Should Central Banks target Consumer Prices or the
Exchange Rate?
∗
Tatiana Kirsanova
University of Exeter
Campbell Leith
Glasgow University
Simon Wren-Lewis
University of Exeter
January 31, 2006
Abstract
In this paper we consider two arguments suggesting that monetary authorities in an open economy should target output price inflation and not consumer price inflation. Thefirst suggests that output price inflation corresponds to the distortions caused by price rigidity. The second shows how policy rules involving consumer price inflation can induce instability because of the feedback from interest rates to consumer price inflation via the exchange rate. We examine both arguments in the context of an open economy which is subject to a range of shocks. We show that both arguments remain robust, but that there is a case for including a terms of trade or real exchange rate gap term in the authorities welfare function alongside the output gap and output price inflation.
∗Our thanks to Mike Artis, Chris Allsopp, David Cobham, Alan Sutherland and an anonymous referee for helpful comments, but any mistakes are ours alone.
1
Introduction
In all relatively open economies where the monetary authorities have an explicit inflation objective, such as the U.K., that objective involves some measure of consumer price inflation. Recently two arguments have been put forward that suggest that this is the wrong inflation measure to target. Thefirst asserts that output price inflation, rather than consumer price inflation, more accurately reflects the welfare loss associated with changing prices. A second argues that simple policy rules based on consumer price inflation may lead to instability because of the influence of interest rates on the exchange rate, and its feedback to consumer price inflation.
The relationship between consumer and output price inflation depends on the ex-change rate, and so these arguments are part of a more long-standing debate about what role the exchange rate should have in influencing monetary policy. It has frequently been argued that interest rate setting should be influenced by exchange rate movements, or the distance between the exchange rate and some concept of its equilibrium level (e.g. Wren-Lewis (1997)). This argument became intense in the UK at the end of the 1990s, as sterling appreciated substantially but inflation remained close to its target level. However adding an exchange rate target alongside output and inflation as a policy objective re-mains unorthodox, and some of the recent literature that explicitly derives social welfare functions suggest it is unnecessary.
In this paper we critically examine both arguments against the use of consumer price inflation as a policy objective. Section 2 outlines the welfare arguments for focusing on output price inflation. We generalise earlier analysis by explicitly introducing Interna-tional Risk Sharing (IRS) shocks into the model. The focus on output price inflation remains, but in the general case there is now a potential role for some form of real ex-change rate objective analogous to the output objective. Section 3 uses the same model to illustrate the instability that may emerge with simple rules involving consumer price inflation, and shows how the chances of this instability increase as the economy becomes more open and monetary policy becomes more active. Section 4 concludes.
2
Social Welfare with UIP shocks
Suppose an economy is made up of a number of identical individuals, each of which have utility
max
{Cs,ys}∞s=t
Et
∞
X
s=t
whereC is consumption,yis output for goodz,ξis a taste shock, andβis a discount rate. (The functionv(·)embodies both the disutility of labour and the production technology.) While this or a similar assumption forms the basis of most theoretical macromodels, when analysing policy these models have traditionally postulated a quite separate objective function for policy makers, typically involving quadratic terms in output and inflation. The absence of any link between utility functions and social welfare functions is a major embarrassment for models that emphasise their microfoundations.
It is therefore not surprising that Woodford’s analysis (Rotemberg and Woodford (1997) and Woodford (2003)) showing how such welfare functions could be derived as second order approximations to (1) has been rapidly adopted in the literature. There are now a large number of papers that derive policy makers’ objective functions for an open economy explicitly from consumers’ utility. (These include, besides those papers cited below, Sutherland (2005) , Batini et al. (2003), and Beetsma and Jensen (2004).) A key point about such derivations is that they are model specific: although the utility function (1) may be fairly general, the welfare function derived as a second order approximation to (1) depends on the structure of the model. Woodford shows that if nominal inertia in the model is represented by Calvo contracts and there are sufficient government subsidies to offset monopoly distortions that remain in a zero inflation steady-state, then a second order approximation to social welfare can be represented by quadratic terms in inflation and the deviation of output from its ‘natural’ level. The natural level is the level that would occur without the nominal inertia distortion, and the difference between actual and natural output is described as the ‘output gap’.
In the closed economy analysed by Woodford, there is no distinction between output prices and consumer prices. Aoki (2002) considers a two sector model, where prices in one sector are completelyflexible, and shows that it is only inflation in the non-flexible sector that is relevant for welfare. He explicitly suggests that imported goods in an open economy are akin to theflexible price sector, and that therefore the price of imported goods should not appear in the inflation measure representing welfare. Gali and Monacelli (2005) argue that, under certain conditions, the benevolent policy makers’ objective function in an open economy can be collapsed to quadratic terms involving output price inflation and output gaps, and is therefore isomorphic to the closed economy case. Clarida et al. (2001), using the same framework, show that consumption gaps and terms of trade gaps will move proportionately with the output gap, and so the output gap can replace these other gaps in any objective function. In contrast, Beningno and Benigno (2004), De Paoli (2004) and Pappa (2002) derive measures of social welfare for open economies that do involve terms in the terms of trade gap. Whilst comparing all these derivations is beyond the scope of
the current paper, we will attempt to clarify some of the issues, and show how IRS shocks can introduce a role for terms of trade disequilibrium in welfare once we abandon one of the key assumptions used in Gali and Monacelli (2005).
As International Risk Sharing implies Uncovered Interest Parity (UIP), then IRS shocks embody UIP shocks. The reason for adding IRS shocks is straightforward. One argument that is frequently invoked in favour of exchange rate targeting (and its limit, monetary union) is that markets often drive the exchange rate well away from levels im-plied by fundamentals, and that this has damaging effects on the economy as a whole. (For example, see Buiter and Grafe (2003) in evidence submitted to the U.K. Treasury enquiry into joining EMU.) It is therefore interesting to note whether these shocks intro-duce a role for the exchange rate in the welfare function, and what form that might take. We formally derive a second order approximation to social welfare for this model, noting the implications of special cases.
To focus on the economics behind this derivation, we present the detailed description of the model, and the details of the second order approximation to welfare, in two appendices, and only present key relationships in the main text. In terms of log-linear deviations from a zero inflation steady state, the model of a small open economy can be represented by the following equations. Here we present only first order approximations for clarity, although the appendix derives some second order approximations that are required for welfare analysis. (For any variableXt,Xˆt= ln(Xt/X), where X is the steady state value
of Xt, and we refer to this asX disequilibrium.)
The New Keynesian Phillips curve derived from the existence of Calvo contracts is given by
πH,t=βEt[πH,t+1] +λmcct, (2)
where πH is output price inflation (output prices = price of domestic production = pH),
andλ is a function of model parameters (see Appendix A). Real marginal costs (mc) are given by
c
mc= ˆW −PˆH, (3)
where W are nominal wages. (Where variables are undated they are all at time t.) We assume that a subsidy exists in steady state to exactly offset the monopoly mark-up, so that steady state real marginal cost is unity.
International risk sharing implies ˆ
where Cˆ is consumption, Yˆ∗ overseas output, Sˆ are the terms of trade (the ratio of the price of overseas produced goods to domestically produced goods), 0 < α < 1 andη are demand curve parameters (α is the share of foreign goods in the consumption bundle). Overseas output Yˆ∗
t is equal to overseas consumption (our country is small), and ˆζ is a
distortionary IRS shock.
