µ− −= + + −   where 1 1( ; , , )xef x xβαββ αα−−= .  Equation (7) is a non-linear monetary policy rule that encompasses the various models that have been proposed in the literature with non-quadratic preferences as special cases.     4 The loss function in (6) can take on a number of forms depending on the values of the parameters.  The loss function simplifies to the quadratic when 0πα → , 0yα →  and 1yπβ β= = .  In this case, we obtain the Taylor rule in (5).  Figure 1) depicts *ti i−  as a function of the inflation gap *tπ π− .  In this case, the relationship is a straight line.    Considering the response to inflation, if 1πβ = , we have the Linex preferences of Ruge-Murcia (2003) and Nobay and Peel (2003).  The response of interest rates to inflation is a convex function of inflation if 0πα >  and a concave function if 0πα < .  The relationships between *ti i−  and *tπ π−  in these cases are also depicted in figure 1); the relationship is convex when 0πα >  as a greater concern for excessive inflation leads to a more vigorous adjustment of interest rates when inflation is above the target; the relationship is concave when 0πα < .   Linex preferences imply the optimal policy rule is asymmetric.  An asymmetric response is arguably implausible, not least because many countries have a target zone for inflation rather than a point target (see Petursson, 2004, for details of the target zone in over 25 countries).  Following Orphanides and Wieland (2000), a target zone implies a small response to inflation when this is clearly within the zone and stronger response when inflation is outside the zone.  This type of response is obtained when 1πβ > . Interest rates are unresponsive to inflation in a zone around the desired level.  Outside the zone, interest rates respond more strongly; the strength of the response is an increasing function of πα .   The response is symmetric if πβ  is an even number (figure 2 illustrates this case).  If πβ  is an odd number, the response is asymmetric; the sign of the asymmetry is determined by πα  (there is stronger response when inflation is above rather than below the zone target if 0πα > ).  A higher value of πβ  widens the zone.  This discussion has focussed on the response of interest rates to inflation.  Similar considerations apply to the response to output.  This response is convex if 0yα >  and concave if 0yα < .  A convex response is consistent with arguments that policymakers are more averse to slumps than to booms (Cukierman and Gerlach, 2003).  It has also been suggested that policymakers should only respond strongly to output when this is clearly some way from the equilibrium level (Meyer et al, 2001).    5This suggests a zone-like response to output, which is obtained when 1yβ > .  As with the response to inflation, this response is symmetric if yβ  is even and asymmetric if yβ  is odd.  4) Optimal monetary policy with a non-linear Phillips curve In this section we revert to quadratic preferences but assume the Phillips curve is    (8)   1 11PCPC tPCyst t t t tPCey Eβαβπ γ θ π εα−+−= + +   The optimal monetary policy rule in this case is   (9)   ( )* *1 1 1( )t t t t t t ti i E y E g y Eργρλ π πµ µ− − −= + + −   where 2 1( ; , ) (( 1) )pcpc pcpc pc pcyypc pc pc pcpceg y y y eββαβ β αβ α β βα−−= − +  is proportional to the slope of the Phillips curve. The response of interest rates to inflation in (9) depends on the slope of the Phillips curve, and is thus a function of the output gap.  The policy rule is a generalisation of the  Taylor rule, which is obtained if the Phillips curve is linear, in which case ( ) 1tg y = .  The response of interest rates to inflation reflects the shape of the Phillips curve.   The Phillips curve is convex if 1PCβ =  and 0pcα > ; convexity was proposed by Laxton et al (1995) and is often assumed in recent models. The response of interest rates to inflation is a convex function of output in this case. The concave Phillips curve, suggested by Stiglitz (1997), is obtained where 1PCβ =  and 0pcα < ; in this case the response to inflation is a concave function of output.  Figure 3) illustrates the responses of interest rate to inflation in these cases.  Finally, it has been suggested that the Phillips curve may be concave when the output gap is negative and convex when the output gap is positive (Dupasquier and Ricketts, 1998).   The Phillips curve in (8) has this property when 1PCβ > .  In this case, as figure 4) shows, the response to inflation is close to zero in a zone around the point where the output   6gap is zero, but stronger when the output gap is larger in either direction.   The Phillips curve is symmetric when PCβ  is an even number; reflecting this, the response to symmetric.  