Optimal monetary policy is price level t

Optimal monetary policy: is price-level targeting
the next step?
Patrick Minford and David Peel (Cardi® University)¤
revised 16 July 2003

Abstract
We examine whether in°ation targeting should be regarded as optimal. Targeting in°ation implies (undesirably) that price level variance
tends to in¯nity: we produce some evidence from both a representative
agent model and a long-used forecasting model that, once an endogenous
indexation response is allowed for, price level targeting imposes no extra
costs of macro variability, indeed gives signi¯cant gains.

The 1990s have been a rather successful period for monetary policy. In°ation
has been low and the world economy rather stable, at least since the modest
recession of 1990{92. Some (e.g. Clarida et al, 1999) have declared victory,
lauding the `science' of using interest rate manipulation to achieve both low
in°ation and output stability. However, concerns remain. The principal one is
whether monetary policy should neglect the price-level as opposed to in°ation |
and a number of papers have addressed this issue using a variety of frameworks.
A second is whether we can treat Phillips Curves as stable relationships in terms
of Lucas' critique (Lucas, 1976). This paper is an attempt to round up some of

these issues and consider where we are in our search for the optimal monetary
policy.
One issue we will not pursue here is that of de°ationary environments and the
zero bound on interest rates. Price level targeting (around a rising deterministic
trend) could, it has been argued, be a help in such environments in that it creates
an automatic expectation of future in°ation when prices fall, so lowering real
interest rates in a de°ation. Others have argued that it increases the risk of
hitting the zero bound because after in°ationary episodes prices have to be
forced downwards in a potentially de°ationary way back to the target trajectory.
Our simulations throughout assume an environment in which the zero bound is
avoided and so we cannot shed light on this issue here; we hope to return to it
in future work.
¤ We are grateful for useful comments, without implicating them in our errors, to Harris
Dellas, Kent Matthews, Charles Nolan, Frank Smets, Michael Wickens, two referees of this
journal and also to other participants in the Cardi® University economics workshop, the
European Central Bank visitors' seminar, the 2003 Macroeconomics Conference at Rethymno,
Crete and the European Monetary Forum.

1

In probing the idea of stationarity in targeting | that is, targeting the level
of money rather than its growth rate, and the level of prices rather than in°ation | we are joining a growing literature that is investigating the possibility
of `price stability', de¯ned to mean prices whose long-term variance is kept low
(as opposed to `low in°ation' under which price variance tends to in¯nity as
the horizon lengthens). As noted by Svensson (1999a, b), the consensus of earlier authors (Hall, 1984; Duguay, 1994; Bank of Canada, 1994; Fischer, 1994)
has been that targeting prices rather than in°ation (and presumably also by
implication money rather than its growth rate) would lower long-term price
variance at the expense of higher short-term in°ation and output variance. As
he puts it `The intuition is straightforward: in order to stabilize the price level
under price-level targeting, higher-than-average in°ation must be succeeded by
lower-than-average in°ation (apparently implying) higher in°ation variability
(which) via nominal rigidities would then seem to result in higher output variability.' One could add that the consensus on what monetary policy should be
is an interest rate rule that targets in°ation and the output gap or variants of
these such as nominal GDP growth | for example, Taylor (1993) who began
this vogue following Henderson and McKibbin (1993), Clarida, Gali and Gertler
(1999) and McCallum and Nelson (1999a). Svensson argues however that the
consensus may be wrong: he gives an example where discretionary monetary
policy facing a Phillips Curve with persistence in output actually raises in°ation
variability (without bene¯ting output stability) if it targets in°ation rather than
the price level | in e®ect the price level target is preventing some of the `useless'

in°ationary discretion (useless because private agents fully anticipate it so that
it has no e®ect on output). This result admittedly depends on there being no
commitment that would rule out such useless discretion by other means | with
such commitment price-targeting is inferior to in°ation-targeting. Kiley (1998)
also notes that Svensson's set-up is one of `policy ine®ectiveness' on output (as
in Sargent and Wallace's original paper, 1975) and hence no trade-o®s arise
with output stability; there is, as Svensson puts it, a free lunch in the model as
price-level targeting reduces the in°ation bias with no cost in output stability.
Svensson's approach has been followed up by others, including Vestin (2000)
and most recently Nessen and Vestin (2000); they ¯nd that within a Phillips
Curve with forward- and backward-looking expectations (where there is no such
free lunch) average in°ation targeting (a weighted average of a price level and
in°ation target) can improve on both in°ation and price level targeting, again
because of the e®ect that such targeting has in e®ectively limiting discretion.
We do not employ here the commitment argument for price level targeting;
our reason is policymakers themselves acting within a general consensus about
optimal rules of monetary behaviour can put an end to welfare-reducing behaviour (e.g. by legislation about the proper behaviour of the central bank).
Minford (1995) noted that the principal in all such policy frameworks is the
electorate itself; it is reasonable to assume that sooner or later appropriate institutions for achieving its best interests, which in this context would include
commitment, will be found. Hence the argument seems somewhat fragile to