The demand curve, combined with the international risk sharing expression (see Ap-pendix A.8) is
ˆ
Yt= ˆYt∗+
¡
α2σ−α2η−2α σ+ 2α η+σ¢Sˆt+ (1−α) ˆζt, (5)
where Yˆ is output.
The first order conditions from consumer optimisation are ˆ
W −PˆH−αSˆ= ˆY
1
ψ + ˆC
1
σ + (
h
ψ+
g
σ)ξ, (6)
ˆ
Ct=Et[ ˆCt+1]−σ(ˆrt−Et[πt+1]), (7)
where
σ=− UC
CUCC
, g = UCξ
CUCC
=−σUCξ
UC
, ψ = vy
yvyy
, h= vyξ
yvyy
= vyξψ
vy
. (8)
The left hand side of (6) is the real consumer wage, and the production technology is log-linear (i.e. y = n). Consumer price inflation π and output price inflation πH are
related by
πt=πH,t+α∆Sbt. (9)
By combining (4), (7) and (9), we can derive the standard UIP condition, withζˆpresent as a distortionary shock.
The utility function (1) represents the utility of a representative consumer who supplies labour to industry z. A benevolent policy maker will attempt to maximise the utility of all workers, and so will maximise the discounted sum of terms which at each time s will be of the form
U(Cs, ξs)−
Z 1 0
v(ys(z), ξs)dz. (10)
Taking a second order approximation of (10) around the steady state gives
Ws=U(Cs, ξs)−
Z 1 0
v(ys(z), ξs)dz =CUC[ ˆCs(1−
g σˆξs) +
1 2(1−
1
σ) ˆC
2
s] (11)
−Y vy[ ˆYs(1 +
h ψˆξs) +
1 2(1 +
1
ψ) ˆY
2
s +Vzyˆs(z)
1 2(
1
ψ +
1
where Vz denotes the variance across goods, and tip(3) represents terms that are
inde-pendent of policy (i.e. terms involving the steady state or shocks alone) and terms higher than second order (see Appendix B). The first two terms come from the utility of con-sumption, and represent the costs of aggregate consumption disequilibrium. The next two terms represent similar magnitudes for labour supply and hence output, while the
final term represents costs associated with the output of individual goods differing from average output.
As Woodford (2003) shows, the term in the variance of output across producers exists because of the distortion due to Calvo contracts and can be replaced with a quadratic term in inflation when we consider the discounted sum of utility. With nominal inertia in the form of Calvo contracts, the only reason for output of individual goods to differ is that somefirms change their prices while others do not. The greater is inflation, the larger the movement in relative prices, and therefore the larger the variance in output across goods. This is a particularly clear representation of one of the standard arguments for costs associated with inflation: that higher inflation is associated with a greater distortion in relative prices, and therefore a larger misallocation of production and labour across goods. However it also apparent from this derivation that it is only the relative price of domestically produced goods, and therefore output price inflation, that is relevant for welfare, because only domestic goods require domestic labour supply. This is the key insight that lies behind the argument that social welfare depends on output price inflation rather than consumer price inflation.
The assumption of Calvo contracts is important in determining the way inflation appears in the welfare function. For example, as Steinsson (2003) shows, if we adapt the Calvo contract formulation such that a proportion of producers are backward looking in their price setting (they set a price based on previous aggregate price changes rather than compute an optimum price) then welfare depends on the change as well as the level of inflation.
This analysis introduces inflation into social welfare. How do we transform the re-maining terms into a single term in the output gap, to obtain the isomorphism result? Gali and Monacelli (2005) simplify the model above in four respects to obtain an expres-sion for welfare: they assume iso-elastic utility, that the demand elasticity η = 1, the intertemporal substitutability of consumption σ = 1 (i.e. log utility), and they ignore preference and IRS shocks. We can immediately see how these assumptions can simply the expression for welfare. We can knock out the quadratic terms in consumption ifσ = 1. Without preference shocks, the terms in ξ disappear. So we are left with linear terms
in consumption and output disequilibrium, a term in the variance of output of across goods/inflation and a quadratic term in output disequilibrium.
The role of the η= 1 assumption is less immediately apparent. We show in Appendix A, formulae (36)-(37) that ifη= 1, then the terms of trade and real exchange rate are re-lated in a simple proportional way. As this assumption greatly simplifies the algebra, and is not essential to our argument, we adopt it from now on. We do the same with the as-sumption of iso-elastic utility. The appendix also shows that, combined withσ = 1, these assumptions allow us to write the aggregate demand relationship and IRS relationship as exactly linear. These two relationships involve three endogenous variables (output, con-sumption and the terms of trade), one exogenous variable (world output disequilibrium) and potentially the IRS shock.
Suppose we define ‘natural’ variables as the level of those variables that would occur if prices were fullyflexible (soYˆn is natural output disequilibrium, for example). The same
two linear relationships for aggregate demand and risk sharing will hold for natural levels. We can also show in this case (see equation (63) in appendix B) that any shock to world demand will leave natural output unchanged: only the terms of trade and consumption will change. This allows us to replace the quadratic term in Yˆ in equation (11) with a term in the output ‘gap’ (Yˆ−Yˆn), simply becauseYˆn = 0. More generally, by subtracting natural levels from actual levels, we can use these two relationships to express either the terms of trade gap or the consumption gap in terms of the output gap: gaps move together, as Clarida et al. (2001) note.
This leaves us with linear terms in output and consumption. Of course the two are related through agents optimal consumption/labour supply choice. However, the presence of monopolistically competitive firms in the model means that this choice results in a level of output that is inefficiently low. This in turn will mean that policy makers have an incentive to raise output above the natural level, which will give rise to problems of inflation bias that have been extensively analysed in the literature. To avoid these complications, we follow Gali and Monacelli (2005) and introduce a production subsidy that exactly offsets the monopolistic distortion. This subsidy will mean that we can eliminate the linear terms from the expression for welfare. We are left with quadratic terms in inflation and the output gap, which gives us our isomorphism result with closed economy social welfare.
Introducing either preference shocks or IRS shocks on their own would not fundamen-tally change this analysis, as the appendix shows. The assumption that σ = 1, on the other hand, is crucial. Most calibrations that work with iso-elastic formulations for the utility of consumption assume a value forσsubstantially different from unity. Ifσ6= 1, we
no longer get the isomorphism result, as Appendix B makes clear. However, in the absence of IRS shocks, abandoning this assumption does not directly introduce a role for terms of trade disequilibrium into the expression for welfare. Instead we have an additional term of the form( ˆY −Yˆn)Sbn, which we describe as ‘linear in policy’. This term only begins to
operate if some shock (like a change in world output) shifts the terms of trade that would occur under flexible prices. If such a shift does occur, then welfare could be improved if output departed from its flexible price level. Recall that we introduced a production subsidy to eliminate the steady state monopolistic distortion. In an open economy the relationship between output and consumption can be changed by shifts in preferences or world output, and so the subsidy will no longer always eliminate the monopoly distortion, allowing some scope for a welfare improving movement of output away from its natural level.
