. 4
( 5)


demand sectors only over time, through reduced new investment in the former sectors and
increased new investment in the latter; and the speed with which this occurs is limited by the
assumption of adjustment costs. The resulting model is more realistic, and also represents
a more direct generalization of the constant-capital model developed in chapter 3. (That
model implicitly assumed a constant quantity of capital kt (i) available for the production of
each individual good, rather than a constant aggregate capital stock that would be e¬ciently
reallocated each period among sectors. We show below that the constant-capital model can
be recovered as a limiting case of the present model, in the limit of very high adjustment
costs for investment.)
We assume convex adjustment costs for investment by each ¬rm, of the usual kind as-
sumed in neoclassical investment theory. Increasing the capital stock to the level kt+1 (i)
in period t + 1 requires investment spending in the amount It (i) = I(kt+1 (i)/kt (i))kt (i) in
period t. Here It (i) represents purchases by ¬rm i of the composite good, de¬ned as the
usual Dixit-Stiglitz aggregate over purchases of each of the continuum of goods (with the
same constant elasticity of substitution θ > 1 as for consumption purchases). In this way,
the allocation of investment expenditure across the various goods is in exactly the same
proportion as consumption expenditure, resulting in a demand curve for each producer that
is again of the form
pt (i)
yt (i) = Yt , (5.2)
but where now aggregate demand is given by Yt = Ct +It +Gt , in which expression It denotes
the integral of It (i) over the various ¬rms i. We assume as usual that the function I(·) is
increasing and convex; the convexity implies the existence of costs of adjustment. We further
assume that near a zero growth rate of the capital stock, this function satis¬es I(1) = δ,
I (1) = 1, and I (1) = ψ, where 0 < δ < 1 and > 0 are parameters. This implies that

in the steady state to which the economy converges in the absence of shocks (which involves
a constant capital stock, as we abstract from trend growth), the steady rate of investment
spending required to maintain the capital stock is equal to δ times the steady-state capital
stock (so that δ can be interpreted as the rate of depreciation). It also implies that near the

steady state, a marginal unit of investment spending increases the capital stock by an equal
amount (as there are locally no adjustment costs). Finally, in our log-linear approximation
to the equilibrium dynamics, will be the parameter that indexes the degree of adjustment

Pro¬t-maximization by ¬rm i then implies that the capital stock for period t + 1 will be
chosen in period t to satisfy the ¬rst-order condition

I (kt+1 (i)/kt (i)) = Et Qt,t+1 Πt+1 {ρt+1 (i) +

(kt+2 (i)/kt+1 (i))I (kt+2 (i)/kt+1 (i)) ’ I(kt+2 (i)/kt+1 (i))} ,

where ρt+1 (i) is the (real) shadow value of a marginal unit of additional capital for use by ¬rm
i in period t + 1 production, and Qt,t+1 Πt+1 is the stochastic discount factor for evaluating
real income streams received in period t + 1. Expressing the real stochastic discount factor
as β»t+1 /»t , where »t is the representative household™s marginal utility of real income in
period t, and then log-linearizing this condition around the steady-state values of all state
variables, we obtain

ˆ ˆ ˆ ˆ
»t + ψ (kt+1 (i) ’ kt (i)) = Et »t+1 +

ˆ ˆ
[1 ’ β(1 ’ δ)]Et ρt+1 (i) + β ψ Et (kt+2 (i) ’ kt+1 (i)),
ˆ (5.3)

ˆ ¯ˆ ¯ˆ
where »t ≡ log(»t /»), kt (i) ≡ log(kt (i)/K), ρt (i) ≡ log(ρt (i)/¯), and variables with bars
denote steady-state values.
Note that ρt+1 (i) would correspond to the real “rental price” for capital services if a
market existed for such services, though we do not assume one. It is not possible in the
present model to equate this quantity with the marginal product, or even the marginal
revenue product of capital (using the demand curve (5.2) to compute marginal revenue).
For suppliers are demand-constrained in their sales, given the prices that they have posted;
it is not possible to increase sales by moving down the demand curve. Thus the shadow
value of additional capital must instead be computed as the reduction in labor costs through

