Optimal InflationTargeting Rules
Optimal InflationTargeting Rules
Abstract and Keywords
An increasingly popular approach to the conduct of monetary policy, since the early 1990s, has been inflationforecast targeting. Under this general approach, a central bank is committed to adjust shortterm nominal interest rates periodically so as to ensure that its projection for the economy's evolution satisfies an explicit target criterion. One benefit that is claimed for such an approach—and an important advantage, at least in principle, of inflation targeting over other policy rules—is the possibility of combining reasonable stability of the inflation rate (especially over the medium to long term) with optimal shortrun responses to real disturbances of various sorts. This chapter shows that it is possible to derive robustly optimal monetary policy rule for optimizing models of the monetary transmission mechanism that incorporate a number of common features of recent empirical models: staggered wage and pricesetting; inflation inertia resulting from automatic indexation of wages and prices to a lagged price index; predetermined wagesetting, pricing, and spending decisions; and habit persistence in the level of real private expenditure.
Keywords: inflation targeting, monetary policy, monetary transmission mechanism, inflation inertia, wages, prices, habit persistence, interest rates
An increasingly popular approach to the conduct of monetary policy, since the early 1990s, has been inflationforecast targeting. Under this general approach, a central bank is committed to adjust shortterm nominal interest rates periodically so as to ensure that its projection for the economy's evolution satisfies an explicit target criterion—for example, in the case of the Bank of England, the requirement that the Retail Prices Index minus mortgage interest payments (RPIX) inflation rate be projected to equal 2.5 percent at a horizon two years in the future (Vickers 1998). Such a commitment can overcome the inflationary bias that is likely to follow from discretionary policy guided solely by a concern for social welfare, and can also help to stabilize mediumterm inflation expectations around a level that reduces the output cost to the economy of maintaining low inflation.
Another benefit that is claimed for such an approach (e.g., King 1997; Bernanke et al. 1999)—and an important advantage, at least in principle, of inflation targeting over other policy rules, such as a kpercent rule for monetary growth, that should also achieve a low average rate of inflation—is the possibility of combining reasonable stability of the inflation rate (especially over the medium to long term) with optimal shortrun responses to real disturbances of various sorts. Hence Svensson (1999) argues for the (p.94) desirability of “flexible” inflation targeting, by which it is meant^{1} that the target criterion involves not only the projected path of the inflation rate but one or more other variables, such as a measure of the output gap, as well.
We here consider the question of what sort of additional variables ought to matter—and with what weights, and what dynamic structure—in a target criterion that is intended to implement optimal policy. We wish to use economic theory to address questions such as which measure of inflation is most appropriately targeted (an index of goods prices only, or wage inflation as well?), which sort of output gap, if any, should justify shortrun departures of projected inflation from the longrun target rate (a departure of real gross domestic product [GDP] from a smooth trend path, or from a “natural rate” that varies in response to a variety of disturbances?), and how large a modification of the acceptable inflation projection should result from a given size of projected output gap. We also consider how far in the future the inflation and output projections should extend upon which the current interest rate decision is based, and the degree to which an optimal target criterion should be history dependent—that is, should depend on recent conditions and not simply on the projected paths of inflation and other target variables from now on.
In a recent paper (Giannoni and Woodford 2002a), we expound a general approach to the design of an optimal target criterion. We show, for a fairly general class of linearquadratic policy problems, how it is possible to choose a target criterion that will satisfy several desiderata. First, the target criterion has the property that insofar as the central bank is expected to ensure that it holds at all times, this expectation will imply the existence of a determinate rationalexpectations equilibrium. Second, that equilibrium will be optimal, from the point of view of a specified quadratic loss function, among all possible rationalexpectations equilibria, given one's model of the monetary transmission mechanism.^{2} Thus the policy rule implements the optimal statecontingent evolution of the economy, in the sense of giving it a reason to occur if the private sector is convinced of the (p.95) central bank's commitment to the rule and fully understands its implications.
Third, the rule is robustly optimal, in the sense that the same target criterion brings about an optimal statecontingent evolution of the economy regardless of the assumed statistical properties of the exogenous disturbances, despite the fact that the target criterion makes no explicit reference to the particular types of disturbances that may occur (except insofar as these may be involved in the definition of the target variables—the variables appearing in the loss function that defines the stabilization objectives). This robustness greatly increases the practical interest in the computation of a target criterion that is intended to implement optimal statecontingent responses to disturbances, for actual economies are affected by an innumerable variety of types of disturbances, and central banks always have a great deal of specific information about the ones that have most recently occurred. The demand that the target criterion be robustly optimal also allows us to obtain much sharper conclusions as to the form of an optimal target criterion. For while there would be a very large number of alternative relations among the paths of inflation and other variables that are equally consistent with the optimal statecontingent evolution in the case of a particular type of assumed disturbances, only relations of a very special sort continue to describe the optimal statecontingent evolution even if one changes the assumed character of the exogenous disturbances affecting the economy.
Our general characterization in Giannoni and Woodford (2002a) is in terms of a fairly abstract notation, involving eigenvectors and matrix lag polynomials. Here we offer examples of the specific character of the optimally flexible inflation targets that can be derived using that theory. Our results are of two sorts. First, we illustrate the implications of the theory in the context of a series of simple models that incorporate important features of realistic models of the monetary transmission mechanism. Such features include wage and price stickiness, inflation inertia, habit persistence, and predeterminedness of pricing and spending decisions. In the models considered, there is a tension between two or more of the central bank's stabilization objectives, which cannot simultaneously be achieved in full; in the simplest case, this is a tension between inflation and outputgap stabilization, but we also consider models in which it is reasonable to seek to stabilize interest rates or wage inflation as well. These results in the context of very simple models are intended to give insight into the way in which the character of the optimal target criterion should depend on one's model of the economy, and they should be of interest even to readers who are not persuaded of the empirical realism of our estimated model.
Second, we apply the theory to a small quantitative model of the U.S. monetary transmission mechanism, the numerical parameters of which are fit to vector autoregression (VAR) estimates of the impulse responses of (p.96) several aggregate variables to identified monetary policy shocks. While the model remains an extremely simple one, this exercise makes an attempt to judge the likely quantitative significance of the types of effects that have previously been discussed in more general terms. It also offers a tentative evaluation of the extent to which U.S. policy over the past two decades has differed from what an optimal inflationtargeting regime would have called for.
3.1 Model Specification and Optimal Targets
Here we offer a few simple examples of the way in which the optimal target criterion will depend on the details of one's model of the monetary transmission mechanism. (The optimal target criterion also depends, of course, on one's assumed stabilization objectives. But here we shall take the view that the appropriate stabilization objectives follow from one's assumptions about the way in which policy affects the economy, although the welfaretheoretic stabilization objectives implied by our various simple models are here simply asserted rather than derived.) The examples that we select illustrate the consequences of features that are often present in quantitative optimizing models of the monetary transmission mechanism. They are also features of the small quantitative model presented in section 3.2; hence, our analytical results in this section are intended to provide intuition for the numerical results presented for the empirical model in section 3.3.
The analysis of Giannoni and Woodford (2002a) derives a robustly optimal target criterion from the firstorder conditions that characterize the optimal statecontingent evolution of the economy. Here we illustrate this method by directly applying it to our simple examples, without any need to recapitulate the general theory.
3.1.1 An InflationOutput Stabilization TradeOff
We first consider the central issue addressed in previous literature on flexible inflation targeting, which is the extent to which a departure from complete (and immediate) stabilization of inflation is justifiable in the case of real disturbances that prevent joint stabilization of both inflation and the (welfarerelevant) output gap.^{3} We illustrate how this question would be answered in the case of a simple optimizing model of the monetary transmission mechanism that allows for the existence of such “costpush shocks” (to use the language of Clarida, Galí, and Gertler 1999).
As is well known, a discretetime version of the optimizing model of staggered pricesetting proposed by Calvo (1983) results in a loglinear aggregate supply relation of the form (p.97)
We shall assume that the objective of monetary policy is to minimize the expected value of a loss function of the form
The presence of disturbances of the kind represented by u_{t} in equation (1) creates a tension between the two stabilization goals reflected in equation (3) of inflation stabilization on the one hand and outputgap stabilization (around the value x*) on the other; under an optimal policy, the paths of both variables will be affected by costpush shocks. The optimal responses can be found by computing the statecontingent paths {π_{t}, x_{t}} that minimize equation (2) with loss function (3) subject to the sequence of constraints in equation (1).^{6} The Lagrangian for this problem, looking forward from any date t_{0}, is of the form
Differentiating equation (5) with respect to the levels of inflation and output each period, we obtain a pair of firstorder conditions
As an example, figure 3.1 plots the impulse responses to a positive costpush shock, in the simple case that the costpush shock is purely transitory, and unforecastable before the period in which it occurs (so that E_{t}u_{t+j} = 0 for all j≥1). Here the assumed values of β, κ, and λ are those given in table 3.1,^{8} and the shock in period zero is of size u_{0} = 1; the periods represent quarters, and the inflation rate is plotted as an annualized rate, meaning (p.100)
Table 3.1 Calibrated parameter values for the examples in section 3.1
Value 


Structural parameters 

β 
0.99 
κ 
0.024 
θ^{−1} 
0.13 
σ^{−1} 
0.16 
Shock processes 

ρ_{u} 
0 
ρ_{r} 
0.35 
Loss function 

λ_{x} 
0.003 
λ_{u} 
0.236 
This simple example illustrates a very general feature of optimal policy once one takes account of forwardlooking privatesector behavior: optimal policy is almost always history dependent. That is, it depends on the economy's recent history and not simply on the set of possible statecontingent paths for the target variables (here, inflation and the output gap) that are possible from now on. (In the example shown in the figure, the set of possible rationalexpectations equilibrium paths for inflation and output from period t onward depends only on the value of u_{t}, but under an optimal policy the actually realized inflation rate and output gap depend on past disturbances as well.) This is because a commitment to respond later to past conditions can shift expectations at the earlier date in a way that helps to achieve the central bank's stabilization objectives. In the present example, if price setters are forward looking, the anticipation that a current increase in the general price level will predictably be “undone” soon gives suppliers a reason not to increase their own prices currently as much as they otherwise would. This leads to smaller equilibrium deviations (p.101) from the longrun inflation target at the time of the costpush shock, without requiring such a large change in the output gap as would be required to stabilize inflation to the same degree without a change in expectations regarding future inflation. (The impulse responses under the best possible equilibrium that does not involve history dependence are shown by the dashed lines in the figure.^{9} Note that a larger initial output contraction is required, even though both the initial price increase and the longrun price increase caused by the shock are greater.)
It follows that no purely forwardlooking target criterion—one that involves only the projected paths of the target variables from the present time onward, like the criterion that is officially used by the Bank of England—can possibly determine an equilibrium with the optimal responses to disturbances. Instead, a historydependent target criterion is necessary, as stressed by Svensson and Woodford (chap. 2 in this volume).
A target criterion that works is easily derived from the firstorder conditions (6)–(7). Eliminating the Lagrange multiplier, one is left with a linear relation
This is the optimal target criterion that we are looking for: it indicates that deviations of the projected inflation rate π_{t} from the longrun inflation target (here equal to zero) should be accepted that are proportional to the degree to which the output gap is projected to decline over the same period that prices are projected to rise. Note that this criterion is history dependent, because the acceptability of a given projection (π_{t}, x_{t}) depends on the recent past level of the output gap; it is this feature of the criterion that will result in the output gap's returning only gradually to its normal level following a transitory costpush shock, as shown in figure 3.1.
How much of a projected change in the output gap is needed to justify a (p.102) given degree of departure from the longrun inflation target? If λ is assigned the value that it takes in the welfaretheoretic loss function, then Ф = θ^{−1}, where θ is the elasticity of demand faced by the typical firm. The calibrated value for this parameter given in table 3.1 (based on the estimates of Rotemberg and Woodford 1997) implies that Ф=.13. If we express the target criterion in terms of the annualized inflation rate (4π_{t}) rather than the quarterly rate of price change, the relative weight on the projected quarterly change in the output gap will instead be 4Ф, or about 0.51. Hence, a projection of a decline in real GDP of 2 percentage points relative to the natural rate of output over the coming quarter would justify an increase in the projected (annualized) rate of inflation of slightly more than 1 percentage point.
3.1.2 Inflation Inertia
A feature of the New Keynesian aggregatesupply relation (1) that has come in for substantial criticism in the empirical literature is the fact that past inflation rates play no role in the determination of current equilibrium inflation. Instead, empirical models of the kind used in central banks for policy evaluation often imply that the path of the output gap required in order to achieve a particular path for the inflation rate from now onward depends on what rate of inflation has already been recently experienced, and this aspect of one's model is of obvious importance for the question of how rapidly one should expect that it is optimal to return inflation to its normal level, or even to undo past unexpected pricelevel increases, following a costpush shock.
A simple way of incorporating inflation inertia of the kind that centralbank models often assume into an optimizing model of pricing behavior is to assume, as Christiano, Eichenbaum, and Evans (2001) propose, that individual prices are indexed to an aggregate price index during the intervals between reoptimizations of the individual prices, and that the aggregate price index becomes available for this purpose only with a oneperiod lag. When the Calvo model of staggered pricesetting is modified in this way, the aggregatesupply relation (1) takes the more general form^{11}
The welfaretheoretic stabilization objective corresponding to this alternative structural model is of the form of equation (2) with the period loss function (3) replaced by (p.103)
The solution is therefore also the same, with this substitution.
Figure 3.2 shows the impulse responses of inflation, the output gap, and the price level to the same kind of disturbance as in figure 3.1, under optimal (p.104) policy for economies with alternative values of the indexation parameter γ. (The values assumed for (β, κ, and λ are again as in table 3.1.) Once again, under an optimal commitment, the initial unexpected increase in prices is eventually undone, as long as γ 〈 1, and this once again means that inflation eventually undershoots its longrun level for a time. However, for any large enough value of γ, inflation remains greater than its longrun level for a time even after the disturbance has ceased, and only later undershoots its longrun level; the larger is γ, the longer this period of aboveaverage inflation persists. In the limiting case that γ = 1, the undershooting never occurs; inflation is simply gradually brought back to the longrun target level.^{13} In this last case, a temporary disturbance causes a permanent change in the price level, even under optimal policy. However, the inflation rate is eventually restored to its previously anticipated longrun level under an optimal commitment, even though the rate of inflation (as opposed to the rate of acceleration of inflation) is not welfare relevant in this model. (Note that the optimal responses shown in figure 3.2 for the case γ=1 correspond fairly well to the conventional wisdom of inflationtargeting central banks, but our theoretical analysis allows us to compute an optimal rate at which inflation should be projected to return to its longrun target value following a disturbance.)
As in the previous section, we can derive a target criterion that implements the optimal responses to disturbances regardless of the assumed statistical properties of the disturbances. This optimal target criterion is obtained by replacing π_{t} in equation (8) by yielding
In the special case that γ=1, the optimal target criterion adjusts the current inflation target oneforone with increases in the existing rate of inflation—the target criterion actually involves only the rate of acceleration of inflation. But this does not mean that disturbances are allowed to permanently shift the inflation rate to a new level, as shown in figure 3.2. In fact, in the case of full indexation, an alternative target criterion that also leads to the optimal equilibrium responses to costpush shocks is the simpler criterion (p.105)
The result that the longrun inflation target associated with an optimal target criterion is indeterminate depends, of course, on the fact that we have assumed a model in which no distortions depend on the inflation rate, as opposed to its rate of change. This is logically possible but unlikely to be true in reality. (Distortions that depend on the level of nominal interest rates, considered in the next section, would be one example of a realistic complication that would break this result, even in the case of full indexation.) Because the model considered here with γ = 1 does not determine any particular optimal longrun inflation target (it need not vary with the initially existing inflation rate, for example), even a small perturbation of these assumptions is likely to determine an optimal longrun inflation target, and this will generally be independent of the initially existing rate of inflation. (The monetary frictions considered in the next subsection provide an example of this.)
