Michael Woodford

Princeton University

June 1999

Revised September 2002

Preliminary and Incomplete

c Michael Woodford 2002

Contents

6 In¬‚ation Stabilization and Welfare 1

1 Approximation of Loss Functions and of Optimal Policies . . . . . . . . . . . 4

2 A Utility-Based Welfare Criterion . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Output-Gap Stability and Welfare . . . . . . . . . . . . . . . . . . . . 13

2.2 In¬‚ation and Relative-Price Distortions . . . . . . . . . . . . . . . . . 18

3 The Case for Price Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 The Case of an E¬cient Natural Rate of Output . . . . . . . . . . . . 26

3.2 Consequences of an Ine¬cient Natural Rate of Output . . . . . . . . 31

3.3 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Extensions of the Basic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Transactions Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 The Zero Interest-Rate Lower Bound . . . . . . . . . . . . . . . . . . 49

4.3 Asymmetric Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Sticky Wages and Prices . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Time-Varying Tax Wedges or Markups . . . . . . . . . . . . . . . . . 76

Chapter 6

In¬‚ation Stabilization and Welfare

We turn now to the evaluation of alternative monetary policy rules, using the analytical

framework developed in the previous chapters. A ¬rst question, of course, is what the goals

of monetary policy should be. Most discussions give primary, if not exclusive, attention to two

goals in terms of which alternative policies should be evaluated. The ¬rst, which has attained

particular prominence in recent discussion, is the goal of maintaining a low and stable rate

of in¬‚ation. It is sometimes argued that this should be the exclusive goal of monetary

policy, and various recent developments “ the explicit responsibility given to several central

banks in the 1990™s for achievement of in¬‚ation targets, or the exclusive concern with price

stability that is speci¬ed in the Maastricht treaty as the goal of the European Central Bank

“ indicate that this view has become in¬‚uential among policymakers.1 At the same time,

most discussions of actual monetary policy, even now, and even in countries with “in¬‚ation

targeting” central banks, assume at least some degree of concern with the stabilization of

economic activity as well; in the early literature (e.g., Poole, 1970), this was often treated

as the primary goal of monetary stabilization policy.

There is thus a fair amount of consensus in the academic literature that a desirable

monetary policy is one that achieves a low expected value of a discounted loss function,

where the losses each period are a weighted average of terms quadratic in the deviation

of in¬‚ation from a target rate and in some measure of output relative to potential. But

1

It may, however, be questioned whether the “in¬‚ation targeting” central banks should be understood to

care solely about in¬‚ation stabilization. See, e.g., Svensson (1999a).

1

2 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

even agreement upon this general form of the objective still allows considerable scope for

disagreement about details, that may well matter a great deal for the design of an optimal

policy. First of all, obviously, there is the question of the relative weight to be placed upon

in¬‚ation stabilization and output stabilization. But this is hardly the only ambiguity in the

conventional prescription. For instance, which kind of output measure should be stabilized?

In particular, should one seek to stabilize output relative to a concept of “potential output”

that varies in response to real disturbances that shift the short-run aggregate supply curve,

or should one seek to stabilize output relative to a smooth trend?2

Similarly, in which sense should price stability be pursued? Should one seek to stabilize

deviations of the price level from a deterministic target path (as proposed, for example,

by Hall and Mankiw, 1994), so that unexpected in¬‚ation in excess of one™s target rate

should subsequently be deliberately counteracted, in order to bring the price level back to

its target path? Or should one seek to stabilize deviations of the in¬‚ation rate from its

target level (as assumed, for example, by Svensson, 1997), so that “ assuming that the

variance of the unforecastable component of in¬‚ation cannot be reduced by policy “ one

should not seek to counteract past in¬‚ation ¬‚uctuations, in order to minimize variation in

the forecastable component of in¬‚ation? Should greater priority perhaps be given to reducing

the variability of unforecastable in¬‚ation, on the ground that this is what causes unexpected

modi¬cations of the real consequences of pre-existing nominal contracts, while forecastable

variations in in¬‚ation can simply be incorporated into contracts? Or should greater priority

be given to stabilization of forecastable in¬‚ation, on the ground that expected in¬‚ation

distorts incentives (like an anticipated tax), while unforecastable in¬‚ation has no incentive

e¬ects (like an unanticipated wealth levy)?

The aim of the present chapter is to show how economic analysis can be brought to bear

upon these questions. An important advantage of using a model founded upon private-sector

optimization to analyze the consequences of alternative policy rules is that there is a natural

2

Di¬erent answers to this question lead Bean (1983) and West (1986) to reach diametrically opposite

conclusions about the case in which nominal GDP targeting would be preferable to money-supply targeting.

3

welfare criterion in the context of such a model, provided by the preferences of private agents

that are displayed in the structural relations that determine the e¬ects of alternative policies.

Such a utility-based approach to welfare analysis has long been standard in the theory of

public ¬nance. It is not yet too common in analyses of monetary policy, perhaps because

it is believed that the main concerns of monetary stabilization policy are assumed away in

models with explicit micro-foundations. But we have seen that models founded on individual

optimization can be constructed that, thanks to the presence of nominal rigidities, allow for

realistic e¬ects of monetary policy upon real variables. Here we shall see those same nominal

rigidities provide welfare-economic justi¬cation for central bankers™ traditional concern for

price stability.

Individuals are not assumed, of course, to care directly about prices; their economic

welfare depends directly only upon the goods they consume and the amount of e¬ort they

expend upon production. But just as taxes can cause deadweight losses because of their ef-

fects upon the equilibrium allocation of resources, so can in¬‚ation. In a model with nominal

rigidities “ more speci¬cally, in one in which it is recognized that prices are not adjusted in

perfect synchronization with one another (which requires, but is stronger than, the observa-

tion that they are not all adjusted continually) “ instability of the general price level leads to

unnecessary and undesired variation in the relative prices of goods whose prices are adjusted

at di¬erent times. These relative price distortions result in deadweight losses, just as in the

case of distorting taxes. We shall see that this e¬ect can justify not only a loss function that

penalizes in¬‚ation variations, but indeed “ if one assumes parameter values implied by the

apparent degree of nominal rigidity in actual economies “ a much larger relative weight on

in¬‚ation variation than upon output variation than is assumed in the loss functions used in

many monetary policy evaluation exercises.

Derivation of a utility-based welfare criterion in this way can not only allow us to jus-

tify a general concern with price stability, but can furthermore provide exact answers to

the questions raised above about the precise formulation of the appropriate loss function.

These answers, of course, depend upon the assumptions we make about the structure of the

4 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

economy; for example, they depend crucially upon the nature of the nominal rigidities that

are present. Insofar as the correct structural relations of our model of the economy remain

controversial, the proper welfare criterion to use in evaluating policy will remain controver-

sial as well; and our goal here is more to illustrate a method than to reach ¬nal conclusions.

But insofar as particular parameter values are found to be empirically justi¬ed in that they

are required in order for our structural equations to ¬t historical data, they will contain

important information about the proper welfare criterion as well.

1 Approximation of Loss Functions and of Optimal

Policies

The method that we shall employ in the analysis below derives a quadratic loss function, that

represents a quadratic (second-order Taylor series) approximation to the level of expected

utility of the representative household in the rational expectations equilibrum associated

with a given policy. There are several reasons for our resort to this approximation. One is

simply mathematical convenience; with a quadratic approximation to our objective function

and linear approximations to our structural equations, we can address the nature of opti-

mal policy within a linear-quadratic optimal control framework that has been extensively

studied, and numerical computation of optimal policy is relatively simple. This convenience

is especially great when we turn to questions such as the optimal use of indicator variables

under circumstances of partial information.

But there are other advantages as well. One is comparability of our results with those

of the traditional literature on monetary policy evaluation, which almost always assumes a

quadratic loss function of one sort or another. Casting our own results in this familiar form

allows us to discuss similarities and di¬erences between our utility-based welfare criterion and

those assumed in other studies without letting matters be obscured by super¬cial di¬erences

in functional form that may have relatively little consequence for the results obtained.