substitution of capital inputs for labor, while still supplying the quantity of output that
happens to be demanded. We thus obtain
˜ ˜ ˜
f (ht (i)) ’ ht (i)f (ht (i))
ρt (i) = wt (i) ,
At f (ht (i))
where wt (i) is the real wage for labor of the kind hired by ¬rm i and ht (i) ≡ At ht (i)/kt (i)
is ¬rm i™s e¬ective labor-capital input ratio.75 We can alternatively express this in terms of
the output-capital ratio for ¬rm i (in order to derive an “accelerator” model of investment
demand), by substituting (5.1) to obtain
wt (i) ’1
ρt (i) = f (yt (i)/kt (i))[φ(yt (i)/kt (i)) ’ 1], (5.4)
where φ(y/k) is the reciprocal of the elasticity of the function f , evaluated at the argument
f ’1 (y/k).
We the recall from chapter 3 the ¬rst-order condition for optimizing labor supply, which
we may write in the form
vh (f ’1 (yt (i)/kt (i))kt (i)/At ; ξt )
wt (i) = , (5.5)
again writing labor demand in terms of the demand for good i. Substituting this into (5.4)
and log-linearizing, we obtain
φ ˆ ˆ ˆ
ρt (i) = νφ +
ˆ ωp (ˆt (i) ’ kt (i)) + ν kt (i) ’ »t ’ ωqt ,
y (5.6)
where φ > 1 is the steady-state value of φ(y/k), i.e., the reciprocal of the elasticity of the
production function with respect to the labor input, ωp > 0 is the negative of the elasticity
of f (f ’1 (y/k)) with respect to y/k, and ν > 0 is once again the elasticity of the marginal
disutility of labor with respect to labor supply. (The composite exogenous disturbance qt
is de¬ned as in equation (2.2).) Substituting this into (5.3), we then have an equation to
solve for the dynamics of ¬rm i™s capital stock, given the evolution of demand yt (i) for its
product, the marginal utility of income »t , and the exogenous disturbance qt .
Note that in the case of a ¬‚exible-price model, the ratio of wt (i) to the denominator would always equal
marginal revenue, and so this expression would equal the marginal revenue product of capital, though it
would be a relatively cumbersome way of writing it.

As the coe¬cients of these equations are the same for each ¬rm, an equation of the same
form holds for the dynamics of the aggregate capital stock (in our log-linear approximation).
Our equilibrium condition for the dynamics of the capital stock is thus of the form

ˆ ˆ
ˆ ˆ
»t + ψ (Kt+1 ’ Kt ) = β(1 ’ δ)Et »t+1 +

ˆ ˆ ˆ ˆ
[1 ’ β(1 ’ δ)][ρy Et Yt+1 ’ ρk Kt+1 ’ ωqt ] + β ψ Et (Kt+2 ’ Kt+1 ), (5.7)

where the elasticities of the marginal valuation of capital are given by
ρy ≡ νφ + ωp > ρk ≡ ρy ’ ν > 0.
The implied dynamics of investment spending are then given by

ˆ ˆ ˆ
It = k[Kt+1 ’ (1 ’ δ)Kt ], (5.8)

where It is de¬ned as the percentage deviation of investment from its steady-state level, as
a share of steady-state output, and k ≡ K/Y is the steady-state capital-output ratio.
We have here derived investment dynamics as a function of the evolution of the marginal
utility of real income of the representative household. This is in turn related to aggregate
spending through the relation »t = uc (Yt ’ It ’ Gt ; ξt ), which we may log-linearize as

ˆ ˆ ˆ
»t = ’σ ’1 (Yt ’ It ’ gt ), (5.9)

where the composite disturbance gt once again re¬‚ects the e¬ects both of government pur-
chases and of shifts in private impatience to consume.76 Finally, recalling the relation between
the marginal utility of income process and the stochastic discount factor that prices bonds,
the nominal interest rate must satisfy

1 + it = {βEt [»t+1 /(»t Πt+1 )]}’1 ,

which we may log-linearize as

ˆ ˆ
ˆt = Et πt+1 + »t ’ Et »t+1 .
± (5.10)
Note that the parameter σ in this equation is no longer the intertemporal elasticity of substitution in
consumption, but rather C/Y times that elasticity. In a model with investment, these quantities are not
exactly the same, even in the absence of government purchases.