This ‘linear in policy’ term is important, because it implies that it is no longer optimal for policy to aim to reproduce the flexible price equilibrium. (This point has been noted by Beningno and Benigno (2003) among others.). However it does not directly introduce a role for the actual terms of trade (Sˆrather thanSˆn), and therefore the actual real exchange
rate, in welfare. The reason for this is noted in Clarida et al. (2001). In the absence of IRS shocks, any term involving the terms of trade gap or the consumption gap can be replaced by the output gap, as they move proportionately together (see Appendix B.2). So, although allowingσ6= 1means that the quadratic term in consumption disequilbrium remains in equation (11), it can be transformed into a quadratic term in the consumption gap, and then merged with the quadratic term in output disequilibrium.1
When we allow for IRS shocks, then it is no longer the case that theflex price equilib-rium is the efficient (non-distortionary) level, even with an output subsidy. We therefore have a choice of how we define that natural level: is it the non-distortionary efficient level (and therefore independent of the IRS shock), or is it the flex price level (and therefore subject to the IRS distortion). In what follows we continue to use the term ‘natural’ to correspond to theflex price level (i.e. removing the nominal inertia distortion), but denote levels that remove both the nominal inertia and IRS distortion as ‘efficient’, so that Yˆe
denotes the efficient level of output for example. In these circumstances it is natural to focus on ‘gaps’ in terms of the difference between actual and efficient levels, as efficient levels are obvious targets for policy. We can also think of these gaps as a combination of a conventional notion of the ‘gap’ i.e. the difference between the variable and its natural 1This linear in policy term could be eliminated by taking a second order expansion of part of the model, and using this to substitute out this and other linear terms (see Sutherland (2005) and Benigno and Woodford (2004)). Beningno and Benigno (2004) show that this will introduce a quadratic term in the terms of trade gap.
level and a target for the gap reflecting the extent to which the natural level is not efficient, ˆ
X−Xˆe = ( ˆX
−Xˆn)
−( ˆXe
−Xˆn) for any variable X. From this point on we focus on
these efficiency gaps, Xˆ −Xˆe, but note the possibility of this decomposition throughout.
Crucially, gaps defined in terms of efficient levels no longer obey the proportionality noted in Clarida et al. (2001). IRS shocks mean that output gaps can move differently from consumption or terms of trade gaps, so that whenσ 6= 1 we will not be able to eliminate the quadratic term in the consumption gap from (11), and additional quadratic terms will arise because the demand curve is no longer linear. This introduces a role for quadratic terms in the terms of trade gap into social welfare.2 In this sense, IRS shocks are a means of establishing a role for terms of trade (and therefore real exchange rate) disequilibrium in welfare.
The appendix shows that, whenη= 1,utility is iso-elastic but we place no restrictions on σ and include preference and IRS shocks, then the per-period loss function can be rewritten in the form (see formula (69))
Ls =CuC{A1( ˆYs−Yˆse)
2+A2( ˆS
s−Sˆse)
2+A3( ˆS
s−Sˆse)( ˆYs−Yˆse)
+A4(Ss−Sse)S e
s +A5π2H,s}. (12)
where P∞s=tβs−tWs = −P∞s=tβs−tLs+tip(3). A key point to note about the terms of
trade term is that, like output, it is in the form of deviations from the efficient level i.e. the terms of trade that would occur with no nominal inertia or IRS shocks. A number of studies have experimented with simple feedback rules which include some form of exchange rate targeting, but generally not in terms of deviations from its efficient level. For example, Kollmann (2002) finds that adding the exchange rate to a feedback rule (with optimised parameters) that already includes output price inflation and output disequilibrium terms adds virtually nothing to welfare.3 This result is interesting, because CPI inflation targeting can be roughly ‘recovered’ from separate terms on output price inflation and the change in the exchange rate. However our analysis suggests terms in the terms of trade ‘gap’: the difference between actual terms of trade disequilibrium and the disequilibrium that would occur with no distortions. Not only is the dimension of this expression different from the change in the exchange rate, but a change in exchange rate term makes no attempt to allow for ‘warranted’ exchange rate movements i.e. efficient disequilibrium. For this reason CPI inflation targeting cannot be considered as an attempt 2Alternatively we could work with consumption gaps rather than terms of trade gaps, but targets in consumption are not familiar to policy makers.
3In Kollmann (2003), it is argued that exchange rate targeting may be of greater value if it helps reduce the size of UIP shocks.
to combine output price inflation targeting and exchange rate targeting in this welfare framework.
One early example that does come close to trying to capture the concept of a real exchange rate gap is the Target Zone proposal of Williamson and Miller (Williamson and Miller (1987), see also Currie and Wren-Lewis (1989) for an evaluation), where interest rates differentials were assigned entirely to stabilising the real exchange rate around its medium term equilibrium level (FEER), and fiscal policy was assigned to inflation sta-bilisation. While this particular policy assignment may no longer be on the agenda, the FEER measure of an equilibrium exchange rate is close to the idea of an efficient level. In particular, the FEER is the real exchange rate that would occur if the economy was in ‘internal balance’, which can be interpreted as abstracting from business cycle effects generated by nominal inertia as well as IRS shocks. Unlike PPP, the FEER is influenced by medium term net saving (private or public), which can be thought of as one example of a preference shock.
While our analysis confirms the result that the monetary authorities should target output price inflation rather than consumer price inflation4, it also suggests that there is an additional case for a separate terms of trade target, in the form of the difference between the terms of trade and its efficient level. The intuition behind both results is straightforward. Nominal inertia leads to the staggering of price changes, generating variation in individual goods prices and therefore a misallocation in the distribution of production across goods. This misallocation will be directly related to the overall level of inflation. However the fact that the distortion relates to the allocation of output, rather than the pattern of consumption, means that it is output price inflation rather than consumer price inflation that is the relevant proxy for this inefficiency. The intuition behind the presence of terms of trade disequilibrium terms is that the intertemporal profile of consumption is not identical to the profile of output in an open economy, and so ensuring that the latter is at its efficient level will not guarantee the former is as well. In particular, IRS shocks can move the terms of trade away from their efficient level, which 4Meade (1951) argued for targeting output price inflation rather than consumer price inflation, because of a concern that terms of trade shocks will require offsetting domestic price changes (given some CPI target), and that this will be costly because domestic prices are sticky. Kara and Nelson (2002) suggest the result is vulnerable to the assumption that all imported goods are finished consumer goods that contain no domestic value added. They suggest an alternative framework, where all imported goods are intermediate goods, and they argue that this framework is more consistent with empirical evidence on aggregate pricing. It is certainly the case that the final purchase price of finished imported consumer goods contains a large proportion of domestic value added, because of wholesale and distributor margins. However, the logic of the analysis above is that the target inflation index should in fact relate to a measure of domestic value added, such as the GDP deflator. The GDP deflator will include the element that domestic retailers add to imported consumer goods prices, but not the element of price that is generated overseas.
Welfare measure/Shock Cost-push IRS Preference World
Traditional 0.08 -1.35 0 0
Social, but output and inflation only 1.41 -1.31 0 0 Social, complete 3.50 -1.05 0.13 -0.13
Table 1: Optimal response of interest rate gap to different shocks using alternative welfare measures
may move consumption off its optimal intertemporal path without necessarily creating output gaps.