And ¬nally, it does not make sense to be concerned with a higher-order approximation

to our welfare criterion if we do not plan to characterize the e¬ects of alternative policies

1. APPROXIMATION OF LOSS FUNCTIONS AND OF OPTIMAL POLICIES 5

with a degree of precision su¬cient to allow computation of those higher-order terms. In

the ¬rst part of this study, we have shown how to derive a log-linear approximation to the

equilibrium ¬‚uctuations in in¬‚ation and output under alternative policies, using a log-linear

approximation to the exact structural equations of our model. Using this method, we com-

pute the equilibrium ¬‚uctuations in these variables only up to a residual of order O(||ξ||2 ),

where ||ξ|| is a bound on the amplitude of the exogenous disturbances. Given that we do

not compute the terms of second order in ||ξ|| in characterizing equilibrium ¬‚uctuations, we

cannot expect to compute the terms of third or higher order in ||ξ|| in evaluating the ex-

pected utility of the representative household.3 Of course, one might also wish to undertake

a more accurate approximation of the predicted evolution of the endogenous variables under

alternative regimes. However, such a study would introduce a large number of additional

free parameters, to which numerical values would have to be assigned for purposes of com-

putation; and there is likely to be little empirical basis for the assignment of such values

in most cases, given the degree to which the empirical study of macroeconomic time series

makes use of linear models.4

However, even a second-order approximation to utility can be computed on the basis

of a merely linear approximation to our model structural equation only under special cir-

cumstances. We shall assume that these hold in our calculations here, but it is important

to be clear about the scope of validity of our results. Let x represent a vector of endoge-

nous variables, and suppose that we wish to evaluate E[U (x; ξ)] under alternative policies,

where ξ is a vector of random exogenous disturbances, and U (·; ξ) is a concave function for

3

There is thus no obvious advantage to the approach sometimes adopted in utility-based welfare analyses,

such as Ireland (1997) or Collard et al. (1998), of evaluating an exact utility function but using a log-linear

approximation to the model™s structural equations in order to compute the equilibrium.

4

A common approach in the quantitative equilibrium business cycle literature, of course, is to assume

special functional forms for preferences and technology that allow the higher derivatives of these functions

to be inferred from the same small number of parameters as determine the lower-order derivatives, which

may then be inferred from ¬rst and second moments of the time series alone. This approach often obscures

the relation between the properties of the time series and the model parameters that are identi¬ed by them,

and allows “identi¬cations” that are in fact quite sensitive to the arbitrary functional form assumption. We

prefer instead to assume special functional forms as little as possible, but to be clear about the order of

approximation that is involved in our calculations.

6 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

each possible realization of ξ, and at least twice di¬erentiable. Now suppose that we are

able to compute a linear (or log-linear) approximation to the equilibrium responses of the

endogenous variables, under a given policy regime, of the form

x = x0 + a ξ + O(||ξ||2 ), (1.1)

where the vectors of coe¬cients xss and a may both depend upon policy. (This represents a

¬rst-order Taylor series approximation to the exact equilibrium responses x(ξ), assumed to

be nonlinear but di¬erentiable, taken around the mean values ξ = 0. The conditions under

which the solution to the linearized structural equations yield a valid approximation of this

kind to the solution to the exact structural equations are discussed in Appendix A.)

Under the assumption that the constant term x0 in (1.1) is itself of at most order O(||ξ||),

we can take a similar Taylor series expansion of the utility function U (x), and be con¬dent

that terms that are of at most order O(||x||3 ) are of at most order O(||ξ||3 ). We then can

write

1 1

¯

U (x; ξ) = U + Ux x + Uξ ξ + x Uxx x + x Uxξ ξ + ξ Uξξ ξ + O(||ξ||3 ),

˜ ˜ ˜˜ (1.2)

2 2

¯

where U ≡ U (¯; 0), x ≡ x ’ x, and all partial derivatives of U are evaluated at (¯; 0). We

x ˜ ¯ x

wish to use approximation (1.1) to the equilibrium ¬‚uctuations in x to compute the terms

of second or lower order in approximation (1.2) to utility. But the term Ux x in (1.2) will

˜

generally contain terms of second order that depend upon the neglected second order terms

in (1.1). In order to be able to neglect these terms, we must also assume that Ux (¯; 0) is at

x

most of order O(||ξ||). In that case, the neglected terms of order O(||ξ||2 ) contribute only to

terms of order O(||ξ||3 ) in Ux x.

˜

Assuming this, substitution of (1.1) minus the residual into (1.2) minus the residual yields

a correct quadratic approximation to U (x; ξ). Taking the expected value of this expression,

and using the fact that we normalize ξ so that E(ξ) = 0, we obtain the approximate welfare

criterion

1 1

¯

E[U ] = U + Ux E[˜] + tr{Uxx var[x]} + tr{Uxξ cov(ξ, x)} + tr{Uξξ var[ξ]} + O(||ξ||3 ). (1.3)

x ˜

2 2

1. APPROXIMATION OF LOSS FUNCTIONS AND OF OPTIMAL POLICIES 7

Here we use the notation E[z] for the expectation of a random vector z, var[z] denotes the

variance-covariance matrix, and cov(z1 , z2 ) the matrix of covariances between two random

vectors z1 , z2 . In expression (1.3) it is understood that the various ¬rst and second moments

are those that one computes using the linear approximation (1.1).

The validity of this last expression, when the ¬rst and second moments are computed

using (1.1), thus depends upon two special assumptions. These are that x0 is only of order

O(||ξ||), and that Ux (¯; 0) is similarly only of order O(||ξ||). Technically, we shall suppose

x

that ||ξ|| is a bound both upon the amplitude of the exogenous disturbances, and upon the

size of x0 and Ux (¯; 0). Our approximation result then refers to a sequence of economies in

x

which ||ξ|| eventually becomes arbitrarily small; as we progress along this sequence, both the

distribution of the disturbances, and certain other parameters of the model that determine

x0 and Ux , are varied so as to respect the changing bound, while keeping the speci¬cation of

the policy rule the same. What the Taylor theorem guarantees is then that if (1.3)minus the

residual yields a higher value for one policy than for another, it will be true for all economies

far enough out in this sequence that the ¬rst policy yields higher expected utility than the

other in the equilibrium of the exact model.

The stipulation regarding x0 is an assumption about the kind of policy regime which we

seek to evaluate, while the stipulation regarding Ux (¯; 0) is an assumption about the point

x

around which we choose to compute the Taylor expansion in in (1.3). The latter assumption

implies that we expand around a state of a¬airs x that is close to being optimal, not simply

¯

in the sense of being the best we can do using the set of policies under consideration, but in

the sense of being near the maximum of U (x; 0) over all possible values of x.5 Of course, it

is really only necessary that Ux be small in directions in which it is possible for the average

value of x to di¬er under alternative policies. Thus it is not necessary for households to

5

One way to guarantee this would be to stipulate that x is in fact the value that maximizes U (x; 0).

¯

We do not wish, however, to insist upon this. In some cases, those “¬rst-best” values do not correspond

to a possible equilibrium, even in the absence of disturbances. We could linearize the model™s structural

equations around such values nonetheless, but we prefer to follow convention in linearizing around values

that represent a particular equilibrium in the absence of shocks. One advantage of this convention is that

our linearized structural equations always have zero constant terms.

8 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

be nearly satiated both in consumption and in leisure in order for x to be optimal in the

¯

necessary sense; it is enough that it not be possible to greatly increase utility by varying

both consumption and work e¬ort in a way that is feasible given the economy™s production

function. But it is somewhat delicate to draw conclusions about the directions in which it

is possible for policy to vary the second-order terms in x without actually computing the

second-order terms in (1.1), and so we prefer to substitute constraints of this kind into our

de¬nition of the objective function U (x), and then require all elements of the vector Ux to

be small.

Kim and Kim (1998) provide an example of a problem, relating to the welfare gains

from risk-sharing, where this requirement for validity of a welfare calculation based upon

the linear approximation (1.1) is not satis¬ed. They consider the expected utility E[U (Ci )]

obtained by a household i in each of two cases. In the ¬rst case (autarchy), each household

consumes its own random income Yi , while in the second case (perfect risk-sharing), two

households pool their incomes, so that Ci = (Y1 + Y2 )/2 for each. Kim and Kim consider

the validity of a log-linear approximation to the relation between consumption and income.

In the case of autarchy, the log-linear relation (which is exact) is given by

ˆ ˆ

Ci = Yi ,

¯

where as usual hatted variables denote deviations of the logs from the value log Y around

which one log-linearizes. In the case of perfect risk-sharing, the log-linear approximation is

instead

ˆ ˆ ˆ

Ci = (Y1 + Y2 )/2.

Substitution of these two log-linear expressions into a quadratic approximation to the utility

function,

1 ’ γ ˆ2

¯ ¯ ¯ˆ

U (Ci ) = U (Y ) + Y U (Y ) Ci + Ci ,

2

¯

where γ > 0 is the coe¬cient of relative risk aversion (evaluated at consumption level Y ),

ˆ ˆ

does not yield a correct quadratic approximation to utility as a function of Y1 and Y2 . Indeed,

if 0 < γ < 1, this approximation implies that expected utility is higher under autarchy.