The system of equations (5.7) “ (5.10) then comprise the “IS block” of our model. These
jointly su¬ce to determine the paths of the variables {Yt , It , Kt , »t }, given an initial capital
stock and the evolution of short-term real interest rates {ˆt ’ Et πt+1 }. The nature of the
e¬ects of real interest-rate expectations on these variables is discussed further in section 3.3

5.2 Optimal Price-Setting with Endogenous Capital

We turn next to the implications of an endogenous capital stock for the price-setting decisions
of ¬rms. The capital stock a¬ects a ¬rm™s marginal cost, of course; but more subtly, a ¬rm
considering how its future pro¬ts will be a¬ected by the price it sets must also consider how
its capital stock will evolve over the time that its price remains ¬xed.
We begin with the consequences for the relation between marginal cost and output. Real
marginal cost can be expressed as the ratio of the real wage to the marginal product of labor.
Again writing the factor input ratio as a function of the capital/output ratio, and using (5.5)
for the real wage, we obtain

vh (f ’1 (yt (i)/kt (i))kt (i)/At ; ξt )
st (i) =
»t At f (f ’1 (yt (i)/kt (i)))

for the real marginal cost of supplying good i. This can be log-linearized to yield

ˆ ˆ ˆ ¯
st (i) = ω(ˆt (i) ’ kt (i)) + ν kt (i) ’ »t ’ [ν ht + (1 + ν)at ],
ˆ y (5.11)

where once again ω ≡ ωw + ωp ≡ νφ + ωp > 0 is the elasticity of marginal cost with
respect to a ¬rm™s own output, and

qt ≡ ω ’1 [ν ht + (1 + ν)at ]

is the percentage change in output required to maintain a constant marginal disutility of
output supply, in the case that the ¬rm™s capital remains at its steady-state level.77
That is, qt measures the output change that would be required to maintain a ¬xed marginal disutility
of supply given possible ¬‚uctuations in preferences and technology, but not taking account of the e¬ect of
possible ¬‚uctuations in the ¬rm™s capital stock; thus qt is again an exogenous disturbance term. Note that

Letting st without the index i denote the average level of real marginal cost in the
economy as a whole, we note that (5.11) implies that

ˆ ˆ
st (i) = st + ω(ˆt (i) ’ Yt ) ’ (ω ’ ν)(kt (i) ’ Kt ).
ˆ ˆ y

Then using (5.2) to substitute for the relative output of ¬rm i, we obtain

st (i) = st ’ (ω ’ ν)kt (i) ’ ωθpt (i),
ˆ ˆ ˆ (5.12)

˜ ˆ ˆ
where pt (i) ≡ log(pt (i)/Pt ) is the ¬rm™s relative price, and kt (i) ≡ kt (i) ’ Kt is its relative
capital stock.
As in chapter 3, the Calvo price-setting framework implies that if ¬rm i resets its price
in period t, it chooses a price to satisfy the (log-linear approximate) ¬rst-order condition

(±β)k Et [ˆt+k (i) ’ st+k (i)] = 0.
p ˆ

Substituting (5.12) for st+k (i) in this expression, we obtain

(±β)k Et [(1 + ωθ)ˆt+k (i) ’ st+k + (ω ’ ν)kt+k (i)] = 0.
p ˆ (5.13)

We can as before express the entire sequence of values {ˆt+k (i)} as a linear function of the
relative price p— chosen at date t and aggregate variables (namely, the overall rate of price
in¬‚ation over various future horizons). However, we cannot yet solve for the optimal choice
of p— , because (5.13) also involves the relative capital stock of ¬rm i at a sequence of future
dates, and this depends upon the investment policy of the ¬rm.
We must therefore use the investment theory of the previous section to model the evolu-
tion of ¬rm i™s relative capital stock. Equation (5.7) implies that

˜ ˜ ˜ ˜ ˜
ψ (kt+1 (i) ’ kt (i)) = [1 ’ β(1 ’ δ)][ρy Et (ˆt+1 (i) ’ Yt ) ’ ρk kt+1 (i)] + β ψ Et (kt+2 (i) ’ kt+1 (i)).

the expression given here for qt in terms of the underlying disturbances di¬ers from that in section 2.1 above,
because of our di¬ering speci¬cation here of how the technology factor At shifts the production function.
Nonetheless, this de¬nition of qt is directly analogous to that used in the case of the constant-capital model;
it is actually our use of the notation at that is di¬erent here.

Again using the demand curve to express relative output as a function of the ¬rm™s relative
price, this can be written as

Et [Q(L)kt+2 (i)] = ΞEt pt+1 (i),
ˆ (5.14)

where the lag polynomial is

+ L2 ,
Q(L) ≡ β ’ [1 + β + (1 ’ β(1 ’ δ))ρk ψ ]L

Ξ ≡ (1 ’ β(1 ’ δ))ρy θ > 0.