It is worth noting that even though the optimal dynamic responses shown in figure 3.2 for the case of large γ confirm the conventional wisdom of inflationtargeting central bankers with regard the desirability of a gradual return of the inflation rate to its longrun target level following a costpush shock, the optimal target criterion for this model does not involve a “mediumterm” inflation forecast rather than a shorterrun projection.
(p.106) Even in the case that we suppose that the central bank will often have advance information about disturbances that will shift the aggregatesupply relation only a year or more in the future, the robust description of optimal policy is one that indicates how shortrun outputgap projections should modify the acceptable shortrun inflation projection, rather than one that checks only that some more distant inflation forecast is still on track. Of course, a commitment to the achievement of the target criterion in equation (12) each period does imply that the projection of inflation several quarters in the future should never depart much from the longrun inflation target, but the latter stipulation is not an equally useful guide to what should actually be done with interest rates at a given point in time.
3.1.3 An Interest Rate Stabilization Objective
The policy problems considered above assume that central banks care only about the paths of inflation and the output gap and not about the behavior of nominal interest rates that may be required to bring about a given evolution of inflation and output that is consistent with the aggregatesupply relation. However, actual central banks generally appear to care about reducing the volatility of nominal interest rates as well (Goodfriend 1991). Such a concern can also be justified in terms of microeconomic foundations that are consistent with the kind of aggregatesupply relations assumed above, as discussed in Woodford (2003, chap. 6).
For example, the transaction frictions that account for money demand imply a distortion that should be an increasing function of the nominal interest rate, as stressed by Friedman (1969); the deadweight loss resulting from a positive opportunity cost of holding money should also be a convex function of the interest rate, at least for interest rates close enough to the optimal one (the interest rate paid on base money). Alternatively, the existence of a zero lower bound on nominal interest rates can make it desirable to accept somewhat greater variability of inflation and the output gap for the sake of reducing the required variability of nominal interest rates, given that the smaller range of variation in the nominal interest rate allows the average nominal interest rate (and hence the average inflation rate) to be lower. A quadratic penalty for deviations of the nominal interest rate from a target level may then be justified as a proxy for a constraint that links the feasible average level of nominal interest rates to the variability of the nominal interest rate.
For any of these reasons, we may be interested in a policy that minimizes a loss function of the form
In a simple optimizing model that has been used in many recent analyses of optimal monetary policy (e.g., McCallum and Nelson 1999; Clarida, Galí, and Gertler 1999; and Woodford 1999b), the aggregatesupply relation (1) is combined with an intertemporal Euler equation for the timing of private expenditure of the form
This leads us to consider the problem of finding the statecontingent evolution of inflation, output, and interest rates to minimize the expected discounted value of equation (14) subject to the constraints of equations (1) and (15). A similar Lagrangian method as in section 3.1.1 leads to firstorder conditions of the form
A notable feature of figure 3.3 is that once again optimal policy must be history dependent, for the optimal responses to the disturbance are more persistent than the disturbance itself. As discussed in Woodford (1999b), optimal interest rate policy is inertial, in the sense that interest rates are (p.108)
A historydependent target criterion that can bring about the desired impulse responses, again regardless of the statistical properties of the disturbances and u_{t} (including any assumptions about the degree of correlation (p.109) between these disturbances), can be derived once more from the firstorder conditions (16)–(18). Using the last two equations to substitute for the two Lagrange multipliers in the first equation, we are left with a linear relation of the form
One can furthermore show that not only is this a necessary feature of an optimal equilibrium, but it also suffices to characterize it, in the sense that the system consisting of equation (19) together with the structural equations (1) and (15) has a unique nonexplosive solution, in which the equilibrium responses to shocks are optimal.^{16}
Requirement (19) can be interpreted as an inertial Taylor rule, as discussed in Giannoni and Woodford (2003). However, this requirement can also be equivalently expressed in a forwardintegrated form, that more directly generalizes the optimal target criterion derived in section 3.1.1. It is easily seen that our sign assumptions on the model parameters imply that A(L) can be factored as
This optimal target criterion can be expressed in the form
Thus the optimal conditional forecast is one that places positive weight on the projection for each future period, beginning with the current period, with weights that decline exponentially as the horizon increases. The mean distance in the future of the projections that are relevant to the target criterion is equal to
In the case of the calibrated parameter values in table 3.1, the rate at which these weights decay per quarter is , so that the mean forecast horizon in the optimal target criterion is 2.1 quarters. Thus, while the optimal target criterion in this case involves projections of inflation and output beyond the current quarter, the forecast horizon remains quite short compared to the actual practice of inflationforecasttargeting central banks. For these same parameter values, the optimal relative weight on the outputgap forecast is Ф = .04,^{18} indicating that the target criterion is largely an inflation target. The remaining optimal coefficients are θ_{x} = .04, θ_{i} = .24, and θ_{∆} = .51, indicating a substantial degree of history dependence (p.111) of the optimal flexible inflation target. The fact that θ_{x} = Ф indicates that it is the forecasted increase in the output gap relative to the previous quarter's level, rather than the absolute level of the gap, that should modify the inflation target, just as in section 3.1.1. The signs of θ_{i} and θ_{∆} imply that policy will be made tighter (in the sense of demanding a lower modified inflation forecast) when interest rates have been high and/or increasing in the recent past; this is a way of committing to interest rate inertia of the kind shown in figure 3.3.
Note that in the limiting case in which λ_{i} = 0, this target criterion reduces to equation (8). In that limit, θ_{i}, θ_{∆} and the decay factor become equal to zero, while Ф and θ_{x} have a welldefined (common) positive limit. Thus in this limiting case, the optimal targeting rule is one in which the inflation target must be modified in proportion to the projected change in the output gap, but it is no longer also dependent on lagged interest rates, and the relevant inflation and outputgap projections do not involve periods beyond the current one. This will also be nearly true in the case of small enough positive values of λ_{i}.
We may similarly introduce an interest rate stabilization objective in the case of the model with inflation inertia considered in section 3.1.2. In this case, the loss function (10) is generalized to
Again using the latter two equations to eliminate the Lagrange multipliers, we obtain a relation of the form
The lag polynomial A(L) can be factored as A(L) = (1 − λ_{1}L)L^{2}B(L^{−1}), where B(L^{−1}) is a quadratic polynomial, and under our sign assumptions one can further show^{20} that 0 〈 λ_{1} 〈 1, while both roots of B(L) are outside the unit circle. Relation (25) is then equivalent^{21} to a relation of the form
This provides us with a robustly optimal target criterion that can be expressed in the form
Furthermore, for fixed values of the other parameters, as γ → 0, θ_{π} approaches zero and the other parameters approach the nonzero values associated with the target criterion (22). Instead, as γ → 1, θ_{π} approaches 1, so that the target criterion involves only the projected change in the rate of inflation relative to its already existing level, just as we found in section 3.1.2 when there was assumed to be no interest rate stabilization objective.
The effects of increasing γ on the coefficients of the optimal target criterion (28) is illustrated in figure 3.4, where the coefficients are plotted against γ, assuming the same calibrated values for the other parameters as before. It is interesting to note that each of the coefficients indicating history dependence (θ_{π}, θ_{x}, θ_{i}, and θ_{∆}) increases with γ (except perhaps when γ is near one). Thus if there is substantial inflation inertia, it is even more important for the inflationforecast target to vary with changes in recent economic conditions. It is also worth noting that the degree to which the inflation target should be modified in response to changes in the outputgap projection (indicated by the coefficient Ф) increases with γ. While our conclusion for the case γ = 0 above (Ф = .04) might have suggested that this (p.113)
The panels of figure 3.5 correspondingly show the relative weights α_{z, j}/α_{z,0} on the forecasts at different horizons in the optimal target criterion (28), for each of several alternative values of γ. As above, the inclusion of an interestrate stabilization objective makes the optimal target criterion more forward looking than was the case in section 3.1.2. Indeed, we now find, at least for high enough values of γ, that the optimal target criterion places nonnegligible weight on forecasts more than a year in the future. But it is not necessarily true that a greater degree of inflation inertia justifies a target criterion with a longer forecast horizon. Increases in γ increase the optimal weights on the currentquarter projections of both inflation and the output gap (normalizing the weights to sum to one), and instead make the weights on the projections for quarters more than two quarters in (p.114)
For higher values of γ, increases in γ do increase the absolute value of the weights on forecasts for dates one to two years in the future (these become more negative). But even in this case, the existence of inflation inertia does not justify the kind of response to longerhorizon forecasts that is typical of inflationtargeting central banks. An increase in the forecast level of inflation and/or the output gap during the second year of a bank's current projection should justify a loosening of current policy, in the sense of a policy intended to raise projected inflation and/or the output gap in the next few quarters. This is because in the model with large γ, welfare losses result from inflation variation rather than high inflation as such; a forecast of higher inflation a year from now is then a reason to accept somewhat higher inflation in the nearer term than one otherwise would.
3.1.4 Wages and Prices Both Sticky
A number of studies have found that the joint dynamics of real and nominal variables are best explained by a model in which wages as well as prices are sticky (e.g., Amato and Laubach 2003; Christiano, Eichenbaum, and Evans 2001; (p.115) Smets and Wouters, 2002; Altig et al., 2002; and Woodford, 2003, chap. 3). This is often modeled in the way suggested by Erceg, Henderson, and Levin (2000), with monopolistic competition among the suppliers of different types of labor, and staggered wage setting analogous to the Calvo (1983) model of price setting. The structural equations of the supply side of this model can be written in the form
We note furthermore that κ_{p} ≡ ξ_{p}ω_{p} and κ_{w} ≡ ξ_{w}(ω_{w} + σ^{−1}), where ω_{p} 〉 0 measures the elasticity of marginal cost with respect to the quantity supplied, at a given wage; ω_{w} 〉 0 measures the elasticity of the supply wage with respect to quantity produced, holding fixed households' marginal utility of income; and σ 〉 0 is the same intertemporal elasticity of substitution as in equation (15). In the limit of perfectly flexible wages, ξ_{w} is unboundedly large, and equation (30) reduces to the contemporaneous relation . Using this to substitute for w_{t} in equation (29), the latter relation then reduces to equation (1), where
Given the proposed microeconomic foundations for these relations, Erceg, Henderson, and Levin (2000) show that the appropriate welfaretheoretic stabilization objective is a discounted criterion of the form of equation (2), with a period loss function of the form
(p.116) Here the relative weights on the various stabilization objectives are given by
Here we again abstract from the motives for interest rate stabilization discussed in the previous section. As a result, we need not specify the demand side of the model. We then wish to consider policies that minimize the criterion defined by equations (2) and (33), subject to the constraints (29)–(31).
The Lagrangian method illustrated above now yields a system of firstorder conditions
(p.117) In the special case that κ_{w} = κ_{p} = κ 〉 0, which empirical studies such as that of Amato and Laubach (2003) find to be not far from the truth,^{24} the optimal target criterion (40) reduces simply to q_{t} = 0, or
This is of the same form as the optimal target criterion (8) for the case in which only prices are sticky, with the exception that the index of goods price inflation π_{t} is now replaced by an index that takes account of both price and wage inflation. Of course, the weight that should be placed on wages in the inflation target depends on the relative weight on wage stabilization in the loss function (33). If one assumes a “traditional” stabilization objective of the form of equation (3), so that λ_{w} = 0, then equation (42) is again identical to equation (8). However, one can show that expected utility maximization corresponds to minimization of a discounted loss criterion in which the relative weight on wageinflation stabilization depends on the relative stickiness of wages and prices, as discussed by Erceg, Henderson, and Levin (2000).^{27}
3.1.5 Habit Persistence
In the simple models thus far, the intertemporal IS relation (15) implies that aggregate demand is determined as a purely forwardlooking function of the expected path of real interest rates and exogenous disturbances. Many empirical models of the monetary transmission mechanism instead (p.118) imply that the current level of aggregate real expenditure should depend positively on the recent past level of expenditure, so that aggregate demand should change only gradually even in the case of an abrupt change in the path of interest rates. A simple way of introducing this is to assume that private expenditure exhibits “habit persistence” of the sort assumed in the case of consumption expenditure by authors such as Fuhrer (2000), Edge (2000), Christiano, Eichenbaum, and Evans (2001), Smets and Wouters (2002), and Altig et al. (2002).
Here, as in the models above, we model all interestsensitive private expenditure as if it were nondurable consumption; that is, we abstract from the effects of variations in private expenditure on the evolution of productive capacity.^{28} Hence, we assume habit persistence in the level of aggregate private expenditure, and not solely in consumption, as in the models of Amato and Laubach (2001) and Boivin and Giannoni (2003). This might seem odd, given that we do not really interpret the C_{t} in our model as referring mainly to consumption expenditure. But quantitative models that treat consumption and investment spending separately often find that the dynamics of investment spending are also best captured by specifications of adjustment costs that imply inertia in the rate of investment spending (e.g., Edge 2000; Christiano, Eichenbaum, and Evans 2001; Altig et al. 2002; Basu and Kimball 2002). The “habit persistence” assumed here should be understood as a proxy for adjustment costs in investment expenditure of that sort, and not solely (or even primarily) as a description of household preferences with regard to personal consumption.^{29}
Following Boivin and Giannoni (2003), let us suppose that the utility flow of any household h in period t depends not only on its real expenditure in that period, but also on that household's level of expenditure in the previous period.^{30} Specifically, we assume that the utility flow from expenditure is given by a function of the form
In this extension of our model, the marginal utility for the representative household of additional real income in period t is no longer equal to the marginal utility of consumption in that period, but rather to (p.119)
The marginal utility of income in different periods continues to be linked to the expected return on financial assets in the usual way, so that equilibrium requires that
Using equation (43) to substitute for λ in equation (44), we obtain a generalization of the usual Euler equation for the intertemporal allocation of aggregate expenditure given expected rates of return.
Loglinearization of this Euler equation yields a generalization of our previous IS relation (15), of the form
This modification of preferences changes the form of the aggregatesupply relation (1) as well. (For simplicity, we here consider only the case of a model with flexible wages and Calvo pricing.) In the derivation of equation (1), we have assumed that the log marginal utility of real income (which affects real supply costs owing to its effect on real wage demands) can be replaced by a linear function of x_{t}, but just as in the case of the IS relation, this now must be written as a linear function of instead. We then obtain an aggregatesupply relation of the form
Again taking a secondorder Taylor series expansion of the expected utility of the representative household,^{32} we again obtain a discounted criterion of the form of equation (2), but now with a period loss function of the form
An optimal target criterion is easily derived, even in the presence of habit persistence, in the case that there are no transactions frictions, nor any other grounds for an interest rate stabilization objective. In this case an optimal policy seeks to minimize the discounted sum of losses in equation (50) subject to the sequence of constraints in equation (47). The same Lagrangian method as above yields firstorder conditions
Thus the optimal target criterion is exactly the same as in our baseline model and is unaffected by the estimated value of η. The estimated degree of habit persistence does matter for the central bank's judgment about which inflation or output paths are feasible, and also about the interest rate path that will be necessary in order to achieve them. But it has no consequences for the target criterion that should be used to judge whether a given inflation or output projection is acceptable.