1. APPROXIMATION OF LOSS FUNCTIONS AND OF OPTIMAL POLICIES 9

The problem is that the partial derivative of U with respect to log Ci (which is equal

¯ ¯

to Y U (Y )) is non-zero, so that the correct quadratic approximation to expected utility in

ˆ

the case of risk-sharing involves quadratic terms in the Taylor series expansion for Ci . (The

omitted terms raise expected utility in that case, since less variable consumption means

a higher expected value for log consumption, by Jensen™s inequality.) It does not make

sense to assume that this derivative can be made arbitrarily close to zero as we make the

bound ||ξ|| on the amplitude of income variations smaller, either. This could be done only

by varying preferences and/or average income as we make ||ξ|| smaller in such a way that

proportional variations in consumption cease to matter much; but this would mean that in

the limit, no comparisons between alternative consumption processes would be possible. This

di¬culty does not arise in the case of our analysis of the welfare gains from macroeconomic

stabilization, below, as long as we log-linearize around a level of economic activity (in each

sector) that is su¬ciently near to being e¬cient; since an interior optimum does exist in our

case, this is possible. But it is important that we check that the derivatives in question are

indeed small, under our assumptions, and that the quali¬cations that this requires to our

results be noted.

The assumption that x0 is small means that the policies considered are all ones with

the property that in the absence of shocks, the equilibrium value of x would be near the

linearization point x ” or alternatively, that in equilibrium the mean value of x is near x.

¯ ¯

Given our assumption about x, this means that the policies considered are ones under which

¯

the equilibrium value x is nearly optimal, in the sense discussed above. As we are primarily

interested in whether our approximate welfare criterion correctly identi¬es the optimal policy,

the essential requirement is that our model (and the family of available policies) be such

that the best available policy can achieve an outcome that is su¬ciently close to being fully

optimal. Thus the unavoidable frictions “ the ones that cannot be ameliorated through an

appropriate choice of policy “ must be small, even if there exist frictions that imply that

outcomes under a bad policy could be signi¬cantly worse.

In the case of our baseline model, the only frictions that prevent equilibrium from being

10 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

e¬cient are (i) the market power possessed by suppliers of goods, as a result of monopolistic

competition, and (ii) the failure to adjust all goods prices each period. We log-linearize our

structural equations around the steady state with zero in¬‚ation each period, that represents

a possible equilibrium in the absence of real disturbances. In this equilibrium, the failure to

adjust prices constantly results in no distortion of the allocation of resources; this allocation

is thus nearly optimal as long as the distortion due to market power is su¬ciently small. In

our calculations below, we assume that it is.

The other assumption required for the validity of the quadratic approximation obtained

from our log-linear structural equations is that the policy rules considered be ones under

which the equilibrium rate of in¬‚ation in the absence of shocks would in fact be near zero “ or

alternatively, that these policies be ones under which the average rate of in¬‚ation is low. This

also is assumed; since we conclude that it is optimal for a country with characteristics like

those of the U.S. to choose a policy under which the average rate of in¬‚ation is slightly, but

only slightly, positive, this last assumption is relatively innocuous. However, it is important

to realize that under other circumstances “ say, an analysis of optimal monetary policy in

the presence of a need for signi¬cant seignorage revenues “ this assumption as well might be

inappropriate.

Finally, it is important to note that the conditions required for validity of a quadratic

approximation to welfare obtained from log-linear approximations to the structural equations

do not relate solely to the structure of the economy; it also matters in which form we choose

to express our approximate loss function. Alternative quadratic approximations to U , each

equally valid second-order Taylor series expansions (but in terms of di¬erent variables), may

not yield equally valid approximations to welfare when evaluated using a log-linear solution

(1.1) for the model™s endogenous variables.

For example, consider a model like that of the next section. Each period™s contribution

to the utility of the representative household can be approximated by an expression of the

form

ˆ ˆ

U = aC ’ bH + Q1 + R1 , (1.4)

1. APPROXIMATION OF LOSS FUNCTIONS AND OF OPTIMAL POLICIES 11

ˆˆ

where C, H denote the percentage deviations in consumption and hours worked respectively,

Q1 is a set of quadratic terms in the log deviations, and the residual R1 contains terms

of third or higher order, or terms that are independent of the policy chosen, that can be

neglected. Alternatively, one may eliminate hours using the necessary relation between

aggregate consumption and aggregate hours implied by the production function, and obtain

a Taylor series expansion of the form

ˆ

U = cC + Q2 + R2 , (1.5)

where Q2 , R2 are other quadratic terms and residual. Under the assumptions just described,

the coe¬cient c is of order O(||ξ||). It follows that substitution into (1.5) of the solution to

our log-linear structural equations yields a valid second-order approximation to utility.

However, the same is not true of substitution of the same solution into (1.4). The

ˆ ˆ

structural equations include a production-function relation between C and H, of the form

ˆ ˆ

C = f H + Q3 + R 3 , (1.6)

and one of the log-linear structural relations is given by the linear terms in this. In ap-

proximation (1.5), c = a ’ bf ’1 , so that the ¬rst-order terms in the two approximations

would have the same value. But the terms Q2 are not equal to Q1 , because of the presence

of non-zero quadratic terms Q3 in (1.6).6 Hence (1.4) will not yield a correct second-order

ˆ ˆ

approximation to welfare, if one substitutes the solutions for C and H implied by the log-

linear structural relations, including the linear part of (1.6). In terms of the criterion set

out above, substitution of the log-linear approximate solutions into (1.4) yields an incorrect

result because the coe¬cients a and b are not individually of order O(||ξ||), even though

the linear combination c is. Thus it is not enough that one expand around a near-optimal

equilibrium; the expansion must be written in a form that contains no ¬rst-order terms that

do not involve coe¬cients of order O(||ξ||).

6

Even if the production function is of a constant-elasticity form, the log-linear approximation (1.6) will

contain non-zero quadratic terms in the event of variations in government purchases.

12 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

2 A Utility-Based Welfare Criterion

We turn now to the computation of a utility-based approximate welfare criterion, of the kind

discussed in the previous section, for the case of our baseline model.7 The natural welfare

criterion in our model is the level of expected utility

∞

β t Ut

E (2.1)

t=0

associated with a given equilibrium, where the period contribution to utility Ut is given

by expression (xxx) of chapter 3. Here we shall abstract from the welfare consequences of

monetary frictions.8 Then, substituting the equilibrium condition Ct = Yt ,9 we can express

the period utility of the representative household as a function solely of the level of production

of each of the goods in period t,

1

˜

Ut = u(Yt ; ξt ) ’ v (yt (i); ξt )di.

˜ (2.2)

0

Here θ

1 θ’1

θ’1

Yt ≡ yt (i) di (2.3)

θ

0

is again the index of aggregate demand, where yt (i) is production (and consumption) in

period t of di¬erentiated good i.10

˜˜

The function v (y; ξ) indicates the disutility of supplying quantity y. If we assume a “yeo-

man farmer” model, as in Rotemberg and Woodford (1997, 1999a), this can be interpreted

directly as the household™s disutility of supplying output. Alternatively, if we assume ¬rms

and a labor market, as in section xx of chapter 3, we can de¬ne

˜˜

v (y; ξ) ≡ v(f ’1 (y/A); ξ), (2.4)

7

The derivation follows essentially the lines of that presented in Rotemberg and Woodford (1997, 1999a)

in the context of a variant model with additional decision lags.

8

The welfare criterion derived in this section thus applies in the “cashless limit” introduced in section

3.3 of chapter 2. The additional term that must be added in the case of non-negligible monetary frictions is

derived in section 4.1 below.

9

Recall that we can interpret the model as allowing for government purchases, treating these as a shift

term in the utility function u(Y ; ξ).

10

If there is trend growth in productivity and output, the variables Yt , yt (i), and At (introduced in the

next paragraph) should all be interpreted as having been de¬‚ated by a common exponential trend factor, to

render them stationary.

2. A UTILITY-BASED WELFARE CRITERION 13

where v(h; ξ) is the disutility of working h hours in any given production activity, and Af (h)

˜

is the output produced using that labor input. Here ξ ≡ (ξ, a) denotes the complete vector of

exogenous disturbances, including both the preference shocks ξ 11 and the technology shock

a ≡ log A.12

˜

Our Taylor series expansions group terms of di¬erent powers in the elements of ξ, though

we shall continue to use the notation ||ξ|| for the bound on the magnitude of the entire vector

of disturbances.

2.1 Output-Gap Stability and Welfare

Under our previous assumptions, u is an increasing, concave function of Y for each possible

˜

value of ξ, while v is an increasing, convex function of y for each possible value of ξ. Thus

˜

(2.2) implies that Ut is a concave function of the entire vector of levels of production of

the various goods. Note also that in terms of this notation, the real marginal cost function

introduced in chapter 3 is given by

˜

vy (y; ξ)

˜

˜

s(y, Y ; ξ) = . (2.5)

uc (Y ; ξ)

It follows that the elasticity of vy with respect to y is given by ω > 0, the elasticity of real

˜

marginal cost with respect to own output, introduced in chapter 3. One may also observe

that the elasticity of real marginal cost with respect to aggregate output is given by σ ’1 ,

where once again

uc

σ≡’¯

Y ucc

measures the intertemporal elasticity of substitution in private expenditure.

The steady-state level of output associated with zero in¬‚ation, in the absence of real

11

Once again, the vector ξ contains many elements, so that the disturbances to the utility of consumption

may or may not be correlated with the disturbances to the disutility of working.

