. 3
( 5)


Figure 4.12: E¬ects of anticipated natural-rate ¬‚uctuations under an inertial Taylor rule,
allowing for real-balance e¬ects.

Let us again consider a policy rule of the form (2.22), with coe¬cients φπ = 0.6, φx = 0.3,
and ρ = 0.7, as in the baseline case of Figure 4.7. The impulse responses to an unexpected
monetary tightening, both in the case that χ = 0 (as in Figure 4.7) and in the case that
χ = 0.02, are shown in Figure 4.14 below. (The responses shown in that ¬gure are actually
for a variant model in which there is no e¬ect on in¬‚ation or output during the quarter of the
policy shock; but the responses shown for quarter 1 and later are identical to those implied
by the model with structural equations (3.9) and (3.13). Note that the responses indicated
by the solid lines in Figure 4.14 are identical, for periods 1 and later, to those shown by the
solid lines in Figure 4.7.) We observe that the responses of none of the variables are very
di¬erent when real-balance e¬ects are considered, relative to the overall scale of variation in
the variables in question.50


χ = 0.02




0 1 2 3 4 5 6 7 8 9 10


χ = 0.02




0 1 2 3 4 5 6 7 8 9 10

Figure 4.13: E¬ects of anticipated policy shifts under an inertial Taylor rule, allowing for
real-balance e¬ects.

Thus far we have only considered how large real-balance e¬ects might be expected to
be, according to our theory, when the model parameters are calibrated on the basis of the
observed demand for money in the U.S. One might also wonder whether there is much
evidence for real-balance e¬ects in IS or AS relations from econometric estimates of those
relations. To answer this question, Ireland (2000) presents maximum-likelihood estimates of
a structural model consisting of equations (3.3), (3.9) and (3.13), together with a monetary
policy rule, estimated using quarterly U.S. data on in¬‚ation, output, nominal interest rates
In the case of the nominal interest rate, allowance for real-balance e¬ects would predict a substantially
greater decline, relative to the size of the decline predicted in the baseline case (which is very small), if we did
not assume a one-quarter delay in the e¬ects of monetary policy shocks on spending and pricing decisions.
However, this is of little importance, since empirical estimates (see Figure xx below) indicate that most of
the variation in nominal interest rates due to identi¬ed monetary policy shocks represents responses that
occur prior to any e¬ect on real expenditure ” like the response in period zero of Figure xx, rather than
the responses in periods one and later. A substantial proportionate e¬ect on the response predicted in those
later periods still makes very little di¬erence to the overall predicted variation in nominal interest rates.

and money growth. His estimated value for χσ, the coe¬cient indicating the size of the
e¬ect of variations in real money balances on aggregate demand, is -.02, with a standard
error of nearly .04; it is thus not signi¬cantly di¬erent from zero. Nor is this result obtained
only because the e¬ect is not precisely estimated; the 95 percent con¬dence interval implied
by Ireland™s results would allow the coe¬cient to have as large a positive value as .05 or as
negative value as ’.09, but it would not allow values as large as the coe¬cient of .13 assumed
in the case shown in Figures xx-xx. Thus Ireland™s estimates would imply real-balance e¬ects
even more modest than those exhibited in those ¬gures.

3.3 Monetary Policy in a “Liquidity Trap”


4 Delayed E¬ects of Monetary Policy

The simple optimizing model of the e¬ects of interest-rate changes on aggregate demand
presented above di¬ers in a number of respects from the kind of speci¬cation that is common
in macroeconometric models. One is the degree to which the model is forward-looking:
(1.4) implies that changes in expected future interest rates should a¬ect current aggregate
expenditure as much as do changes in current (short-term) interest rates. However, in
practice, expected future interest rates (at least, insofar as these expectations are based on
atheoretical time series models) co-move very closely with current short-term interest rates,
as shown by Fuhrer and Moore (1995b). Thus the coe¬cients multiplying current short
rates in the expenditure equations of econometric models might simply be proxies for a long
distributed lead of expected future short rates. While the two speci¬cations may allow a
similar ¬t to historical data, the forward-looking model has a much clearer rationale in terms
of optimizing behavior (for any of a variety of reasons, as noted in chapter 1); and taking
account of the forward-looking character of aggregate demand has important consequences

for our analysis of optimal policy in Part II of this study.
Another di¬erence between the optimizing model and common econometric speci¬cations,
however, gives us more reason to doubt the empirical realism of the theoretical model. This is
the fact that our model implies that aggregate expenditure responds to current as opposed to
lagged interest rates. Hence our model predicts that a monetary policy disturbance should
immediately e¬ect real expenditure, as shown for example in Figure xx above (where, in
fact, the maximum e¬ect on real activity occurs in the quarter of the shock). Instead, the
conventional wisdom in central banks is that monetary policy can have little immediate
e¬ect on either real activity or in¬‚ation, and VAR studies typically con¬rm this view. For
example, the estimated impulse response of real GDP shown in Figure xx of chapter 3 shows
no non-negligible e¬ect until two quarters following the quarter of the monetary policy shock.
This would be consistent with an equation for aggregate expenditure involving only interest
rates two or more quarters in the past, but not with the theory that we have derived from
consideration of the optimal timing of expenditure.
In this section, we brie¬‚y discuss ways in which our basic model can be extended to allow
for a more realistic delay in the predicted e¬ects of a change in monetary policy. As we shall
see, optimization-based theories incorporating such delays continue to imply that interest-
rate expectations should matter as much as do actual interest rates. And they continue to
imply that the key to in¬‚ation stabilization should be a policy under which the interest rate
controlled by the central bank tracks variations in the natural rate of interest as accurately
as is possible. Thus central insights derived from our basic analysis above continue to apply
in more realistic settings.

4.1 Consequences of Predetermined Expenditure

One simple way in which one can explain the observed delay in the e¬ect of monetary
policy shocks on aggregate expenditure is to assume that expenditure decisions are, to some
important extent, made in advance, just as some or all prices are determined in advance
in the pricing models introduced in chapter 3. Alternatively, we might assume that many

expenditure decisions are based on old information. We shall ¬rst present a model in which
expenditure decisions are predetermined, i.e., in which aggregate real expenditure in period
t must be decided upon in period t ’ d. Later we discuss a closely related model in which
expenditure decisions are not based on completely up-to-date information about ¬nancial
market conditions. In either case, the level of expenditure in period t must be a function
only of period t ’ d information about interest rates, and so can depend on monetary policy
shocks in period t ’ d and earlier, but not on any more recent shocks.

How plausible are such assumptions? In the strict forms in which they have just been
stated, both assumptions are obviously too extreme to be entirely realistic; nonetheless,
they capture in a simple way a feature of many actual expenditure decisions. Many of the
most interest-sensitive components of expenditure, such as investment spending, are in fact
predetermined to an important extent, owing both to the existence of planning lags and to the
fact that individual projects require expenditure over a period of time (“time to build”, in the
terminology of Kydland and Prescott, 1982), which expenditure will in most cases be worth
continuing once the project has been started. (See, e.g., Edge, 2000, for discussion of both
types of delay and their likely quantitative magnitudes.) Once again, while we here model
all interest-sensitive expenditure as if it were household consumption (abstracting from any
e¬ects of this expenditure on productive capacity), our model should really be interpreted as
a model of the timing of private expenditure more generally, and the relevant delays are the
ones that apply to the most interest-sensitive components of such expenditure. And even in
the case of household consumption, Gabaix and Laibson (2002) have argued that it makes
sense to model households as changing their planned consumption levels only intermittently;
they show that this hypothesis can help to reconcile the behavior of aggregate consumption
with the behavior of asset prices. The simple model of predetermined expenditure decisions
proposed here is in the spirit of such a model, though it involves a cruder hypothesis. (The
consequences of intermittent adjustment of the kind proposed by Gabaix and Laibson are
discussed in section xx below.)

