The non-linear Phillips curve and inflation forecast targeting

The non-linear Phillips curve and
inflation forecast targeting
Eric Schaling
Correspondence to: Professor Eric Schaling, Department of Economics, RAU, PO Box 524,
2006 Auckland Park, Johannesburg, Republic of South Africa. Phone + 27 11 489 2927; fax +
27 11 489 3039; email ESC@EB.RAU.ac.za
The views expressed are those of the author and do not necessarily reflect those of the Bank of
England.
Issued by the Bank of England, London, EC2R 8AH to which requests for individual copies
should be addressed: envelopes should be marked for the attention of the Publications Group.
(Telephone 0171-601 4030). Working papers are also available on the Bank’s Internet site at
http://www.bankofengland.co.uk.
Bank of England 1999
ISSN Number 1368-5562

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Contents
Abstract
5
1
Introduction
7
2
A non-linear Phillips curve
9
2.1 Non-linear output inflation dynamics
10
2.2 Optimal monetary policy
13
3
A non-linear policy rule
17
4
Uncertainty about the output gap
23
5
Brainard uncertainty and non-linearities
30
6
Summary and concluding remarks
37
Appendix: the minimum of equation (3.2)
39
References
41
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Abstract
This paper extends the Svensson (1997a) inflation forecast targeting
framework with a convex Phillips curve. An asymmetric target rule is
derived, which implies a higher level of nominal interest rates than the
Svensson (1997a) forward-looking version of the reaction function
popularised by Taylor (1993). Extending the analysis with uncertainty
about the output gap, it is found that uncertainty induces a further upward
bias in nominal interest rates.
Keywords: inflation targets, non-linearities, asymmetries, stochastic
control
JEL Codes: E31, E42, E52, E58
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1 Introduction(1)
The 1990s saw the introduction of explicit inflation targets for monetary
policy in a number of countries: New Zealand, Canada, the United
Kingdom, Sweden, Finland and Spain. Inflation targeting has been
introduced as a way of further reducing inflation and to influence market
expectations, after disappointment with monetary targeting (New Zealand
and Canada) or fixed exchange rates (United Kingdom, Sweden and
Finland).
The relation between inflation targets and central bank preferences has
been thoroughly investigated. On the one hand there is a theoretical
literature (Walsh (1995), Svensson (1997)) that concludes that inflation
targets can be used as a way of overcoming credibility problems because
they can mimic optimal performance incentive contracts.(2) On the other
hand there is an empirical literature that tests whether inflation targets
have been instrumental in reducing the policy-implied short-term trend
rate of inflation (Leiderman and Svensson (1995)). Broadly speaking, the
evidence is that inflation targets have indeed brought about a change in
policymakers’ inflation preferences.
Unlike the relation between inflation targets and central bank
preferences, a relatively underexplored issue is how to translate inflation
targets into short-term interest rates. This is the issue of how to map
explicit targets for monetary policy into monetary policy instruments, or
how to implement an inflation targeting framework. An exception is a
recent and important contribution by Svensson (1997a). He shows that
— because of lags in the transmission process of short-term interest rates
to inflation — inflation targeting implies inflation forecast targeting. In
his analysis the central bank’s forecast becomes an explicit intermediate
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(1) This paper was written while Schaling was an Economist in the Monetary Assessment and
Strategy Division of the Bank of England. The author is grateful for helpful comments by
Marco Hoeberichts, Alison Stuart, Tony Yates, Andy Haldane, Mike Joyce, Douglas Laxton,
Lavan Mahadeva, Peter Westaway, Jagjit Chadha, Paul Tucker, Alistair Milne, Peter Pauly and
seminar participants at the Bank of England, the South African Reserve Bank, CentER, RAU,
the University of the Witwatersrand and attendants at the third Econometrics Conference at the
University of Pretoria. Bruce Devile and Martin Cleaves helped to prepare the paper.
(2) This literature is surveyed in Schaling (1995). Also, by increasing the accountability of
monetary policy, inflation targeting may reduce the inflation bias of discretionary policy. See
Svensson (1997), and Nolan and Schaling (1996).
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target and its optimal reaction function has the same form as the Taylor
rule (1993).(3) Recently, Clarida, Gali and Gertler (1997b) have shown
that this type of reaction function does quite a good job of characterising
monetary policy for the G3. The kind of rule that emerges is what they
call ‘soft-hearted’ inflation targeting. In response to a rise in expected
inflation relative to target, each central bank raises nominal interest rates
sufficiently to push up real rates, but there is also a modest pure
stabilisation component to each rule.
The 1990s have also seen the development of the literature on the so-
called non-linear Phillips curve. (Chadha, Masson and Meredith (1992),
Laxton, Meredith and Rose (1995), Clark, Laxton and Rose (1995,1996),
and Bean (1996).) This recent literature puts the time-honoured inflation
output trade-off debate in a fresh perspective by allowing for convexities
in the transmission mechanism between the output gap and inflation.
More specifically, according to this literature, positive deviations of
aggregate demand from potential (the case of an upswing or ‘boom’) are
more inflationary than negative deviations (downswings) are
disinflationary.(4)
This paper marries both strands of the literature. The Svensson (1997a)
inflation forecast targeting framework is extended with a convex Phillips
curve. Using optimal control techniques, an asymmetric policy rule is
derived that implies higher nominal interest rates than the Svensson (1997a)
forward-looking version of the reaction function popularised by Taylor (1993).
This means that, if the economy is characterised by asymmetries, the
Svensson (1997a) linear target rule may underestimate the correct level of
interest rates.
The rest of the paper is organised into five sections followed by an Appendix.
The model is set out in Section 2. The asymmetric policy rule in the
deterministic case is presented in Section 3. In Section 4 we extend the
analysis with uncertainty about the output gap. Section 5 compares the
implications of multiplicative parameter uncertainty for policy with those of
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(3) For an interesting recent study of the Taylor rule in a UK context, see Stuart (1996).
(4) There is also the view that the Phillips curve is concave (Stiglitz (1997)). It can be modelled
by changing the sign of ϕ in equation (2.1). Obviously, all policy conclusions are reversed.
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the classic Brainard (1967) analysis. Section 6 concludes, and the Appendix
provides proofs behind key results.
2 A non-linear Phillips curve
As stated by Laxton et al (1995, pages 345-46) the broad acceptance of
the expectations-augmented Phillips curve — and the associated ‘natural
rate’ hypothesis — led to the important conclusion that a long-run trade-
off between activity and inflation did not exist. Subsequent research on
output-inflation linkages has focused on how expectations are formed and
the reasons for price ‘stickiness’ that cause real variables to respond to
nominal shocks. Almost all of this work, however, has been predicated
on the assumption that the trade-off between activity and inflation is
linear, that is the response of inflation to a positive gap between actual
and potential output is identical to a negative gap of the same size.
Though analytically convenient, the linear model ignores much of the
historical context underlying the original split between classical and
Keynesian economics: under conditions of full employment, inflation
appeared to respond strongly to demand conditions, whereas in deep
recessions, it was relatively insensitive to changes in activity.(5)
Many of the tests for non-linearity that have been performed have been
uninformative because the filters that people have chosen have been
fundamentally inconsistent with the existence of convexity. However,
when properly tested, there is some evidence for asymmetries. Laxton et
al (1995) find that by pooling data from the major seven OECD countries
the Phillips curve is non-linear. Clark et al (1996) — using quarterly
data from 1964–90 — find that the US inflation-output trade-off is non-
linear. Debelle and Laxton (1997) find that the unemployment-inflation
trade-off is non-linear in the United Kingdom, the United States and
Canada. Finally, recent research at the Bank of England (Fisher et al
(1997)) also finds that a Phillips curve that embodies a mild asymmetry is
consistent with UK data.
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(5) Indeed, as pointed out by Laxton et al (1995), the original article by Phillips emphasised
such an asymmetry, with excess demand having had a much stronger effect in raising inflation
than excess supply had in lowering it.
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2.1 Non-linear output inflation dynamics
The main purpose of this section is to combine a convex Phillips curve
along the lines of Laxton, Meredith, and Rose (1995) with the Svensson
(1997a) model of inflation targeting, to allow for lags in the transmission
process of short-term interest rates. We use this model to analyse the
effects of delaying monetary policy measures on the future levels of
inflation and nominal interest rates.
The functional form we employ to represent the non-linearity in the
inflation-output relationship is
α
∆π
y
1 t
t
f
+ 1 = (•
) =