12

We assume a normalization of the productivity measure A such that the unconditional expectation of a

˜

is zero. We include a rather than A in our de¬nition of ξ because it is a rather than A that is assumed to be

always su¬ciently close to zero (in order for our Taylor series approximation to be accurate), and because

we wish to approximate the production-function relationship by one that is linear in a rather than linear in

A.

14 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

˜ ¯

disturbances (i.e., when ξt = 0 at all times) is the quantity Y that satis¬es

1’„

¯¯

s(Y , Y ; 0) = ≡ 1 ’ ¦. (2.6)

µ

Here „ is the constant proportional tax rate on sales proceeds, and µ ≡ θ/(θ ’ 1) is the

desired markup as a result of suppliers™ market power. The parameter ¦ then summarizes

the overall distortion in the steady-state output level as a result of both taxes and market

¯

power. Since the e¬cient output level Y e for all goods in the absence of shocks satis¬es

¯¯ ¯¯

s(Y e , Y e ; 0) = 1, we observe that Y /Y e is a decreasing function of ¦, equal to one when

¦ = 0. When ¦ is small, we may make use of the log-linear approximation

¯¯

log(Y /Y e ) = ’(ω + σ ’1 )’1 ¦. (2.7)

It is plainly realistic to assume that ¦ > 0. However, we shall assume that ¦ is small, specif-

ically of order O(||ξ||), so that (2.7) is accurate up to a residual of order O(||ξ||2 ). This is

the assumption of near-e¬ciency of the steady state level of output with zero in¬‚ation that

is made in order to allow us to use our log-linear approximations to the model structural

equations in welfare comparisons, as explained in the previous section. Note that the intro-

duction of the distorting tax rate „ allows us to contemplate a series of economies in which

¦ is made progressively smaller, without this having to involve any change in the size of θ,

a parameter that also a¬ects the coe¬cients of the log-linearized equilibrium conditions.13

We now proceed to compute a quadratic Taylor series approximation to (2.2). The ¬rst

term can be approximated as

1˜ 1

˜ ˜

u(Yt ; ξt ) = u + uc Yt + uξ ξt + ucc Yt2 + ucξ ξt Yt + ξt uξξ ξt + O(||ξ||3 )

¯

2 2

13

Rotemberg and Woodford (1997, 1999a) instead assume that „ is of exactly the (negative) size required

to o¬set the distortion due to market power, so that ¦ = 0. The intention is to consider optimal monetary

stabilization policy as part of a broader analysis of optimal policy, in which another instrument (tax policy)

is assigned responsibility for achieving the optimal average level of economic activity, while monetary policy

is used to ameliorate the economy™s response to shocks. However, it is clear that monetary policy must in

practice be chosen in an environment in which such an output subsidy does not, and probably cannot, exist.

Furthermore, the fact that the “natural rate” of output is ine¬ciently low is of importance for certain issues,

notably the in¬‚ationary bias associated with discretionary policymaking, treated in chapter 7. Hence we

here allow for ¦ > 0, while still assuming that ¦ is of order O(||ξ||).

2. A UTILITY-BASED WELFARE CRITERION 15

1ˆ 1¯

¯¯ ˆ ˆ

= u + Y uc · (Yt + Yt2 ) + uξ ξt + Y 2 ucc Yt2

2 2

1

¯ ˆ

+Y ucξ ξt Yt + ξt uξξ ξt + O(||ξ||3 )

2

1¯

¯ˆ ¯ ˆ ¯ ˆ

= Y uc Yt + [Y uc + Y 2 ucc ]Yt2 ’ Y 2 ucc gt Yt + t.i.p. + O(||ξ||3 )

2

1

¯ ˆ ˆ ˆ

= Y uc Yt + (1 ’ σ ’1 )Yt2 + σ ’1 gt Yt + t.i.p. + O(||ξ||3 ). (2.8)

2

¯ ˜

Here the ¬rst line represents the usual Taylor expansion, in which u ≡ u(Y ; 0) and Yt ≡

¯

¯ ˜

Yt ’ Y , and we assume that the ¬‚uctuations in Yt are only of order O(||ξ||). The second line

˜ ˆ ¯

substitutes for Yt in terms of Yt ≡ log(Yt /Y ), using the Taylor series expansion

1ˆ

¯ ˆ

Yt /Y = 1 + Yt + Yt2 + O(||ξ||3 ).

2

The third line collects together in the term “t.i.p.” all of the terms that are independent of

policy, as they involve only constants and exogenous variables, and uses again the notation

ucξ ξt

gt ≡ ’ ¯

Y ucc

for the percentage variation in output required in order to keep the marginal utility of

expenditure uc at its steady-state level, given the preference shock.14 The ¬nal line collects

terms in a useful way; note that the only part of this expression that di¬ers across policies

is the expression inside the curly braces.

We may similarly approximate v (yt (i); ξt ) by

˜

1

¯˜ ˆ

v (yt (i); ξt ) = Y vy yt (i) + (1 + ω)ˆt (i)2 ’ ωqt yt (i) + t.i.p. + O(||ξ||3 )

˜ y ˆ

2

1

¯

= Y uc (1 ’ ¦)ˆt (i) + (1 + ω)ˆt (i)2 ’ ωqt yt (i) + t.i.p. + O(||ξ||3 ), (2.9)

y y ˆ

2

¯

where yt (i) ≡ log(yt (i)/Y ), ω is the elasticity of real marginal cost with respect to own

ˆ

output discussed above, and

vyξ ξt

˜

qt ≡ ’ ¯

Y vyy

˜

14

As in chapters 2 and 4, the exogenous shifts ξt in the relation between the marginal utility of expenditure

uc and aggregate demand Yt should be understood to include variations in the level of government purchases

as well as taste shocks. The approximation derived here continues to apply in the presence of government

purchases, as long as the function u in (2.8) is understood to refer to the indirect utility function called

˜

u(Yt ; ξt ) in chapter 2.

˜

16 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

is the percentage variation in output required to keep the marginal disutility of supply vy

˜

at its steady-state level, given the preference shock. The second line uses (2.5) and (2.6) to

replace vy by (1 ’ ¦)uc , and the assumption that ¦ is of order O(||ξ||) to simplify. Note that

˜

the term premultiplying the expression in curly braces is now the same as in (2.8).

Integrating this expression over the di¬erentiated goods i, we obtain

1

1

¯

v (yt (i); ξt ) = Y uc (1 ’ ¦)Ei yt (i) + (1 + ω)[(Ei yt (i))2 + vari yt (i)] ’ ωqt Ei yt (i)

˜ ˆ ˆ ˆ ˆ

2

0

+t.i.p. + O(||ξ||3 )

1 1

¯ ˆ ˆ ˆ

= Y uc (1 ’ ¦)Yt + (1 + ω)Yt2 ’ ωqt Yt + (θ’1 + ω)vari yt (i) ˆ

2 2

+t.i.p. + O(||ξ||3 ), (2.10)

using the notation Ei yt (i) for the mean value of yt (i) across all di¬erentiated goods at date

ˆ ˆ

t, and vari yt (i) for the corresponding variance. In the second line, we use the Taylor series

ˆ

approximation to (2.3),

1

ˆ

Yt = Ei yt (i) + (1 ’ θ’1 )vari yt (i) + O(||ξ||3 ),

ˆ ˆ (2.11)

2

to eliminate Ei yt (i).

ˆ

Combining (2.8) and (2.10), we ¬nally obtain

1 1

¯ ˆ ˆ ˆ

Ut = Y uc ¦Yt ’ (σ ’1 + ω)Yt2 + (σ ’1 gt + ωqt )Yt ’ (θ’1 + ω)vari yt (i)

ˆ

2 2

+t.i.p. + O(||ξ||3 )

¯

Y uc

(σ ’1 + ω)(xt ’ x— )2 + (θ’1 + ω)vari yt (i) + t.i.p. + O(||ξ||3 ). (2.12)

=’ ˆ

2

ˆ ˆ

Here the second line rewrites the expression in terms of the output gap xt ≡ Yt ’ Ytn , where

ˆ

Ytn again denotes the (log of the) natural rate of output, the equilibrium level of output

under complete price ¬‚exibility, given by

’1

ˆtn ≡ σ gt + ωqt ,

Y (2.13)

σ ’1 + ω

¯¯

and in terms of the e¬cient level of the output gap, x— ≡ log(Y e /Y ), given by (2.7). (Note

that if ¦ is positive and of order O(||ξ||), the same is true of x— .)

2. A UTILITY-BASED WELFARE CRITERION 17

Expression (2.12) represents a quadratic approximation to (2.2), under the assumption

that ¦ (and hence the ine¬ciency of the steady-state level of output) is of order O(||ξ||).