As a simple example, let us suppose that the state-contingent consumption plan from

date t onward is chosen to maximize

β s’t u(Cs ; ξs )

subject to an intertemporal budget constraint of the form
∞ ∞
Et’d Qt’d,s Ps Cs ¤ Et’d Qt’d,t Wt + Et’d Qt’d,s [Ps Ys ’ Ts ], (4.1)
s=t s=t

where {Qt,s } is the same system of stochastic discount factors as in chapter 2. Here d is the
assumed length of the delay (in periods) between expenditure decisions and the time that
the expenditure actually occurs (or alternatively, the delay in the receipt of new information
about aggregate conditions). The intertemporal budget constraint (4.1) is required to hold
only in present value discounting back to the state of the economy at date t ’ d, because we
assume as before (sequentially) complete ¬nancial markets, allowing a household to insure
itself at date t ’ d against the realization of state at date t in which the present value of
its subsequent after-tax income is unusually low. The present value Et’d Qt’d,t Wt of period
t initial wealth is given as an initial condition, as it follows from period t ’ d wealth, the
consumption path already chosen for periods t’d through t’1, after-tax income expectations
for those same periods, and ¬nancial-market prices.
A necessary condition for an interior solution to this optimization problem is that

β d Et’d uc (Ct ; ξt ) = Λt’d Et’d [Qt’d,t Pt ] (4.2)

at each date t ’ d ≥ 0 and each possible state at that date, where Λt’d > 0 is the Lagrange
multiplier on the household™s budget constraint (4.1) looking forward from that date, indi-
cating the shadow value (in terms of period t ’ d utility) of additional nominal income at
date t ’ d. Another necessary condition, given the existence of complete ¬nancial markets,
is that
Λt Qt,s = β s’t Λs , (4.3)

for any two dates 0 ¤ t < s and any possible state at date s. Note further that using (4.3),
we can equivalently write (4.2) as

Et’d uc (Ct ; ξt ) = Et’d [Λt Pt ]. (4.4)

One can furthermore show that a system of Lagrange multipliers {Λt } and a consumption
plan {Ct } such that (i) for each date, Ct depends only on the state of the world at t ’ d;
(ii) conditions (4.3) “ (4.4) are satis¬ed in each possible state at each date; and (iii) the
intertemporal budget constraint (4.1) holds with equality, looking forward from the initial
date; su¬ce to characterize an optimal plan.51
As in section 1, it is useful to give an approximate characterization of the optimal timing
of private expenditure in the case of small disturbances by log-linearizing these equilibrium
conditions around the deterministic steady-state consumption plan that is optimal in the case
of no real disturbances and a monetary policy consistent with zero in¬‚ation and a nominal
interest rate equal to the rate of time preference. Substituting Yt ’ Gt for Ct in (4.4) and
log-linearizing, we obtain
Yt = gt ’ σEt’d »t , (4.5)
ˆ ¯
where »t ≡ log(Λt Pt /uc (C; 0)). Here the composite disturbance term is de¬ned as

ˆ ¯
gt = Gt + sC Et’d Ct ,

generalizing (2.3); note that this need not be predetermined at date t ’ d, if government
purchases are not determined as far in advance as is interest-sensitive private expenditure.52
The disturbances Gt , Ct and the coe¬cient σ > 0 are de¬ned as before.
Equation (4.3) implies that

1 + it = β ’1 Λt [Et Λt+1 ]’1 ,

which when log-linearized becomes

ˆ ˆ
it = »t + Et [πt+1 ’ »t+1 ].
Here it is assumed that the household faces prices such that the right-hand side of (4.1) is well-de¬ned
and ¬nite; otherwise, no optimal plan is possible, as before. Thus this condition is also a requirement for
equilibrium, as in chapter 2. Similarly, if we assume that households can choose to hold a non-negative
quantity of non-interest-bearing currency if they wish, then it is also a requirement for the existence of an
optimal plan that the household face a non-negative nominal interest at all times.
The assumption that gt is not public knowledge at t ’ d does not require that the government has better
information about macroeconomic conditions or a shorter planning horizon than the private sector. This
simply indicates exogenous random variation in government purchases that cannot be forecasted d periods
in advance on the basis of public information.

This can be solved forward to yield

ˆ ˆ
»t = »∞ + Et (it+j ’ πt+j+1 ),

which when substituted into (4.5) yields

ˆ ˆ
Yt = Y∞ + gt ’ σ Et’d (ˆt+j ’ πt+j+1 ),
± (4.6)

generalizing (1.4). Alternatively, our intertemporal “IS relation” can be written in di¬erenced
form as
ˆ ˆ
Yt = gt + Et’d (Yt+1 ’ gt+1 ) ’ σEt’d (it ’ πt+1 ). (4.7)

This is essentially the same form of IS relation as in the basic model, except that now it
is past expectations of current and future real interest rates that matter, rather than current
interest rates or current expectations. Such a speci¬cation implies that an unexpected change
in monetary policy in period t can have no e¬ect on aggregate real expenditure before period
t + d. Yet the delayed e¬ect of a monetary policy shock does not occur because it is actual
past interest rates, rather than current or expected future rates, that matter for current
expenditure; only past expectations regarding real rates of return from now on are relevant
to the desired substitution between current and future expenditure, even when the decision
itself has been made at a past date.
The di¬erence between this speci¬cation and an IS relation according to which current
real expenditure depends on lagged (but not current) interest rates has quite important
implications for the conduct of monetary policy. If expenditure depends on actual lagged
interest rates, it would follow that interest rates should be adjusted now to o¬set disturbances
that are expected to a¬ect the output gap in the future, even if these disturbances have no
immediate e¬ect; for once the disturbances have their e¬ect, it may be too late for an
interest-rate change to have any countervailing e¬ect on expenditure. Thus policy should
be forward-looking, and respond immediately to news that a¬ects the economic outlook
some quarters in the future. If instead, as in this optimizing model, expenditure depends
on past expectations of current and future rates, it follows that interest-rate policy a¬ects

expenditure only to the extent that it is forecastable in advance. There would therefore be no
advantage (from the point of view of output-gap stabilization) in responding at all to news
except after it has been known to the public for d periods. Instead, it would be important
to base current interest rates on past conditions (possibly including past perceptions of the
outlook for the future), in order to bring about forecastable interest-rate variations that
could be used to o¬set the e¬ects of predictable disturbances.
Under this alternative speci¬cation of our forward-looking model of the determinants of
real expenditure, the e¬ects of an (anticipated future) monetary disturbance are similar to
those analyzed earlier, only delayed in time. Suppose, for example, that there is a lag of the
same length before new price decisions take e¬ect, so that the aggregate-supply relation is
of the form
πt = κEt’d xt + βEt’d πt+1 , ) (4.8)

ˆˆ ˆ
where once again xt ≡ Yt ’ Ytn , and Ytn is de¬ned as before. (Recall equation (xx) of chapter
3.) The component of the output gap that is forecastable d periods in advance furthermore
Et’d xt = Et’d xt+1 ’ σEt’d (ˆt ’ πt+1 ’ rt ),
± (4.9)

as a consequence of (4.7). This relation together with

ˆ ˆ
xt = Et’d xt + (gt ’ Ytn ) ’ Et’d (gt ’ Ytn ) (4.10)

is in fact equivalent to (4.7).
Finally, let monetary policy be speci¬ed by a rule of the form (2.22), as in Figure 4.7.
Then for the same parameter values as are assumed in the baseline case (ρ = 0.7) of that
¬gure, but with a delay of d = 1 quarter, the predicted impulse responses to a contractionary
monetary policy shock are those shown by the solid lines in Figure 4.14. Note that for
quarters 1 and later following the shock, the responses are identical to those shown in Figure
4.7. The only di¬erence is that there are no e¬ects on output or in¬‚ation in the quarter of
the shock itself; as a consequence, the initial increase in the nominal interest rate is much
larger, as the increase in ¯t is not o¬set by the reaction to any immediate declines in output

nominal interest inflation
1 0
0.6 ’0.06
0 5 10 15 20 0 5 10 15 20

real interest output
1.5 0


χ = 0.02

0 ’0.5
0 5 10 15 20 0 5 10 15 20

Figure 4.14: Impulse responses to a contractionary monetary policy shock, when both ex-
penditure and prices are predetermined for one quarter.