likely institutional change. A number of authors have on the basis of such ar2

guments examined optimal monetary policy under commitment. Smets (2000)
uses a model with Calvo-style Phillips Curve to examine the optimal horizon
for bringing in°ation or the price-level back to their targets; he ¯nds that the
optimal length becomes shorter the more forward-looking are the price expectations and the steeper is the Phillips Curve. Williams (1999) evaluates a variety of such rules in the FRB large-scale model of the US in which there is
a forward/backward-looking Phillips Curve and inertial pricing dynamics as in
Fuhrer and Moore (1995). He ¯nds that multi-period in°ation targeting ranks
highly and that price-level targeting only causes minor output instability. With
this family of models, interest rate rules of the Taylor-Henderson-McKibbin type
give good results, with the optimal degree of inertia in response (the lagged interest rate coe±cient) depending on the degree of forward-lookingness. In this
commitment context the addition of price level targeting at a suitably long horizon has little e®ect on the optimal trade-o® | as such it an innocuous optional
extra for policy-makers desiring to anchor the long-run price level.
Commitment thus basically removes the bene¯ts of price-level targeting.
However, little attention in this work has been paid to the e®ect of changes
in wage contract structure in evaluating price level targeting | an `interesting
issue', Svensson notes, as yet unexplored. Minford, Nowell and Webb (2003)
set out a model in which the structure of wage contracts responded to the
behaviour of the economy. Ascari (2000) and Casares (2002) have examined
respectively how the parameters of Taylor-type overlapping ¯xed-wage contract

and of Calvo-type price-setters respond. Minford et al use instead a model
in which an overlapping wage contract sets wages that do not have to be the
same in every contract period; this has the attractive `natural rate property'
(Minford and Peel, 2003) according to which expected in°ation at the time of
contract-signing does not a®ect expected real wages and so expected output.
This property is not possessed by Taylor-type ¯xed-wage or Calvo contracts.
Within the model of Minford et al it turned out that if monetary policy
targeted, at the shortest time horizon, the level of a nominal variable (there it
was the money supply), then wage indexation would fall because the persistence
of shocks to real wages would be reduced; the slope of the Phillips Curve would
thus become °atter (while that of the aggregate demand curve would become
steeper). These two things in turn caused output stability to increase in the
face of real shocks; thus provided monetary shocks could be kept within tight
limits such a nominal-level target improved welfare. In this paper we adopt this
model to examine the question of whether such level-targeting is optimal and if
so whether it should take the form of price-level targeting (as pioneered by the
Bank of Sweden in the inter-war period, Berg and Jonung 1999).
We begin by reviewing the standard current approach to optimal monetary
policy under commitment; in it we assume as is normal that the central bank
can hit an in°ation target exactly. We then go on to consider more realistic

rules where the central bank cannot hit targets exactly but rather conducts
monetary policy in terms of an intermediate money supply target which it hits
inexactly (with a random error). We review the performance of both in°ation
and price level targeting rules of this sort; we do so within a representative agent
3

model ¯rst and secondly within a long-used forecasting model of the UK. Within
both we allow the degree of indexation to respond to the monetary regime. We
conclude by suggesting that price level targeting could be a natural next step
for monetary rules.

1

The standard approach

One familiar approach to monetary policy optimisation is to assume some sort
of Phillips Curve together with a quadratic social objective function in terms
of output and in°ation. The latter is justi¯ed as an approximation to the
welfare of the representative agent; when expectations of it are taken in forming
the best intertemporal plan for monetary policy, it implies a trade-o® between

the variances of in°ation and output. Since monetary policy cannot raise the
expected level of output and provided there is commitment, the expected rate
of in°ation can be set equal to the in°ation target, so that this trade-o® is the
focus of policy.
Let us follow the set-up in Svensson (1997), for example. Let the Phillips
curve (model) with persistence be:
y t = ½yt¡ 1 + ®(¼ t ¡ ¼ et) + ²t

(1)

The set-up is that the central bank has scope to react to shocks | implicitly
because the wage contract underlying this Phillips Curve is longer than the
publication/reaction time to the shock. Hence although this appears to be a
non-contract supply curve, we are to think of the `period' as being one equal to
the length of a contract so that within it policy can react (one can also extend
the analysis to explicit overlapping contracts without much di±culty). We also
assume that the central bank can set the in°ation rate exactly using its tools of
monetary policy; this is obviously unrealistic but it does remind us usefully that
it would be optimal if it could | as we shall see it implies that the central bank
should totally o®set `demand' shocks occurring elsewhere than in the Phillips