˜

It is interesting to observe that the preference and technology shocks ξt matter, in this

approximation, only through their e¬ects upon a single exogenous state variable, the natural

ˆ

rate of output Ytn . Furthermore, output variability as such does not matter for our utility-

based welfare criterion; rather, it is the variability of the output gap that matters, and the

measure of potential output with respect to which the gap should be measured for purposes

of the welfare criterion is the same “natural rate” of output that (as shown in chapter 3)

determines the short-run relation between output and in¬‚ation. Thus we can already o¬er an

answer to one question posed in the introduction to this chapter: it is the output gap, rather

than output relative to trend, that one should seek to stabilize, and (if the only distortions

in the economy are those associated with monopolistic competition and sticky prices) the

relevant output gap is the same one that appears in the short-run aggregate supply curve.

However, (2.12) implies that stabilization of the output gap should not be the sole concern

of policy, since the dispersion of output levels across sectors matters as well.15 In fact, in

our baseline framework, there is no reason for equilibrium output to be di¬erent for di¬erent

goods except as a result of relative price distortions that result from sticky prices in an

environment where the overall price level is unstable. It is through this channel that price

stability turns out to be relevant for welfare, in a way that goes beyond the mere association

between in¬‚ation and the level of the aggregate output gap.

Speci¬cally, our assumed CES (Dixit-Stiglitz) preferences over di¬erentiated goods imply

that each supplier faces a constant-elasticity demand curve of the form

log yt (i) = log Yt ’ θ(log pt (i) ’ log Pt ). (2.14)

15

More generally, it is the dispersion of output gaps across sectors that matters, along with the aggregate

˜

output gap. We here assume that the only disturbances ξt that a¬ect the natural rate of output have identical

e¬ects upon all sectors, so that the dispersion of output gaps across sectors is identical to the dispersion

of output levels. In section xx below, we consider the consequences of allowing for shocks with asymmetric

e¬ects on di¬erent sectors.

18 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

It follows from this that

vari log yt (i) = θ2 vari log pt (i),

so that (2.12) may equivalently be written

¯

Y uc

(σ ’1 + ω)(xt ’ x— )2 + θ(1 + ωθ)vari log pt (i) + t.i.p. + O(||ξ||3 ).

Ut = ’ (2.15)

2

Thus we ¬nd that, in addition to stabilization of the output gap, it is also appropriate for

policy to aim to reduce price dispersion. In our framework, this is achieved by stabilizing the

general price level; but the exact way in which ¬‚uctuations in the general price level a¬ect

price dispersion, and hence welfare, depend upon the details of price-setting.

2.2 In¬‚ation and Relative-Price Distortions

The approximation (2.15) to the utility of the representative household applies to any model

with no frictions other than those due to monopolistic competition and sticky prices, re-

gardless of the nature of the delays involved in price-setting. The relation between the price

dispersion term and the stability of the general price level depends, instead upon the details

of price-setting. Here we do not attempt a general treatment, but illustrate the form of the

relation in three simple examples, including our baseline model of staggered price-setting

from chapter 3.

As a ¬rst example, consider again the case, discussed above in section xx of chapter 3,

of an economy in which a fraction 0 < ι < 1 of goods prices are fully ¬‚exible, while the

remaining 1 ’ ι must be ¬xed a period in advance. In such an economy, as shown above, the

aggregate supply relation takes the familiar “New Classical” form

πt = κxt + Et’1 πt , (2.16)

where the slope coe¬cient is given by

ι σ ’1 + ω

κ≡ > 0.

1 ’ ι 1 + ωθ

2. A UTILITY-BASED WELFARE CRITERION 19

In this model, in any period all ¬‚exible-price goods have the same price, p1 , and all

t

sticky-price goods have the same price, p2 , which satis¬es

t

log p2 = Et’1 log p1 + O(||ξ||2 ). (2.17)

t t

The overall price index (de¬ned by equation (xx) of chapter 3) furthermore satis¬es

log Pt = ι log p1 + (1 ’ ι) log p2 + O(||ξ||2 ),

t t

so that

πt ’ Et’1 πt = ι[log p1 ’ Et’1 log p1 ] + O(||ξ||2 )

t t

= ι[log p1 ’ log p2 ] + O(||ξ||2 ),

t t

using (2.17). It follows that under this assumption about pricing,

vari log pt (i) = ι(1 ’ ι)(log p1 ’ log p2 )2

t t

1’ι

(πt ’ Et’1 πt )2 .

=

ι

As asserted above, equilibrium price dispersion is closely connected with the stability of

the general price level; but in this special case, it is only the volatility of the unexpected

component of in¬‚ation that matters.

Substituting this expression into (2.15), we obtain

Ut = ’„¦Lt + t.i.p. + O(||ξ||3 ),

where „¦ is a positive constant and Lt is a quadratic loss function of the form

Lt = (πt ’ Et’1 πt )2 + »(xt ’ x— )2 , (2.18)

with a relative weight on output gap variability of » = κ/θ. We thus obtain precise con-

clusions regarding both the sense in which aggregate output and in¬‚ation variations matter

for welfare (it is the output gap that matters, and the unexpected component of in¬‚ation),

20 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

and the relative weight that should be placed upon the two concerns (the relative weight on

output gap variations is proportional to the slope κ of the short-run Phillips curve).

In fact, in the context of this model, there is no tension between the goals represented by

the two terms of (2.18). For (2.16) implies that the output gap is itself proportional to the

surprise component of in¬‚ation. Thus we can simplify (2.18) further, and say that the sole

goal of policy should be to minimize the variability of unexpected in¬‚ation, or alternatively,

that the sole goal should be to stabilize the output gap (when properly measured).16

While we obtain a simple result in this case, the model is not a very realistic one, since,

as discussed earlier, it is unable to account for the persistence of the observed output e¬ects

of monetary disturbances. Let us consider instead, then, the consequences of the kind of

staggered pricing assumed in our baseline model, a discrete-time version of the Calvo (1983)

pricing model. In this model, a fraction 0 < ± < 1 of all prices remain unchanged each

period, with the probability of a price change assumed to be independent of both the length

of time since the price was last changed and of the degree to which that good™s price is out

of line with others. This implies that each period, the distribution of prices {pt (i)} consists

of ± times the distribution of prices in the previous period, plus an atom of size (1 ’ ±) at

the price p— that is chosen at date t by all suppliers who choose a new price at that date. As

t

shown in chapter 3, the aggregate supply relation takes in this case the “New Keynesian”

form

πt = κxt + βEt πt+1 , (2.19)

where now the slope coe¬cient is given by

(1 ’ ±)(1 ’ ±β) (σ ’1 + ω)

κ≡ > 0. (2.20)

± 1 + ωθ

Letting

¯

Pt ≡ Ei log pt (i), ∆t ≡ vari log pt (i),

16

However, if we allow for disturbances to the short-run aggregate supply relation (2.16) that “ unlike the

preference, technology, or government-purchase shocks considered in chapter 3 “ do not shift the e¬cient

level of output to the same extent, then the loss function (2.18) would still be correct, while the output gap

that appears in this formula would no longer coincide perfectly with unexpected in¬‚ation. In that extension

of the model, it would be quite important to know the correct relative weight » to place on output-gap

variations. See section xx below.

2. A UTILITY-BASED WELFARE CRITERION 21

we observe from the above recursive characterization of the distribution of prices at date t

that

¯ ¯ ¯

Pt ’ Pt’1 = Ei [log pt (i) ’ Pt’1 ]

¯ ¯

= ±Ei [log pt’1 (i) ’ Pt’1 ] + (1 ’ ±)(log p— ’ Pt’1 )

t

¯

= (1 ’ ±)(log p— ’ Pt’1 ).

t

Similar reasoning about the dispersion measure ∆t yields

¯

∆t = vari [log pt (i) ’ Pt’1 ]

¯ ¯

= Ei {[log pt (i) ’ Pt’1 ]2 } ’ (Ei log pt (i) ’ Pt’1 )2

¯ ¯ ¯ ¯

= ±Ei {[log pt’1 (i) ’ Pt’1 ]2 } + (1 ’ ±)(log p— ’ Pt’1 )2 ’ (Pt ’ Pt’1 )2

t

± ¯ ¯

(Pt ’ Pt’1 )2 .

= ±∆t’1 +

1’±

Finally, substituting the log-linear approximation

¯

Pt = log Pt + O(||ξ||2 ),

we obtain

±

πt + O(||ξ||3 )

2

∆t = ±∆t’1 + (2.21)

1’±

as a law of motion for the dispersion of prices. Note that price dispersion is again a function

of the degree of instability of the general price level, though now the relation is a dynamic

one. Note also that under this assumptions about pricing, both expected and unexpected

in¬‚ation contribute equally to increases in price dispersion.

Integrating forward (2.21) starting from any initial degree of price dispersion ∆’1 in the

period before the ¬rst period for which a new policy is contemplated, the degree of price

dispersion in any period t ≥ 0 under the new policy will be given by

t

±

t+1

±t’s πs + O(||ξ||3 ).