How important is this generalisation of the closed economy social welfare function likely to be for policy in practice? To analyse this we looked at a calibrated version of our model, the details of which are in Appendix B (they include σ = 0.5, but η = 1). We looked at four types of shock that can hit the economy: a preference shock, a shock to world demand, a cost-push shock and an IRS shock. As there is no backward looking inertia in our model, and the shocks are not persistent, then optimal discretionary policy can be expressed as a rule that relates the real interest rate gap (the difference between the real interest rate and its efficient level) to the current value of these four shocks. (The solution to this type of problem is discussed in Söderlind (1999) among others.) Table 1 looks at the parameters on this optimum rule for three different welfare measures.
The first welfare measure is a traditional combination of quadratic terms in inflation and the output gap, where the relative weight on the output gap is 0.5. The second has the same structure, but where the relative weight on the output gap comes from our derived social welfare function. The third is the rule implied by our derived social welfare function in its complete form, including terms of trade and linear in policy terms. When welfare only involves inflation and output gaps, then policy does not react to world or preference shocks. The reason is that changes in preferences or world output have no effect on gaps or inflation when monetary policy is optimal. With the complete social welfare function, there are linear in policy terms of the form described above (we do not reproduce theflex price outcome), and so policy will react to these shocks. With changing world demand or a preference shock, the relationship between output and consumption changes, so the subsidy no longer eliminates all monopolistic distortions. Policy can therefore change consumption and output to reduce this monopolistic distortion. The scale of the reaction is significant, but considerably smaller than to the cost-push shock. (Equation (64) in Appendix A shows why the reaction to world and preference shocks will be equal and opposite.)
welfare function is much smaller than the traditional 0.5. This is a standard result from closed economy analysis (which is discussed further below), and it accounts for the greater response to cost push shocks in the second and third rows compared to the first. The impact of our analysis of an open economy can be seen from comparing the second and third cases. We have already noted how the linear in policy terms introduce some reaction to preference and world shocks. The other noticeable difference that introducing terms in the terms of trade into the welfare function makes is to significantly change the reaction to both the cost-push and IRS shock. In a welfare function that only contains the output gap, the IRS shock matters only in so far as it influences that gap. In the complete welfare function, there are additional terms in the terms of trade gap, and as a result the reaction to shocks changes considerably.
The intuition for why these additional terms of trade terms might reduce the reaction to an IRS shock is as follows. Interest rates influence the economy only through their effect on consumption, so we can in effect treat consumption as an exogenous instrument for a one-offshock of this sort. An IRS shock will, for given consumption, appreciate the exchange rate which reduces output. Consumption is increased to offset this (requiring a fall in interest rates - hence the negative coefficient in Table 1), but the output gap cannot be completely eliminated because a positive consumption gap will increase inflation. In the complete social welfare function the terms in the terms of trade appear in part to represent the impact of consumption gaps on welfare in so far as these move differently from output gaps. Thus allowing for these consumption gaps moderates the extent to which consumption will be increased to prevent output falling. The overall implication is that using a ‘closed economy’ welfare function in an open economy setting is not only wrong conceptually (unless the utility of consumption is logarithmic), but it can also lead to policy errors of a significant size.
How does this relate to the UK experience of the late 1990s? We can reasonably conjecture that the UK experienced a large terms of trade gap over this period. (See Wren-Lewis (2003) on the extent of sterling’s ‘overvaluation’.) Given that UK consumption was buoyant at this time, then in the context of this model this would imply that the UK experienced a positive IRS shock. One interpretation of policy is that it reacted to this IRS shock only to the extent that it influenced inflation or the output gap. (See the companion paper by Cobham in this issue.) Our analysis of welfare, and the calibration of optimal policy above, suggests that this could have been a significant mistake. However, as our calibration also shows, we cannot presume that correcting this mistake would have implied lower UK interest rates. In addition, our analysis suggests that policy may also
have been distorted by targeting consumer rather than output price inflation.5
One limitation of these welfare results is that they focus on just one cost of inflation. Business cycles only matter here because they disrupt the optimal intertemporal allocation of consumption and labour supply and because they distort relative prices. The model ignores money, and therefore the cost of inflation that arises from foregone interest in holding the medium of exchange. However, Woodford (2003) adds money to the utility function in a non-separable way and shows that it implies that a quadratic term in the level of the nominal interest rate should be present in the welfare function, because the nominal interest rate represents the cost of holding money. It does not imply a move from output price inflation to CPI targeting.
Another clear limitation of this and the other models that have been used so far is that business cycles affect all consumers/workers in a similar way. The representative agent assumption means that everyone keeps working, although some will be working too little or too much. There is no unemployment, and no bankruptcy. It is therefore perhaps not surprising that typical calibrationsfind that the relative size of the output disequilibrium term is small compared to the inflation term: this reflects a point made by Lucas and Johnson (1987) that the cost of the intertemporal misallocation of consumption generated by business cycles appears relatively small. Even without attaching additional disutility to becoming unemployed or bankrupt, the fact that in reality business cycles leave some consumers/workers untouched while others are severely affected will increase the overall impact on welfare unless utility is linear, as Lewis (2003) clearly illustrates. However this criticism is about the relative weight to attach to the output gap relative to inflation, and does not appear to have any bearing on the choice of inflation target.
3
Stability
There has been another concern with targeting consumer price inflation that has been occasionally raised in the literature. This is that targeting consumer price inflation might be destabilising, because of the feedback from interest rates to the exchange rate. We can illustrate this point by considering the small open economy model outlined above. We ignore shocks, and combine (6) with (3), and then substitute into (2) to give
πH,t=βEt[πH,t+1] +λ( ˆCt/σ+ ˆYt/ψ+αSˆt). (13)
5A common complaint surrounding overvaluation is that this hits some sectors of industry more than others, depending on their export orientation. This issue needs to be addressed using a model where firms are not all identical in the amount they sell overseas.
Substituting (9) into (7) gives ˆ
Ct =Et[ ˆCt+1]−σ(rt−Et[πH,t+1+α∆Sˆt+1]). (14)
We can go on to use the risk sharing equation (ignoring shocks and world output) and the demand curve to substitute for both output and the terms of trade in (13), to give
πH,t=βEt[πH,t+1] +ACˆt, (15)
whereAis a composite of structural parameters (and is positive). These last two equations represent a dynamic system in consumption and inflation. We consider the two most simple monetary policy rules:
forecast output inflation targeting
ˆ
rt= (1 +m)Et[πH,t+1], (16)
forecast consumer inflation targeting
ˆ
rt= (1 +m)Et[πH,t+1+α∆Sˆt+1]. (17)
Both consumption and inflation jump, so the dynamic system in these two variables must have unstable roots. In Appendix C we show that with output inflation targeting, we require the Taylor principle to hold (m > 0). In addition, since the rule is targeting expected inflation, there is an upper bound on the aggressiveness of monetary policy (see also Batini et al. (2004)), but for reasonable parameter values it is unlikely to bite, and it tends to infinity asα (the share of overseas goods in the consumption bundle) tends to unity.