One can easily show78 that the lag polynomial can be factored as

Q(L) = β(1 ’ µ1 L)(1 ’ µ2 L),

where the two roots satisfy 0 < µ1 < 1 < β ’1 < µ2 . We also note that

ρy ρy
Q(1) = βθ µ2 (1 ’ µ1 )(1 ’ µ’1 ).
Ξ = ’θ 2
ρk ρk

It then follows that in the case of any bounded process {ˆt (i)}, (5.14) has a unique bounded
solution for the evolution of {kt (i)}, given an initial capital stock for the ¬rm. This solution
is given by
˜ ˜
kt+1 (i) = µ1 kt (i) ’ zt (i), (5.15)

which we may integrate forward starting from an initial condition kt (i); here we de¬ne

µ’j Et pt+j (i).
zt (i) ≡ β Ξ ˆ

∞ k
Condition (5.13) requires that we evaluate the in¬nite sum k=0 (±β) Et kt+k (i). We note
that (5.15) implies that

˜ ˜
Et kt+k+1 (i) = µ1 Et kt+k (i) ’ Et zt+k (i)
The properties asserted follow directly from the observations that Q(0) = β > 0, Q(β) < 0, Q(1) < 0,
and that Q(z) > 0 for all large enough z > 0. These conditions imply that Q(z) has two real roots, one
between 0 and β and another that is greater than 1.

for all k ≥ 0. Integrating this law of motion we then ¬nd that
˜ ˜ µk’1’j Et zt+j (i),
µk kt (i)
Et kt+k (i) = ’ 1

from which it follows that
∞ ∞
1 ±β
˜ ˜
(±β)j Et zt+j (i).
(±β) Et kt+k (i) = kt (i) ’ (5.16)
1 ’ ±βµ1 1 ’ ±βµ1 j=0

The ¬nal term in this last relation can furthermore be expressed in terms of expected relative
prices, yielding
® 
∞ ∞ ∞
µ’j Et pt+j (i) ’ (±β)j Et pt+j (i)» .
(±β)j Et zt+j (i) = ° ˆ ˆ (5.17)
β(1 ’ ±βµ2 ) j=1 2
j=0 j=1

Now substituting (5.16) “ (5.17) for the sum of expected relative capital stocks in (5.13),
we obtain a relation that involves only the initial relative capital stock kt (i). This relation
can furthermore be simpli¬ed if we average it over all of the ¬rms i that choose new prices
at date t. Because the Calvo model assumes that all ¬rms are equally likely to choose new
˜ ˜
prices at date t, the average value of kt (i) is zero (even though the average value of Et kt+k (i)
need not be zero for horizons k > 0). The average value of Et pt+k (i) can also be expressed
ˆt ’ Et πt+j ,

where p— denotes the average relative price (average value of log pt (i)/Pt ) for the ¬rms that
choose new prices at date t. With these substitutions, (5.13) yields an equation for p— of the
∞ ∞ ∞
b)ˆ— k k
µ’k Et πt+k ,
(a ’ pt = (±β) Et st+k + a
ˆ (±β) Et πt+k ’ b (5.18)
k=0 k=1 k=1

1 + ωθ ± Ξ
a≡ + (ω ’ ν) > 0,
1 ’ ±β 1 ’ ±β (1 ’ ±βµ1 )(1 ’ ±βµ2 )
± Ξ
b ≡ (ω ’ ν) > 0.
1 ’ µ’1 (1 ’ ±βµ1 )(1 ’ ±βµ2 )

This allows us to solve for the average relative price chosen at date t by optimizing price-
setters, as a function of information at that date about the future evolution of average real
marginal costs and the overall rate of price in¬‚ation.
As in chapter 3, it is useful to quasi-di¬erence this pricing relation in order to obtain an
aggregate supply relation. Equation (5.18) implies that

(a ’ b)Et [(1 ’ ±βL’1 )(1 ’ µ’1 L’1 )ˆ— ] =

Et [(1 ’ µ’1 L’1 )ˆt ] + a±βEt [(1 ’ µ’1 L’1 )πt+1 ] ’ bµ’1 Et [(1 ’ ±βL’1 )πt+1 ]. (5.19)
2 2 2

We then recall that in the Calvo pricing model the overall rate of price in¬‚ation will be given
1’± —
πt = p.
Using this to substitute for p— in (5.19), we obtain an in¬‚ation equation of the form

πt = ξ0 st ’ ξ1 Et st+1 + ψ1 Et πt+1 ’ ψ2 Et πt+2 ,
ˆ ˆ (5.20)