The degree of habit persistence does matter for the optimal target criterion in the case of an interest rate stabilization objective. Suppose that the loss function (50) is generalized to the form
Using two of these relations to eliminate the Lagrange multipliers from the other, we obtain a target criterion of the form
3.2 A Small Quantitative Model of the U.S. Economy
We now turn to the question of the likely quantitative importance of the various considerations discussed in section 3.1 in the actual conduct of monetary policy. In order to do this, we first estimate the numerical parameters of a model that, while still very stylized, is intended to capture important features of the monetary transmission mechanism in the U.S. economy. We present an updated version of the analysis in Rotemberg and Woodford (1997), incorporating a number of additional complications—habit persistence, wage stickiness, and inflation inertia—that have been argued in the subsequent empirical literature to afford important improvements in the realism of this sort of optimizing model of the transmission mechanism, as discussed in section 3.1. The model that we use is similar to the one estimated by Boivin and Giannoni (2003), extended to allow for sticky wages.
Our approach to estimation of the model parameters follows the lines proposed in Rotemberg and Woodford (1997) and also used in Boivin and Giannoni (2003). First, we estimate an unconstrained vector autoregression (VAR) model of a small number of U.S. aggregate time series. This VAR is used (along with weak identifying assumptions) both to identify the coefficients of the Federal Reserve's reaction function in the historical period, and to estimate the impulse responses of our variables to an identified monetary policy shock under that historical policy. In a second step, we develop a simple optimizing model that can replicate the effects of identified monetary policy shocks, as implied by the VAR. We estimate the structural parameters of the model by minimizing the weighted distance between the estimated VAR impulse responses to a monetary policy shock and the model's predicted responses to the same shock. We are then able to recover the historical sequence of structural disturbances and to estimate a law of motion for them, which we use for certain exercises in section 3.3. However, for purposes of the sort of characterization of optimal policy offered here (as opposed to those proposed by Rotemberg and Woodford [1997, 1999]), our conclusions about the character of the historical disturbance processes are much less important than our conclusions about the coefficients of the structural relations that relate the endogenous variables to one another.
In a third step, discussed in section 3.3, we derive a welfaretheoretic loss function for the evaluation of alternative monetary policy rules, by computing a secondorder approximation to the expected utility of the representative household in our model. We then proceed along the lines of Giannoni and Woodford (2002a, b) (p.123) to derive a robustly optimal inflationtargeting rule for monetary policy.
3.2.1 The Effects of Monetary Disturbances
Here we briefly present the VAR that we use to estimate the actual monetary policy rule as well as the effects of monetary policy disturbances. We assume that the recent U.S. monetary policy can be described by the following feedback rule for the federal funds rate
To identify the monetary policy shocks and estimate the coefficients in equation (57) we assume as in the studies of Bernanke and Blinder (1992), Rotemberg and Woodford (1997), Bernanke and Mihov (1998), and Christiano, Eichenbaum, and Evans (2001), among others, that a monetary policy shock at date t has no effect on inflation, output, or the real wage in that period. It follows that equation (57) can be estimated by ordinary least squares (OLS) and that the residuals of the estimated equation will represent a historical sequence of monetary policy shocks.
We model the dynamics of the vector by a structural VAR of with three lags. This can then be written in companion form as
Table 3.2 Estimated monetary policy rule (1980:1–2002:2)
Estimates 


Ф_{i1} 
0.572 (0.104) 
Ф_{i2} 
−0.085 (0.127) 
Ф_{i3} 
0.192 (0.090) 
Ф_{w0} 
0.365 (0.202) 
Ф_{w1} 
−0.008 (0.302) 
Ф_{w2} 
−0.406 (0.191) 
Ф_{π0} 
0.071 (0.098) 
Ф_{π1} 
0.146 (0.115) 
Ф_{π2} 
0.472 (0.115) 
Ф_{y0} 
0.333 (0.176) 
Ф_{y1} 
−0.038 (0.241) 
Ф_{y2} 
−0.118 (0.169) 
R^{2} 
0.956 
DW 
2.033 
Note: Standard errors are in parentheses.
To estimate the VAR, we consider quarterly U.S. data on the sample period 1980:1–2002:2. As in Rotemberg and Woodford (1997) and Amato and Laubach (2003), we begin the sample in the first quarter of 1980 because several empirical studies have identified a significant change in monetary policy around that period (see, e.g., Clarida, Galí, and Gertler 2000; Boivin 2003; Boivin and Giannoni 2003; Sargent 2001, 2002).^{34}
Table 3.2 reports the coefficients of the estimated policy rule. While these coefficients are difficult to interpret as such, we note that the estimated rule (p.125) implies that the interest rate would eventually increase by 2.14 percentage points in the long run, in response to a 1 percentage point permanent increase in inflation, and that it would increase by 0.55 percentage point in response to a 1 percent permanent increase in output. These are similar longrun response coefficients to those obtained by authors such as Taylor (1993, 1999), Judd and Rudebusch (1998), and Clarida, Galí, and Gertler (2000). The estimated realwage response coefficients at different lags are close to cancelling; the estimated reaction function is quite similar to one in which the central bank responds only to the rate of realwage growth rather than to the level of real wages. The response to real wage growth is strongly positive, indicating that increases in wages lead to a stronger and more immediate increase in nominal interest rates than do increases in prices of the same magnitude. While wages are not often included as an explanatory variable in estimated federal reaction functions, our results here suggest that wage growth is also an important explanatory variable.
Figure 3.6 shows the estimated impulse response functions of output, the real wage, inflation, and the interest rate. Here the dashed lines indicate 90 percent confidence intervals, obtained using Kilian's (1998) bootstrap procedure. Because of our identifying assumption, output, inflation, and
3.2.2 A Quantitative Model of the Transmission Mechanism
We now describe a simple optimizing model that we use to explain the effects of monetary policy on output, inflation, the real wage, and interest rates. While the model is still very stylized, it contains several ingredients that allow it to replicate important features of the impulse response functions estimated using our VAR. We assume that there exists a continuum of households indexed by h and distributed uniformly on the [0, 1] interval. Each household h seeks, at date t, to maximize a lifetime expected utility of the form
Optimal Consumption Decisions
While the optimal allocation consumption at date t is chosen at date t and is determined by the usual DixitStiglitz demand relations, we assume as in Rotemberg and Woodford (1997) that households must choose their index of consumption at date t − 2. Equivalently, we assume that is determined at the beginning of period t − 1 (i.e., before the monetary policy shock in t − 1 is known). We assume that financial markets are complete so that risks are efficiently shared. As a result, each household faces a single intertemporal budget constraint.
The firstorder conditions for optimal timing of consumption by the representative household require that (p.127)
We make use of loglinear approximations of these relationships about a steady state equilibrium in which there is no inflation. Loglinearization of equation (44) yields
For our welfare analysis, it is convenient to rewrite this relation in terms of the output gap
Expressing equation (63) in terms of the output gap, we obtain
Optimal Wage and Price Setting
As in Erceg, Henderson, and Levin (2000), Amato and Laubach (2003), and Woodford (2003, chap. 3), we assume that there is a single economywide labor market. The producers of all goods hire the same kinds of labor and face the same wages. Firm z is a monopolistic supplier of good z, which it produces according to the production function
(p.129) We assume that the wage for each type of labor is set by the supplier of that type, who is in a situation of monopolistic competition and who is ready to supply as many hours of work as may be demanded at that wage. We assume that each wage is reoptimized with a fixed probability 1 − α_{w} each period. However, as in Woodford (2003, chap. 3), if a wage is not reoptimized, it is adjusted according to the indexation rule
As shown in Woodford (2003, chap. 3), this setup yields as a firstorder approximation, a wage inflation equation of the form
Integrating equation (67) forward, we note that nominal wages at date t tend to increase (above lag inflation) when expected future positive output gaps are positive and when real wages are expected to be below their natural rate. The variable , which corresponds to the discrepancy between the (log) marginal utility of real income and the (log) marginal utility of consumption, satisfies
The presence of E_{t−1}μ_{t} in equation (67) indicates a moderating effect on nominal wage inflation of an expectation at date t − 1 of real rates of return (p.130) between t and t + 1 that are higher then those that were anticipated at t − 2—that is, at the time that consumption decisions were made for period t. In fact, unexpectedly high real rates of return increase the value of income in period t and thus lower average wage demands.
Similarly, we assume that the suppliers of goods are in monopolistic competition and that each price is reoptimized with a fixed probability 1 − α _{p} each period. However, as in Woodford (2003, chap. 3), if a price is not reoptimized, it is again adjusted according to the indexation rule
Finally, the evolution of the real wage is linked to wage inflation and price inflation through the identity (31). Our structural model can then be summarized by a demand block of equations (64) and (65) and a supply block consisting of equations (67)–(70) together with equation (31). We finally close the model with an equation such as (57) that characterizes the behavior of the central bank. These equations then allow us to determine the equilibrium evolution of the variables of interest:, π_{t}, , x_{t}, , and w_{t}.
3.2.3 Estimated Parameter Values
We turn now to the estimation of the parameters of the structural model just set out. As mentioned above, we are looking for structural parameters that allow the model to describe as well as possible the transmission mechanism of monetary policy. Following Rotemberg and Woodford (1997), we choose the structural parameters that minimize the distance between the estimated VAR impulse response functions to a monetary policy shock and the model's predicted response to the same shock. As discussed in Amato and Laubach (2003), Boivin and Giannoni (2003), and Christiano, Eichenbaum, and Evans (2001), this is quite generally an estimation procedure (p.131) that allows for statistical inference on the model's estimated structural parameters. Note also that the model that we consider is constructed so as to be consistent with the identifying assumptions made for the estimation of the VAR impulse response functions. In particular, both the model and the VAR have the feature that output, inflation, and the real wage respond to unexpected changes in the interest rate with a lag of at least one quarter. In addition, to the extent that we estimate the structural parameters on the basis of impulse responses to monetary shocks, our estimation method has the advantage of providing parameter estimates that are robust to potential misspecifications of the remaining shock processes in the model. This is because in order to compute the impulse responses we do not need to specify the stochastic process of the shocks such as .
As in the studies mentioned above, we set β = 0.99 so that β^{−1} corresponds approximately to the steadystate real gross rate of interest, which is about 1.01. In addition, we calibrate the elasticity to 0.33 as in Rotemberg and Woodford (1997). This would be implied by a CobbDouglas production function in which the elasticity of output with respect to hours is 0.75. Such a production function would yield a share of wages in the value of output of 0.75/μ_{p} where μ_{p} ≡ θ_{p}/(θ_{p} − 1) is the average gross markup of prices over marginal cost due to market power in the goods markets. (This means a labor share of 0.74, given the markup estimate reported below.)
We estimate the vector of the remaining seven structural parameters υ ≡ [φ, η, ξ_{p}, ξ_{w}, ω_{w}, γ_{p}, γ_{w}]′ by minimizing the distance
The estimated parameter values are shown in table 3.3. Standard errors are in parentheses; an asterisk next to the reported standard error indicates that the standard error may not be reliable as the estimated parameter lies on the boundary of the allowed parameter space. Here we report estimates
Table 3.3 Estimated structural parameters for the baseline case and restricted models
Baseline 
No habit · = 0 
No indexation γ_{P} = γ_{w} = 0 
Flexible wages ζ 


Estimated parameters 


0.6715 (0.3330) 
4.3144 (1.0253) 
1.5026 (0.4221) 
0.7564 (0.2823) 

0.5025 (0.0692)* 
0 (—) 
0.5025 (0.1121)* 
0.5025 (0.0515)* 
ξ_{p} 
0.0020 (0.0009) 
0.0015 (0.0005) 
0.0072 (0.0039) 
0.0015 (0.0012) 
ξ_{w} 
0.0042 (0.1343) 
0.0042 (0.0612) 
0.0046 (0.0310) 
+∞ (—) 
ω_{w} 
19.551 (595.1) 
19.991 (269.5) 
19.072 (122.6) 
0.5642 (0.1253) 
γ_{p} 
1 (0.3800)? 
1 (0.3484)? 
0 (—) 
1 (0.5374)? 
γ_{W} 
1 (10.908)* 
1 (12.4613)* 
0 (—) 
0 (—) 
Implied parameters 

φ 
0.7483 
0.2318 
0.3344 
0.6643 
η 
1 
0 
1 
1 
κ_{p} ≡ ξ_{p}ω_{p} 
0.0007 
0.0005 
0.0024 
0.0004 
ω ≡ ω_{p} + ω_{w} 
19.884 
20.325 
19.405 
0.8975 

14.663 
14.994 
14.304 
0.4231 

1.0039 
1.0027 
1.0143 
1.0029 

1.5361 
1.5731 
1.6113 
n.a. 
Objective function value 
13.110 
15.886 
16.580 
18.837 
Wald test (pvalue) 
n.a. 
0.000 
0.000 
0.000 
Note: Standard errors in parentheses. Asterisk indicates that standard error lies on boundary of parameter space and may be unreliable. n.a. = not available.
Table 3.4 Additional calibrated parameter values
Value 


β 
0.99 
ω_{p} 
1/3 
α_{p} 
2/3 
α_{w} 
2/3 
φ 
4/3 
While some of the model parameters cannot be estimated at all precisely, as indicated by the large standard errors, our estimation results are consistent with our theory insofar as we estimate positive values for the response coefficients φ, ξ_{p}, ξ_{w}, and ω_{w} in our structural equations. The values of measuring the interest sensitivity of aggregate expenditure,^{39} and ξ_{p}, measuring the response of inflation to the realwage gap, are both significantly positive, although the estimates of ξ_{w}, and ω_{w} are instead quite imprecise. We also find small enough standard errors on the estimates of ῆ, measuring the degree of habit persistence, and γ_{w}, measuring the degree of indexation of prices, to allow some inference about the magnitudes of those parameters (for example, both are significantly positive), while the value of γ_{w} is very imprecisely estimated. In general, the parameters of our wage equation are poorly estimated, while both our IS relation and our inflation equation are much better estimated.^{40}
The second through fourth columns of table 3.3 report the corresponding estimates, using the same method, of various restricted versions of our model. In column (2), we assume zero habit persistence, as in the models of Rotemberg and Woodford (1997) and Amato and Laubach (2003); in column (3), no inflation inertia (i.e., no indexation of either wages or prices to (p.134) the lagged price index), also like the two models just mentioned; and in column (4), flexible wages, as in the models of Rotemberg and Woodford (1997) and Boivin and Giannoni (2003).^{41} In each case, the objective function value is reported for the restricted model—that is, the weighted distance D(υ) defined above. The pvalues reported on the last line refer to Wald tests of the null hypothesis that the restricted model is correct. In the last column, the parameter γ_{w} is set to zero as it is not identified in the case of flexible wages. We see that each of these restrictions assumed in earlier studies can be individually rejected, although the assumption of flexible wages is the one that would reduce the model's ability to fit the estimated impulse response functions to the greatest extent.^{42} Hence each of the complications introduced here is found to be justified: in this respect, our findings agree with those of Christiano, Eichenbaum, and Evans (2001), Altig et al. (2002), and Smets and Wouters (2002), although these authors all also introduce additional complications in order to explain a larger set of time series.