or in¬‚ation (as occurred in Figure 4.7). This modi¬cation of the model greatly increases
its realism, not only because the e¬ects on output and in¬‚ation are delayed, but because
of the prediction of a transitory “liquidity e¬ect” of a monetary tightening. Note that the
predicted increase in nominal interest rates is weaker and more persistent (or even absent,
as in the cases ρ = 0.7 or 0.8) in Figure 4.7. The responses shown in Figure 4.14 are much
more similar to empirical estimates, discussed further in the next section.53
Under this extension of our basic model, the key to in¬‚ation and output-gap stabilization
continues to be the adjustment of interest rates so as to track the variations in the natural
rate of interest due to exogenous real disturbances. For example, suppose that monetary

The ¬gure also shows the predicted impulse responses in the case that we allow for real-balance e¬ects,
parameterizing monetary frictions in the way discussed in section xx above. These are not substantially
di¬erent from those predicted in the cashless case. Hence we continue to abstract from real-balance e¬ects
in the remainder of this section and in our discussion of empirical models.

policy is described by a Taylor rule of the form

ˆt = ¯t + Et’d [φπ (πt ’ π ) + φx (xt ’ x)],
± ± ¯ ¯ (4.11)

where x is consistent with π as assumed earlier, and ¯t is again an exogenous intercept term.54
¯ ¯ ±
The evolution of the forecastable components of in¬‚ation, the output gap, and the nominal
interest rate is then determined by the system of equations consisting of (4.8), (4.9) and
the equation obtained by taking the conditional expectation of both sides of (4.11) at date
t ’ d. This system of equations has a structure that is precisely analogous to the system
consisting of (1.8), (1.9) and (1.10) in the case of the basic model. It implies a determinate
rational-expectations equilibrium in exactly the same case as with the previous system, i.e.,
if and only if the coe¬cients satisfy the Taylor Principle.
In this case, the unique bounded solution is given by

πt = π +
¯ ψj Et’d (ˆt+j ’ ¯t+j + π ),
± ¯ (4.12)

ˆ ˆ
ψj Et’d (ˆt+j ’ ¯t+j + π ) + (gt ’ Ytn ) ’ Et’d (gt ’ Ytn ),
xt = x +
¯ ± ¯ (4.13)

ˆt = ¯t +
± ± ψj Et’d (ˆt+j ’ ¯t+j + π ),
± ¯ (4.14)

where the coe¬cients {ψj } for y = π, x, i are exactly the same as before. Here the solutions
for the forecastable components of all three variables are given by the forecasts d periods in
advance of the solutions previously presented in (2.18) ” (2.20). The variables πt and ˆt ’¯t
have no unforecastable components, while the unforecastable component of xt is given by
This solution implies, once again, that in¬‚ation and the output gap are both stabilized,
to the greatest extent possible, by commitment to a Taylor rule in which the intercept term
tracks variation in the natural rate of interest. The only di¬erence is that now it is only
necessary to track the ¬‚uctuations in the natural rate that can be forecasted d periods in
The assumption that the endogenous terms involve only the components of in¬‚ation and the output gap
that are forecastable at t ’ d is purely notational. For it follows from (4.8) that πt is entirely forecastable at
t ’ d, while it follows from (4.10) that the unforecastable component of xt is purely exogenous.

advance. A rule of the form (4.11) with ¯t = Et’d rt and coe¬cients φπ , φx satisfying (2.7)
implies a determinate equilibrium in which

πt = π ,
¯ Et’d xt = x

at all times. This obviously stabilizes in¬‚ation to the greatest extent possible. Since the
forecastable and unforecastable components of the output gap are necessarily uncorrelated,

var{xt } = var{Et’d xt } + var{xt ’ Et’d xt }
ˆ ˆ
= var{Et’d xt } + var{(gt ’ Ytn ) ’ Et’d (gt ’ Ytn )},

as a consequence of (4.10). Given that the second term is independent of monetary policy,
the most that can be done to stabilize the output gap is to reduce the variance of the
forecastable component to zero; the proposed policy achieves this.
Certain complications arise when we seek to combine this alternative model of the e¬ects
of interest rates on aggregate demand with aggregate-supply relations of the sort derived in
chapter 3. The derivations in chapter 3 assume that period t expenditure is chosen optimally
on the basis of period t information, so that (4.5) holds with d = 0. The place at which we
have used this assumption in chapter 3 is in replacing the marginal utility of income by a
function of current consumption, in our account of optimizing labor supply (and hence of the
marginal cost of supplying goods). More generally, the (log-linearized) real marginal cost
function for the model with ¬‚exible wages is given by

ˆ ˆ ˆ
st (i) = ω(ˆt (i) ’ Ytn ) + σ ’1 (Yt ’ Ytn ) ’ µt ,
ˆ y (4.15)

ˆ ˆ
µt ≡ »t ’ σ ’1 (gt ’ Yt ) (4.16)

is the discrepancy between the (log) marginal utility of real income and the (log) marginal
utility of consumption. When period t expenditure is chosen optimally at date t, µt = 0, and
the aggregate supply relations derived in chapter 3 are correct. But when interest-sensitive

private expenditure must be chosen in advance, (4.5) implies only that Et’d µt = 0, whereas
µt need not equal zero.
Now suppose that even though the aggregate index of demand (i.e., the demand for the
composite good) is determined d periods in advance, the way that this demand is allocated
across the various di¬erentiated goods is not committed in advance. It follows that a supplier
who considers a price change that will take e¬ect in less than d periods will still calculate
the optimal price taking into account the e¬ect on demand for its good from the ¬rst period
in which the price change takes e¬ect. Then in the case of random intervals between price
changes of the kind assumed by Calvo, and a delay of s periods before a newly chosen price
takes e¬ect, the aggregate-supply relation should be of the form

πt = κEt’s xt ’ Et’s µt + βEt’s πt+1 .) (4.17)

where > 0 is the elasticity of average real marginal cost with respect to the level of

aggregate output.
Only in the case that s ≥ d could the Et’s µt term be neglected (as in (4.8) above). If
s < d, one would instead need to use the form (4.17), together with the relation
ˆ ˆ
Et [ˆt+j ’ πt+j+1 ] + σ ’1 [(Yt ’ gt ) ’ Et (Yt+d ’ gt+d )]
µt = ± (4.18)

relating µt to observables using (4.5).55 The presence of the Et’s µt term indicates a moder-
ating e¬ect on expected supply costs in period t, and hence on in¬‚ationary pressure, of an
expectation at t ’ s of real rates of return between periods t and t + d ’ s that are higher
than those that were anticipated at the time that expenditure was planned for periods t
through t + d ’ s ’ 1. Unexpectedly high real rates of return increase the value of income in
period t, and so lower average wage demands, even if they occur as a result of shocks that
(because unanticipated) are unable to a¬ect aggregate demand.
There appears, however, to be little evidence of an e¬ect of interest rates on supply
costs of the kind implied by the Et’s µt term in (4.17). If anything, unexpected interest-rate
See the empirical model of Rotemberg and Woodford (1997) for an illustration.