2

∆t = ± ∆’1 +

1’±

s=0

Note that the ¬rst term will be independent of the policy that is chosen to apply in periods

t ≥ 0. Thus if we take the discounted value of these terms over all periods t ≥ 0, we obtain

∞ ∞

±

t

β t πt + t.i.p. + O(||ξ||3 ).

2

β ∆t =

(1 ’ ±)(1 ’ ±β) t=0

t=0

22 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

Substitution of this in turn into (2.15), we ¬nd that

∞ ∞

t

β t Lt + t.i.p. + O(||ξ||3 ),

β Ut = ’„¦ (2.22)

t=0 t=0

where in this case the normalized quadratic loss function is given by

Lt = πt + »(xt ’ x— )2 .

2

(2.23)

Here the relative weight on output gap variability is again given by » = κ/θ, but now the

value of κ referred to is that given in (2.20).17

The loss function (2.23) is in fact of a form widely assumed in the literature on monetary

policy evaluation (and also in positive models of central bank behavior).18 Here, however, we

are able to present a theoretical justi¬cation for the attention to variations in in¬‚ation (rather

than, say, variations in the price level), as well as for the common assumption that in¬‚ation

variations are equally costly whether forecastable or not, in terms of the relative-price distor-

tions resulting from price-level instability in the Calvo model of staggered price-setting. We

are also able to derive an optimal rate of in¬‚ation with respect to which deviations should

be measured (namely, zero, as it is in this case that no relative-price distortions result from

imperfect synchronization of price changes). And ¬nally, we are again able to derive an

optimal relative weight upon output-gap variation as opposed to in¬‚ation variation; this

depends upon model parameters, but in a way that makes an estimate of the slope of the

short-run aggregate supply curve directly informative about the proper size of this weight.19

17

Note that the values of „¦ and » obtained here are slightly di¬erent from those that follow from the

derivation presented in Rotemberg and Woodford (1999a). The reason is that we are here interested in

approximating the expected value of the discounted sum of utilities, conditioning upon the pre-existing

degree of price dispersion at date ’1, whereas they compute an unconditional expectation. Note that the

loss measure that we compute here, for a given policy, will not depend upon the initial price dispersion ∆’1 .

Nonetheless, it matters whether one conditions upon the value of ∆’1 in computing the expected utility.

Computing the unconditional expectation, rather than conditioning upon the value of ∆’1 , penalizes policies

that lead to higher average price dispersion also for the higher average value assumed for ∆’1 if one integrates

over the unconditional distribution of values for ∆t associated with a given stationary equilibrium.

18

See, e.g., Walsh (1998, chap. 8), Clarida et al. (1999), or Svensson (1999a).

19

The size of this weight is of greater interest in the case of this model, since aggregate supply relation

(2.19) does not imply that in¬‚ation and the output gap should perfectly co-vary under most circumstances.

It is true that complete stabilization of one implies complete stabilization of the other, as we discuss further

in the next section, and in this sense there is no tension between the two goals if (2.19) holds. But it may not

2. A UTILITY-BASED WELFARE CRITERION 23

As we discuss further in section 4.2 below, the estimate of the slope of the short-run

aggregate supply curve for the U.S. of Rotemberg and Woodford (1997) implies a value for

»x on the order of .05, if the output gap is measured in percentage points and in¬‚ation is

measured as an annualized percentage rate. This value is much lower than the value »x = 1

often assumed in the literature on evaluation of monetary policy rules, on a ground such as

“giving equal weight to in¬‚ation and output” as stabilization objectives.20 Our utility-based

analysis implies instead that if one assumes the degree of price stickiness that is needed

to account for the persistence of the real e¬ects of monetary policy shocks, the distortions

associated with in¬‚ation are more important than those associated with variation in the

aggregate output gap.

As yet another alternative, suppose that prices are indexed to a lagged price index be-

tween the occasions on which they are re-optimized, as in the model with “in¬‚ation inertia”

set out in section xx of chapter 3. In this case, the aggregate supply relation takes the form

πt ’ γπt’1 = κxt + βEt [πt+1 ’ γπt ], (2.24)

generalizing (2.19), where γ measures the degree of indexation to the lagged price index,

and κ is again de¬ned as in (2.20). Recall that in the periods in which a given price is not

re-optimize, it is automatically increased by an amount

log pt (i) = log pt’1 (i) + γπt’1

owing to the change in the lagged price index. It follows that (2.21) generalizes to

±

(πt ’ γπt’1 )2 + O(||ξ||3 ).

∆t = ±∆t’1 + (2.25)

1’±

Price dispersion is increased only when the prices that are re-optimized are increased by

an amount di¬erent than γπt’1 , the amount by which the prices that are not re-optimized

increase. This occurs if and only if the overall rate of in¬‚ation πt di¬ers from γπt’1 .

be possible to achieve complete stabilization, e.g., because of the zero lower bound on nominal interest rates,

or informational restrictions upon feasible policies; and in such cases optimal policy will generally depend

upon the relative weight placed upon the two goals.

20

See, e.g., Rudebusch and Svensson (1999) and Williams (1999).

24 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

We can integrate (2.25) forward as before, and again obtain an expression of the form

(2.22) for discounted utility. But now the normalized quadratic loss function is given by

Lt = (πt ’ γπt’1 )2 + »(xt ’ x— )2 , (2.26)

where » > 0 and x— ≥ 0 are de¬ned as in (2.23). In the case of full indexation of individual

prices to the lagged price index (γ = 1),21 this implies that it is the rate of in¬‚ation accel-

eration, ∆πt , rather than the rate of in¬‚ation itself, that should be stabilized around zero

in order to reduce the distortions associated with price dispersion.22 Owing to the existence

of complete indexation to past in¬‚ation, there are no distortions resulting from constant

in¬‚ation, only from changes in the rate of in¬‚ation. In its implication that steady in¬‚ation

at any rate causes no harm, this model is like the “New Classical” model (with which it

shares the prediction of a vertical “long run Phillips curve”). But in the model with full

indexation, in¬‚ation changes distort the allocation of resources, whether they are predictable

in advance or not.

3 The Case for Price Stability

While the loss measures derived above under the various assumptions about the timing of

pricing decisions are each di¬erent in certain respects, they all share an important common

property. This is that the deadweight losses due to relative price distortions can in each

case be completely eliminated, in principle, by stabilizing the aggregate price level. The

intuition for this result is simple. The aggregate price level is stabilized by creating an

environment in which suppliers who choose a new price have no desire at any time to set a

price di¬erent from the average of existing prices. But if this is so, the average of existing

21

Note that in this case, the aggregate supply relation is essentially identical to the popular empirical

speci¬cation proposed by Fuhrer and Moore (1995a, 1995b).

22

Steinsson (2000) and Amato and Laubach (2001b) similarly ¬nd, in the case of an AS relation with

in¬‚ation inertia owing to the presence of backward-looking “rule-of-thumb” price-setters of the kind proposed

by Gali and Gertler (1999), that the utility-based loss function should penalize variations in ∆πt as well as

variations in πt . The result (2.26) is simpler, however, than the one obtained by these authors, and leads to

simpler conclusions regarding the character of optimal policy in the presence of in¬‚ation inertia. This is one

reason that we emphasize indexation to a lagged price index as a possible source of in¬‚ation inertia in this

study.

3. THE CASE FOR PRICE STABILITY 25

prices never changes, and so the new prices that are chosen at all times are always the same,

and eventually all goods prices are equal to that same, constant value. Thus aggregate price

stability is a su¬cient condition for the absence of price dispersion in our simple framework.

At the same time, in most cases, it is also a necessary condition. This is not true in the

pure “New Classical” case, as in that case it is only necessary that there be no unexpected

changes in the aggregate price level in order for there to be no price dispersion. But this

is clearly a highly special case; even if some prices are fully ¬‚exible, if the sticky goods

prices as in the Calvo model, complete price stability will again be necessary to eliminate

distortions. Similarly, even in the model of Mankiw and Reis (2001), which is like the “New

Classical” model in assuming that each goods price at any date is set optimally conditional

on information available at some prior date (though the dates are di¬erent for di¬erent

goods), complete price stability is necessary to eliminate distortions. This is because in the

Mankiw-Reis model, for any horizon k, there are a positive fraction of goods prices that

are set more than k periods in advance (or on the basis of information that is more than k

periods out of date). This means that relative-price distortions will be created in the case of

any disturbance that a¬ects the price level, even if it has no e¬ect upon the price level until

k periods later. Since this is true for arbitrary k, distortions are completely eliminated only

if exogenous disturbances never imply any change in the price level.

Similarly, in the model with staggered pricing and full indexation to a lagged price index,

price stability is not necessary for the absence of price dispersion; it is simply necessary

that the in¬‚ation rate be constant over time. But again this is a highly special case. If the

indexation parameter γ takes any value other than one, only zero in¬‚ation is consistent with

an absence of price dispersion.23 The same conclusion would be reached if only some prices

are indexed to the lagged price index.