It is striking to note that the model fits the impulse responses best when the degree of inflation indexing (γ_{p}) and wage indexing to inflation (γ_{w}) reach their upper bound at 1. This corresponds to the assumption of full wage and price indexing made by Christiano, Eichenbaum, and Evans (2001). A value of γ_{p} = 1 is also roughly consistent with the weight on lagged inflation in the “hybrid” aggregatesupply relation estimated by Galí and Gertler (1999) and results in an aggregatesupply relation quite similar to the one proposed by Fuhrer and Moore (1995).
The relatively small values of ξ_{p} and ξ_{w} suggest that changes in the output gap and the real wage gap have a relatively small impact on price and wage inflation. However, the estimated value of ω_{w} suggests that a 1 percent increase in economic activity increases workers' desired wages by nearly 20 percent, for given prices. The estimate of φ corresponds to an elasticity of intertemporal substitution (adjusted by the degree of habit formation) of φ^{−1} = 1.3. While authors such as Fuhrer (2000) and Christiano, Eichenbaum, and Evans (2001), among others, have estimated substantial degrees of habit formation, our estimate lies at the upper bound of 1.
While the estimated parameter values for η, γ_{p} and γ_{w} are significantly smaller when we estimate our model using impulse response functions over (p.135) the first six quarters or less following the monetary shock, all parameter estimates are very similar to those reported in table 3.2, when we use impulse response functions that extend longer than six quarters.^{43} This suggests that in order to adequately capture the degree of persistence in the endogenous variables we need to perform our estimation using long enough responses.
Assuming, as in Rotemberg and Woodford (1997), that α_{p} = 2/3,^{44} and similarly that α_{w} = 2/3, together with the other parameter values already mentioned above, it is possible to infer the elasticities of substitution θ_{p} and θ_{w} from the estimated values of ξ_{p} and ξ_{w} respectively, using the definitions (68) and (71). The values of these elasticities implied by our estimates imply a gross markup of prices over marginal costs of only μ_{p} = θ_{p}/(θ_{p} − 1) = 1.004 in the goods market, but a considerably higher gross markup of μ_{w} = θ_{w}/(θ_{w} − 1) = 1.54 in the labor market. The fact that these implied markups are greater than 1 (i.e., that the implied elasticities of substitution are greater than 1) again indicates consistency of our estimates with our theoretical model.
Finally, our estimated value for ω_{w} can be used to derive an implied value of v, the inverse of the Frisch elasticity of labor supply, using the definition ω_{w} = vФ and a calibrated value for Ф, the inverse of the elasticity of output with respect to the labor input. (The calibrated value of φ reported in table 3.4 is implied by the same CobbDouglas production function as was used to calibrate the value of ω_{p}, discussed above.) The Frisch elasticity of labor supply implied by our estimates is thus only on the order of 0.07, less than one onehundredth of the value implied by the estimates of Rotemberg and Woodford (1997), and much more consistent with many estimates in the empirical literature on labor supply. Because of the assumption of sticky wages, our model is able to account for nonnegligible effects of a monetary disturbance on real activity without assuming that voluntary labor supply (under flexible wages) would be highly elastic. (Note that under the restriction of flexible wages, we would obtain estimates implying an elasticity of labor supply greater than 2.) While the values of these implied parameters do not matter for the ability of our model to fit the estimated impulse responses, they do matter for our welfare analysis below.
The solid lines in figure 3.6 indicate the impulse response functions generated by our estimated model. Overall, it appears that the model is able to replicate quite well the impulse responses estimated by the VAR (circled lines), and the impulse responses remain consistently within the 90 percent confidence intervals. The model replicates in particular the estimated humpshaped output and realwage responses. While it does not capture (p.136) the oscillations in the inflation response implied by the VAR, we note that this response is estimated quite imprecisely.
3.3 Optimal Policy for the Estimated Model
Now that we have an estimated structural model that allows us to account for at least certain aspects of the responses of output and of price and wage inflation to monetary disturbances, we turn to the characterization of optimal policy in the context of this model.
3.3.1 A WelfareTheoretic Stabilization Objective
An advantage of having developed a structural model based on optimizing behavior is that it provides a natural objective for the monetary policy, namely, maximization of the expected utility of the representative household. Following the method of Woodford (2003, chap. 6), we can express a secondorder Taylor series approximation to this objective as a quadratic function of (wage and price) inflation, the output gap, and the nominal interest rate. The way in which various aspects of our model specification affect the appropriate welfaretheoretic stabilization objective in simple cases has already been discussed in section 3.1.
In the technical appendix to this paper (see www.nber.org/data/ ), we show that for the model developed in section 3.2, the corresponding welfaretheoretic loss function, abstracting from any grounds for concern with interestrate stabilization, is given by
In this expression, the weights λ_{p}, λ_{w} 〉 0 are again defined as in equation (34); the weight λ_{x} 〉 0 is again defined as in equation (35), but using now the definition (49) for κ in the latter expression; the coefficient 0 ≤ δ ≤ η is again the smaller root of equation (45); and is the same function of the microeconomic distortions affecting the efficiency of the steadystate output level as in equation (50).
This result combines features of several simpler cases discussed in section 3.1. Deadweight loss depends on squared deviations of both price and wage inflation (separately) from the rates that would minimize relative price and relativewage distortions, given that both wages and prices are sticky, as in equation (33). Due to the indexation of both prices and wages to a lagged price index, the lossminimizing rates of wage and price inflation each period are determined by the lagged inflation rate and the indexation coefficients in each case, as in equation (10). And finally, the presence of habit persistence implies that deadweight loss depends not on squared deviations of the output gap from a constant value but rather on squared deviations of x_{t} − δx_{t−1} from a constant value, as in equation (50).
Table 3.5 Lossfunction coefficients implied by our parameter estimates
λp 
λw 
16λx 
λ 

0.9960 
0.0040 
0.0026 
0.035 
These conclusions depend, of course, on our parameter estimates. It may seem surprising that the weight on wage inflation stabilization is so small, given that our estimates do not imply that wages are substantially more flexible than prices (for example, ξ_{w} is larger than ξ_{p} but not by a large factor). The conclusion that λ_{w} is nonetheless very much smaller than λ_{p} reflects mainly the fact that our estimates imply a value for θ_{p} that is much larger than Ф^{−1}θ_{w}. This in turn results from the fact that the estimated value of ω_{w} is much larger than the calibrated value of ω_{p}.^{45} Because it is not plausible to assume a technology for which ω_{p} could be nearly as large as the estimated value of θ_{w}, we are led to assume a value of ω_{p} substantially larger than Ф^{−1}θ_{w}. The result that λ_{p} greatly exceeds λ_{w} then follows, using equation (34).
The conclusion that λ_{x} is small follows, using equation (35), from the small value of κ_{p} and large value of θ_{p} implied by our parameter estimates. Since κ_{p} ≡ ξ_{p}ω_{p} and the value of θ_{p} is inferred from the value of ξ_{p} using equation (71), both of these conclusions depend crucially on the small estimated value for ξ_{p}. Essentially, the observed insensitivity of inflation to variations in output allows us to infer underlying microeconomic parameters that imply that variations in the output gap cause relatively modest distortions—this is the only way, in the context of our other assumptions, to explain the fact that inflation is not more strongly affected (i.e., that the Phillips curve is not steeper).
(p.138) Finally, the conclusion that δ is small (despite the fact that η = 1) follows, using equation (48), from the fact that the value of ω implied by our estimates is large relative to the estimated value of φ. Essentially, the observed sensitivity of wages to variations in real activity on the one hand (implying a large value for ω_{w}) and the sensitivity of aggregate expenditure to interest rate changes on the other (implying that φ cannot be too large) indicate preferences under which variations in the level of real activity will create greater distortions than variations in the rate of growth of real activity. Even when η = 1, the level of output matters to the representative household because of its consequences for the amount that the household must work; if the marginal disutility of output supply increases sharply with the level of real activity (as implied by a large value of ω), it will still be relatively more important to stabilize the level of real activity than its rate of change.^{46}
3.3.2 An Optimal Target Criterion
The method illustrated in section 3.1 for the derivation of optimal target criteria under alternative assumptions can be applied as well in the case of the empirical model described in section 3.2. Details of the relevant calculations are included in the technical appendix to this paper; here we simply present the quantitative implications of our estimated parameter values.
A first observation about optimal policy in our estimated model follows from the fact that wages, prices, and output are all predetermined for one quarter or longer in the model. It follows that in our structural equations, any variations in the shortterm nominal interest rate i_{t} that are not fore castable a quarter earlier are irrelevant to the determination of wages, prices, or output. Hence this component of interest rate policy cannot be relevant for welfare except through its consequences for the expected discounted value of the λ_{i}(i_{t} − i*)^{2} term that must be added to equation (72) if we take account of monetary frictions. But this last term is obviously minimized (in the case of any λ_{i} 〉 0) by a policy under which the nominal interest rate is completely forecastable a quarter in advance. Even in the case that λ_{i} = 0, there is no harm to any other stabilization objectives in eliminating unforecastable interest rate variations; and so it seems plausible to assumes at least some tiny concern with interest rate stabilization, so that it is optimal to suppress such variation in the interest rate.^{47}
To simplify, we shall restrict attention to the case of a model in which γ_{p} = γ_{w} = 1, as assumed by Christiano, Eichenbaum, and Evans (2001), and as indicated by our estimates in section 3.2. In the appendix, we show that the firstorder conditions for an optimal statecontingent evolution of the endogenous variables can be manipulated, after the fashion illustrated in section 3.1, to yield a characterization of optimal policy in terms of the projected paths of the target variables alone. However, in the present case, unlike the simpler ones discussed in section 3.1, the most convenient representation of these conditions is not in terms of a single target criterion, but two distinct ones. First of all, optimality requires that projections in any period t satisfy a condition of the form^{49}
Here for each of the variables z = π, w, the expression F_{t}(z + 1) refers to a weighted average of forecasts of the variable z at various future horizons, conditional on information at date t,
In general, this optimal target will not be constant over time.
In addition to the above requirement (which amounts to the condition that the lefthand side of equation [74] be forecastable a quarter in advance), optimality also requires that projections at date t satisfy another condition as well, of the form
The optimal target criteria in equations (74)–(76) and (77)–(78) generalize, for the estimated model, the simple criterion in equation (13) obtained in the case of inflation inertia, γ = 1, flexible wage, no habit persistence and no delays. To make this comparison more apparent, and to get some intuition about the two optimal target criteria, it is useful to consider the special case in which wages are flexible. As we show in the technical appendix, the shortrun optimal target criterion of equations (74)–(76) reduces in this case to
As in section 3.1.2, a commitment to ensure that equations (77) and (78) hold in each period t ≥ t_{0} for a particular value of the constant π* is equivalent to a commitment to ensure that a firstdifferenced form of these equations holds in each period.^{52} Such a firstdifferenced form would have the advantage that it could be expressed entirely in terms of projections of the first differences of the three variables—the inflation rate, the real wage, and the output gap—with no dependence on the absolute levels of any of the variables. The target criterion of equations (77) and (78), instead, has the advantage of being simpler, as it only involves a comparison of projections made in the current period with certain other projections in the previous period.
It may be wondered how we can specify optimal policy in terms of two distinct target criteria involving different linear combinations of projections, when the central bank has only one instrument at its disposal. The key to this is to observe that the target criterion specified by equations (74), (75), (76) restricts only the surprise components of the quarter t projections—that is, the way in which they may differ from the projections that were made in quarter t − 1 for the same variables. Hence it is only the surprise component of the central bank's interestrate decision—the difference between the E_{t}i_{t+1} announced in quarter t and E_{t−1}i_{t+1}—that can be determined by this criterion for optimal policy. The evolution of the (twoperiodahead) predetermined component of policy, E_{t−2}i_{t}, can instead be chosen so as to ensure that the second target criterion, specified by equations (77) and (78), is satisfied each period.
We may thus imagine the implementation of the optimal targeting rule to occur in the following way.^{53} First, in each quarter t, the central bank intervenes in the money markets (through openmarket operations, repurchases, standing facilities in the interbank market for centralbank balances, etc.) so as to implement the interest rate target i_{t} announced in (p.142) quarter t − 1. Second, as part of the quartert decision cycle, the bank must choose an operating target i_{t+1} to announce for the following quarter. This is chosen in order to imply a projected evolution of (wage and price) inflation from quarter t + 1 onward that satisfies the target criterion (74), where is a target value that had been determined in quarter t − 1. Third, it is also necessary, as part of the quartert decision cycle, for the central bank to choose the target for the following quarter. This is chosen so as to ensure that future policy will be conducted in a way that allows the bank to project (conditional on its current information) that the target criterion of equations (77) and (78) should be satisfied. In practice, this means that the central bank should use its model of the transmission mechanism to determine the future evolution of the economy under the assumption that equations (77) and (78) will hold in all future periods; this forecast then determines the target value using equation (76).^{54}
Algebraic expressions for each of the coefficients in the optimal target criteria, as functions of the underlying model parameters, are given in the appendix. Here we discuss only the numerical coefficients implied by our estimated parameter values. In the case of the shortterm criterion (74), the coefficient Ф_{w} is equal to 0.565.^{55} Thus, if unexpected developments in quarter t are projected to imply a higher future level of real wages than had previously been anticipated, policy must ensure that projected future price inflation is correspondingly reduced. This is because of a desire to stabilize (nominal) wage inflation as well as price inflation, and under circumstances of expected real wage growth, inflation must be curbed in order for nominal wage growth to not be even higher.
The relative weights that this criterion places on projections at different future horizons are shown in figure 3.7. The two panels plot the coefficients α_{πj}, α_{wj} respectively, as functions of the horizon j. Note that the quarter for which the projections receive greatest weight is one quarter in the future, in each case. However, while the realwage projection that matters is primarily the projected growth in real wages between the present quarter and the next one, substantial weight is also placed on projected inflation farther in the future; in fact, the mean lead ∑jα_{πj}j is between ten and eleven quarters in the future in the case of the inflation projection F_{t}(π_{+1}). Thus the shortrun target criterion is a (timevarying) target for the average rate of inflation that is projected over the next several years, adjusted to take account of expected wage growth, mainly over the coming quarter. Roughly speaking, optimal policy requires the central bank to choose E_{t}i_{t+1} in quarter t so (p.143)
In the case of the longterm criterion (77), instead, the numerical coefficients of the target criterion are given by
In this case, outputgap projections matter as well; a higher projected future output gap will require a reduction in the projected future rate of inflation, just as will a higher projected future real wage. The numerical size of the weight placed on the outputgap projection may appear modest; but as we shall see in the next section, the degree of variability of outputgap projections in practice are likely to make this a quite significant correction to the path of the target criterion.
The relative weights on forecasts at different horizons in this criterion (p.144)
Finally, the coefficients of the rule (78) determining the target value for the longterm criterion are given by
(p.145) The weights in the projections (conditional on information in the previous quarter) at various horizons are plotted in the second row of figure 3.8. Here too, it is primarily projections for two quarters in the future that matter in each case. Roughly speaking, then, the target value for the wage and outputadjusted inflation projection two quarters in the future is high when a similar adjusted inflation projection (again for a time two quarters in the future) was high in the previous quarter.