In the case of consumer price inflation targeting, we obtain the following condition in place of the Taylor principle,
m
(1− 1mα−α) >0. (18)
Ifαis small then this condition is likely to hold. However ifα is large, then the condition could well fail with a moderately aggressive monetary policy. (For example, if m = 1, such that real interest rates rise by 1 point for each 1 point increase in inflation, then the system will be indeterminate for α > 0.5. )6 In fact, we also show in Appendix C that an additional, if more complex, condition places an even stronger cap on determinate values of m when the rule targets expected inflation. A similar result (for a continuous 6Estimated values of m are generally less than unity, but exercises computing the optimal value of m often result in values much larger than one (e.g. Kollman, 2002).
time version of a small open economy model) is demonstrated in Linnemann and Schabert (2001).
This stability problem appears to be fairly generic: it would apply, for example, to an economy with a more traditional static IS curve, but with dynamics provided by UIP7. Leith and Wren-Lewis (2003) show that it also extends to a two-country world. Their set-up involves some differences from the model above: in particular, international risk sharing is dropped and instead consumers are of the Blanchard/Yaari type.
The intuition behind these instability results is fairly straightforward. As α becomes large, most consumer goods are produced overseas, and if the country is small these prices will be immune to the domestic nominal inertia distortion. Following a domestic demand shock, the CPI will be largely unaffected, but any increase in domestic interest rates will appreciate the exchange rate which will lower the CPI.
Even if the parameters of the model are such that complete instability is avoided, it seems likely that the feedback from interest rates to the CPI through the exchange rate may cause problems for any policy based on simple rules. For simplicity the model above postulated rules that reacted to one period ahead forecast inflation. If, in practice, policy reacts to actual inflation, then the rule may generate erratic paths for inflation. For example, consider a domestic demand shock which will raise future inflation. Markets, knowing the rule and that future inflation will rise, will anticipate future interest rate increases, and this will generate an immediate appreciation. This will lower current CPI inflation, leading the monetary authorities to lower interest rates. If underlying welfare depends on domestic inflation and output gaps, then this policy response is unhelpful. (See Leith and Wren-Lewis (2001) for some further examples.)
4
Conclusions
In this paper we have considered two arguments why monetary authorities in an open economy should not target consumer price inflation or the exchange rate, but instead target output price inflation alone. The first derives second order approximations to the authorities’ objective function from the representative consumer’s utility, and shows that it is output price inflation, rather than consumer price inflation or the exchange rate, that appears in this objective function. The second shows how policy rules involving consumer price inflation can induce instability because of the feedback from interest rates to consumer price inflation via the exchange rate.
7However any reduced pass through of exchange rate changes, generated by pricing to market for example, will reduce the risk of this instability occurring.
This second argument, concerning stability, may not be critical in economies with a large proportion of non-traded goods as long as monetary policy is not too aggressive. Problems of timing, where the exchange rate reacts to future expected increases in interest rates leading to changes in the CPI that may not be coincident with domestic demand pressure, are a nuisance that can be dealt with if policy avoids sticking to a fixed simple rule. However these difficulties need only be faced if policy requires stabilising consumer rather than output price inflation, which brings in the first argument.
In this paper we have shown that in the general case policy makers should be con-cerned about the real exchange rate as well as output price inflation and the output gap. In particular, International Risk Sharing or Uncovered Interest Parity shocks may lead to distortions in the real exchange rate which impact on the intertemporal pattern of con-sumption, without necessarily producing output gaps. It may therefore be appropriate for monetary authorities to respond to the ‘terms of trade gap’ as well as the ‘output gap’, where the terms of trade gap is the difference between the actual level and the level that would occur in the absence of IRS shocks and nominal inertia. The terms of trade gap concept has many similarities to the deviation of the real exchange rate from the Fundamental Equilibrium Exchange Rate (FEER).
However, these arguments do not influence the case for targeting output rather than consumer price inflation. In welfare terms, inflation captures the distortion in the pattern of output generated by asynchronised price adjustment caused by menu costs, and here it is output price inflation rather than consumer price inflation that is relevant. A consumer price inflation target can be thought of as the combination of an output price inflation target and a target for changes in the nominal exchange rate. Our welfare analysis suggests targeting the terms of trade gap, and not the change in the nominal exchange rate. Therefore the argument that the inflation target should be output price inflation rather than consumer price inflation appears robust.
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A
Deriving the Model
A.1
Home and Foreign goods
The representative household maximises (1) where the aggregate consumption bundle is given by
C = [(1−α)1ηC η−1
η
H +α
1
ηC η−1
η
F ]
η
η−1, (19)
whereCH andCF are indices of consumption of domestic and foreign goods, and we drop
the time subscript whenever all variables are dated at t. The parameter α is related to the share of imported goods in domestic consumption. In turn
CH =
·Z 1 0
c −
1
H (z)dz
¸ −1
, CF =
·Z 1 0
c −
1
F (z)dz
¸ −1
, >1. (20)
The optimal allocation of any given expenditure within each category of goods yields the demand functions:
cH(z) =
µ
pH(z)
PH
¶−
CH, cF(z) =
µ
pF(z)
PF
¶−
CF, z ∈[0,1], (21)
where
PH =
·Z 1 0
p1H−(z)dz
¸ 1 1−
, PF =
·Z 1 0
p1F−(z)dz
¸ 1 1−
. (22)
The optimal allocation of expenditures between domestic and foreign goods implies, given the law of one price:
CH = (1−α)
µ
PH
P
¶−η
C, CF =α
µ
PF
P
¶−η
C, (23)
where the consumer price index (CPI) is:
P = ((1−α)PH1−η+αPF1−η)1−1η. (24)
We can also define the home output indexY as
Y = [
Z 1 0
A.2
Intertemporal optimisation
Each household faces the sameflow budget constraint:
PtCt+Et[Rt,t+1Bt+1]≤Bt+WtNt+Tt, (26)
where Bt+1 is the nominal payoff in periodt+ 1 of a portfolio held at the end of period
t, R is a stochastic discount factor for nominal payoffs,W is the nominal wage, N labour supply andT denotes lump sum transfers/taxes. The riskless short term nominal interest rate, rt, is given by
1 1 +rt
=Et(Rt,t+1).
Together with transversality conditions, the budget constraint can be solved forward to yield:
∞
X
s=t
Et(Rt,sPsCs)≤Bt+
∞
X
s=t
Et(Rt,s[WsNs+Ts]).