Curve. As for `supply' shocks in the Phillips Curve, here represented by ²; we
now discover how they should be optimally reacted to.
Using the usual utility function the problem under commitment is:
n
o
2
2
V (y t¡1 ) = M ax (wrt¼ t ; ¼ et ) E t¡1 ¡0:5 (¼ t ¡ ¼ ¤ ) ¡ 0:5¸ (yt ¡ y ¤ ) + ¯ V (yt )

(2)

The reaction function that emerges is:
¼t = ¼¤ ¡

1+

®¸
²t
¡ ¯½2

®2 ¸

Notice that the supply shock response in now increased by the persistence
term owing to the need to stabilise future output.

4

Figure 1: Optimal Monetary Policy
We can usefully illustrate the resulting set-up in the familiar form of IS/LM
and AS/AD curves. What we see is that it is as if there was under optimal
policy an AD curve that reacts to the supply shock shifting the AS/Phillips
curve. This AD curve simply traces the points of tangency between the Phillips
Curve and the social indi®erence ellipses as the supply shock varies. We can then
trace out in the upper IS/LM quadrant the implied behaviour of the monetary
instruments. E®ectively we have a vertical LM curve that shifts appropriately
with the supply shock as dictated by the lower quadrant; when however there
is a demand shock shifting the IS curve, this must be o®set by a rise in interest
rates so that output is una®ected | i.e. the LM curve is vertical. Notice that
the slope of the AD curve corresponds to the implied trade-o® between in°ation
and output variance; the steeper the slope the lower the output variance and

the higher the in°ation variance.

5

Under the assumption that the central bank can observe current shocks we
can then write down the optimal central bank deployment of its instruments.
Let the IS curve be:
y t = ¡¯ rt + ut

(3)

and let the money demand function be:
Mt = ¡° (rt + ¼ ¤ ) + ±y t + pt + v t

(4)

where p t = ¼ t + p t¡1 is the log of prices, r is the real rate of interest and
u; v are respectively the aggregate demand and the money demand shocks.
Then the optimal real interest rate target will be obtained by substituting
for optimal output into (3) as:

rt =
b

1
fu t ¡ ½yt¡ 1 ¡ (1 + ®b)² tg
¯

(5)

and the optimal money supply target will be obtained by substituting the
optimal output, real interest rate and price level into (4) to obtain
½
¾
½·
¸
¾
° + ¯±
°
ct = ° + ¯± ½yt¡ 1 +
M
(1 + ®b) + b ² t + pt¡ 1 + ¼ ¤ ¡ ut + vt
¯
¯
¯
(6)
Of course in practice the above set-up needs careful reinterpretation in terms
of both the exact speci¯cation of overlapping contracts and the information
available to the monetary authorities. The central bank does not observe these
shocks and so the practical debate over monetary policy rules is in terms of
`e±cient surrogates' for these shocks. For example one could rationalise a `Taylor
Rule' under which real interest rates are optimally related (all positively) to the
output gap, the in°ation gap (over target) and past real interest rates as follows
from (5): the current output gap is a good indicator of the demand shock, u;
the current in°ation gap is a good inverse indicator of the supply shock; ¯nally
the lagged output gap will also be inversely related (see Fig. 1) to lagged real
interest rates. The di±culty in validating this rationalisation is that every model
di®ers in detail over these indicator relations. This therefore leads naturally to
a discussion of the practical assessment of alternative monetary policy rules,
which we will pass over at this stage.

1.1

Optimal monetary policy under Calvo contracts |
another basis for price targeting?

The whole of this discussion assumes that the social objective function set out
above in terms of in°ation and output is correct. This function can be derived
(e.g. Rotemberg and Woodford, 1997) by a second-order Taylor series approximation around a standard representative agent's utility function in terms of
6