The argument for the necessity of stability of the general price index for the elimination of

23

To be precise, an absence of price dispersion will require that prices change at a common rate πt satisfying

the di¬erence equation πt = γπt’1 , given some arbitrary initial rate of in¬‚ation. But when γ < 1, this

implies zero in¬‚ation every period, at least asymptotically. A stationary policy regime that fully eliminates

distortions resulting from price dispersion would have to be one with zero in¬‚ation at all times.

26 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

price dispersion is also independent of the speci¬c details of the way that the timing of price

changes is modeled by Calvo (1983). One need not assume that the probability of revision

of any given price in a given period is the same for all prices; one might instead assume that

prices are revised at ¬xed time intervals, as in the models of Taylor (1980), Blanchard and

Fischer (1989, sec. 8.2), King and Wolman (1999), or Chari et al. (2000), among others, or

one might endogenize the timing of price revisions as in the model of Dotsey et al. (1999). In

any of these cases, price dispersion is eliminated only by a policy that completely stabilizes

the general price index.

Moreover, price stability is not only the case in which the distortions associated with

ine¬cient output composition are eliminated. As we shall see, it is also the route to min-

imization of the distortions associated with an ine¬cient level of output; and so, in the

context of the kind of simple model considered thus far, it is an unambiguously desirable

goal for monetary policy. The argument for this is simplest in the case that the equilibrium

level of output under ¬‚exible prices is optimal, so we take up this case ¬rst. But as we shall

see, our conclusions require only minor modi¬cation even when we allow for the possibility

that the natural rate of output is ine¬ciently low.

3.1 The Case of an E¬cient Natural Rate of Output

Here we assume not merely that the ine¬ciency wedge ¦ de¬ned in (2.6) is of order O(||ξ||),

but that it is equal to zero (or at any rate, that it is of order O(||ξ||2 ), so that we may

¯ ¯

neglect it in our quadratic approximation to expected utility). This implies that Y = Y e

(or at least that their log di¬erence x— is of order O(||ξ||2 )), so that the steady-state level

of output under ¬‚exible prices is e¬cient (at least to second order). Since we have already

veri¬ed, above, that percentage ¬‚uctuations in the natural rate are equal (to second order)

to the percentage ¬‚uctuations in the e¬cient level of output, this actually implies that (to

second order) the natural rate of output coincides with the e¬cient level of output at all

times.

In this case, we easily obtain a very simple conclusion about the nature of optimal

3. THE CASE FOR PRICE STABILITY 27

monetary policy. For each of the individual terms in our quadratic loss function can be

shown to achieve its minimum possible value, zero, if in¬‚ation is zero at all times. We have

just discussed the fact that this is true of the terms that measure the deadweight loss due

to an ine¬cient composition of output. But in the present case, x— = 0, so that the term

in the loss function that involves the aggregate output gap is also minimized (and equal to

zero) if and only if xt = 0 at all times. Each of the aggregate supply relations (2.16), (2.19)

and (2.24) implies that this will be true in the case of zero in¬‚ation at all times.

In the informal argument just given, we have ignored the role of initial conditions that

may not be consistent with an equilibrium with zero in¬‚ation and zero output gap at all

times. If we wish to consider optimal policy from some initial date, however, the initial

conditions that happen to exist ” not necessarily ones that result from an anticipation of

the policy that will be followed from now on ” may constrain possible outcomes from now

on. Thus it may not be possible to completely stabilize both in¬‚ation and the output gap

at zero each period from the initial period onward.

In the case of the “New Keynesian” aggregate supply relation (2.19), there is no problem;

because this AS relation is purely forward-looking, the set of paths for in¬‚ation and the

output gap that are feasible from date zero onward are independent of anything that has

happened previously. In this case, πt = 0 and xt = 0 for all t ≥ 0 is a possible equilibrium

outcome regardless of initial conditions, and so this is plainly the optimal outcome given

a loss function (2.23). On the other hand, in the case of the “New Classical” AS relation

(2.16), such an outcome represents a possible equilibrium only in the case of initial conditions

under which E’1 π0 = 0. In the case of other initial conditions, it is obvious that an optimal

policy will involve π0 = E’1 π0 = 0 and x0 = 0, which reduces the period zero loss (2.18) to

its minimum possible value, zero. A loss of zero is also possible in all later periods, through

commitment to any path for in¬‚ation that is completely predictable a period or longer in

advance; in particular, a commitment to πt = 0 in all periods t > 0 is one way of achieving

the minimum possible discounted stream of losses.

Thus a commitment to price stability would again represent an optimal policy, except

28 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

possibly during a brief transition period. If we seek to choose a time-invariant policy rule

that achieves the pattern of behavior that it would be optimal to commit eventually to

follow ” as it is argued in chapters 7 and 8 that we should24 ” then a commitment to

price stability would be an example of such a policy. One reason why it might be sensible to

commit to follow such a rule from the beginning, even when E’1 π0 = 0, is that a willingness

to deliver whatever rate of in¬‚ation may have been expected “just this once” may allow the

private sector to expect that the same principle would be followed in the future as well, if it

ever turned out that Et’1 πt di¬ered from zero, in which case there would be no determinate

equilibrium level of in¬‚ation for the private sector to expect.

In the case of indexation to a lagged price index, resulting in an AS relation (2.24) that

implies in¬‚ation inertia, matters are still more complex. An equilibrium with πt = 0 and

xt = 0 for all t ≥ 0 is possible only in the case of initial conditions under which π’1 = 0.

However, it is evident from the form of the AS relation that the set of possible paths for

d

the variables πt ≡ πt ’ γπt’1 and xt from period zero onward is independent of the initial

conditions, and one also notes that the utility-based loss function (2.26) depends only on the

paths of these two variables. It is thus obvious that the optimal policy, from the point of

view of minimizing the discounted sum of losses from period zero onward given the economy™s

prior state, is one under which πt = γπt’1 and xt = 0 for all t ≥ 0. Thus the resulting path

of in¬‚ation will depend on the economy™s initial rate of in¬‚ation, prior to adoption of the

optimal policy; speci¬cally it will satisfy πt = γ t+1 π’1 .

As long as γ < 1, optimal policy in this case is again one under which the in¬‚ation

will asymptotically equal zero ” and not just on average, but in all possible states of the

world. Thus optimal policy will involve a commitment to eventual stabilization of the price

level. Furthermore, even in the transition period, the optimal path of in¬‚ation will be

deterministic, which implies that will be una¬ected by any random disturbances that may

occur from period zero onward. Thus optimal policy has the immediate consequence that the

price level should no longer be a¬ected at all by any random disturbances that may occur,

24

See the discussion in those chapters of policymaking “from a timeless perspective”.

3. THE CASE FOR PRICE STABILITY 29

be they disturbances to technology, to preferences, or to the level of government purchases.

The existence of in¬‚ation inertia only a¬ects the rate at which it is optimal for the central

bank to commit to lowering the rate of in¬‚ation (assuming an initial positive level) to its

long-run target of zero.

It is only in the special case that γ = 1 (full indexation to the lagged price index) that this

result is not obtained. In that case, the optimal commitment given initial conditions is to a

constant in¬‚ation rate πt = π’1 for all t ≥ 0; it is therefore optimal never to disin¬‚ate, once

in¬‚ation has been allowed to begin. But such a result plainly depends on the absence, in this

simple model, of any distortions associated with steady in¬‚ation. If the prices of even a few

goods are not fully indexed to the lagged price index, zero in¬‚ation is required to eliminate

relative-price distortions. Similarly, if the adjustment of prices as a result of indexation is

not continuous, but occurs only occasionally (though more frequently than re-optimizations

of pricing policy), relative-price distortions will again be completely eliminated only in the

case of zero in¬‚ation. In either case, optimal policy will steadily reduce in¬‚ation over time,

to a long-run rate of zero, as there is always a motive for some reduction in the in¬‚ation rate

(as long as the current rate remains positive) but never a motive for increasing it (as opposed

to keeping it near its recent level). Thus one ¬nds quite generally that optimal policy will

involve a commitment to price stability, at least eventually. And this means that eventually

in¬‚ation will not only equal zero on average, but zero regardless of the real disturbances that

may a¬ect the economy.

This strong conclusion regarding the optimality of complete price stability depends upon

various details of our model, as we discuss further in section xx. Nonetheless, it is interesting

to remark that it holds despite our having allowed for several di¬erent kinds of stochastic

disturbances. In particular, our framework allows for exogenous disturbances to technology,

to government purchases, to households™ impatience to consume, to their willingness to

supply labor, or to the transactions technology that determines their demand for money

balances. In the face of each of these types of disturbance, it remains optimal, under the

circumstances assumed here, for the general level of prices to be held ¬xed.