Assume a linear production technology, such that Nt =yt(z). The Lagrangian and first
order conditions are
L=Et
∞
X
s=t
βs−t[u(Cs, ξs)−v(ys(z), ξs)]
−λ[
∞
X
s=t
Et(Rt,sPsCs)−Bt−
∞
X
s=t
Et(Rt,s[Wsy(z)s+Ts]),
∂L ∂ys(z)
=−βs−tvh(ys, ξs) +λRt,sWs = 0, (27)
∂L ∂Cs
=βs−tuC(Cs, ξs)−λRt,sPs = 0. (28)
The consumption first order condition can be rewritten as:
βEt
µ
uC(Ct+1, ξt+1)
uC(Cti, ξt)
Pt
Pt+1 ¶
= 1 1 +rt
,
and we can also write the familiar leisure/consumption trade off
vy(yt, ξt)
uC(Ct, ξt)
= Wt
Pt
. (29)
A.3
Terms of trade and real exchange rate
We assume that the law of one price holds = PF
P∗
F
We define terms of trade as
S = PF
PH
= P
∗
F
P∗
H
, (31)
and the real exchange rate
Q= P
∗
P . (32)
We also assume that prices are equal in equilibrium and the second economy is large so that
PF∗ =P∗, (33)
Y∗ =C∗, (34)
For future reference, the formula for the real exchange rate can be rewritten in terms of the terms of trade:
Q= P
∗
P =
PF∗
((1−α)PH1−η+αPF1−η)1−1η
=PF((1−α)P1− η H +αP
1−η F )−
1
1−η (35)
= ((1−α)S−(1−η)+α)−1−1η.
Note that for η= 1 we have
Q=S1−α, (36)
which is a multiplicative relationship and its log-linear approximation is an exact linear relationship:
ˆ
Q= (1−α) ˆS. (37)
A.4
Price setting
Price setting follows the usual Calvo set-up with 1−γ of firms changing price in a given period. The log-linear pricing rule for prices changed in period t is given by
˜
pH,t=µ+ (1−βγ)
∞
X
k=0
(βγ)kEt{mct+k},
whereµ= log(ε−ε1)is the gross mark-up in the steady-state, andmcare nominal marginal costs. The following log-linear approximation
allows us to write (set pH,t−1 =pH,t−πH,t to obtain an equation for γπH,t and subtract
βγπH,t+1)
πH,t=βEt[πH,t+1] +
(1−γ)(1−βγ)
γ mcct, (39)
which is (2) in the main text.
A.5
International risk sharing
With complete securities markets, the Euler equation
βuC(C
i
t+1, ξt+1)
uC(Cti, ξt)
Pt
Pt+1
=Rt,t+1 (40)
must also hold for the rest of the world, i.e.
βuC(C ∗
t+1, ξt+1)
uC(Ct∗, ξt)
P∗
t
P∗
t+1
=R∗t,t+1. (41)
Perfect arbitrage would suggest
Et[
R∗t,t+1 t+1
t
] =Et[Rt,t+1], (42)
which is Uncovered Interest Parity when 1+1r
t =E[Rt,t+1] and
1 1+r∗
t =E[R
∗
t,t+1]. We want to introduce a distortionζt such that
tζt
Et[ t+1ζt+1](1 +rt∗)
= 1 1 +rt
. (43)
The empirical importance of distortions to UIP are discussed in Kollmann (2003) and Kollmann (2002). Using this relationship with the two Euler equations gives
uC(Ct+1, ξt+1)uC(Ct∗, ξ∗t)
uC(Ct∗+1, ξ∗t+1)uC(Ct, ξt)
ζt ζt+1
= Qt
Qt+1
, (44)
whereQ is the real exchange rate defined above. The terms in ξandξ∗ enter in a similar way to ζt , but with the crucial difference that ζt does not enter the efficient level of
variables because it is distortionary. By integrating, and assuming complete markets, we can show that home consumption will be related to world consumption, the real exchange rate, and the UIP distortion (Gali and Monacelli (2005)):
uC(Ct∗, ξt)ζt=ϑQtuC(Ct, ξt). (45)
Assuming isoelastic utility
Ut=
(Ctξt)
1−σ1
formula (45) becomes:
Ct =ϑσCt∗Q σ t
µ
ξt ξ∗t
¶(1−1
σ)σ
ζt. (46)
Note that if η= 1 then
Ct =ϑσCt∗S σ(1−α)
t
µ
ξt
ξ∗t
¶(1−σ1)σ
ζt
is a multiplicative form with respect to S and its log linearised form up to the second order is in fact an exact linear relationship:
ˆ
Ct = ˆCt∗+σ(1−α) ˆSt−(1−σ)
³ ˆ
ξt−ˆξ
∗
t
´ + ˆζt.
A.6
Aggregate Demand
Market clearing implies:
Y =CH+CH∗
Substitute relationships (23) to obtain:
Yt= (1−α)
µ
PHt
Pt
¶−η
Ct+α∗
µ
P∗
Ht
P∗
t
¶−η
Ct∗
= (1−α) µ
PHt
Pt
¶−η
Ct+α∗
µ
tPHt∗
Pt
Pt tPt∗
¶−η
Ct∗
= (1−α) µ
PHt
Pt
¶−η
Ct+α∗
µ
PHt
Pt
¶−η
QηtCt∗
= ((1−α) +αS1−η)1−ηη((1−α)C
t+α∗QηtCt∗). (47)
It is useful to substitute in the risk sharing relationship at this stage in order to obtain:
Yt=
µ
PHt
Pt
¶−η
QηtCt∗ϑ σ
Ã
(1−α)Qσt−η
µ
ξt
ξ∗t
¶(1−σ1)σ
ζt+α∗ϑ− σ
!
.
As Gali and Monacelli (2005) discuss, the assumption of zero trade balance in equi-librium implies α∗ϑ−σ =α and so:
Yt=
µ
PHt
Pt
¶−η
QηtCt∗ϑσ
Ã
(1−α)Qσt−η
µ
ξt ξ∗t
¶(1−1
σ)σ
ζt+α
!
. (48)
From this formula it is clear that ifσ=ηthen output can be written as a multiplicative relationship:
Yt=ϑσ
µ
PHt
Pt
¶−η
QηtCt∗
"
(1−α) µ
ξt
ξ∗t
¶(1−σ1)σ
ζt+α
#
where the expression in square brackets depends on shocks only.
Note that ifη=σ = 1(this is the case considered by Gali and Monacelli (2005)) then:
Yt=ϑσStCt∗[(1−α)ζt+α],
which does not contain non-loglinear terms in St.
A.7
Labour Supply Equation in Flex-price equilibrium
Thefirst order condition (29) can be written in theflexible-price equilibrium (superscript
n) as :
vh(ysn, ξs)
uC(Csn, ξs)
= Ws
Ps
= µ
w
µ PHs
Ps
. (49)
whereµw is a steady state employment subsidy. Substituting isoelastic utility, we obtain:
Ynψ1Cn
1
σξn
1
ψ+
1
σ = µ
w
µ ((1−α) +αS
f1−η)−1−1η. (50)
Again, if η = 1 then the relationship is multiplicative:
Ynψ1Cnσ1ξ
1
ψ+
1
σ = µ
w
µ S
n−α, (51)
and its linearisation results in the exact relationship: 1
ψYˆ
n+ 1
σCˆ
n+αSˆn+
µ 1
ψ +
1
σ
¶ ˆ
ξ= 0. (52)
A.8
Second-Order Expansion
We work with the system of two equations, aggregate demand (47) and the risk sharing condition (46). We use definition (35) to represent the real exchange rate as a function of the terms of trade. It is convenient to substitute (35) into both (47) and (46)first and also use (46) to substitute out consumption in (47). We come to the following system:
Yt =S η t
Ã
(1−α)((1−α)St−(1−η)+α)−
(σ−η) 1−η
µ
ξt
ξ∗t
¶(1−σ1)σ
ζt+α
!