consumption and leisure. The usual derivation also assumes a Calvo contract
set-up which gives rise to a rather di®erent Phillips Curve. In this Calvo set-up
a similar in°ation-output optimal response can be found as above under such a
contract set-up | though there is an issue about the inclusion of lagged output
in the response, see also Svensson and Woodford, 2003; McCallum, 2003); in this
case in°ation deviations from the target represent distortions of relative prices
while output deviations from the natural rate (the °exible-price equilibrium,
made optimal by o®setting imperfect competition distortions via a production
subsidy) represent output distortions, so that half their squared values represent
lost consumer surplus
If one suppresses the error term in the Calvo Phillips Curve on the grounds
that it has no easy interpretation, then the usual dilemma for monetary policy
| trading o® in°ation versus output variance | disappears. Optimal monetary
policy with commitment should then stabilise the price level at its existing level
provided there are no other distortions than those associated with shocks to
prices and hence also output relative to its °exprice equilibrium | Clarida
et al, 1999; Woodford, 2000, Goodfriend and King (2001). Khan, King and
Wolman (2002). The reason is that this will imply no further distortions due
to price changes: existing distortions, due to past in°ation, cannot be a®ected
since each period those changing price are chosen randomly hence one is as likely
to add a new distortion by changing prices as one is to o®set a previous price
change.
Recent work has explored how sensitive this optimising strategy is to the
introduction of capital and adjustment costs and of a demand for cash (e.g. via
money in the utility function) and to the removal of the production subsidy so
that the °exible price equilibrium is non-optimal | most recently Collard and
Dellas (2003) ¯nd that it remains close to optimal, when the non-linearity of
the solution is allowed for by higher order Taylor series approximation.
The di±culty with such rules is that there is a potential for time-inconsistency.
If for example the history of price shocks has generated a highly skewed relative
price distribution or the capital stock is depressed, there is an incentive to produce price changes that would at least partially unwind this history. In general
how much can one rely on the policy of price ¯xing being carried out without
error? Should prices rise for some reason (e.g. a control failure by the central bank), then carrying out the commitment by reducing prices subsequently
implies that those whose relative prices are already out of equilibrium (due to
the price error) will not necessarily be bene¯ted because they may not have
the chance to change prices back, while those others still in equilibrium whose
prices did not change will be driven out of it if it is their turn to change prices.
The time-inconsistent optimum is now to validate the price rise | implying
base drift, an in°ation not a price level ¯xing rule. Therefore the commitment
optimum of price level stability is fragile | perhaps incapable of credibility. If
so, then the Calvo set-up most naturally gives rise to in°ation rules not price
level ones.
However one may wish to question the representation of consumer welfare
implied by the Calvo set-up. At a purely intuitive level it is odd that people
7

do not care about price stability. They clearly must, in the sense that if only
in°ation is stabilised, this makes the price level a random walk, and so causes
the variance of prices to tend to in¯nity the longer the time horizon. This
means that the value of longer term contracts set in nominal terms is subject to
unacceptable variance; so nominal contracts will be shortened. It does not seem
likely that such a forced shortening (or equivalently indexation) of contracts is
optimal; indexation is known to su®er from imperfections.
There is also a theoretical reason for questioning the usual Calvo set-up, as
argued by Minford and Peel (2003). Calvo contracts must be `indexed up' by
agents to allow for ongoing in°ation. Thus:
¼ t = ¹E t¼ t+1 + ¸(yt ¡ yt¤ ) + u t

(7)

is the Calvo forward-looking Phillips Curve in which e®ectively the whole
path of future output (marginal costs) a®ects current price rises. What this
implies is that prices are rising because some (relative) prices are rising | which
given that there is no general expected in°ation implies that they also rise by
this much in nominal terms | and the other prices are held ¯xed in nominal
terms (because the menu cost is greater than the cost of them staying out of
equilibrium). Thus the basic Calvo equation is conditioned on the assumption
that the expected general in°ation rate is zero. To extend the Calvo model
it is assumed (e.g. Erceg, Henderson and Levin, 2000; Gali and Monacelli,
2002; Christiano et al, 2002) that it is costless for all agents to uprate prices or
wages by the generally expected (or `core') rate of in°ation | this being like
a `relabelling' or `indexing' of all prices on the `menu'. Call this core rate of
in°ation ¼et . Then the equation should be rewritten:
¼t ¡ e
¼ t = ¹(E t¼ t+1 ¡ Et e
¼ t+1 ) + ¸(yt ¡ y ¤t ) + ut

(8)

Following di®erent practices of di®erent authors, we could emerge with three
forms of Phillips Curve:
(a) e
¼ t = ¼ t , core in°ation:

¼ t = ¹E t ¼ t+1 + (1 ¡ ¹)¼ t + ¸(y t ¡ yt¤ ) + u t

(b) e
¼ t = ¼ t¡1 , lagged in°ation:
¼t =

¹
1
¸
1
E t ¼ t+1 +
¼ t¡1 +
(yt ¡ y ¤t ) +
ut
1+¹
1+¹
1+¹
1+¹

(9)

(10)

(c) e
¼ t = E t¡1 ¼ t ; rational expectations using available (lagged) information:
¼ t = E t¡1 ¼ t + ¸(y t ¡ yt¤ ) + u t

8

(11)

Of these three only (c) would be adopted by agents with rational expectations. Yet notice that it is merely the original Sargent and Wallace (1975)
`surprise Phillips Curve' in which output responds to unexpected prices (or
equivalently unexpected in°ation). It no longer implies that in°ation deviations
from target produce distorted relative prices; only the current surprise one does
so.
Now it remains true under such a rewritten Calvo contract that by the same
logic as above we can represent welfare losses by the squared in°ation surprise
and squared output deviation. However now we have truly thrown out the baby
with the bathwater; not only is there no case for price level targeting, there is
also none for in°ation targeting since any in°ation rate is as good as any other
| only its predictability matters.