increases probably increase supply costs in the short run, as found by Barth and Ramey
(2000) and Christiano et al. (2001). It is thus important to note that this e¬ect appears in
(4.17), even under the assumption that s < d, only in the case that the delay in the e¬ect
of interest rates on expenditure is derived from a planning delay rather than in delay in
obtaining up-to-date information about ¬nancial conditions.
Under an alternative interpretation of (4.6), period t spending decisions are made at
date t, but on the basis of an estimate of the household™s marginal utility of real income that
re¬‚ects information available at date t ’ d rather than the complete information available to
¬nancial market participants at date t. Let us suppose that each household has an agent that
optimally manages its investments. This agent, with full information about ¬nancial market
conditions (as well as about the household™s tastes and labor income prospects) produces an
estimate of the marginal utility of additional wealth; individual household members who go
into the goods markets then purchase individual goods to the point at which the marginal
utility from an additional dollar of spending on a given good equals the current estimate of
the marginal utility of wealth. But suppose there is a time delay in the transmission of the
¬nancial advisor™s estimate, so that in period t the household™s spending decisions are based
on the ¬nancial advisor™s period t ’ d estimate of what the household™s marginal utility of
real wealth in period t would be.
Suppose further that the household members simply use this estimate, rather than up-
dating it on the basis of what they should be able to infer about unexpected changes in
¬nancial conditions from the prices that they observe.56 The level of expenditure Ct is then
determined at date t to satisfy the ¬rst-order condition

uc (Ct ; ξt ) = Et’d [Λt Pt ], (4.19)

where the right-hand side is the signal transmitted by the ¬nancial advisor at date t ’ d.
This has identical implications to (4.4), except that taste shocks ξt that are not forecastable
at date t ’ d can still a¬ect private expenditure in this version.57 Equating Ct with Yt ’ Gt
There is admittedly an element of bounded rationality in this assumption, that is not required if we
assume that the household has committed itself in advance to a particular level of real expenditure.

and log-linearizing, we again obtain (4.5) except that in this case gt is again de¬ned as in
(2.3). We then again obtain an IS relation of the form (4.6).
Under this alternative interpretation, however, each household™s labor supply decisions
should also be a¬ected by its imperfect information about current ¬nancial-market condi-
tions. Thus real wages, and hence real marginal costs, should depend on Et’d »t rather than
upon the true value of »t , as this would be evaluated by the household™s ¬nancial advisor.
ˆ ˆ
But (4.5) implies that Et’d »t can be written as ’σ ’1 (Yt ’ gt ), even if »t cannot. Thus
we obtain (4.15) without the µt term, and correspondingly (4.17) without the Et’s µt term.
That is, we obtain exactly the form of AS relation derived in chapter 3 for the case d = 0.

4.2 Small Quantitative Models of the E¬ects of U.S. Monetary

A model only slightly more complex than those just described is used by Rotemberg and
Woodford (1997, 1999a) as a basis for quantitative analysis of alternative interest-rate rules
for the U.S. economy. Rotemberg and Woodford assume an intertemporal IS equation of the
form (4.7) with a delay of d = 2 quarters, and interpret the delay as due to predeterminedness
of interest-sensitive private expenditure. Their AS equation is instead a more complex version
of (4.8), in which the delay required before revised prices take e¬ect is not the same for all
goods. Instead, it is assumed that for a fraction • of all goods, a new price that is chosen in
period t (or at any rate, on the basis of public information in period t) applies to purchases
beginning in period t + 1, while for the remaining goods, a new price chosen in period t takes
e¬ect only beginning in period t + 2. In the case of both types of goods, it is assumed (as in
the Calvo model) that a fraction 1 ’ ± of all goods prices are revised each period, with the
price of each good having the same probability of being revised in any given period.
In this case, the aggregate supply relation (4.17) generalizes to
1 κ ψ
πt = κEt’1 xt ’ Et’1 µt + βEt’1 πt+1 + {κEt’2 xt + βEt’2 πt+1 } , ) (4.20)
1+ψ 1+ψ
An appealing feature of this alternative derivation is that it makes it clear why expenditure at date t on
individual goods should still depend on the prices of the individual goods, as we have assumed, rather than
only on the forecast of their prices at date t ’ d.

where ψ ≡ (1 ’ •)/•± > 0, while (4.18) implies that

ˆ ˆ
Et’1 µt = Et’1 [ˆt ’ πt+1 ] + σ ’1 [(Yt ’ gt ) ’ (Yt+1 ’ gt+1 )].
± (4.21)

(Note that when d = 2, (4.7) implies that the value of the second term in square brackets
here is known at date t ’ 1, so that we may omit the conditional expectation operator
for that term.) The log-linearized structural equations of the Rotemberg-Woodford model
then consist of the intertemporal IS relation (4.7) with d = 2, the aggregate supply relation
obtained by substituting (4.21) into (4.20), a speci¬cation of the exogenous disturbance
processes {gt , Ytn }, and a monetary policy rule specifying ˆt as function of its own history,
current and lagged values of in¬‚ation and output, and a serially uncorrelated58 exogenous
monetary policy shock. (Once the processes {gt , Ytn } have been speci¬ed, the evolution of
the natural rate of interest rt that appears in (4.7) is given by (1.11).)
The monetary policy rule, the laws of motion for the exogenous disturbances, and certain
parameters of the structural equations are also speci¬ed so as to allow the model to ¬t as well
as possible the joint evolution of short-term nominal interest rates, in¬‚ation and output in
the U.S. economy. Rotemberg and Woodford characterize the co-movements of these latter
three variables by estimating an unrestricted VAR model for the federal funds rate, the rate
of growth of the GDP de¬‚ator, and the linearly detrended log of real GDP, using quarterly
data for the sample period 1980:1-1995:2. These particular measures of the interest rate,
in¬‚ation and output are used following Taylor (1993), as it is desired that one equation of the
VAR should represent an estimate of the Fed™s reaction function. The sample period begins
at the beginning of 1980 because of the general recognition that an important change in the
way that monetary policy was conducted in the U.S. occurred around this time. (Recall the
discussion in chapter 1 of the alternative interest-rate rules estimated for di¬erent sample
periods. An even shorter sample period might be preferred on the same ground ” say, a
post-1987 sample as in Taylor, 1993, 1999b ” but this would allow even less precise estimates
Note that if a su¬cient number of lags of the endogenous variables are included, this speci¬cation is
equivalent to one in which the monetary policy disturbance is allowed to be an arbitrary autoregressive

of the impulse responses to a monetary policy shock.)
In the VAR model, ˆt , πt+1 , and Yt+1 are regressed on three lags of each of these variables,
with the coe¬cients otherwise unrestricted. (Additional lags are not included as they were
not found to be signi¬cant.) The particular lags that are included in the case of each variable
are chosen in this way because the model implies that ˆt , πt+1 and Yt+1 are all part of the same
information sets: these variables are known by period t (in particular, before the period t + 1
interest-rate decision is made), because both in¬‚ation and output are predetermined,59 but
not yet known in period t’1 (i.e., before the period t interest-rate decision is made). Because
the model implies that the period t interest-rate decision cannot a¬ect the determination
of either period t output or period t in¬‚ation, an OLS regression of ˆt on the lags of all
three variables (which include πt and Yt ) should identify the coe¬cients of the monetary
policy rule, and the residual of this equation should identify the sequence of monetary policy
shocks.60 The two VAR residuals orthogonal to this one are instead interpreted as the two
innovations in the joint exogenous process for the disturbances {gt , Ytn }.
With this identi¬cation of the historical monetary policy shocks, the just-identi¬ed VAR
model can be used to estimate the impulse responses of all three variables to an monetary
policy shock. Figure 4.15 plots the estimated responses, together with the associated (+/- 2
s.e.) con¬dence intervals, in the case of a one-standard-error innovation in the federal funds
rate, i.e., an unexpected monetary tightening. By construction, the funds rate increases,
while there is no e¬ect on either output or in¬‚ation in the quarter of the shock. However,
the results also indicate (as in Figure xx of chapter 3) that there is no noticeable e¬ect on
output in the following quarter, either, though output sharply declines in the second quarter
following the shock; this is the reason for the inclusion of a delay d = 2 quarters in the
determination of private expenditure in the Rotemberg-Woodford model. The contraction
of output relative to trend persists for several quarters, though Rotemberg and Woodford ¬nd
Note that it is assumed that the composite exogenous disturbance gt is known at date t ’ 1, i.e., prior
to the determination of the period t interest rate. Since government purchases are in fact typically budgeted
in advance, this is not implausible.
The assumptions used to identify the monetary policy shock here are common in the structural VAR
literature on this question; see, e.g., Christiano et al. (1999).