ϑσCt∗, (53)
Ct =ϑσCt∗((1−α)S
−(1−η)
t +α)−
σ
1−η
µ
ξt ξ∗t
¶(1−1
σ)σ
The log-linearisation up to first order of this system is ˆ
Yt = ˆCt∗+
¡
α2σ−α2η−2α σ+ 2α η+σ¢Sˆt (55)
+ (1−α)³(σ−1)³ˆξt−ˆξ∗t´+ ˆζt´,
ˆ
Ct = (σ−1)
³ ˆ
ξt−ˆξ
∗
t
´
+ ˆζt+ ˆCt∗+σ (1−α) ˆSt. (56)
The second-order log-linear expansion is (from now on we assume η= 1) ˆ
Ct = ˆCt∗+σ (1−α) ˆSt+ (σ−1)
³ ˆ
ξt−ˆξ∗t´+ ˆζt+1
2σ (α−1)σ(α−1) ˆS 2
t (57)
− 12Cˆt2−(σ−1)σ (α−1) ˆSt
³ ˆ
ξt −ˆξ∗t´−σ (α−1) ˆStˆζt−σ (α−1) ˆStCˆt∗
+ (σ−1) ˆζt
³ ˆ
ξt−ˆξ
∗
t
´
+ (σ−1) ˆCt∗ ³ˆξt− ˆξ
∗
t
´ +1
2(σ−1) 2³ˆ
ξt −ˆξ
∗
t
´2 +1
2ˆζ 2
t
+ ˆζtCˆt∗+1 2Cˆ
∗2
t +O(3),
ˆ
Yt= ˆCt∗+
¡
α2σ−α2−2α σ+ 2α +σ¢Sˆt+ (1−α)
³
(σ−1)³ˆξt−ˆξ∗t´+ ˆζt´ (58)
− 12¡α3−2ασ−α2−σ2−α+ 3ασ2+ 4α2σ−2α3σ−3α2σ2+α3σ2¢Sˆt2
− 12Yˆt2+ (α σ−α −σ) (α−1) ˆSt
³
(σ−1) ³
ˆ
ξt−ˆξ
∗
t
´ + ˆζt
´ +¡α2σ−α2 −2α σ+ 2α +σ¢ SˆtCˆt∗−
1
2(α−1) ˆζ 2
t + (1−α) ˆζtCˆt∗
−(σ−1) (α−1) µ
1
2 (σ−1) ³
ˆ
ξt−ˆξ
∗
t
´
+ ˆζt+ ˆCt∗
¶ ³ ˆ
ξt−ˆξ
∗
t
´ +1
2Cˆ
∗2
t +O(3).
We can now express the term of trade from equation (58) and substitute it into equa-tion (57). Assumingˆξt= ˆξ
∗
t we obtain:
ˆ
Yt=
(α(2−α) (1−σ) +σ) (1−α)σ Cˆt−
α (2−α) (1−α)σ ˆζt+
α (−σ+ 2 +α σ−α) (α−1)σ Cˆ
∗
t (59)
+ 1 2
α(1−α)3(1−σ)2 (σ+α(2−α) (1−σ))2
³ ˆ
Yt−Cˆt∗
´2
− (1−α)
2
(1−σ)α
(σ+α(2−α) (1−σ))2 ³
ˆ
Yt−Cˆt∗
´ ˆ
ζt
+ 1 2
(1−α)α
(σ+α(2−α) (1−σ))2 ˆ
A.9
E
ffi
cient Equilibrium
We can use the relationships above ( (55), (56) and (52)) to derive the following expressions for efficient levels valid tofirst order:
ˆ
Yte = ˆCt∗+
¡
α2σ−α2−2α σ+ 2α +σ¢Sˆte, (60)
ˆ
Cte = ˆCt∗+σ (1−α) ˆSte, (61)
ˆ
ξt =− σ
(σ+ψ)Yˆ
e t −
ψ
(σ+ψ)Cˆ
e t −
αψσ
(σ+ψ)Sˆ
e
t. (62)
This system can be solved to yield ˆ
Yte = (α(α−2) (σ−1) +σ) (ψ+σ)
σ(α(α−2) (σ−1) +σ+ψ) ˆξ
t+
(1−σ) (α−2)αψ
σ(α(α−2) (σ−1) +σ+ψ)Cˆ
∗
t, (63)
ˆ
Ste =
(ψ+σ)
σ(α(α−2) (σ−1) +σ+ψ) ³
ˆξ
t−Cˆt∗
´
, (64)
ˆ
Cte = (ψ+σ) (1−α)
(α(α−2) (σ−1) +σ+ψ)ˆξt+
α(2−α+ασ−σ+ψ) (α(α−2) (σ−1) +σ+ψ)Cˆ
∗
t. (65)
B
Second Order Representation of Welfare
B.1
General Formula
We assume that utility is isolelastic, that is
U(Cξ) = (Cξ) 1−σ1
1− 1
σ
, v(yξ) = (yξ)
1+1
ψ
1 + 1
ψ
,
The latter assumption implies g = 1−σ, h = 1 +ψ. The log-linear Taylor expansion of welfare (10) is then
Wt=UcC
½ ˆ
C− 1−σ
σ Cˆtˆξt+
1 2(1−
1
σ) ˆC
2
t
−UY vy
cC
· ˆ
Yt(1 +
1 +ψ ψ ˆξt) +
1 2(1 +
1
ψ) ˆY
2
t +Vzyˆt(z)
1 2( 1 ψ+ 1 ) ¸¾
where tip(3) contains terms independent of policy and of third order and above. Substi-tuting (59) into the expression for welfare gives
Wt=UcC
½µ
1 + Y vy
UcC
(α(α−2) (σ−1) +σ) (α−1)σ
¶ ˆ
Ct−
1−σ
σ Cˆtˆξt (66)
− 12(1−σ
σ ) ˆC
2
t +
1 2
Y vy
UcC
α(α−1)3(σ−1)2 (σ+α(α−2) (σ−1))2
³ ˆ
Yt−Cˆt∗
´2
− Y vy UcC
(α−1)2(σ−1)α
(σ+α(α−2) (σ−1))2 ³
ˆ
Yt−Cˆt∗
´ ˆ
ζt− Y vy UcC
1 +ψ ψ Yˆtˆξt −UY vy
cC
1 2(1 +
1
ψ) ˆY
2
t −
Y vy
UcC
Vzyˆt(z)
1 2( 1 ψ + 1 ) ¾
+tip(3).
We assume that a production subsidy exists such that the following holds in steady state 1 + Y vy
UcC
(α(α−2) (σ−1) +σ) (α−1)σ = 0.
This allows us to eliminate the linear term in consumption from the expression for welfare. The remaining terms are quadratic terms in output and consumption, or cross-product terms involving three shocks, Cˆ∗
t,ˆζt andˆξt.