2

Assessing targeting rules in terms of consumer
welfare

It seems unlikely that unconditional variability cannot matter in either in°ation
or prices. The rate of in°ation is routinely included in social objective functions;
we would wish to use a model in which in°ation did indeed matter for agents'
welfare. We would also argue that people must care about price level stability.
Chadha and Nolan (2003) argue that macro outcomes are not too di®erent if a
price level rule is substituted for a low in°ation rule; e®ectively this is what we
wish to examine here. To anticipate our results, we tend to ¯nd the contrary,
especially when the contract structure is varied endogenously; but while we
would accept that their result comes partly from legitimate di®erences of model
speci¯cation, it is this endogeneity of contracts that we believe is likely to be
important in making price level targeting a signi¯cant macro factor in most
models.
So far we have considered optimal monetary policy in a world where the
central bank could exactly control prices. However, of course such is not the
world we live in. In practice central banks only control either the (base) money
supply or interest rates and do not observe the current shocks hitting the economy. Hence they must choose rules for responses of their chosen instrument to
the available observed data. Usually it is assumed they observe approximately
current (quarterly) in°ation and output. Thus the well-known `Taylor Rule'
(named after John Taylor, previously Chairman of George Bush Senior's Council of Economic Advisers) involves, as noted above, interest rates reacting to the
current output gap, the in°ation deviation from target and the lagged interest
rate. However, it must be stressed that we cannot derive this as a closed-form
optimising relation in the way we did above with the in°ation setting; the reason
is that a) the relation between the errors and these observables is stochastic b)
the model in which policy is embedded is complex and nonlinear. Hence we are
forced to use stochastic simulation analysis to evaluate rival rules in essentially a
`black box' fashion | by which we mean that while we may have some hunches

9

informed by simpler models of what rules can work well, we can only check by
trial and error with these stochastic simulations.
The rules we investigate are those that target either the price level or in°ation
in the coming period, using information in the current period. The rules set the
money supply to hit the planned target exactly; but there is a stochastic error
in execution. The money supply therefore does not respond to other current
shocks, as would occur for example in an interest-rate setting regime.
To approach optimal monetary policy from this angle we use two sorts of
model. First we set out a representative-agent model based on micro foundations
and experiment with this; while still too complex to evaluate analytically it is
simple enough to develop some insights into the mechanisms involved. Second,
we use a UK forecasting model, the Liverpool Model, which has both been in use
for two decades reasonably successfully and is based on an IS/LM approximation
to the same micro-foundation used in our representative-agent model. Our
results are, as above, only preliminary; but we do have some indications of
where this approach may lead.

2.1

Targeting within a representative-agent model:

In a recent paper Minford, Nowell and Webb (2003) set up a model where an
employed representative agent chooses an optimal degree of wage indexation
(to prices and the auction wage) in response to the monetary regime. The
model has two exogenous shocks driving it, a demand shock (to the money
supply presumed to originate from monetary policy), and a supply (productivity) shock. The productivity shock is (rather naturally) modelled as a random
walk throughout. Of course whether the money supply shock is transitory or
permanent depends on the monetary rule; if it targets for example the level
of money it will be transitory, if targets the money supply growth rate, it will
be turned into a random walk The authors then asked whether the monetary
regime should target the growth rate or the level of the money supply; or of
prices? And should the current money supply be exogenous (as in `monetary
base control'), or endogenously ¯xed by an interest rate rule, as is the general
practice of central banks? They suggested that these familiar choices appear in
an unfamiliar light when indexation is endogenous. When the monetary regime
moves to a price level rule with exogenous stationary money supply shocks, the
aggregate supply curve °attens (as we have seen already above in our Phillips
Curve set-up) and the aggregate demand curve steepens, generating a high degree of macro stability (i.e. in the face of supply shocks) provided that money
supply shocks themselves are low-variance and stationary.
What they found is that there is a general class of monetary rules that target
the level of a nominal variable, whether money or prices, whether via interestrate setting or via money supply control, that is markedly superior to the class
currently in use, that target rates of change of the nominal variables. The reason
for this superiority is strong shift in contract structure away from indexation
to nominal. This shift is strongest when a price-level rather than money-level
target is in place | but in welfare terms it is hard to choose between the two, as
10