’3 Output
x 10



0 1 2 3 4 5 6 7 8 9 10
’3 Inflation Rate
x 10



0 1 2 3 4 5 6 7 8 9 10
’3 Interest Rate
x 10




0 1 2 3 4 5 6 7 8 9 10

Figure 4.15: Predicted [solid line] and estimated [dashed line] impulse responses to a mon-
etary policy shock. Dash-dotted lines indicate bounds of the con¬dence intervals for the
estimated responses. Source: Rotemberg and Woodford (1997).

a peak output e¬ect in the second quarter following the monetary shock, which is sooner
than what is indicated by most studies using a longer sample.61 (Compare Figure xx of
chapter 3.) The e¬ects of a monetary policy shock on in¬‚ation are less well-estimated, but
the point estimates indicate lower in¬‚ation for many subsequent quarters as a result of the
policy tightening.

One equation of the VAR can also be interpreted as indicating the Fed™s reaction function
over this period. The estimated coe¬cients on the three lags of the funds rate itself are all

Boivin and Giannoni (2001) compare the results that would be obtained using a similar method to analyze
the response to monetary policy shocks in a pre-1980 sample. They argue that much of the di¬erence can
be accounted for by the change in their estimated monetary policy rule between the pre-1980 and post-1980
samples, though they also ¬nd that a better ¬t to the pre-1980 responses is possible in the case of a model
that incorporates additional grounds for persistence in the e¬ects of interest-rate changes on private-sector
expenditure, as discussed below.

positive, and they sum to 0.69; this implies substantial inertia in the Fed™s interest-rate
policy, as in the estimated rules discussed in chapter 1. The sums of the coe¬cients on
current and lagged values of in¬‚ation and detrended output are also both positive, and imply
long-run interest-rate responses to sustained increases in these two variables of ¦π = 2.13 and
¦y = 0.47 respectively. Thus except for the additional dynamics implied by the inclusion
of lags of all three variables, the estimated reaction function is similar to Taylor™s (1993)
characterization of Fed policy under Greenspan™s chairmanship.
Rotemberg and Woodford then estimate certain parameters of their model so as to ¬t the
estimated responses to a monetary policy shock as closely as possible, given that monetary
policy is described by the estimated interest-rate rule. It can be shown that the model™s
predicted responses depend only on the parameters β, σ, κ, and a certain function of ω and
ψ, in addition to the coe¬cients of the monetary policy rule.62 There are thus at most four
free parameters that can be chosen (within certain a priori bounds) so as to improve the
model™s ¬t. Furthermore, because the model implies that β ’1 ’ 1 should equal the long-run
average real rate of interest, Rotemberg and Woodford calibrate the discount factor β = 0.99
on this ground, rather than using information about the responses to shocks to estimate this
parameter. They then estimate the values of the other three parameters that minimize the
distance between the predicted impulse responses and those implied by the unrestricted VAR
estimates. The predicted impulse responses in the case of these parameter values are shown
by the solid lines in Figure 4.15.
Note that the model can account quite well for both the size and persistence of the
estimated response of real GDP. The predicted output response is in fact essentially the same
as the one shown in Figure 4.7 for the inertial policy rule with ρ = 0.7, with the magnitude
of the policy shock appropriately rescaled63 and the response delayed for two quarters. For
the IS and AS relations of the Rotemberg-Woodford model are identical to those of the
Parameter identi¬cation, along with other aspects of the estimation strategy, are discussed in detail in
the Appendix to the 1998 working paper version of their paper.
The size of the e¬ective shift in the intercept term of the policy rule in the second period following the
shock is found by substituting the predicted responses in the previous two periods for the lagged interest-rate
and in¬‚ation terms in the rule.

baseline model if one takes expectations conditional upon information two quarters earlier.
It follows that the predicted responses of all variables two or more quarters following the
shock are the same as in our earlier analysis, if the initial conditions that apply two quarters
later are appropriately adjusted; the result follows for the same reason as in the case of
the one-quarter delay shown in Figure 4.14. Thus the predicted output response would be
exactly the same as in Figures 4.7 or 4.14, but with a two-quarter delay, if the policy rule
were of the simpler form assumed in those ¬gures.

The model also accounts well for the estimated response of the funds rate itself to a
monetary policy shock: the theoretical model predicts, as the VAR indicates, that the funds
rate returns essentially to the level that would have been predicted in the absence of any
shock within two quarters following the shock. (This occurs for the same reason that in
Figure 4.7 there is very little response of the nominal interest rate to the policy shock, and
that in Figure 4.14 the interest rate returns to its normal level after one quarter.) Note that
this fact does not mean, as is sometimes supposed, that an “interest-rate channel” cannot
account for the timing of the observed real e¬ects of monetary policy disturbances, which
instead occur only beginning two quarters after the shock. The reason is that it is the real
rate of interest, and not the nominal rate, that primarily matters for aggregate demand; and
if a monetary tightening is expected to lower in¬‚ation for several quarters (as predicted both
by the theoretical responses and the VAR), this implies a higher real rate of interest that
persists for several quarters, even if the nominal rate has returned to its normal level. The
predicted and estimated in¬‚ation responses do not match as well as in the case of the other
two variables, but it should be noted that the estimated responses are highly imprecise.

The parameter values required for this degree of ¬t are consistent with the a priori
restrictions implied by the model™s microeconomic foundations. The values estimated by
Rotemberg and Woodford are shown in Table 4.1. The table also includes their “calibrated”
values for several model parameters that cannot be identi¬ed from the impulse responses;
these are included so as to provide further insight into the economic signi¬cance (and plau-
sibility) of the estimated parameters, and also because some of the additional parameters

Table 4.1: Parameter values in the quarterly model of Rotemberg and Woodford (1997).

± 0.66
β 0.99
• 0.63
ψ 0.88
σ ’1 0.16
ν 0.11
φ’1 0.75
ωw 0.14
ωp 0.33
ω 0.47
(θ ’ 1)’1 0.15
ζ 0.14
κ .024

matter for the welfare evaluation of alternative policies.

Like most of the equilibrium business-cycle literature, they assume a Cobb-Douglas ag-
gregate production function, f (h) = h» , and calibrate the elasticity » to be 0.75 on the basis
of the observed labor share in national income.64 This implies that the component of the
elasticity of real marginal cost with respect to output that is due to the diminishing marginal
product of labor should equal ωp = 0.33. They furthermore propose, on the basis of other
structural VAR studies that estimate real wage responses, that the elasticity of the real wage
with respect to output changes not associated with any change in production possibilities is
approximately of the magnitude 0.3. This would imply an overall elasticity of average real
marginal cost with respect to aggregate output of = 0.63.

= ω + σ ’1 , the estimated value of σ ’1 implies that
Since in their cashless model, mc

ω = ωw + ωp = 0.47, and hence that ωw = 0.14. Since ωw = νφ, where ν measures the
curvature of the disutility of labor function and φ is the inverse of the labor elasticity, the

Because the assumed value of θ implies a ratio of price to marginal cost of 1.15, a labor elasticity of
0.75 implies that one should observe a labor share of 0.75/1.15 = 0.65, which is about what is observed on
average for the U.S.

implied value of ν is 0.75(0.14) = 0.11. This is a very low degree of curvature, but still
a positive one, and so the implied preferences for the representative household satisfy the
standard concavity restrictions. While the implication of highly elastic responses of voluntary
labor supply to real wage variations may be judged implausible, the problem is a familiar
one for equilibrium business-cycle models (such as standard RBC models) that incorporate
a wage-taking representative-household model of labor supply.65
Given this value for ω, it is then possible to estimate a value for ψ, namely 0.88. This
value is also positive, as required by the model. A variety of values of ± and • would be
consistent with this value for ψ. Rotemberg and Woodford calibrate ± = 0.66 on the basis
of survey evidence on the typical frequency of price changes in the U.S. economy, such as
that of Blinder et al. (1998, Table 4.1). (This value of ± implies a mean time between price
changes of 3 quarters.) The estimate of ψ then implies the value • = 0.63, which is between
zero and one as required by the theory.
Finally, given these values for ± and β, the estimated value of κ is consistent with the
theoretical prediction that66
(1 ’ ±)(1 ’ ±β)
κ≡ ζ
in the case that ζ (the measure of “real rigidity” from a variety of sources, discussed in
chapter 3) is equal to 0.14. Under the Rotemberg-Woodford assumptions regarding the
structure of production costs and demand, the degree of real rigidity should be given by