Thus we find that forecasting exercises, in which the central bank projects the evolution of both inflation and real variables many years into the future under alternative hypothetical policies on its own part, play a central role in a natural approach to the implementation of optimal policy. A forecast of inflation several years into the future is required in each (quarterly) decision cycle in order to check whether the intended interest rate operating target for the following quarter is consistent with the criterion (74). In addition, the timevarying mediumterm inflation target, must be chosen each period on the basis of yet another forecasting exercise. While the longrun target criterion (77) primarily involves projections for a time only two quarters in the future, the choice of requires that the central bank solve for a projected path of the economy in which criterion (77) is satisfied not only in the current period but in all future periods as well. Hence, this exercise as well requires the construction of projected paths for inflation and real variables extending many years into the future. The relevant paths, however, will not be constant–interest rate projections (of the kind currently published by the Bank of England) but rather projections of the economy's future evolution given how policy is expected to evolve. Indeed, the projections are used to select constraints upon the bank's own actions in future decision cycles (by choosing both the interest rate operating target E_{t}i_{t+1} and the adjusted inflation target in period t).
3.3.3 A Comparison with Actual U.S. Policy
An interesting question about this policy rule is the extent to which it would prescribe policy different from that which the Federal Reserve has actually pursued during our sample period. A simple way of considering this is to ask to what extent, under actual policy, projections of the evolution of inflation and output have satisfied the optimal target criteria stated above. Answering this question requires, of course, that we estimate what the projected future paths of the target variables should have been at various past dates. However, our VAR characterization of the data over our sample period provides one way of generating such projections. Here we propose to appraise how close actual policy has been to being optimal by asking to what extent projections based on the VAR would have satisfied the target criterion.
In our characterization of optimal policy above, there are actually three criteria that must be satisfied each period—one relating to the component (p.146)
The next condition for optimality that we consider is the shortterm target criterion (74)–(76). Figure 3.10 shows a plot of the historical path of the wageadjusted inflation projection that is targeted under this criterion, using the VAR forecasts to form this projection each quarter, together with the path for the target value given by equation (76), also using the VAR forecasts for the projections in the previous quarter. Figure 3.11 decomposes the variation in both the adjusted inflation projection (74) and the timevarying target into the parts that are due to variation in the inflation projections (at various horizons) on the one hand and the parts that (p.148)
Once again, the data are fairly consistent with this criterion for optimal policy. While the wageadjusted inflation projection has varied (according to the VAR) over a range of a few percentage points, these variations have been fairly forecastable based on the previous quarter's information set, as required by the target criterion. The gap between the projection and the target value has a standard deviation of only 48 basis points over this sample.
Of course, passing this test requires only that wage and price inflation, like the federal funds rate, be highly forecastable a quarter in advance. It may accordingly be felt that it is the inertial character of wage and price inflation (p.149) that is confirmed by figure 3.10, rather than something that depends much on monetary policy. It should also be noted that the “target” series plotted in the figure only indicates how the righthand side of equation (76) has varied over the sample period, under actual U.S. monetary policy, rather than the way in which the target would have evolved under optimal monetary policy, given that the inflation projections that determine this target would have been different under a different sort of monetary policy. This latter sort of exercise would require that we solve for the counterfactual equilibrium paths of the endogenous variables under optimal policy, given the historical sequence of exogenous shocks, as undertaken by Rotemberg and Woodford (1997). We do not attempt such an exercise here.
Testing the extent to which the historical data have satisfied the longrun target criterion (77)–(78) is more complicated, because it requires the construction of projections for the path of the output gap. The output gap is not directly observed, and our approach to the estimation of the model in section 3.2 does not require us to commit ourselves to an empirical proxy for the gap, despite the appearance of this variable in the model structural equations. In order to estimate the model parameters needed for our calculations thus far, we had only to be able to compute the predicted impulse responses of prices, wages, output, and interest rates to a monetary disturbance. For this purpose, we could rely on the fact that, according to our model, the output gap should equal Ŷ_{t} (detrended log output) minus a term that is unaffected by monetary disturbances; there was no need to identify the time variation in that latter term. Yet in order to evaluate the longrun target criterion at each date, we need to be able to do so.
One possible approach is to use our estimated structural equations to infer the historical sequence of disturbances from the residuals of the structural equations, using VAR forecasts of the endogenous variables as proxies for the expectation terms in these equations, as do Rotemberg and Woodford (1997). This approach can be used, however, only under strong assumptions of debatable validity. The “natural rate of output” process that we are able to infer from the residuals of our structural equations corresponds to the equilibrium level of output under complete wage and price flexibility.^{59} But this may or may not be the concept of exogenously given potential output that should be used to define the welfarerelevant “output gap” that appears in the loss function (72).
Under certain assumptions that are made precise in the appendix (and that have been tacitly maintained thus far in our exposition), the “output (p.150) gap” that appears in the structural equations (57) and (60) as a source of inflationary pressure—without any additional costpush shock term of the kind routinely included in the models of section 3.1—is exactly the same variable as the distortion measure appearing in equation (72). Yet this need not be true in general; time variation in distorting taxes or in the degree of market power in either labor markets or goods markets, for example, will result in a timevarying wedge between the flexiblewageandprice equilibrium level of output and the efficient level of output, with the result that the relevant output gap for the two purposes ceases to be the same.^{60} We can allow for this extension of our framework by letting the gap between actual output and the flexiblewageandprice equilibrium output be denoted x_{t} + μ_{t}, as in equations (29)–(30) above, where x_{t} is the welfarerelevant outputgap concept (the variable that appears in the welfaretheoretic loss function), while μ_{t} is a costpush disturbance term.
In the case of the extended model, the method of Rotemberg and Woodford allows us to construct an empirical proxy for the evolution of the series x_{t} + μ_{t}, as this is what appears in the wage and pricesetting equations. However, the projections that are required for checking whether the target criterion is satisfied are projections for x_{t}, the variable that appears in the loss function (72). Further assumptions must be made in order to infer what the projected variations in the welfarerelevant output gap should have been. These assumptions are not testable within the context of the model and the small set of time series used here.
One simple, though extreme, assumption, would be that the welfare relevant concept of potential output is a smooth trend, so that cyclical variation in should be almost entirely attributed to transitory variation in the costpush term μ_{t}.^{61} In this case, it should be more accurate to identify the welfarerelevant output gap with Ŷ_{t}, detrended output, than with the series x_{t} + μ_{t} inferred from the residuals of the structural equations. Under this assumption, we can construct our outputgap projections using the VAR alone, without any need to reconstruct disturbances using the equation residuals.
We first consider the conformity of historical policy with the optimal target criteria when detrended output is considered an adequate proxy for the output gap. In figure 3.12, we plot the historical series for the wage and outputadjusted inflation projection that is targeted under the longterm criterion (77) over our sample period, using the VAR forecasts for inflation, the real wage, and detrended output, and the numerical weights given in section 3.3.2. (Since the constant π* in equation [78] is arbitrary, we assume a longrun inflation target equal to 2.39 percent per annum, which (p.151)
This alternative (longerrun) adjusted inflation projection has also been relatively stable over our historical sample, and once again the gap between the target and the current projection has never been large; the standard deviation of target misses in the case of this criterion is only 52 basis points. However, target misses under this criterion have been somewhat persistent, with a quarterly autocorrelation of 0.19. Thus we can identify periods in which policy was consistently too loose or too tight for quarters at a time, according to this criterion, although federal policy never violated the criterion (p.152)
If, instead, we use the residuals from our structural equations to infer the evolution of the output gap, the plots corresponding to figures 3.12 and 3.13 instead look like those shown in figures 3.15 and 3.16. In this case, historical paths of both the adjusted inflation projection and its optimal target (p.153)
In this case, the gap between the adjusted inflation projection and its optimal target value (plotted in figure 3.17) is also found to be fairly large in many individual quarters. The standard deviation of the discrepancy using this measure of the output gap is nearly 1.80 percentage points. However, the target misses are extremely transitory in this case; their autocorrelation is actually negative (−0.53), indicating that a target overshoot one quarter (p.155)
Overall, a comparison between U.S. time series over the past twenty years and the criteria for optimal policy discussed in the previous section do not indicate any gross discrepancy. However, this may simply mean that the diagnostics proposed here are not very useful as a way of diagnosing deviations from optimal policy in the historical record. We have plotted only the time variation in the optimal target criteria that would be implied by the variation in lagged projections that has occurred, given the actual evolution of the U.S. time series, rather than attempting to determine the variation in the target values that would have occurred under optimal policy, given the historical disturbance processes. These two ways of judging the historical time series might yield quite different pictures. Our optimal target criteria demand that certain adjusted inflation projections not be too (p.156) different than similar projections have been in the quarter before; this will result in plots of projections and target values that look fairly similar, regardless of the paths of the U.S. time series, as long as each of our four variables has been relatively smooth (as is the case). Nonetheless, inflation and other variables might have wandered for years at some distance from the levels that they would have had under fully optimal responses to the historical disturbances.
3.4 Conclusions
We have shown that it is possible to derive robustly optimal monetary policy rule for optimizing models of the monetary transmission mechanism that incorporate a number of common features of recent empirical models: staggered wage and pricesetting; inflation inertia resulting from automatic indexation of wages and prices to a lagged price index; predetermined wagesetting, pricing, and spending decisions; and habit persistence in the level of real private expenditure. In this way, we have sought to show that the approach to the design of optimal policy rules proposed by Giannoni and Woodford (2002a) can be applied to models of practical interest.
In each of the cases that we have discussed, the optimal policy rule is a modified inflationforecast targeting rule. The optimal rule differs from a simple (or “strict”) inflation target in that projections of the future paths of variables other than goodsprice inflation also receive some weight in the target criterion—in particular, wage inflation, a measure of the output gap, and nominal interest rates. Nonetheless, according to our numerical analysis in the case of an estimated model of the U.S. monetary transmission mechanism, the weight on the inflation projection (in each of the two target criteria involved in our characterization of optimal policy for that model) is strong enough that it makes sense to speak of optimal policy as a (flexible) inflationforecasttargeting procedure.
In our examples, the optimal rule also differs from a simple inflation target (and even from many simple examples of “flexible inflation targeting” rules discussed in the literature) in that the optimal target value for the modified inflation forecast should vary over time, depending on current and recent past macroeconomic conditions. We have illustrated the possible degree of history dependence of an optimal inflation target by showing how our two optimal target criteria would have varied in the United States over the past two decades, given our VAR characterization of the U.S. time series and the parameters of our estimated structural model. Even when we use detrended output as our proxy for the output gap (which results in a less volatile outputgap series than the one implied by the residuals of our structural equations), and even over the relatively uneventful period 1984–2000, our analysis implies that the optimal target criterion (p.157) has varied from quarter to quarter over a range of several percentage points.
Finally, we have shown, in the context of our empirical model, that an optimal policy may be too complex in structure to be conveniently described by a single target criterion. Our estimated model of the U.S. monetary transmission mechanism implies that optimal policy must satisfy three distinct criteria: one that governs the way that interest rates in a given quarter should respond to unexpected developments during that quarter; one that governs the way in which the central bank's commitment regarding interest rates in that quarter, announced the quarter before, should respond to unexpected developments in the quarter when the commitment is made; and still a third criterion that determines the component of interest rate policy that can be anticipated two quarters in advance. Nonetheless, the decision procedure takes the form of an inflationforecasttargeting procedure, in which (a) the instrument used to ensure satisfaction of the target criterion is the central bank's commitment regarding its interest rate operating target for the following quarter, and (b) the inflation target each quarter is itself the product of a policy decision in the previous quarter, also aimed at ensuring that a certain adjusted inflation projection satisfies a target criterion.
Our optimal target criteria are a good bit more complex than the sort used by actual inflationtargeting central banks, which typically specify a timeinvariant inflation target and a particular horizon at which it is to be reached (for example, RPIX inflation of 2.5 percent at a horizon of eight quarters in the future, in the case of the Bank of England). Our advocacy of a more complex form of targeting rule is not meant to deny the desirability of having a mediumterm inflation target that remains the same even if the actual inflation rate may depart from it temporarily. In the examples that we have considered, optimal policy almost always involves a welldefined longrun inflation target, to which the inflation rate should be expected to return after each disturbance, and it is surely desirable for a central bank to be explicit about this aspect of its policy commitment, in order to anchor the public's mediumterm inflation expectations.
Rather, we wish to suggest that it is insufficient to specify no more of a policy commitment than this. The mere fact that a central bank wishes to see inflation return to a rate of 2.5 percent at a horizon two years in the future is not sufficient to say which of the various possible transition paths that reach that endpoint should be preferred. There will always be a range of possible scenarios consistent with the terminal condition: for example, looser policy this year to be compensated for by tighter policy next year, or alternatively the reverse.
In practice, the Bank of England, like many other forecasttargeting banks, deals with this problem by demanding that a constantinterest rate forecast satisfy the terminal condition. That is, the current level of overnight (p.158) interest rates is held to be justified if a projection under the assumption that that level of interest rates will be maintained implies that RPIX inflation should equal 2.5 percent eight quarters in the future. However, this implies no commitment to actually maintain interest rates at the current level over that period, or even that interest rates are currently expected to remain at that level on average. (It is frequently the case that the published constant–interest rate projection would itself imply that interest rates will need to be changed over the coming year, in order for the target criterion to be satisfied by a constant–interest rate projection under the conditions that are forecasted to obtain by then.) It is thus hard to see how basing policy decisions on a forecasttargeting exercise of this particular kind can be expected to serve the goals of making monetary policy more transparent or improving the degree to which policy is correctly anticipated by the private sector.
The conceptually superior approach, surely, is to base policy on a projection that is computed under the assumption that policy will be made in accordance with the targeting rule in the future as well,^{64} so that the projection that is used to justify current policy will correspond to the bank's own best forecast of how it should act in the future, as in the case of the projections used to justify policy decisions by the Reserve Bank of New Zealand. It will, of course, be necessary to stress that the bank's only commitment is to the rule embodied in this projection, not to the particular time path of interest rates indicated as most likely. But given the use of “fan charts” to show that a variety of possible future scenarios can be envisioned, depending on how various types of uncertainty happen to be resolved, it is not clear why it should not be possible to talk about probability distributions for future interest rates along with those for inflation and real activity without giving rise to the appearance of a more specific commitment than is intended.
Once this is done, however, it becomes necessary to specify a target criterion that can determine the appropriate shortrun dynamics for the economy, and not simply a terminal condition for a date some years in the future. Such a criterion will accordingly place substantial weight on projections of the target variables over the coming year, as in the case of the optimal target criteria derived in this chapter. It will also have to take a stand as to the kinds of projected departures of real variables from their longrun average values that justify shortrun departures of the inflation projection from its longrun target value; it will no longer suffice simply to specify what the (unchanging) longrun inflation target is. None of the inflationtargeting central banks actually believe that it is desirable to keep inflation as close as possible to the longrun target value at all times; this is (p.159) why forecasttargeting procedures only seek to ensure that inflation is projected to return to the target value after many quarters.^{65} But by formulating no explicit doctrine as to the way in which one should choose among alternative transition paths to that mediumterm goal, they avoid having to clarify the nature of acceptable tradeoffs among competing stabilization goals.^{66}
A coherent approach—and, in particular, one that could be justified as seeking to implement the conditions for optimal policy discussed in this chapter—would instead have to make explicit the kind of projections for output and other real variables that should justify a modification of the shortrun inflation target, and the degree to which they should affect it. In all likelihood, the inflationtargeting banks have shied away from such explicitness out of a suspicion that the types of circumstances that might reasonably justify shortterm departures from the inflation target are too various to be catalogued. But the theory developed here has sought to show that it is possible to state shortrun target criteria (criteria that apply to the shortest horizon at which current policy decisions can still have an effect) that will be robustly optimal, meaning that the same criterion continues to determine the correct degree of shortrun departure from the longrun inflation target regardless of the nature of the disturbance that may have occurred.