there is a trade-o® between the interests of the employed (who prefer the price
rule) and the unemployed (who prefer the money rule).
They embed the representative household in an environment of pro¯t-maximizing competitive ¯rms which on a large proportion of their capital sto ck face
a long lag before installation (a simple time-to-build set-up) and a government
that levies taxes and pays unemployment bene¯ts (which distort households'
leisure decisions and introduce a `social welfare' element into monetary policy). Firms and governments use the ¯nancial markets costlessly and settle
mutual cash demands through index-linked loans; since there is no binding cash
constraint on these agents, these loans are assumed to be una®ected by the
imperfections of the price index which are short term in nature. This model is
too simpli¯ed in many ways to match the data of a modern economy whether
in trend or dynamics; however its focus is purely on the wage contract decision
and its simplicity is justi¯ed in terms of its ability to match the OECD facts
about wage contracts.
In calibrating the model the authors chose parameters perceived as plausible
for modern OECD economies. The contract length is set at 4 quarters; the
elasticity of leisure supply at 3; the share of stocks and other `short-term' capital
at 0.3; the average life of other capital at 20 quarters; the share of labour income
in value-added at 0.7 (the production function is Cobb-Douglas); the elasticity
of the o±cial price index to unanticipated in°ation at 0.2 (implying that a 1%
unexpected rise in in°ation would result in a 0.2% temporary overstatement of
the price level faced by the representative consumer). The initial values assume
10% unemployment; a capital-output ratio of 6; an average (=marginal) tax
rate of 0.10; a real interest rate of 5%.
The government is assumed to smooth both the tax rate and the growth rate
of the money supply by borrowing (from ¯rms). Nevertheless it cannot avoid
noise in its money supply setting | the source of this could be its inability to
monitor the money supply quickly or even at all (for example in the USA the
use of dollars by foreigners around the world makes it impossible to know what
the domestic issue of dollars is).
Money supply raises prices in the long run, and in the short run also raises
output, with persistence extending up to 15 quarters but with most e®ect over
after 10. In the high-indexed case there is less real e®ect and less persistence
than in the high-nominal case.
These fairly standard properties stem from the model's deliberate drawing
on elements that have been shown by past work to be useful in explaining the
business cycle and also natural rates as discussed for example by Parkin (1998),
though he notes we are still some way from building dynamic stochastic general
equilibrium models that can fully explain the business cycle. The elements
here include: time-to-build investment, cash-in-advance, nominal contracting
(as noted above), household liquidity constraints, and (on the natural rate side)
the in°uence of unemployment bene¯ts on labour supply. With suitable countryby-country calibration one would expect to be able to mo del OECD countries'
business cycle and natural rate experience with at least some modest success.
One example of a model with many of these elements is the Liverpool Model of
11

the UK which we discus and use in the next section.
Minford et al found that in the face of stationary productivity and money
supply shocks indexation would be minimal with only a slight tendency to rise
as the variance of money shocks rose dramatically. However when shocks to either became highly persistent indexation to prices or to their close competitor,
auction wages, (which together we term `real wage protection') become large,
becoming largest when both shocks are persistent. The reason was that productivity shocks would disturb prices and so the real worth of nominal wage
contracts; indexation was of little use in remedying this disturbance if it was
temporary because by the time the indexation element was spent the shock
would have disappeared, but with a permanent disturbance indexation can help
o®set it with a lag. If into this already-indexed world of persistent productivity shocks, monetary persistence is also injected, indexation rises further, to
help alleviate the increased disturbance to real wages. This higher indexation
also helps to alleviate the instability in unemployment which accompanies the
greater shock persistence of money | the point being that this persistence induces persistence in the economy's departure from its baseline and so disturbs
unemployment too for longer.
The authors looked at experience in the OECD in the 1970s where it is wellknown that real wage protection was substantial; their calibrated model, when
estimated variances and persistence of money and productivity shocks were fed
into it, predicted high protection in all countries they could cover, apparently
in line with the facts. They also found, contrary to much casual comment, that
there was little evidence of any diminution of real wage protection in the 1990s;
the model also predicted as much, for even though the variance of money supply
shocks fell by then, their persistence remained essentially unchanged.
In order to assess the welfare of society from di®erent monetary policy rules
the model gives the average household the standard Constant Relative Risk
Aversion utility function with Cobb-Douglas preferences across consumption
and leisure. The resulting welfare function for the policy-maker to maximize is
therefore:

EUt = E

1
X

¿ =t

±¿ ¡ t

½

(cº¿ [¸ + a¿ ]1¡º )1¡ ½ ¡ 1
1¡½

¾

where
ct =

W t¡ 1
(1 ¡ at¡ 1 ); ¸ = 1
pt

implying that leisure time is equal to working time when unemployment at is
zero. We set v = 0:7, based on the marginal valuation of leisure at wages net of
unemployment bene¯t. Because households get unemployment bene¯t on their
spells of eligible unemployment, at , this implies that their choice is distorted;
they choose leisure (¸, which we in practice set at unity, is the ineligible part of
leisure) in response to the di®erential between wages and bene¯ts. But then of
12

course they must pay for the bene¯t burden via taxes; the present discounted
value of this tax burden is the same as this bene¯t bill and so we deduct this
from their consumption to obtain total private utility.
In subsequent work Minford and Nowell, 2003, use this model to examine
the relative merits of in°ation | and price-level targeting. The target rule holds
current money supply exogenous at last period's target setting and chooses a
money supply for next period that forces the expected in°ation (price level) to be
on target in this next period; this money supply plan is however executed with
an error, the model's 'monetary shock.' (which can in practice be interpreted
as a shock on either the supply or demand side of the money market; it is the
model's demand shock).. There is thus no current response of money supply to
shocks; nor any implied interest rate smoothing in the current period.
Their results can be summarised simply. In°ation-targeting generates a high
degree of indexation. When price-level targeting is undertaken but indexation
is assumed constant, welfare falls, because the variability of unemployment rises
sharply. But when indexation is allowed to change endogenously, it drops to
nil and the result is a rise in welfare, with the variance of consumption down
somewhat and that of unemployment down substantially.
Table 1: Price-level and in°ation targeting compared within a calibrated representative agent model
standard error in parenthesis +
In°ation-target = 100
Indexation
Welfare
Var
Var
(%)
#1
#2
(cons.)
(unemp.)
In°ation-targeting
71
100
100
100
100
(3)
(3)
(3)
Price-level targeting
71
98
91¤¤
99
119 ¤¤
(holding indexation ¯xed)
Price-level targeting
0
102 125 ¤¤
95¤
69¤¤
(indexation endogenous)
q¡ ¢
+
2
standard error of Montecarlo sample variance = est. variance £
where n
n
is the number of sample observations (here 2000) | source Wallis, 1995
De¯nition: #1 is the standard CRRA formula in the text; #2 is the weighted average
(weight on consumption = 0:7 , on unemployment = 1:0) of the two (inverted) variances.
¤
signi¯cant at 10% level
¤
signi¯cant at 1% level

2.2

Targeting within the Liverpool forecasting model of
the UK

Plainly the work of this last section has been carried out within a deliberately
simpli¯ed structure. Yet claims are being made about the potential bene¯ts for
policy. Before policy-makers would be likely to be tempted to try such rules,

13

they would need to see evidence that they perform adequately in a variety of
possible macro structures. Thus we would like a full menu of shocks within a
model that has been estimated empirically and that has withstood the test of
forecasting as well as applied policy analysis; preferably this model should be
one that is derivable from the sort of micro-foundations of our simple model.
Fortunately we have available a model that answers roughly to this description:
the Liverpool Model of the UK. This is an open economy version of a rational
expectations IS-LM model, such as can be derived from a micro-founded model
by suitable approximations (McCallum and Nelson, 1999b) | thus for example
the Liverpool Model IS curve has the expectation of future output in it, the
hallmark of this approximation. The model's Phillips or Supply curve assumes
overlapping wage contracts as in our simple model. The labour market underpinning it is explicit and the model solves for equilibrium or natural rates of
output, unemployment and relative prices. In recent work a new FIML algorithm developed in Cardi® University (Minford and Webb, 2000) has been used
to reestimate the model parameters: it turns out that the new estimates are
little di®erent from the model's original ones, based partly on single-equation
estimates, partly on calibration from simulation properties.
The model has been used in forecasting continuously since 1979, and is now
one of only two in that category. The other is the NIESR model, which however
has been frequently changed in that 20-year period: the only changes in the
Liverpool Model were the introduction of the explicit natural rate supply-side
equations in the early 1980s and the shift from annual data to a quarterly version
in the mid-1980s. In an exhaustive comparative test of forecasting ability over
the 1980s, Andrews et al (1996) showed that out of three models extant in that
decade | Liverpool, NIESR, and LBS | the forecasting performance of none
of them could `reject' that of the others in non-nested tests, suggesting that
the Liverpool Model during this period was, though a newcomer, at least no
worse than the major models of that time. For 1990s forecasts no formal test is
available, but the LBS model stopped forecasts and in annual forecasting postmortem contests the NIESR came top in two years, Liverpool in three. Thus we
would suggest that the Liverpool Model has a respectable forecasting record,
at least on a par with the only other model available of the general type we
seek | viz. micro-founded and suitably estimated. Comparative work on the
NIESR model would also be of interest; so far it has not been possible. There
are also models in the public sector | those of the Treasury and the Bank of
England | however, we would question the extent to which they have suitable
micro-foundations for our purpose and their forecasting record is also unclear.
Lastly, in respect of simulation properties and use of these for policy analysis,
we note that the Liverpool Model has been extensively used in policy analysis
bearing on the `monetarist' and `supply-side' reforms of the Conservative governments of Margaret Thatcher. It is now generally conceded that these reforms
have been broadly successful; the Liverpool Model acted to some degree as intellectual underpinning for them at a time of general academic hostility from
UK macroeconomists.
We therefore suggest that the Liverpool Model could be regarded as a suit14