(2.1)
1− α ϕy
1
t
where π is p − p
t
t − 1 , ie the inflation (rate) in year t, pt is the (log) price
level, y is an endogenous variable output, α >
≤ϕ <
1
0 and 0
1 are
parameters, and ∆ is the backward difference operator. We normalise the
natural rate of output in the absence of uncertainty to zero.(6) This means that
y is the (log) of output relative to potential, ie the output gap. Equation (2.1)
is graphed in Figure 2.1. Its relevant properties can be derived by looking at
the first derivative of f ( )
• — ie the slope of the output-inflation trade-off:
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(6) With uncertainty, the natural rate of output in the non-linear model will always be below that
of the linear model. See, for instance, Clark et al (1995). The reason is that if output were
maintained, on average, equal to the natural rate of the linear model, then the asymmetry in the
response of inflation to demand shocks would make it impossible to maintain inflation at a
constant inflation target. To see this formally, lead the Phillips curve one period and take
expectations at time t, which yields Et ∆πt
E
+ =
2
t α
[
y
1 t
/
α ϕy
+
−
1 1
1
t + 1] . In a sustainable
equilibrium with a constant rate of inflation equal to the inflation target, Et∆πt+ =
2
0 .
2
Taking account of Jensen’s inequality we get 0 = f (Et yt+ 1) + ϕ / 2 f ' ' (Et yt+ 1)σε . This
equality then (implicitly) defines Et yt+ 1 , the average level of output in the presence of shocks.
With the convexity parameter value used in this paper ( ϕ = 0 5
. ) this level lies about 0.1 percent
below the corresponding level of output in the absence of shocks. Since several empirical papers
— see for instance Debelle and Laxton (1997) — suggest a larger gap between the stochastic
and deterministic equilibrium.
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