ζ= ,
1 + ωθ

as shown in chapter 3. Hence the required value of ζ is consistent with the values obtained for
and ω in the case that the elasticity of substitution among alternative di¬erentiated goods

is equal to θ = 7.88. This value is greater than one, as required by the theoretical model, and
Rotemberg and Woodford point out that the assumption of extremely elastic labor supply is not necessary
to make their estimated low value of κ consistent with the theoretical restrictions of the model, but only
to make the model consistent with the observation of an only modestly procyclical real wage response to
monetary policy shocks.
See equation (xx) of chapter 3. This corresponds to the case of speci¬c factor markets, CES preferences
over di¬erentiated goods, and no intermediate inputs, as in the baseline model developed in that chapter.

pi, pi(’k) pi, R(’k)
pi, y(’k)
0.03 0.015 0.03

0.02 0.02
0.01 0.01
0 0

’0.01 0 ’0.01
0 10 20 0 10 20 0 10 20
’3 y, pi(’k) y, y(’k) y, R(’k)
x 10
5 0.06 0.02

0 0.04 0.01

’5 0.02 0

’10 0 ’0.01

’15 ’0.02 ’0.02
0 10 20 0 10 20 0 10 20
R, pi(’k) R, y(’k) R, R(’k)
0.04 0.02 0.08

0.03 0.06
0.02 0.04
0.01 0.02

0 0.005 0
0 10 20 0 10 20 0 10 20

Figure 4.16: Predicted [solid line] and estimated [dashed line] second moments of quarterly
U.S. data, 1980:1-1995:4. Source: Rotemberg and Woodford (1997).

is also a fairly plausible magnitude; it is neither so small as to imply an implausible degree
of market power for the typical producer in the U.S. economy (it implies an average markup
of prices over marginal cost of less than 15 percent), nor so large as to make it implausible
that suppliers should leave their prices unchanged for a period of nine months on average.

It is thus possible to account for the estimated impulse responses to an identi¬ed monetary
policy shock fairly accurately, assuming parameter values that are not only theoretically
possible, but that are also consistent with a variety of other observations about the U.S.
economy. Once the parameters of the structural equations have been assigned numerical
values in this way, it is then possible to specify the joint stochastic process for the two
composite real disturbances {gt , Ytn } so as to match other features of the estimated joint
distribution of the three time series characterized by the Rotemberg-Woodford VAR. Figure
4.16 shows the extent to which this is possible. The ¬gure plots the estimated autocovariance

and cross-covariance functions for the three variables (the ones implied by the estimated
VAR model), and together with these the predictions of the theoretical model, given the
Rotemberg-Woodford speci¬cation of the exogenous disturbance processes. These match
quite closely. For example, it is worth noting that the model has no di¬culty accounting for
the observed degree of persistence of either the ¬‚uctuations in in¬‚ation or in the deviations
of output from trend over this sample period. Nor does it have di¬culty accounting for such
often-remarked features of the data as the negative correlation between output and lagged
interest rates and the positive correlation between output and interest-rate leads.
Of course, these last successes of the theoretical model are mainly a result of assuming
real disturbances with the proper statistical properties; Rotemberg and Woodford impose no
a priori restrictions upon the joint law of motion for the two composite disturbances, except
that they be stationary processes that evolve independently of the monetary policy shocks.
Because the structural VAR model implies that only a small amount of the variability of any
of the three variables is ultimately caused by the identi¬ed monetary policy disturbances,
the restriction that the assumed real disturbances be independent of the identi¬ed monetary
policy shocks constrains only slightly their ability to choose disturbance processes that imply
the desired second moments for the data series. If one were instead to start with tightly
parameterized a priori assumptions about the laws of motion of the disturbance processes,
as is common both in the literature on maximum-likelihood estimation of structural macroe-
conometric models and in the literature on calibrated equilibrium business-cycle models, one
might instead ¬nd that the model would ¬t the properties of the time series less well.67 Yet
the theory developed here gives one no reason to assume particular kinds of “simple” laws
of motion for the real disturbances. Indeed, it implies that each of the real disturbances is
actually a composite of many di¬erent sorts of underlying real disturbances, and that many
kinds of real disturbances should a¬ect both gt and Ytn , albeit with di¬erent dynamics. Thus
For criticism of the model on this ground, see Fuhrer (1997), who assumes that the real disturbances
should be serially uncorrelated. McGrattan (1999) also ¬nds considerably worse performance when the real
disturbance processes are assumed to consist solely of a production-function residual and variation in real
government purchases. But the theoretical model allows for preference disturbances as well, which may play
an important role in accounting for the historical time series.

there is no reason to expect the two processes to have simple serial correlation properties, or
to be uncorrelated with one another.
Because the approach to the design of optimal monetary policy rules emphasized in
chapter 8 below makes the form of an optimal policy rule independent of the statistical
properties of the real disturbances, I do not here further discuss the details of the disturbance
processes estimated by Rotemberg and Woodford. However, it is perhaps of some interest
to note the implications of their estimates for the question of the variability of the natural
rate of interest. In their IS relation (4.7), it is only the forecastable component Et rt+2 that
matters, owing to the assumption that interest-sensitive private expenditure is predetermined
two quarters in advance. Thus it seems most appropriate to ask about the variability of this
exogenous term that is implied by the residuals of the estimated IS relation. As reported
in Woodford (1999a), this (annualized) series has a standard deviation of 3.72 percentage
points. Since the estimated long-run average funds rate and in¬‚ation rate imply a mean
natural rate of interest of only 2.99 percentage points, this implies that even a one-standard
deviation decline in the natural rate involves a natural rate well below zero.68
Thus these estimates imply that the natural rate of interest should be negative fairly
ˆn ¯
often. This means that a policy under which the funds rate would always equal Et’2 rt + π ”
as would be necessary, according to the Rotemberg-Woodford model, in order to completely
stabilize both Et’2 πt and Et’2 xt ” would be consistent with the zero lower bound for the
funds rate only if the in¬‚ation target π is well above zero. Thus the zero bound creates a
tension between in¬‚ation stabilization and the pursuit of a low average rate of in¬‚ation, as
is discussed further in section xx of chapter 6.69
Using a quite di¬erent approach, Laubach and Williams (2001) estimate a natural-rate series for the
U.S. that exhibits substantial variability, though not quite so much. When they assume that the component
of the natural rate of interest due to factors other than variation in the economy™s long-run growth rate is
a stationary AR(2) process, they estimate the standard deviation of the natural rate to be 1.98 percentage
points. Their estimate of low-frequency variation implies that the natural rate was as low as only 10 basis
points in late 1994. Since their method seeks only to isolate a low-frequency component of natural-rate
variations, while theory indicates that higher-frequency variations are likely as well, it is quite plausible that
the overall variability of the natural rate should be greater than that estimated by Laubach and Williams,
even assuming that they correctly identify low-frequency variations.
The tradeo¬ between these two objectives is demonstrated quantitatively in the context of the estimated
model by Figure 5 in Rotemberg and Woodford (1997).





0 1 2 3 4 5 6 7 8
Inflation Rate



0 1 2 3 4 5 6 7 8
Real Wage

0 1 2 3 4 5 6 7 8
Interest Rate
0 1 2 3 4 5 6 7 8

Figure 4.17: Predicted [solid line] and estimated [dashed line] impulse responses to a mone-
tary policy shock. Source: Amato and Laubach (2001).