able vehicle for checking the `realism' of our policy conclusion on the simple
model that level rules are preferable to rate of change rules for monetary policy.
Our method is as before to run these rules in the model in the face of stochastic
simulations (further details can be found in Matthews, Meenagh, Minford and
Webb, 2003) . We shock the full range of endogenous and exogenous errors,
exactly as in the model speci¯cation. The model's wage equation is written in
terms of the real wage reacting to the real bene¯t rate and to unemployment,
which are the auction wage components (implicitly the auction wage element
has a weight of 0.2), and negatively to the di®erence of the price level from the
average forecast of it at the times of wage contracting, then positively to this
di®erence lagged.
For our purposes here we adapt it as follows:
Wt = vP t + wE t¡j Pt + ®(Pt ) + w ¤ : : : =
(® + v)(Pt ¡ E t¡j Pt ) + E t¡j P t + w ¤ : : :
Now we lag the auction and indexed elements two periods because of a delay
due to the ¯rm's internal checking procedures in adjusting pay to the unexpected
change in the price level and obtain the real wage as:
W t ¡ Pt = ¡(Pt ¡ E t¡j P t) + (® + v)(Pt¡ 2 ¡ E t¡j¡2 P t¡2 ) + w ¤ : : :
Hence the two-period-lagged term carries the extent of real wage protection.
It is this part that is adjusted endogenously by the employed to minimise the
variance of their real wage.
The results within the Liverpool Model (Table 2) are about as favourable
to price-level targeting as in the representative-agent model. Again, they show
that under in°ation-targeting there is a high degree of indexation and that this
would drop to nil under price-level targeting. Similarly, too, they show that
if indexation is assumed endogenous, welfare will rise signi¯cantly if price-level
targeting is introduced; in the Liverpool Model the variance of consumption falls
more and that of unemployment falls less than in the representative agent model
but both fall signi¯cantly. The di®erence is that when indexation is held ¯xed,
the variances behave very much the same in the Liverpool Model as when indexation is endogenous; there is still a substantial gain over in°ation targeting. One
reason for this seems to be that in the Liverpool wage equation the indexation
we have assumed as feasible is rather imperfect. Thus real wages even under
the fullest indexation are disrupted by a persistent price shock; eliminating this
persistence gives gains that are almost as large as under zero indexation. A further reason for the improved stability under price-level targeting is that, on the
demand side, ¯nancial wealth e®ects are dampened since price shocks (to the
real value of long-term nominal bonds) are temporary rather than permanent.

2.3

Conclusions

We have examined whether monetary policy could be imporved by targeting the
price level. We lo oked at the operation of rules targeting either the in°ation rate
15

Table 2: Price-level and in°ation targeting compared within the Liverpool
Model
standard error in parenthesis +
In°ation-target = 100
Indexation Welfare
Var
Var
(%)
#1
#2
(cons.) (unemp.)
In°ation-targeting
80
100
100
100
100
(1.3)
(1.3)
(1.3)
Price-level targeting
80
102:3
120 ¤¤
71 ¤¤
91 ¤¤
(holding indexation ¯xed)
Price-level targeting
0
102:4
120 ¤¤
70 ¤¤
92 ¤¤
(indexation endogenous)
q¡ ¢
2
+
standard error of Montecarlo sample variance = est. variance £
n where n

is the number of sample observations (here 12078) | source Wallis, 1995
De¯nition: #1 is the standard CRRA formula in the text; #2 is the weighted average
(weight on consumption = 0:7, on unemployment = 1:0 ) of the two (inverted) variances.
¤
¤

signi¯cant at 5% level
signi¯cant at 1% level

or the price level both within a representative-agent model and within a `live'
forecasting model of the UK. We found in both some con¯rmation of the idea:
that targeting the price level would bring a signi¯cant gain in output stability,
rather than the loss usually assumed,. because the endogenous response of
reduced indexation would enhance the economy's stability..

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