Much work remains to be done, of course, before a quantitative characterization of optimal policy of the kind that we offer in section 3.3 could be used in practical policy deliberations. One of the most obvious issues requiring further study concerns the way in which a central bank should take account of uncertainty about the correct model of the transmission mechanism, as well as uncertainty in its evaluation of current macroeconomic conditions. Uncertainty about the current state of the economy is relatively straightforward to deal with, at least in principle. One can allow for partial information on the part of the central bank in characterizing the optimal equilibrium responses to shocks, using methods similar to those employed here, and derive an optimal target criterion that is valid in the presence of partial information (Svensson and Woodford 2003, 2004; Giannoni and Woodford 2002b). Because of the principle of certainty equivalence in linearquadratic policy problems of this kind (discussed in detail by Svensson and Woodford 2003, 2004), the optimal target criterion (once correctly expressed) involves coefficients that are independent of the degree of uncertainty (p.160) in centralbank estimates of the current state of the economy; however, the target may involve variables that are not directly observed by the central bank and that must instead be estimated using a Kalman filter.
Dealing with uncertainty about the numerical values of structural parameters (to say nothing of more fundamental doubts about model specification) is a much harder problem, for which few general guidelines exist at present. Giannoni (2001, 2002) illustrates one approach to the problem, for the case of uncertainty about the numerical values of the elasticities κ and σ in a model similar to our baseline model (but in which an interest rate stabilization objective is assumed). For the particular kind of parameter uncertainty considered, Giannoni finds that a concern for robustness (in the sense of guarding against bad outcomes in the least favorable case) should lead a central bank to choose a Taylorstyle interest rate rule with stronger response coefficients than it would choose on the basis of its preferred estimates of the model parameters; this means allowing less variability of inflation in equilibrium, at the cost of greater variability in nominal interest rates. This suggests that a concern for robustness might justify targeting rules that are even closer to strict inflation targeting than the optimal rules obtained in this paper; the question is surely one that deserves further analysis.
References
Bibliography references:
Altig, David, Lawrence J. Christiano, Martin S. Eichenbaum, and Jesper Linde. 2002. Technology shocks and aggregate fluctuations. Cleveland, Ohio: Federal Reserve Bank of Cleveland. Unpublished manuscript.
Amato, Jeffrey D., and Thomas Laubach. 2001. Implications of habit formation for optimal monetary policy. Finance and Economics Discussion Series Paper no. 2001–58. Washington, D.C.: Federal Reserve Board, August.
———. 2003. Estimation and control of an optimizationbased model with sticky wages and prices. Journal of Economic Dynamics and Control 27 (May): 1181–1215.
Ball, Laurence, and David Romer. 1990. Real rigidities and the nonneutrality of money. Review of Economic Studies 57:183–203.
Basu, Susanto, and Miles S. Kimball. 2002. Investment planning costs and the effects of fiscal and monetary policy. University of Michigan, Department of Economics. Manuscript, November.
Bernanke, Ben S., and Alan S. Blinder. 1992. The federal funds rate and the transmission of monetary policy. American Economic Review 82:901–21.
Bernanke, Ben S., Thomas Laubach, Frederic S. Mishkin, and Adam S. Posen. 1999. Inflation targeting. Princeton, N.J.: Princeton University Press.
Bernanke, Ben S., Jean Boivin, and Piotr Eliasz. 2004. Measuring the effects of monetary policy: A factoraugmented vector autoregressive FAVAR approach. NBER Working Paper no. 10220. Cambridge, Mass.: National Bureau of Economic Research, January.
(p.161) Bernanke, Ben S., and Ilian Mihov. 1998. Measuring monetary policy. Quarterly Journal of Economics 113 (3): 869–902.
Blinder, Alan S. 1994. On sticky prices: Academic theories meet the real world. In Monetary policy, ed. N. G. Mankiw, 117–50. Chicago: University of Chicago Press.
Boivin, Jean. 2003. Has U.S. monetary policy changed? Evidence from drifting coefficients and realtime data. Unpublished Manuscript. Columbia University, Graduate School of Business.
Boivin, Jean, and Marc P. Giannoni. 2003. Has monetary policy become more effective? NBER Working Paper no. 9459. Cambridge, Mass.: National Bureau of Economic Research, January.
Calvo, Guillermo A. 1983. Staggered prices in a utilitymaximizing framework. Journal of Monetary Economics 12 (3): 383–98.
Christiano, Lawrence J., Martin S. Eichenbaum, and Charles L. Evans. 2001. Nominal rigidities and the dynamic effects of a shock to monetary policy. NBER Working Paper no. 8403. Cambridge, Mass.: National Bureau of Economic Research, July.
Clarida, Richard, Jordi Galí, and Mark Gertler. 1999. The science of monetary policy: A new Keynesian perspective. Journal of Economic Literature 37:1661–707.
———. 2000. Monetary policy rules and macroeconomic stability: Evidence and some theory. Quarterly Journal of Economics 115:147–80.
Cogley, Timothy, and Thomas Sargent. 2001. Evolving post–World War II U.S. inflation dynamics. NBER macroeconomics annual 2001, ed. Ben S. Bernanke and Kenneth Rogoff, 331–73. Cambridge: MIT Press.
———. 2002. Drifts and volatilities: Monetary policies and outcomes in the postWWII U.S. University of California, Davis, Department of Economics, and New York University, Department of Economics. Manuscript, August.
Dixit, Avinash K., and Joseph E. Stiglitz. 1977. Monopolistic competition and optimum product diversity. American Economic Review 67:297–308.
Edge, Rochelle. 2000. Timetobuild, timetoplan, habitpersistence, and the liquidity effect. International Finance Discussion Paper no. 2000–673. Washington, D.C.
Erceg, Christopher J., Dale W. Henderson, and Andrew T. Levin. 2000. Optimal monetary policy with staggered wage and price contracts. Journal of Monetary Economics 46:281–313.
Friedman, Milton. 1969. The optimum quantity of money. In The optimum quantity of money and other essays. Chicago: Aldine.
Fuhrer, Jeffrey C. 2000. Habit formation in consumption and its implications for monetarypolicy models. American Economic Review 90 (3): 367–90.
Fuhrer, Jeffrey C., and Geoffrey R. Moore. 1995. Inflation persistence. Quarterly Journal of Economics 110 (1): 127–59.
Galí, Jordi, and Mark Gertler. 1999. Inflation dynamics: A structural econometric analysis. Journal of Monetary Economics 44:195–222.
Giannoni, Marc P. 2000. Optimal interestrate rules in a forwardlooking model, and inflation stabilization versus pricelevel stabilization. Princeton University, Department of Economics. Unpublished manuscript.
———. 2001. Robust optimal monetary policy in a forwardlooking model with parameter and shock uncertainty. Federal Reserve Bank of New York. Unpublished manuscript.
———. 2002. Does model uncertainty justify caution? Robust optimal monetary policy in a forwardlooking model. Macroeconomic Dynamics 6 (1): 111–44.
Giannoni, Marc P., and Michael Woodford. 2002a. Optimal interestrate rules: I. (p.162) General theory. NBER Working Paper no. 9419. Cambridge, Mass.: National Bureau of Economic Research, December.
———. 2002b. Optimal interestrate rules: II. Applications. NBER Working Paper no. 9420. Cambridge, Mass.: National Bureau of Economic Research, December.
———. 2003. How forwardlooking is optimal monetary policy? Journal of Money, Credit, and Banking 35 (6): 1425–69 (part II).
Goodfriend, Marvin. 1991. Interest rate smoothing in the conduct of monetary policy. CarnegieRochester Conference Series on Public Policy 34:7–30.
Judd, John F., and Glenn D. Rudebusch. 1998. Taylor's rule and the Fed: 1970–1997. Federal Reserve Bank of San Francisco Economic Review 1998 (3): 3–16.
Kilian, Lutz. 1998. Smallsample confidence intervals for impulse response functions. Review of Economics and Statistics 80 (2): 218–30.
King, Mervyn. 1997. Changes in UK monetary policy: Rules and discretion in practice. Journal of Monetary Economics 39:81–97.
McCallum, Bennett T., and Edward Nelson. 1999. Performance of operational policy rules in an estimated semiclassical structural model. In Monetary policy rules, ed. J. B. Taylor, 15–56. Chicago: University of Chicago Press.
Roberts, John M. 1995. New Keynesian economics and the Phillips curve. Journal of Money, Credit and Banking 27 (4): 975–84.
Rotemberg, Julio J., and Michael Woodford. 1997. An optimizationbased econometric framework for the evaluation of monetary policy. In NBER macroeconomics annual, ed. Ben S. Bernanke and Kenneth Rogoff, 297–346. Cambridge: MIT Press.
———. 1999. Interestrate rules in an estimated stickyprice model. In Monetary policy rules, ed. J. B. Taylor, 57–119. Chicago: University of Chicago Press.
Smets, Frank, and Raf Wouters. 2002. Sources of business cycle fluctuations in the U.S.: A Bayesian DSGE approach. Seminar presentation. Princeton University, 1 November.
Steinsson, Jon. 2003. Optimal monetary policy in an economy with inflation persistence. Journal of Monetary Economics 50:1425–56.
Svensson, Lars E. O. 1999. Inflation targeting as a monetary policy rule. Journal of Monetary Economics 43:607–54.
Svensson, Lars E. O., and Michael Woodford. 2003. Indicator variables for optimal policy. Journal of Monetary Economics 50:691–720.
———. 2004. Indicator variables for optimal policy under asymmetric information. Journal of Economic Dynamics and Control 28:661–90.
Taylor, John B. 1993. Discretion versus policy rules in practice. CarnegieRochester Conference Series on Public Policy 39:195–214.
———, ed. 1999. Monetary policy rules. Chicago: University of Chicago Press.
Vickers, John. 1998. Inflation targeting in practice: The UK experience. Bank of England Quarterly Bulletin 38:368–75.
Woodford, Michael. 1998. Doing without money: Controlling inflation in a postmonetary world. Review of Economic Dynamics 1:173–219.
———. 1999a. Commentary: How should monetary policy be conducted in an era of price stability? In New challenges for monetary policy, 277–316. Kansas City, Mo.: Federal Reserve Bank of Kansas City.
———. 1999b. Optimal monetary policy inertia. NBER Working Paper no. 7261. Cambridge, Mass.: National Bureau of Economic Research, August.
———. 2003. Interest and prices: Foundations of a theory of monetary policy. Princeton, N.J.: Princeton University Press.
General Remarks
The contribution to this volume by Marc Giannoni and Michael Woodford is of a very important, wideranging, and innovative nature. Building on their earlier work,^{1} the authors estimate a New Keynesian model featuring a significant amount of dynamics from the specification of preferences, the indexation structure, and wage stickiness. They derive the aggregate welfare function that is consistent with their model and compare actual U.S. inflation outcomes with the recommended inflation path that arises from the model.
Giannoni and Woodford's main finding is that the optimal inflation target—that is, the periodbyperiod rate of quarterly inflation consistent with the maximum attainable value of households' intertemporal utility function—varies sharply over time, with values ranging over several percent under several different assumptions about the shock processes. And this finding occurs despite seemingly small weights on terms beside inflation variability in the model's welfare function.
Giannoni and Woodford's finding illustrates that tradeoffs between inflation stability and other policy goals matter very much in their model. These tradeoffs come from the generalizations that the authors contemplate of a basic New Keynesian model with price stickiness and few sources of intrinsic dynamics. As Woodford (2003, chap. 6) has shown, this basic model provides a rationalization for an approximately quadratic loss function that penalizes variations in inflation and in output relative to potential—Giannoni and Woodford's initial loss function (3). Giannoni and Woodford's modifications to the basic model, suggested by other recent work with optimizing models for monetary policy, include habit formation in preferences over consumption, nominal wage stickiness, automatic indexation of nominal wages and prices, and a timevarying wedge between the socially desirable output level and the level of potential output. In keeping with results reported in other papers, each modification has an effect on the implied social welfare function. Habit formation puts volatility in the quasidifference, rather than in the level, of the output gap into the period loss function; wage stickiness makes nominal wage growth variability welfarerelevant; indexation means that fluctuations in the quasidifference of inflation and wage growth, rather than the level fluctuations, matter for (p.164) utility; and distortions to potential GDP have the effect of making some deviations in output relative to potential optimal.
Not all these modifications to the social welfare function produce sizable fluctuations in the optimal inflation target. As Giannoni and Woodford note (section 3.1.5), habit formation by itself has no material effect on the firstorder condition for optimal policy, because the habitformation parameters cancel after substitution. Indexation of nominal prices does have a substantial effect, because—with the priceindexation parameter estimated to be at the boundary of its parameter space—the period loss function now penalizes the variability of the first difference of price inflation rather than the level of inflation. But, as Giannoni and Woodford observe, this actually takes the policy implications of the model closer to inflation targeting in practice, since the model then recommends that the price level be permitted to exhibit nonstationary behavior.
These considerations suggest that the principal sources of the time variation in the optimal inflation target come from (a) nominal wage stickiness (including wage indexation), and the consequent tradeoff between inflation and nominal wagegrowth volatility; and (b) the tradeoff between stabilizing inflation and stabilizing the welfarerelevant output gap.
I will suggest in the remainder of this comment that these two factors imply a less severe tradeoff than the authors' findings suggest. This more benign outlook arises from considering two separate issues. The first issue involves a free lunch: I will suggest that observed nominal wage rigidity may give an exaggerated picture of the tradeoff faced by welfaremaximizing policymakers. Consequently, there is not a compelling reason to give up stability in price inflation in order to moderate wagegrowth volatility. The second issue does not imply a free lunch: given the setting of other policy instruments, greater stability of price inflation may well, as the authors suggest, come at a cost of larger swings of output around its socially desirable level. Nevertheless, I argue that a strategy of targeting price inflation alone may be a sensible one for a central bank even in the presence of this tradeoff.
The TradeOff between Inflation and WageGrowth Volatility
The possibility that stickiness in nominal wages creates a distortion, for which the monetary policy remedy is to stabilize nominal wage fluctuations, was recognized explicitly by Friedman (1967, n. 11) and formalized in a dynamic general equilibrium context by Erceg, Henderson, and Levin (2000). The latter paper established that, in conditions of staggered contracts for both nominal prices and nominal wages, optimal monetary policy involves a tradeoff between stabilization of price inflation and nominal wage growth. Thus, wage stickiness considerably complicates the welfaremaximization problem for monetary policy. It is therefore vital to establish (p.165) that the nominal wage behavior observed in practice corresponds to the kind of wage stickiness that monetary policy should be concerned about.
Giannoni and Woodford's parameter estimates are consistent with wage and not just price stickiness being empirically important. These estimates are obtained by achieving as close a match as possible for the model with vector autoregression impulse responses to a monetary policy shock in the U.S. data, for four variables including the real wage. This procedure could exaggerate the tradeoff situation in an environment that, following Mankiw (1987) and Goodfriend and King (2001), I consider to be empirically relevant: namely, one where observed nominal wages display considerable rigidity, yet output and inflation behavior are consistent with a stickyprice, flexiblewage model.