Amato and Laubach (2002) extend the analysis of Rotemberg and Woodford by adding a
real wage series to the VAR, and estimating the impulse response to wages as well prices to a
monetary policy shock. Their estimated impulse responses to an identi¬ed monetary policy
shock are shown in Figure 4.17. They ¬nd that the real wage response is never signi¬cantly
di¬erent from zero, just as in the estimates of Christiano et al. (2001) reported in Figure xx
of chapter 3. As a result, the estimated responses (taking account of the real wage response
along with the others, which are changed very little by the inclusion of the additional series in
the VAR) are no longer consistent with the simple Rotemberg-Woodford model, with wage-
taking households, for any assumed preference parameters. Amato and Laubach extend the
model to allow wages as well as prices to be sticky, along the lines of the models discussed in
section xx of chapter 3. The aggregate-supply block of their model consists of a pair of wage
and price in¬‚ation equations that generalize equations (xx) “ (xx) of chapter 3 to incorporate

Table 4.2: Parameter values in the quarterly model of Amato and Laubach (2002).

±w , ±p 0.66
β 0.99
•w , •p 0.56
σ ’1 0.26
ωw 0.27
ωp 0.33
(θw ’ 1)’1 0.13
(θp ’ 1)’1 0.19
ξw .066
ξp .058
κw .035
κp .019

predetermined wage and price changes like those in the Rotemberg-Woodford model.
The numerical parameter values in the Amato-Laubach model are given for comparison
purposes in Table 4.2. Here the meaning of the parameters is the same as in the presentation
of the model of Erceg et al. (2000) in chapter 3, except for the introduction of the parameters
•w , •p (assumed for simplicity to take a common value) that indicate the fraction of newly
revised wages and prices respectively that take e¬ect after only one quarter. The values of
±p , β, and ωp are calibrated as in Rotemberg and Woodford, and ±w = ±p and γw = γp are
assumed to reduce the number of free parameters. Given the calibrated value for ωp , the
ratio κp /ξp is also ¬xed. The free parameters σ, ξw , κw , κp and the common value of ψ in
both in¬‚ation equations are then estimated to minimize a measure of the distance between
the theoretical and estimated impulse responses. These estimates then imply the remaining
parameter values listed in the table.
The Amato-Laubach model parameter values imply predicted responses of output, in¬‚a-
tion and the nominal interest rate quite similar to those of the Rotemberg-Woodford model,
but in addition the model now implies a smaller real wage response, and one that is more
gradual and more persistent than the output response. (The responses predicted by their
model, given the estimated parameter values, are shown in Figure 4.17.) The model yields

these predictions as a result of substantial wage as well as price stickiness, indicated by the
small estimated value for κw . While κw is estimated to be larger than κp , so that the real wage
is predicted to move slightly procyclically (see Figure xx of chapter 3), the two coe¬cients
are similar in magnitude, and the predicted real wage response is quite small. Note that
despite the fact that wages are estimated to be relatively sticky, the estimated model still
implies very low curvature of the disutility of labor supply. Even though ωw can no longer be
directly inferred from the relative magnitudes of the output and real-wage responses, it can
be inferred from the relative magnitudes of κw and ξw ; the fact that Amato and Laubach
estimate that ξw is large relative to the size of κw can be reconciled with the underlying
microeconomic foundations of the model only if the marginal disutility of working does not
rise much with increases in employment.
Another weakness of the simple Rotemberg-Woodford model is its failure to predict an
in¬‚ation response as persistent as the one indicated by the VAR. This response is estimated
quite imprecisely by their VAR; but as discussed in chapter 3, a large number of other
studies also conclude that in¬‚ation responses are both more delayed and more persistent
than predicted by their model. A simple way to increase the realism of this aspect of their
model is to assume that prices are indexed to past in¬‚ation between the occasions on which
they are re-optimized, as discussed in section xx of chapter 3.70 Let us assume ¬‚exible
wages and staggered price-setting, with individual prices partially indexed to a lagged price
index, and again let the parameter 0 ¤ γ ¤ 1 indicate the degree of indexation between
the occasions on which prices are revised. Suppose that there is a delay of s = 1 quarter
before price revisions take e¬ect, but once again a delay of d = 2 quarters before interest-rate
disturbances can a¬ect real expenditure. Finally, suppose that this latter delay is interpreted
as an information delay rather than actual precommitment of real expenditure, so that the
µt term disappears from (4.15) as discussed at the end of the previous section.71 With the
Boivin and Giannoni (2001) also estimate a variant of the Rotemberg-Woodford model that allows for
in¬‚ation inertia of a similar sort to that discussed here, but derived from the hypothesis of backward-looking
“rule-of-thumb” price-setters proposed by Gali and Gertler (1999). Their estimated fraction of backward-
looking price-setters for the post-1980 sample period implies a degree of in¬‚ation inertia that is only very
slightly greater than that implied by the model discussed in the text in the case that γ = 1.

’3 Output
x 10



0 1 2 3 4 5 6 7 8 9 10
’3 Inflation Rate
x 10



0 1 2 3 4 5 6 7 8 9 10
’3 Interest Rate
x 10




0 1 2 3 4 5 6 7 8 9 10

Figure 4.18: Predicted [solid line] and estimated [dashed line] impulse responses to a mon-
etary policy shock, in the case of a model with in¬‚ation inertia. Estimated responses are
again taken from Rotemberg and Woodford (1997).

addition of the partial indexation, and specializing to the case just described, the aggregate
supply relation (4.17) becomes

πt ’ γπt’1 = κ Et’1 xt+1 + + ±β Et’1 (πt+1 ’ γπt ). (4.22)

In this variant model, it is no longer necessary to assume that some goods price changes
must be determined two quarters in advance (as in the Rotemberg-Woodford model) in order
to account for the fact that the maximum decline in in¬‚ation does not occur in the quarter
If one instead adopts the alternative interpretation, as in Rotemberg and Woodford, Amato and Laubach,
and Boivin and Giannoni, one ¬nds that the model best ¬ts the estimated impulse responses when the
coe¬cient on Et’1 µt in the AS relation is made as small as possible. This indicates the superiority of the
interpretation proposed here. Note that the problem does not appear in Rotemberg and Woodford (1997)
because the Et’1 µt term a¬ects only the predicted response of in¬‚ation in the ¬rst quarter following the
quarter of the shock, and in their model that particular response can be matched by assuming a su¬ciently
large value for ψ > 0. Here instead the assumption a¬ects the degree of ¬t because we impose ψ = 0.

Table 4.3: Parameter values in an estimated quarterly model with in¬‚ation inertia.

± 0.83
β 0.99
γ 1.00
ψ 0
σ ’1 0.22
ζ 0.15
κ .0052

following the monetary policy shock. As shown in Figure 4.18, the model proposed here
predicts a further decline in in¬‚ation in the second quarter even when ψ = 0; hence we
impose the assumption that ψ = 0 for the sake of simplicity.
Except for the replacement of (4.20) by (4.22), we continue to assume the same model
as in Rotemberg and Woodford. The model parameters σ, κ, and γ are estimated so as
to match the same three impulse responses as in Figure 4.8, assuming β = 0.99 as before,
and imposing the constraint that 0 ¤ γ ¤ 1. The estimated parameter values are shown in
Table 4.3.72 We ¬nd the best ¬t when γ = 1, implying full indexation to lagged in¬‚ation,
as assumed by Christiano et al. (2001).73 The resulting aggregate supply relation is fairly
similar to the one proposed by Fuhrer and Moore (1995a, 1995b), and used in other small
econometric models of the U.S. monetary transmission mechanism, such as Orphanides et
al. (1997).74
These parameters are those that minimize the simple sum of squared deviations between the predicted
and estimated responses of the three variables in the ¬rst ten quarters following the shock, i.e., the statistic
shown in the last column of Table 4.4.
This implies roughly equal weights on lagged in¬‚ation and expected future in¬‚ation in the aggregate
supply relation (4.22), which is similar to what Boivin and Giannoni (2001) ¬nd when the estimate a model
with a similar, but not quite identical, speci¬cation. In addition to introducing in¬‚ation inertia through
“rule-of-thumb” price-setting of the kind hypothesized by Gali and Gertler (1999), the model of Boivin and
Giannoni di¬ers from this one in assuming that only aggregate purchases are predetermined two quarters in
advance, and not the purchases of individual goods.
Indeed, the three-equation model, consisting of (4.7), (4.22), and the estimated federal funds-rate reaction
function, is quite similar in form to the three-equation model of Fuhrer and Moore (1995b). The main
di¬erence is in the way that delays in the e¬ects of monetary policy are introduced here; Fuhrer and Moore
assume an AS relation that involves no delay, and an IS relation in which output responds to a long-term
real interest rate one quarter earlier, as opposed to the expectation two quarters earlier of the present
quarter™s long-term real rate (as in Rotemberg and Woodford and here). Other di¬erences in the Fuhrer and

Table 4.4: Root-mean-square deviations between predicted and estimated responses for each
of three variables, averaged over quarters 1 through 10 following the monetary policy shock.
(Each reported value should be multiplied by 10’3 .) The ¬nal column indicates the sum of
squared deviations over the ten quarters, summing over the three variables. (These values
should be multiplied by 10’6 .)