30 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

The generality of the conclusion results from a simple intuition, stressed by Goodfriend

and King (1997). Under the circumstances assumed here, the failure of prices to be con-

tinually adjusted is the only distortion that prevents rational expectations equilibrium from

achieving an optimal allocation of resources. Thus an optimal monetary policy is one that

achieves the same allocation of resources as would occur with ¬‚exible prices, if this is pos-

sible. Flexible-price equilibrium models of aggregate ¬‚uctuations (i.e., real business cycle

models25 ) are then of practical interest, not as descriptions of what aggregate ¬‚uctuations

should be like regardless of the monetary policy regime, but as descriptions of what they

would be like under an optimal policy regime. Finally, our models of optimal price-setting

imply that price stickiness will have no e¬ects upon equilibrium outcomes in the case that

monetary policy keeps the general price level completely unchanged over time, since in this

case suppliers of goods would not wish to change their prices more frequently even if it were

costless for them to do so. Thus complete price stability achieves the optimal allocation of

resources.

Verifying that it is in fact possible, in principle, to achieve this ¬rst-best allocation

through suitable monetary policy requires that we verify that we can solve the equations

of our model for the evolution of all variables (including the interest-rate instrument of the

central bank) under the assumption that πt = 0 at all times. As discussed in chapter 4,

n

this is possible, as long as the natural rate of interest rt is always non-negative; in this

case, what is required is that output equal the natural rate of output at all times, and that

25

Standard real business cycle models (King and Rebelo, 1999) di¬er from the ¬‚exible-price limit of the

model assumed here in that product markets are competitive, rather than monopolistically competitive; in

that all output is produced using inputs purchased from the same factor markets, so that there is a common

level of marginal cost for all ¬rms at any time; and in that the endogenous dynamics of the capital stock

in response to shocks is modeled, and indeed emphasized (as the only endogenous propagation mechanism

in simple RBC models). However, in the ¬‚exible-price limit of our baseline model, all goods prices move

together, and similarly the levels of production of each good, so that marginal cost is in fact the same for all

¬rms. If we assume, as in this section, that an output or employment subsidy o¬sets the distortion due to

¬rms™ market power, the ¬‚exible-price equilibrium is equivalent to that of a competitive model with a single

good. Finally, if we extend the baseline model to take account of capital-accumulation dynamics (which, as

we have argued in chapter 4, are not so important for our concerns), then the ¬‚exible-price dynamics of our

model are fully equivalent to those of a standard RBC model. Note that these models, like our “cashless”

model, abstract from real-balance e¬ects upon consumption demand, labor supply, and so on.

3. THE CASE FOR PRICE STABILITY 31

the nominal interest rate equal the natural rate of interest at all times. Hence such an

equilibrium is possible (under the quali¬cation stated), and thus such an outcome is the one

that an optimal policy would aim at.

Further discussion of exactly what kinds of output and interest-rate variations in re-

sponse to real disturbances this should imply can be found in section xx of chapter 4. Here

we recall simply that in the case of none of the types of real disturbances discussed there

would it be desirable to use monetary policy to suppress all e¬ects of the real disturbance on

aggregate output. Nor can we support, in general, so simple a conclusion as that reached by

Ireland (1996), who argues that one should use monetary policy to “insulate aggregate out-

put” against “shocks to demand”, while accommodating “shocks to supply”. Many readers

might assume that “shocks to demand” would include disturbances such as our government-

purchase or consumption-demand shocks, but the result just derived here (together with

our discussion in chapter 4 of variation in the natural rate of output) implies that it is not

optimal to stabilize output in response to these shocks. In fact, in Ireland™s theoretical anal-

ysis, “shocks to demand” refer solely to money-demand shocks, as this is the only type of

exogenous disturbances other than technology shocks that he considers.

3.2 Consequences of an Ine¬cient Natural Rate of Output

We now consider the extent to which the above conclusions must be modi¬ed in the case

that (quite realistically) we assume that ¦ > 0, so that the equilibrium rate of output under

¬‚exible prices would be ine¬ciently low. The distortions represented by the coe¬cient ¦, i.e.,

the market power resulting from monopolistic competition and the constant rate of distorting

taxation „ , introduce a wedge between this natural rate of output and the e¬cient output

level. However, this wedge is assumed to be constant over time, so that percentage changes

in the natural rate still correspond precisely (in our log-linear approximation) to percentage

changes in the e¬cient level of output.26 Thus, as shown above, the distortions associated

with a suboptimal aggregate level of economic activity are still measured a quadratic function

26

The consequences of time-variation in the size of this wedge are considered below in section xx.

32 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

of the output gap, »(xt ’ x— )2 , even if now the constant x— is assumed to be positive. (Note

that we assume even in this case that ¦, and hence x— , is of order O(||ξ||).)

While this di¬erence matters for the optimal average levels of in¬‚ation and output “ that

is, for the deterministic part of our above description of the optimal policy commitment “ it

has no e¬ect (in our log-linear approximation to optimal policy) upon the optimal responses

to shocks. We ¬rst demonstrate this in the simple context of our “New Classical” model of

price-setting. In this case, the normalized quadratic loss function (2.18) can be written

Lt = (πt ’ Et’1 πt )2 ’ 2»x— xt + »x2 , (3.1)

t

dropping the term »x—2 that is independent of policy. The second term on the right-hand

side now indicates a welfare gain from an increase in the expected output gap in any period.

However, because x— is of order O(||ξ||), a ¬rst-order approximation to the solution for xt

su¬ces to give us a second-order approximation to this term. Hence we may substitute using

the aggregate supply relation (2.16), to obtain

Lt = (πt ’ Et’1 πt )2 ’ 2θ’1 x— [πt ’ Et’1 πt ] + »x2 ,

t

recalling that » = κ/θ.

Taking the expected discounted value of such terms (and dropping the term E’1 π0 that

is independent of policy), we obtain the utility-based welfare criterion

∞ ∞

t ’1 —

β t [(πt ’ Et’1 πt )2 + »x2 ].

E0 β Lt = ’2θ x π0 + E0 (3.2)

t

t=0 t=0

Note that each of the terms proportional to x— has canceled, except the one indicating a

welfare gain from surprise in¬‚ation at date zero, the time at which a new policy commitment

is adopted. Because it is not possible to commit in advance to an in¬‚ation surprise at any

later date, the corresponding terms for dates t ≥ 1 do not matter. But this means that

allowing for x— > 0 has no e¬ect upon the nature of the optimal policy commitment, except

in the initial (transitional) period, when it is possible to take advantage of the fact that

private sector expectations of period zero in¬‚ation are already given, before the policy is

3. THE CASE FOR PRICE STABILITY 33

adopted.27 It is arguable (as we discuss in the next chapter) that it does not make sense

to behave di¬erently in this initial period than one commits to behave later, if one wants

the commitment to be credible. But regardless of how one manages the transition to the

optimal regime, it is optimal to commit to an eventual zero rate of in¬‚ation, and to a path

for in¬‚ation that is una¬ected by any stochastic disturbances.28

It might be thought that this result depends upon the fact that in the special case in

which all prices are changed every period (though some are committed a period in advance),

only unexpected in¬‚ation has an e¬ect upon output. Yet a similar conclusion is obtained in

our baseline model, with Calvo price-setting. In this case, we can similarly substitute into

(2.23) using the aggregate supply relation (2.19), to obtain

Lt = πt ’ 2θ’1 x— [πt ’ βEt πt+1 ] + »x2 ,

2

t

noting again that » = κ/θ. Taking the expected discounted value, we obtain the utility-based

welfare criterion

∞ ∞

t ’1 —

β t [πt + »x2 ] .

2

E0 β Lt = ’2θ x π0 + E0 (3.3)

t

t=0 t=0

Once again all of the terms proportional to x— cancel, except the one indicating welfare

gains from a surprise in¬‚ation in period zero. Committing in advance to non-zero in¬‚ation

in any later period does not produce any such e¬ect. For the value of the increase in output

in any period t ≥ 1 resulting from higher in¬‚ation in period t must be o¬set by the cost

of the reduction in output in period t ’ 1 as a result of expectation of that higher in¬‚ation

in period t. From the standpoint of the discounted loss criterion (3.2), the costs resulting

from the anticipation of the in¬‚ation are weighted more strongly (by a factor of β ’1 > 1),

as they occur earlier in time. On the other hand, the output e¬ect of anticipated in¬‚ation,

by shifting the short-run aggregate supply curve, is also smaller than the e¬ect of current

27

Of course, this di¬erent prescription in the case of the initial period shows that optimal policy is not

time-consistent in this case. This issue is taken up in the next chapter.

28

Of course, in this model, there is no advantage of complete price stability over any other policy that

makes in¬‚ation completely forecastable a period in advance. But in order to stress the similarity of the

results obtained under the alternative aggregate supply speci¬cations, it is worth noting that also in this

case there is no advantage to any variation in in¬‚ation in response to shocks.

34 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

in¬‚ation, by exactly the factor β < 1, with the result that the two e¬ects exactly cancel, to

¬rst order (which is to say, to second order when multiplied by x— ). Thus once again there