To be concrete, let us neglect the complications of decision lags, indexation, and habit formation, and consider simply the basic stickyprice New Keynesian model considered early in Giannoni and Woodford's paper—namely, their equations (1) and (15). In this standard stickyprice model, the predicted behavior of real unit labor costs is that they are proportional to the output gap:
With a production function of the form y_{t} = αn_{t} + α_{t} (α_{t} being a technology shock), this relationship implies that real wages are a function of current output
In the alternative to this baseline that I want to entertain, observed real wages are a noisy mixture of their lagged value and output:
One rationalization for equation (3) is that, as conjectured by Mankiw and GoodfriendKing, observed wages are not a satisfactory indicator of current labormarket conditions. Equation (3) can also be viewed as holding in a more general situation in which, although inflation is driven by real marginal cost, measurement error in wages renders observed real unit labor costs an imperfect and sluggish indicator of true marginal cost, where the latter is strictly proportional to the output gap. The positive weight on current output^{2} in equation (3) delivers the property that real unit labor costs and true marginal cost are positively correlated, and so it is consistent (p.166)
In a stickyprice model, completed by a standard Taylor rule with smoothing,^{3} the output, inflation, and nominal interest rate responses to a monetary policy shock will necessarily be the same whether wage equation (1) or (3) is used, as figure 3C.1 shows.^{4} But the figure also confirms that if observed real labor costs follow equation (3), real wages will exhibit an inertial response to monetary policy shocks that will appear to confirm the importance of wage stickiness. It is important to bear in mind that the realwage response constitutes approximately onequarter of the criterion function that determines Giannoni and Woodford's parameter estimates. Once the wagestickiness parameter is estimated to be sizable, the volatility of nominal wage growth (or of its first difference) enters Giannoni and Woodford's utilitybased welfare function and the associated tradeoff calculations. But in the example given here, the inertial observed realwage response to the policy shock is solely responsible for the estimated wage stickiness; output and inflation responses are consistent with a purely stickyprice story, and wage variability does not appear in the true social welfare function.
Nor, under certain conditions, do departures of actual output and inflation (p.167) responses from the patterns predicted by a flexiblewage, stickyprice model imply that wage stickiness is of genuine policy significance. Consider a situation where real wages follow a pattern like equation (3), while real wages enter the monetary policy reaction function as they do in Giannoni and Woodford's equation (57). Then matching the persistence of observed real wages will be important in obtaining a good fit of the model with the data responses for the interest rate, output, and inflation. But this improvement in fit will not reflect any importance of wage stickiness in the structure of the Phillips curve or the welfare function; the improved fit comes in effect from a better match to the policy rule in force during the estimation period.
An initial glance at the authors' table 3.3 might lead to the conclusion that the hypothesis that prices are the sole source of nominal stickiness is strongly rejected. For example, the flexiblewage restriction does more damage to the objective function value than other restrictions that are contemplated in the table, such as the nohabit and noindexation settings. This finding does amount to a strong rejection of the joint hypothesis of sticky prices and of wages being allocative and observed without error. But it is not necessarily inconsistent with the hypothesis that prices are the sole source of nominal rigidity that are relevant for inflation dynamics and welfare, with observed nominal wage rigidity reflecting a measurement error of the form implicit in equation (3). Under the latter hypothesis, improvements in fit from modeling wages as sticky are to be expected, for the reasons outlined above. And there is an important feature of the GiannoniWoodford estimates that seems to me highly consistent with the hypothesis that price stickiness is the only inflationrelevant and welfarerelevant form of nominal rigidity. This feature is that several key parameters, notably ω (governing the interest elasticity of aggregate demand), η (the habit formation parameter), and κ_{p} (the elasticity of inflation with respect to the output gap), are virtually identical to their unrestricted values when the restriction of flexible wages is imposed (see the “implied parameters” in the final column of Giannoni and Woodford's table 3.3). If wage stickiness played a decisive role in the structure of the model, and so in the intrinsic dynamics of output and inflation behavior, one would expect these parameter estimates to be highly sensitive to the assumption made about wages. In fact, they do not exhibit such a sensitivity, which lends support to the proposition that separate terms involving real wages or wage growth are not required in the Phillips curve or in the social welfare function. The behavior of nominal wages, in other words, does not justify trading off stability in price inflation against stability in nominal wage growth.
The TradeOff between Inflation and OutputGap Volatility
If the tradeoff between stability in inflation and nominal wage growth is put aside, the key tradeoff in Giannoni and Woodford's model is between (p.168) variability in inflation and variability in output relative to its socially desirable value. This real deviation, denoted x_{t}, is labeled the “output gap” in the paper, but it is important to stress that this is a slightly different outputgap concept from that used in Phillipscurve analysis. The latter, which I will call the “inflationrelevant output gap,” corresponds to x_{t} + u_{t} in Giannoni and Woodford's paper, and represents the percentage difference between (detrended levels of) output and potential output. With (detrended log) potential output (i.e., the level of output prevailing in the absence of any nominal rigidity) denoted by , the following relationship holds between output concepts, the inflationrelevant output gap x_{t} + u_{t}, and what Giannoni and Woodford call the “welfarerelevant output gap,” x_{t}:
One of the key conditions describing optimal policy in Giannoni and Woodford's setup is that projections of endogenous variables under optimal policy satisfy their equation (77), reproduced here:
if, as argued in the previous section, the term involving wages in this condition can be ignored, the expression can be alternatively written as
With
(p.169) And with the Phillips curve (omitting wage terms) implying the relationship , condition (7) may be cast as
Let us consider the merits of making such a modification to the objective function. It would represent a departure from the spirit of assigning to monetary policy the goal of maximizing social welfare. But it would be a departure only in a limited sense: by instructing the central bank that it should not engineer deviations of GDP from potential, even when the latter is distorted relative to the social optimum.
In Rotemberg and Woodford (1997) and in many of the cases in Woodford (2003), fluctuations in are efficient, but the steadystate level of potential output is generally inefficient. These studies assume that a subsidy is provided by the government that eliminates the inefficiency of the steadystate potential output level. While this subsidy is often treated as a convenient assumption, it can be given a normative interpretation as reflecting an optimal assignment of policy instruments. Indeed, Rotemberg and Woodford (1998, 52) offered just such a normative interpretation, arguing that “monetary policy is not an appropriate instrument with which to seek to affect the longrun average level of real economic activity, given the existence of other instruments.” This perspective is closely related to the resolution of the inflationbias issue proposed by King (1996, 61), whereby “the central bank does not use monetary policy as a substitute for microeconomic structural reforms” and is embedded in Svensson's (1999) “flexible inflation targeting” concept. So far, this argument applies to the mean level of output, but there is a clear dynamic analogue to this policy prescription. Specifically, for stabilization policy, the above principle entails trying to limit variability in the inflationrelevant output gap, x_{t} + u_{t}, rather than variation in x alone.
From such a perspective, monetary policy is a natural instrument for eliminating the real distortions (i.e., deviations of y_{t} from ) that arise (p.170) from nominal stickiness, and for pursuing a mean inflation rate that is consistent with insulating the economy from the most serious effects of violations of superneutrality (e.g., downward pressure on potential output arising from the interaction of high inflation and nonindexed tax scales). But the achievement of minimum price inflation and gap variability is conditional on the real shocks and on the steadystate magnitudes that determine the flexibleprice values of output and other real variables. Such an arrangement amounts to a prescription for a “neutral” monetary policy, in the terminology of Goodfriend and King (1997). Movement of output closer to its social optimum is then the task of other policy instruments, which achieve this aim through policies designed to reduce the variance of u_{t} to zero. If this view about instrument delegation is accepted, the tradeoff problem of monetary policy is eased. There is no conflict between minimizing variability in inflation and in the inflationrelevant output gap, and so the sharp fluctuations in the inflation target exhibited in Giannoni and Woodford's section 3.3.3 are no longer called for.
Conclusions
As I noted at the outset, Giannoni and Woodford have produced an important, wideranging, and innovative paper. Its findings on the optimal inflation target for the United States will be a benchmark for future work, and applications to inflationtargeting countries can provide a welfare evaluation of the constant inflation targets typically followed in practice. In my comments here, I have argued that the tradeoff problem—and so the sources of desirable variations in inflation—may be exaggerated by Giannoni and Woodford's emphasis on the importance of wage stickiness for inflation dynamics and optimal policy, and by their assigning to monetary policymakers the duty of offsetting inefficiencies arising from tax and competition arrangements. If, as I have suggested, the nominal wage rigidity observed in practice does not have welfare consequences, and the outputgap concepts that appear in the Phillips curve and in monetary policymakers' objective function coincide, then there is a stronger case for a constant inflation target.
References
Bibliography references:
Boivin, Jean, and Marc P. Giannoni. 2003. Has monetary policy become more effective? NBER Working Paper no. 9459. Cambridge, Mass.: National Bureau of Economic Research.
Erceg, Christopher J., Dale W. Henderson, and Andrew T. Levin. 2000. Optimal monetary policy with staggered wage and price contracts. Journal of Monetary Economics 46:281–313.
Friedman, Milton. 1967. The monetary theory and policy of Henry Simons. Journal of Law and Economics 10:1–13.
Giannoni, Marc P. 2000. Optimal interestrate rules in a forwardlooking model, (p.171) and inflation stabilization versus pricelevel stabilization. Princeton University, Department of Economics. Unpublished manuscript.
Giannoni, Marc P., and Michael Woodford. 2002a. Optimal interestrate rules: I. General theory. NBER Working Paper no. 9419. Cambridge, Mass.: National Bureau of Economic Research.
———. 2002b. Optimal interestrate rules: II. Applications. NBER Working Paper no. 9420. Cambridge, Mass.: National Bureau of Economic Research.
Goodfriend, Marvin, and Robert G. King. 1997. The new neoclassical synthesis and the role of monetary policy. In NBER macroeconomics annual 12, ed. Ben S. Bernanke and Julio J. Rotemberg, 231–83. Cambridge: MIT Press.
———. 2001. The case for price stability. NBER Working Paper no. 8423. Cambridge, Mass.: National Bureau of Economic Research.
King, Mervyn A. 1996. How should central banks reduce inflation?—Conceptual issues. In Achieving price stability, 53–91. Kansas City, Mo.: Federal Reserve Bank of Kansas City.
Mankiw, N. Gregory. 1987. Comment on “The new Keynesian microfoundations.” In NBER macroeconomics annual 2, ed. Stanley Fischer, 105–10. Cambridge: MIT Press.
Rotemberg, Julio J., and Michael Woodford. 1997. An optimizationbased econometric framework for the evaluation of monetary policy. In NBER macroeconomics annual 12, ed. Ben S. Bernanke and Julio J. Rotemberg, 297–346. Cambridge: MIT Press.
———. 1998. An optimizationbased econometric framework for the evaluation of monetary policy: Expanded version. NBER Technical Working Paper no. 233. Cambridge, Mass.: National Bureau of Economic Research.
Rudebusch, Glenn D. 2002. Term structure evidence on interest rate smoothing and monetary policy inertia. Journal of Monetary Economics 49:1161–87.
Sbordone, Argia M. 2002. Prices and unit labor costs: A new test of price stickiness. Journal of Monetary Economics 49:265–92.
Svensson, Lars E. O. 1999. Inflation targeting as a monetary policy rule. Journal of Monetary Economics 43:607–54.
Woodford, Michael. 2003. Interest and prices: Foundations of a theory of monetary policy. Princeton, N.J.: Princeton University Press.
Discussion Summary
Olivier Blanchard commended the paper's concern for performing the analysis within a model that had plausible empirical properties, but expressed skepticism about several features of the model introduced to achieve better empirical performance, notably the use of indexation in price setting and of habit formation in consumption. If these features were indeed a wrong representation of the structure of the economy, then the welfare analysis based on the model would be misleading.
Glenn Rudebusch suggested that the distinction between the new classical and New Keynesian Phillips curve, namely the timing of inflation expectations, might not be an important issue. Although the model used in the analysis was parameterized as a quarterly model, the relevant expectation (p.172) was probably a fourquarterahead expectations of inflation, which was not very sensitive to the timing.
Marvin Goodfriend expressed concern about the degree of inflation control that the model assumed the central bank had. This feature relied on the assumption that the public was able to observe all shocks with precision. It would be important to account in the analysis for the possibility that the public might mistake movements in observed inflation for a change in the central bank's inflation target.
Donald Kohn questioned the feasibility of the central bank's committing one quarter ahead to an inflation target and not altering the previously announced target for the current period regardless of the nature of incoming information.
Laurence Ball expressed skepticism about the strategy of refining optimizationbased models by including frictions that would bring them closer to the data. In the case of the Phillips curve, the specification used in the present paper would imply still counterfactual inflation volatility, while the inflation inertia implied by the presence of lagged inflation would make the model vulnerable to the Lucas critique.
Marc Giannoni responded by stressing that the target criterion remains remarkably similar to the one of the basic model when additional features such as habit formation, inflation indexation, and wages stickiness are introduced into the model. Moreover, adding more structure to the model would not necessarily complicate the target criterion as long as the number of variables in the target criterion does not increase, facilitating communication with the public. He also emphasized the robustness property of the proposed rule with respect to the sources and processes of the stochastic shocks.
In response to the skepticism expressed about the model, Michael Woodford emphasized that the paper did not attempt to recommend a specific rule but rather attempted to provide a disciplined way of translating assumptions about the structure of the economy into a target criterion, and thus into prescriptions of what the policy setting should be.
Notes:
Marc P. Giannoni is an assistant professor of finance and economics at Columbia Business School and a faculty research fellow of the National Bureau of Economic Research (NBER). Michael Woodford is professor of economics at Columbia University and a research associate of NBER.
We would like to thank Jean Boivin, Rick Mishkin, Ed Nelson, and Lars Svensson for helpful discussions, Brad Strum for research assistance, and the National Science Foundation for research support through a grant to the NBER.
Edward Nelson is a research officer at the Federal Reserve Bank of St. Louis.
The views expressed here should not be interpreted as those of the Bank of England or the Monetary Policy Committee.
(1.) Svensson discusses two alternative specifications of an inflationtargeting policy rule, one of which (a “general targeting rule”) involves specification of a loss function that the central bank should use to evaluate alternative paths for the economy, and the other of which (a “specific targeting rule”) involves specification of a target criterion. We are here concerned solely with policy prescriptions of the latter sort. On the implementation of optimal policy through a “general targeting rule,” see Svensson and Woodford (chap. 2 in this volume).
(1.) For example, Boivin and Giannoni (2003), Giannoni and Woodford (2002a, Rotemberg and Woodford (1997), and Woodford (2003).
(2.) Technically, the statecontingent evolution that is implemented by commitment to the policy rule is optimal from a “timeless perspective” of the kind proposed in Woodford (1999a), which means that it would have been chosen as part of an optimal commitment at a date sufficiently far in the past for the policymaker to fully internalize the implications of the anticipation of the specified policy actions, as well as their effects at the time that they are taken. This modification of the concept of optimality typically used in Ramseystyle analyses of optimal policy commitments allows a timeinvariant policy rule to be judged optimal and eliminates the time inconsistency of optimal policy. See Giannoni and Woodford (2002a) and Svensson and Woodford (chap. 2 in this volume) for further discussion.
(2.) Actually this weight is on the current output gap, but potential GDP has been included in the composite disturbance e_{wt} in equation (3).
(3.) Possible sources of disturbances of this sort are discussed in Giannoni (2000), Steinsson (2003), and Woodford (2003, chap. 6).
(3.) I use the estimated specification of Rudebusch (2002, 1164):, where ρ_{R} = 0.73. This rule is shocked in period 1.