Model rms(y) rms(π) rms(i) SSD
γ=0 0.30 0.73 1.02 10.49
γ=1 0.40 0.59 0.58 8.48

Because this speci¬cation predicts a more inertial response of in¬‚ation (and hence of the
real rate of interest), the value of σ required to account for the observed output decline
following a monetary tightening is now somewhat smaller. In addition, because the e¬ects
of low real marginal costs on in¬‚ation cumulate over time under the γ = 1 speci¬cation, the
size of κ required to account for the observed in¬‚ation decline is now much smaller than that
indicated in Table 4.1. If we continue to assume that ± = 0.66, as in the model of Rotemberg
and Woodford, this would imply ζ = 0.03, a value that is perhaps too small to be plausible
(see the discussion in chapter 3). However, survey evidence on the frequency of price changes
no longer requires that ± be this small, as one might interpret some of the observed price
changes as automatic price changes as a result of indexation, rather than reconsiderations of
pricing policy. If, for example, we assume that re-optimizations occur once every six quarters
on average (rather than every three quarters, as assumed by Rotemberg and Woodford), then
± = 0.83. In this case, the estimated value of κ would correspond to ζ = 0.15, as shown
in the table. This requires only real rigidities of a magnitude that can be reconciled with
plausible assumptions about preferences and technology, as shown in chapter 3. If one is
willing to assume an even larger value of ±, less real rigidity would be required; for example,
if ± = 0.9, one would only need ζ = 0.43.
The degree of ¬t to the estimated impulse responses obtained under these parameter
values is shown in Figure 4.18. Table 4.4 gives the root-mean-square deviation between the
Moore speci¬cation include the inclusion of lagged output terms in the IS relation and the assumption of an
interest-rate rule involving only ¬rst di¬erences of the funds rate.

predicted and estimated responses in the case of each of the three panels of the ¬gure, and
compares these with the corresponding measures for Figure 4.15. Despite the imposition
of the restriction that ψ = 0, the model that incorporates in¬‚ation inertia better ¬ts the
estimated responses of both in¬‚ation and the nominal interest rate, though at the price of a
somewhat worse ¬t to the estimated output response. (Assuming that γ = 1 increases the
predicted persistence of the e¬ect of the shock on in¬‚ation, which better ¬ts the estimated
response, but also reduces the predicted persistence of the e¬ect on output, despite the lower
assumed value of κ. Thus there is some tension between the assumptions required to account
for the persistence of in¬‚ation responses and those required to account for the persistence of
output responses.) The overall sum of squared deviations is reduced by 19 percent.
While the model with in¬‚ation inertia better ¬ts the point estimates for the response of
in¬‚ation and interest rates to a monetary policy shock implied by the VAR of Rotemberg
and Woodford (1997), the imprecision of those estimates makes it hard to say, on the basis
of this evidence alone, that a large value of γ is clearly more realistic. However, as discussed
in chapter 3, a number of other studies ” both single-equation estimates and complete
estimated models ” have also found that similar aggregate supply relations ¬t U.S. data
better than do speci¬cations that do not incorporate in¬‚ation inertia. Hence in part II of
this study we consider in some detail the consequences for optimal policy of indexation to
lagged in¬‚ation of this kind.

4.3 Additional Sources of Delay


5 Monetary Policy and Investment Dynamics

One of the more obvious omissions in the baseline model developed above is the absence
of any e¬ect of variations in private spending upon the economy™s productive capacity, and

hence upon supply costs in subsequent periods. This means that we have treated all private
expenditure as if it were non-durable consumption expenditure; and while this has kept our
analysis of the e¬ects of interest rates on aggregate demand quite simple, one may doubt the
accuracy of the conclusions obtained, given the obvious importance of variations in invest-
ment spending both in business ¬‚uctuations generally and in the the transmission mechanism
for monetary policy in particular. We have suggested that the baseline model ought not be
interpreted as one in which investment spending is literally constant; in particular, we have
argued that the parameter σ in that model ought not be “calibrated” on the basis of studies
of intertemporal substitution of consumer expenditure, but should instead be taken to refer
to the degree of intertemporal substitutability of overall private expenditure, largely as a
result of intertemporal substitution in investment spending. In this section we develop an
extended version of the model in which investment spending is explicitly modeled, to see to
what extent such a more detailed model has properties di¬erent than those of the baseline
model, when the latter is calibrated to re¬‚ect an elasticity of intertemporal substitution of
overall spending that is several times as large as the elasticity of non-durable consumption.

5.1 Investment Demand with Sticky Prices

Our ¬rst task is to develop a model of optimizing investment demand by suppliers with sticky
prices, and that are demand-constrained as a result. We begin by modifying our production
function to include an explicit representation of the e¬ects of variation in the capital stock.
The production function for good i is assumed to be of the form

yt (i) = kt (i)f (At ht (i)/kt (i)), (5.1)

where f is an increasing, concave function as before, with f(0) = 0. Note that in the case
that kt (i) is a constant, this reduces to the form of production function assumed in chapter 3,
except that the technology factor At is now assumed to multiply the labor input rather than
the entire production function (“labor-augmenting” technical progress). We now change the
speci¬cation of the technology factor so that a one permanent increase in At will still result

in a one percent long-run increase in equilibrium output, now that the eventual increase in
the capital stock is taken into account. (In our model with endogenous capital accumulation,
the capital used per unit of labor will eventually increase in proportion to the increase in At ,
in order to maintain a constant long-run relation between the marginal product of capital
and the rate of time preference of households.)

We shall assume that each monopolistic supplier makes an independent investment deci-
sion each period; there is a separate capital stock kt (i) for each good, that can be used only
in the production of good i, rather than a single capital stock that can “rented” for use in
any sector, at a single economy-wide “rental rate” for capital services. The latter assump-
tion is instead remarkably common in the literature on intertemporal general-equilibrium
models with sticky prices; important early examples of models of that kind include Hairault
and Portier (1993), Kimball (1995), Yun (1996), King and Watson (1996), King and Wol-
man (1996), and Chari et al. (2000), among others. Simplicity probably accounts for this
(together with the bad example set by early examples of intertemporal general-equilibrium
models with imperfect competition, such as Rotemberg and Woodford, 1992). But the as-
sumption of a single economy-wide rental market for capital is plainly unrealistic, and its
consequences are far from trivial in the present context. For it would imply that di¬erences
in the demand for goods that have set their prices at di¬erent times should result in instan-
taneous reallocation of the economy™s capital stock from lower-demand to higher-demand
sectors, and this in turn has an important e¬ect upon the degree to which marginal cost of
supply should vary with the demand for a given good. We have shown in chapter 3 that the
assumption of economy-wide factor markets greatly reduces the predicted degree of strategic
complementarity of the pricing decisions of di¬erent suppliers, and in so doing increases the
speed of adjustment of the overall level of prices to varying demand conditions. Hence we
here assume instead that while all sectors purchase investment goods from the same suppliers
(i.e., that the investment goods used by the di¬erent sectors are perfect substitutes for their
producers), these goods cease to be substitutable once they have been purchased for use
in production in a particular sector. Capital can be reallocated from low-demand to high-


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