Using expressions (60, 61, 62) and (63, 65, 64) to eliminate terms in the preference and IRS shocks in (66), we obtain (substituting shock ˆξt from (62) and shock ˆζt using
(55) and (56)):
Wt=UcC{−
1 2
(1−α)σ(1 +ψ) (α(α−2) (σ−1) +σ)ψ
³ ˆ
Yt−Yˆte
´2
(67)
− 1
2
(σ−1) (α(α−2) (σ−1) +σ+ 1) (α−1)2ασ
(σ+α(α−2) (σ−1))3
³ ˆ
Yt−Yˆte
´2
+1 2(
σ−1
σ )
³ ˆ
Ct−Cˆte
´2
− (1−α)σ(α−1) (σ−1)α
(σ+α(α−2) (σ−1))2 ³³
ˆ
St−Sˆte
´ ³ ˆ
Yt−Yˆte
´
+³Sˆt−Sˆte
´
Ste´ − (1−α)σ
(α(α−2) (σ−1) +σ)Vzyˆt(z) 1 2(
1
ψ +
1
)}+tip(3).
Finally, note that from the (55), (56), (60) and (61) it follows that ³
ˆ
Yt−Yˆte
´
−α(2−α)³Sˆt−Sˆte
´
= (1−α)³Cˆt−Cˆte
´
Using this we can rewrite the formula in terms of output and the terms of trade only:
Wt=−ΦUcC
(Ã
(1 +ψ) 2ψ +
(1−σ)¡1− ασ (α(2−α) (1−σ) + 1 +σ) (1−α)Φ3¢ 2σ(1−α)2Φ
!³ ˆ
Yt−Yˆte
´2
+1 2
α2(1
−σ) (2−α)2
σΦ(1−α)2 ³
ˆ
St−Sˆte
´2
+Φ(1−σ)α
σ
³ ˆ
St−Sˆte
´ ˆ
Ste (69)
+(1−σ)
σ αΦ
µ
1− (2−α) Φ2(1−α)2
¶ ³ ˆ
St−Sˆte
´ ³ ˆ
Yt−Yˆte
´ +1 2( 1 ψ + 1
)Vzyˆt(z)
¾
+tip(3),
where we denoted
Φ= σ(1−α)
(α(2−α) (1−σ) +σ) ≥0.
Rotemberg and Woodford (1997) have shown that
∞
X
s=t
βs−tVzyˆs(z) =
∞
X
s=t
βs−t
2
(1−γβ)
γ
1−γπ
2
s.
which allows us to write the per period loss function in the form of (12) in the main text.
B.1.1 Logarithmic utility
If we put σ= 1 in formula (69), we obtain
Wt=−
1
2(1−α)UcC ½
(1 +ψ)
ψ
³ ˆ
Yt−Yˆte
´2 + (1
ψ+
1
)Vzyˆt(z)
¾
+tip(3). (70) This is isomorphic to the closed economy expression for welfare derived in Rotemberg and Woodford (1997). Note that this isomorphism holds even though we have generalised Gali and Monacelli (2005) to include preference and IRS shocks.
B.2
Excluding preference or IRS shocks
Suppose we exclude preference and IRS shocks, but allow σ 6= 1. From the linear IRS and AD relationships and (60) and (61), we can now relate the terms of trade gap to the output gap as follows:
³ ˆ
St−Sˆte
´
= 1
(α(α−2) (σ−1) +σ) ³
ˆ
Yt−Yˆte
´
. (71)
As a result, (69) can be simplified to
Wt=−ΦUcC
½ 1 2
µ
(1−σ)
σ Φ
µ
1 +(1−σ) (1−α)α
σ Φ
¶
+ 1 + 1
ψ
¶ ³ ˆ
Yt−Yˆte
´2 (72) +(1−α) (1−σ)α³Yˆt−Yˆte
´ ˆ
Ste+ 1 2(
1
ψ +
1
)Vzyˆt(z)
¾
This is no longer isomorphic to the closed economy case because of the term in(Y −Ye)Se.
This term could be expressed in terms ofYe rather thanSe, because all natural rates are
now perfectly correlated (from system (63)-(65)): ˆ
Cte =
(ψ−σ−α+ασ+ 2)
(2α+σ+ψ−2ασ−α2+α2σ)αCˆ
∗
t, (73)
ˆ
Ste =− (σ+ψ)
(2α+σ+ψ−2ασ−α2+α2σ)σ Cˆ
∗
t, (74)
ˆ
Yte = ψ(1−σ) (α−2)α
(2α+σ+ψ−2ασ−α2+α2σ)σCˆ
∗
t, (75)
so
Wt=−ΦUcC{
1 2
µ
(1−σ)
σ Φ
µ
1 +(1−σ) (1−α)α
σ Φ
¶
+ 1 + 1
ψ
¶ ³ ˆ
Yt−Yˆte
´2
− (σ+ψ(ψα) (1−α) −2)
³ ˆ
Yt−Yˆte
´ ˆ
Yte+Vzyˆt(z)
1 2(
1
ψ +
1
) +tip(3)}. (76)
B.2.1 Calibration
Our calibration used to derive optimal policy rules was as follows. One period is taken as equal to one quarter of a year. We set the discount factor of the private sector (and policy makers) to β = 0.99 generating a real interest rate of around 4% per annum.
γ = 0.75 implies that, on average, prices are fixed for one year. These parameters are taken from Gali and Monacelli (2005). The proportion of imported goods is taken as
α = 0.25, which is slightly lower than the assumed value in Gali and Monacelli (2005). Since we are relaxing the assumption of logarithmic utility, we need to obtain values for
σ andψ. We assumeσ = 0.5, which is in line with the estimated parameter in Leith and Malley (2005). Parameters ψ = 2 and = 6 are taken from Rotemberg and Woodford (1997). The elasticity of substitution between home and foreign goods η is taken as 1. The standard deviation of each shock is set to 0.5. Optimal parameter values for the monetary policy reaction function (determining the interest rate gap) were calculated assuming discretionary policy.
Table 1 shows that the response to world and preference shocks, when non-zero, are of equal and opposite sign. Recall that these shocks only matter because of the linear in policy term involving Sn. We have shown in (64) thatSˆn is proportionate to ˆξ
t−Cˆt∗.
C
Stability Analysis
forecast output inflation targeting
rt= (1 +m)Et[πH,t+1], (77)
forecast consumer inflation targeting
rt= (1 +m)Et[πH,t+1+α∆st+1]. (78)
For a similar analysis using current or past inflation, which comes to similar conclusions, see De Fiore and Lui (2002).
Taking rule (77) first, and substituting it into equations (14)-(13) gives the following dynamic system
Et
·
πt+1
ct+1 ¸
=B
·
πt
ct
¸
, (79)
whereB= " 1
β −A/β m(1−α)σ
β 1−
m(1−α)σ β A
#
. Both variables jump, so we need two unstable roots. Since we are operating in discrete time, we need to assess whether or not the eigenvalues are inside the unit circle. To do so consider the trace,
trB =
1 +β
β −
m(1−α)σA
β ,
and determinant,
|B|= 1
β >1.
The three conditions we need to satisfy for determinacy are as follows (see Woodford (2003) Appendix C),
|B|>1, |B|−trB >−1, and |B|+trB >−1. (80)
The first of these conditions is already satisfied. The second is given by,
|B|−trB =
m(1−α)σA
β −1>−1, (81)
and it is satisfied iffm >0,which is Taylor principle. Finally, the third condition is given by
|B|+trB =
2 +β−m(1−α)σA
β >−1. (82)
This condition is only satisfied if,
m < 2β+ 2