The Phillips curve is asymmetric when PCβ  is odd; the response to inflation in this case is also an asymmetric function of output, with a stronger response when the output gap is positive if 0pcα > .  5) Optimal monetary policy with both non-quadratic preferences and a non-linear Phillips curve  Finally, we consider the case where preferences are non-quadratic and the Phillips curve is non-linear.  Using (6) and (8), the optimal monetary rule in this case is   (10)  * *1 1( ;  , ) ( ; , ) ( ;  , )t t t y y t t pc pc ti i E f y E g y f π πρλ ργβ α β α π π β αµ µ− −= + + −   where the functions (.)f  and (.)g  are as defined above. In this policy rule, the second term reflects non-quadratic preferences over output while the third term reflects both non-quadratic preferences over inflation and a non-linear Phillips curve. This latter interaction suggests that the policy rule can take a number of forms beyond those suggested so far in the literature.  Figures 5) and 6) depict the relationship between *ti i−  and the output gap, for a given inflation gap, when there are zone-symmetric preferences over output and either a concave (figure 5) or a convex-concave (figure 6) Phillips curve.  In both cases, there is a strong response to output when output is some way from equilibrium (the response is asymmetric in figure 5), reflecting the concavity of the Phillips curve).   The most striking feature of these graphs is that there is a “perverse response”, whereby interest rates actually fall when output increases, for some values of the output gap.  In figure 5), the perverse response occurs when the output gap is in a zone close to zero. In figure 6), the perverse response occurs when the output gap is   7negative.  These types of “perverse response” are analysed in more detail in Boinet and Martin (2006).    6) Conclusions This paper has considered optimal monetary policy when non-quadratic preferences and a non-linear Phillips curve imply non-linear responses by policymakers.  We propose a simply yet flexible framework that produces the type of non-linear responses derived in the literature as special cases.  This framework suggests a richer set of non-linear responses than have been considered yet and thus may prompt further work in this area.   The optimal policy rules suggested in this paper can be econometrically estimated. Ultimately, their value will be determined by empirical evidence.   References  Boinet, V, and C Martin, (2006), "The perverse Response of Interest Rates”, unpublished paper, Brunel University  Cukierman, A, and S Gerlach, (2003), "The inflation bias revisited: theory and some international evidence," Manchester School, vol. 71(5), pages 541-565  Dupasquier, C and N Ricketts, (1998), “Non-linearities in the Output-Inflatin Relationship”, Bank of Canada Working Paper 98-14  Laxton, D, Meredith, G and Rose, D (1995), ”Asymmetric Effects of Economic Activity on Inflation: Evidence and Policy Implications”, IMF Staff Papers, Vol 42, No 2.   Meyer, L, E Swanson and V Wieland, (2001). "NAIRU Uncertainty and Nonlinear Policy Rules," American Economic Review, vol. 91(2), pages 226-231, May  Nobay, A. and D Peel, (2000). "Optimal monetary policy with a non-linear Phillips curve," Economics Letters, vol. 67(2), pages 159-164  Nobay A and D Peel, (2003). "Optimal Discretionary Monetary Policy in a Model of Asymmetric Central Bank Preferences," Economic Journal, vol. 113(489), pages 657-665  Orphanides, A, and V Wieland, (2000). "Inflation zone targeting," European Economic Review, vol. 44(7), pages 1351-1387   8 Petursson, T,  (2004), “The effects of inflation targeting on macroeconomic performance”, Economics Department, Central Bank Of Iceland  Ruge-Murcia, Francisco J, 2003. " Inflation Targeting under Asymmetric Preferences," Journal of Money, Credit and Banking, vol. 35(5), pages 763-85  Stiglitz, J, (1997), “Reflections on the Natural Rate Hypothesis”, Journal of Economic Perspectives, 11, 3-10   Taylor, J. (1993). Discretion versus policy rules in practice, Carnegie-Rochester Conference Series on Public Policy, 39, pp 195-214.      9  Figure 1) The response of interest rates  Figure 2) The response of interest  with quadratic and Linex preferences rates with zone-symmetric preferences          Figure 3) The response to inflation   Figure 4) The response to inflation  with linear, convex and concave with a convex-concave Phillips curve Phillips curves         10  Figure 5) The response of interest rates         Figure 6) The response of interest  with zone-symmetric preferences and rates with zone-symmetric preferences  a concave Phillips curve                                  and a convex-concave Phillips curve         

Optimal non-linear monetary policy rules

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