(4.) See Woodford (2003, chap. 3) for a derivation in the context of an explicit intertemporal general equilibrium model of the transmission mechanism. equation (1) represents merely a loglinear approximation to the exact equilibrium relation between inflation and output implied by this pricing model; however, under circumstances discussed in Woodford (2003, chap. 6), such an approximation suffices for a loglinear approximate characterization of the optimal responses of inflation and output to small enough disturbances. Similar remarks apply to the other loglinear models presented below.
(4.) I set the interest elasticity of aggregate demand to 0.20 and the outputgap coefficient in the Phillips curve to 0.10. Equations (1) and (3) are parameterized as follows: h = 2.2, μ = 0.9.
(5.) See Woodford (2003, chaps. 3 and 6) for discussion of how this variable responds to a variety of types of real disturbances. Under conditions discussed in chapter 6, the “natural rate” referred to here corresponds to the equilibrium level of output in the case that all wages and prices were completely flexible. However, our results in this section apply to a broader class of model specifications, under an appropriate definition of the “output gap.”
(6.) Note that the aggregatedemand side of the model does not matter, as long as a nominal interest rate path exists that is consistent with any inflation and output paths that may be selected. This is true if, for example, the relation between interest rates and private expenditure is of the form of equation (15) assumed below, and the required path of nominal interest rates is always nonnegative. We assume here that the nonnegativity constraint never binds, which will be true, under the assumptions of the model, in the case of any small enough real disturbances .
(7.) Obtaining a unique solution requires the specification of an initial value for the Lagrange multiplier See Woodford (2003, chap. 7) for the discussion of alternative possible choices of this initial condition and their significance. Here we note simply that regardless of the value chosen for , the optimal responses to costpush shocks in period t_{0} and later are the same.
(8.) These parameter values are based on the estimates of Rotemberg and Woodford (1997) for a slightly more complex variant of the model used here and in section 3.1.3 .The coefficient λ here corresponds to λ_{x} in the table. Note also that the value of.003 for that coefficient refers to a loss function in which π_{t} represents the quarterly change in the log price level. If we write the loss function in terms of an annualized inflation rate, 4π_{t}, as is conventional in numerical work, then the relative weight on the outputgap stabilization term would actually be 16λ_{x}, or about .048. Of course, this is still quite low compared the relative weights often assumed in the ad hoc stabilization objectives used in the literature on the evaluation of monetary policy rules.
(9.) See Woodford (2003, chap. 7) for derivation of this “optimal noninertial plan.” In the example shown in figure 3.1, this optimal noninertial policy corresponds to the Markov equilibrium resulting from discretionary optimization by the central bank. That equivalence would not obtain, however, in the case of serially correlated disturbances.
(10.) The characteristic equation that determines whether the system of equations consisting of (1) and (8) has a unique nonexplosive solution is the same as for the system of equations solved above for the optimal statecontingent evolution.
(12.) See Woodford (2003, chap. 6) for derivation of this loss function as an approximation to expected utility.
(13.) Note that the impulse response of inflation (for γ=1) in panel A of figure 3.2 is the same as the impulse response of the price level (under optimal policy) in panel C of figure 3.1. The scales are different because the inflation rate plotted is an annualized rate, 4π_{t}, rather than π_{t}.
(14.) Any such policy rule is also optimal from a timeless perspective, under the definition given in Giannoni and Woodford (2002a). Note that alternative rules that result in equilibria that differ only in a transitory, deterministic component of the path of each of the target variables can each be considered optimal in this sense. This ambiguity as to the initial behavior of the target variables cannot be resolved if our concept of optimal policy is to be time consistent. In the present case, ambiguity about the required initial behavior of the target variable, inflation acceleration, implies ambiguity about the required longrun average level of the inflation rate, although there is no ambiguity about how inflation should respond to shocks.
(15.) The real disturbances that cause the natural rate of interest to vary are assumed to create no variation in the costpush term u_{t}; that is, they shift the equilibrium relation between inflation and output only through possible shifts in the natural rate of output. A variety of examples of real disturbances with this property are discussed in Woodford (2003, chap. 6).
(18.) If we write the target criterion in terms of a forecast for the annualized inflation rate (4π_{t}), the relative weight on the outputgap forecast will instead be 4φ, or about .15.
(19.) One easily sees that in the case that γ = 1, the only longrun average inflation rate consistent with these conditions is , where is the unconditional mean of the natural rate of interest. This is true for any λ_{i} 〉 0, no matter how small. Hence, even a slight preference for lower interest rate variability suffices to break the indeterminacy of the optimal longrun inflation target obtained for the case γ = 1 in section 1.2.
(23.) For further discussion of these coefficients, and explicit formulas for them in terms of the frequency of wage and price adjustment, see section 3.2 below.
(24.) See the discussion in Woodford (2003, chap. 3). In this case, the structural equations (29)–(30) imply that the real wage will be unaffected by monetary policy, instead evolving as a function of the real disturbances alone. Empirical studies often find that the estimated response of the real wage to an identified monetary policy shock is quite weak, and not significantly different from zero. Indeed, it is not significantly different from zero in our own analysis in section 3.2, although the point estimates for the impulse response function suggest that wages are not as sticky as prices.
(25.) Here we assume a normalization of the loss function weights in equation (33) in which λ_{p} + λ_{w} = 1, corresponding to the normalization in equation (3).
(26.) This only affects the optimal target criterion, of course, to the extent that the evolution of the real wage is endogenous, which requires that κ_{w} ≠ κ_{p}.
(27.) See also Woodford (2003, chap. 6), which modifies the derivation of Erceg, Henderson, and Levin to take account of the discounting of utility.
(28.) See McCallum and Nelson (1999) and Woodford (2003, chap. 4) for further discussion of this simplification.
(30.) Note that the consumption “habit” is assumed here to depend on the household's own past level of expenditure and not on that of other households.
(31.) In the limiting case in which η = 0, δ = 0, while δ/η approaches the welldefined limit φ(ω + φ), so that κ = ξ_{P}(ω + φ) = ξ_{P}(ω + σ^{−1}). Thus in this limit equation (47) reduces to equation (1), where κ is defined as in equation (32).
(32.) For details of the calculation, see the derivation in the appendix for the full model, incorporating habit persistence, that is introduced in section 3.2.
(33.) Specifically, is the log of real GDP minus a linear trend. Inflation is computed as the quarterly growth of the GDP deflator (chaintype), annualized. The interest rate i_{t} is the quarterly average of the federal funds rate, annualized. The real wage is the log of wages and salaries in the compensation of employees published by the Bureau of Economic Analysis, divided by the GDP deflator; a linear trend is then subtracted from the log real wage to obtain .
(34.) Some studies suggest that monetary policy has changed again around the mid1980s. However, Boivin and Giannoni (2003), following the approach proposed by Bernanke, Boivin, and Eliasz (2004), show that impulse response functions to monetary policy disturbances in a factoraugmented VAR are similar to the ones reported here, when estimated on both the 1980–2002 and 1984–2002 sample periods.
(35.) Because the problem is the same for each household h (the initial level of wealth is assumed to differ for any two households in a way that compensates for any difference in their expected labor incomes, and complete financial markets allow complete pooling of idiosyncratic labor income risk thereafter), all households choose identical statecontingent plans for consumption.
(36.) Up to the loglinear approximation used in our estimation of the model, defined in this way is just the conditional expectation at date t − 1 of the log deviation of the equilibrium level of output when none of these variables are predetermined at all. Because wages and prices are both predetermined a period in advance, it is only the component of the output gap that is forecastable a period in advance that matters in any event for these equations. It is similarly only the variation in the forecastable component of the output gap that need be considered when evaluating welfare under alternative policies, since the unforecastable component of the output gap (defined relative to a concept of the “natural rate” that is not predetermined) would in any event be both exogenous and uncorrelated with the forecastable component. It then simplifies notation to define the output gap as the gap between actual output and the forecastable component of the natural rate. In this way, x_{t} becomes a predetermined state variable.
(37.) The use of the inverse of the complete variancecovariance matrix of impulse responses as a weighting matrix would be more attractive, as this would yield efficient estimates. But such a weighting matrix appears to hinder the stability of the minimization algorithm. The matrix that we propose has the advantage of reducing the weight on responses about which we are less sure, in addition to making our results independent of the units in which we happen to measure the various series.
(38.) Here ψ is estimated to be significantly positive, implying a significant effect of interest rates on aggregate demand, while the corresponding standard error for an estimate of 9 would not allow us to judge that the latter coefficient was significantly positive. Similarly, ῆ is estimated to be significantly positive, implying habit persistence, even though the corresponding standard error for the estimated value of η is much greater than one.
(39.) The parameter ψ is called by Boivin and Giannoni (2003) the “pseudoelasticity of substitution”; it measures the elasticity of expected output growth with respect to changes in the expected real rate of return, holding constant output growth in other periods.
(40.) A Matlab program, available on our webpages, allows readers to check the extent to which our numerical characterization of optimal policy would be different in the case of alternative parameter values.
(41.) The restricted model considered in column (4) corresponds to the model of Boivin and Giannoni, although their method of estimation is different in that they do not fit estimated impulse responses of the real wage along with those of the other three variables, and their model assumes a different form of monetary policy rule. They also calibrate the value of ω = ω_{w} + ω_{p}, rather than only specifying a calibrated value for ω_{p}, and they assume a value of ω much smaller than our estimate. Nonetheless, the estimates for the other parameters reported in column (4) are similar to those obtained by Boivin and Giannoni, providing further evidence regarding the robustness of our conclusions here.
(42.) The implied impulse response functions are compared to the estimated ones in the case of each of the restricted models in the technical appendix to this paper (see http://www.nber.org/dataappendix/giannoni04/).
(43.) Again, see the technical appendix for details.
(44.) Rotemberg and Woodford (1997) base this calibration on Blinder's (1994) survey evidence that prices are maintained constant for an average of nine months, so that 1/(1 − α_{p}) equals three quarters.
(45.) If ξ_{p} and ξ_{w} were assigned equal values, then under our assumption of equal values for α_{p} and α_{w}, (68) and (71) would imply equal values for Ф_{p}θ_{p} and Ф_{w}φ^{−1}θ_{w} (Here we recall that Ф_{w} ≡ υφ.) The implied value of θ_{p} is then larger than Ф^{−1}θ_{w} by exactly the same factor as is larger than Ф_{p}. In fact, our estimated value for ξ_{p} is smaller than our estimate for ξ_{w}, and this further increases the relative size of the implied value of θ_{p}.
(46.) As discussed in section 3.1.3 above, it may also be desirable to reduce the variability of nominal interest rates; in this case, the loss function (72) should include an additional term, proportional to the squared deviation of the nominal interest rate from an optimal value. We do not take up this possible extension of the analysis here.
(47.) For example, even if we assume that monetary frictions are of negligible quantitative significance, we may reasonably assume that the economy is a “cashless limiting economy” of the kind discussed in Woodford (1998), rather than a genuinely cashless economy. In this case, there should in fact exist tiny monetary frictions that suffice to entail a preference for a completely forecastable nominal interest rate in the absence of any offsetting benefit from variations in response to current shocks.
(48.) See further discussion in Svensson and Woodford (chap. 2 in this volume).
(49.) The target criterion could equivalently be expressed in the form , in which case the target criterion would refer solely to projected inflation of different sorts (both price and wage inflation). This would be a representation analogous to the one given in section 3.1.4 above and would make clear that only the projected future paths of target variables (variables that enter the loss function) matter. We feel, however, that the representation proposed here allows a more convenient numerical summary of the content of the target criterion, by collecting the central bank's projections regarding the future level of nominal quantities in a single variable, the projected future price level.
(50.) As with equation (74), we could equivalently express this criterion in terms of a linear function of projections for price inflation, wage inflation, and the output gap.
(51.) Note that in the model considered here, as in section 3.1.2 when γ = 1, there is no welfare significance to any absolute inflation rate, only to changes in the rate of inflation and to wage growth relative to prices. There is therefore no particular inflation rate that could be justified as optimal from a timeless perspective. For purposes of comparison between historical policy and the optimal criterion, discussed below, we assume that steadystate inflation and the steadystate real wage are equal to the longrun values estimated (by the VAR) under historical policy.
(52.) We suppress the details of this alternative optimal targeting rule here. The firstdifferenced formulation is the one described in Woodford (2003, chap. 8). The calculations reported there are further explained in a note that is available on our web pages.
(53.) Because our empirical model is quarterly, it is simplest to discuss the policy process as if a policy decision is also made once per quarter, even though in reality most central banks reconsider their operating targets for overnight interest rates somewhat more frequently than this. Our discussion should not be taken to imply that it is optimal for the policy committee to meet only once per quarter; this would follow from our analysis only if (as in our model) all other markets were also open only once per quarter.
(54.) See Svensson and Woodford (chap. 2 in this volume) for further discussion of the sort of calculations involved in a forecasttargeting decision procedure.
(55.) Here and below, we present the coefficients for a target criterion where the inflation rate is measured in annualized percentage points, rather than as a quarterly rate of change as in the model of section 3.2. When the variables are defined as in the model, the coefficients multiplying the realwage and outputgap terms are only onequarter as large as those given here and below.
(56.) Note that, here and below, the “quartert information set” is taken to include π_{t+1}, w_{t+1}, and Ŷ_{t+1} as well as all variables dated t or earlier, on the ground that prices, wages, and output are all predetermined variables according to our model. See Rotemberg and Woodford (1997) for further discussion.
(57.) Of course, we are judging the forecastability of the funds rate using a VAR that has been fitted to this data set, rather than considering the outofsample forecasting ability of a regression model estimated using only data prior to the quarter for which the funds rate is being forecasted. We are also including variables in the quartert information set the values of which are not announced in quarter t (indeed, not even during quarter t + 1, although the measurements are made during that quarter), which also exaggerates the information actually available in quarter t. But it must also be recognized that decision makers have access to a great deal of information in quarter t that is not included in our data set and that might well allow better forecasting of the funds rate than is possible on the basis of only the variables included in our VAR.
(58.) We have not attempted to quantify the share since the two components are not orthogonal.
(59.) To be precise, it corresponds to the component of this variable that is forecastable a quarter in advance. This is all that can be reconstructed from the paths of the endogenous variables, given that wages, prices, and output are all predetermined according to our model, but this is also what is relevant for the construction of the variable x_{t} that appears in our loss function (72) and, hence, the target criterion stated in the previous section.
(60.) See Giannoni (2000) or Woodford (2003, chap. 6) for further discussion in the context of simpler models.
(61.) This view is implicit in the outputgap measures commonly used in the literature on empirical centralbank reaction functions.
(62.) In the figure, the solid line is a twosided moving average of the dashed line, equal to 1/3 the discrepancy in that quarter, 2/9 of the discrepancy in both the preceding and following quarters, and 1/9 of the discrepancy both two quarters earlier and two quarters later.
(63.) The fact that our model does relatively poorly at matching the dynamics of the estimated response of inflation, as shown in figure 3.6, does not give us much confidence in this regard.
(64.) See Svensson and Woodford (chap. 2 in this volume) for further discussion of what this would mean in practice.
(66.) The fact that a real GDP projection is always included with the projection for RPIX inflation in the introduction to the Bank of England's Inflation Report—and in fact is always discussed first—suggests that some attention is paid to the projected path of output in deciding upon the appropriateness of the current level of interest rates. But the Bank's official target criterion, involving only the constantinterest rate projection of RPIX inflation at the eightquarter horizon, does not make explicit the way in which the output projection should be taken into account.