1’± 1
ξ1 ≡ µ’1 ξ0 ,
ξ0 ≡ , 2
± a’b
a(β + µ’1 ) ’ b(±β + ±’1 µ’1 )
2 2
ψ2 ≡ βµ’1 .
ψ1 ≡ , 2
Once again, this allows us to solve for equilibrium in¬‚ation as a function of the current
and expected future average level of real marginal costs across sectors. (The sign of this
relationship is investigated numerically below.)
It remains to connect the expected evolution of real marginal costs, in turn, with ex-
pectations regarding real activity. Averaging (5.11) over ¬rms i, and substituting (5.9) to
eliminate »t , we obtain

ˆ ˆ ˆ
st = (ω + σ ’1 )Yt ’ σ ’1 It ’ (ω ’ ν)Kt ’ [σ ’1 gt + ωqt ].
ˆ (5.21)

Once again, real marginal costs are increasing in the current level of real activity; but now
this relation is a¬ected not merely by exogenous disturbances to tastes and technology, but

also by ¬‚uctuations in the aggregate capital stock, and by the share of current aggregate
demand that is investment as opposed to consumption demand. Equations (5.20) “ (5.21)
constitute the “aggregate supply block” of our extended model. They jointly replace the
aggregate supply relation of our baseline model, and serve to determine equilibrium in¬‚ation
dynamics as a function of the expected evolution of aggregate real expenditure, the aggregate
capital stock, and aggregate investment spending.

5.3 Comparison with the Baseline Model

Our complete extended model then consists of the system of equations (5.7) “ (5.10) and
(5.20) “ (5.21), together with an interest-rate feedback rule such as (1.7) specifying mon-
etary policy. We have a system of seven expectational di¬erence equations per period
to determine the equilibrium paths of seven endogenous variables, namely the variables
± ˆ ˆ ˆˆˆ
{πt , ˆt , Yt , Kt , It , st , »t }, given the paths of three composite exogenous disturbances {gt , qt , ¯t }.
It is useful to comment upon the extent to which the structure of the extended (variable-
capital) model remains similar, though not identical, to that of the baseline (constant-capital)
We have already noted that the equations of the extended model consist of an “IS
block”(which allows us to solve for the paths of real output and of the capital stock, given
the expected path of real interest rates and the initial capital stock), an “AS block” (which
allows us to solve for the path of in¬‚ation given the paths of real output and of the capital
stock), and a monetary policy rule (which implies a path for nominal interest rates given the
paths of in¬‚ation and output). In this overall structure it is similar to the baseline model,
except that the model involves an additional endogenous variable, the capital stock, which is
determined by the “IS block” and taken as an input to the “AS block”, along with the level
of real activity.79 It also continues to be the case that real disturbances a¬ect the determina-
tion of in¬‚ation and output only through their e¬ects upon the two composite disturbances

The structure of the model is thus similar to rational-expectations IS-LM models such as that of Sargent
and Wallace (1975), which allows for an endogenous capital stock.

gt and qt . Previously, we had emphasized instead the disturbances gt and Ytn , but these
contained the same information as a speci¬cation of gt and qt . (The appropriate de¬nition
of the natural rate of output in the context of the extended model is deferred to the next
subsection.) In the case of in¬‚ation determination alone (and determination of the output
gap) we were previously able to further reduce these to a single composite disturbance, rt .
This is no longer possible in the case of the extended model, although, as we discuss in the
next subsection, it is still possible to explain in¬‚ation determination in terms of the gap
between an actual and a “natural” real rate of interest; the problem is that with endogenous
variation in the capital stock, the natural rate of interest is no longer a purely exogenous
state variable.
We note also that the extended model™s “AS block” continues to be nearly as forward-
looking as that of the baseline model. The in¬‚ation equation (5.20) can once again be “solved
forward”80 to yield a solution of the form

πt = Ψj Et st+j ,
ˆ (5.22)

where the {Ψj } are constant coe¬cients. In the case of the baseline model, the coe¬cients
of this solution are necessarily all positive, and decay exponentially: Ψj = ξβ j , for some
ξ > 0.81 In the extended model, the coe¬cients are not necessarily all positive. Nonetheless,
numerical analysis suggests that for empirically realistic parameter values, one has Ψj > 0
for all small enough values of j.
This is illustrated in Figure 4.19 in the case of parameter values chosen in the following
way. The values used for parameters ±, β, φ, ν, ωp , ω, and θ are those given in Table 4.1,
drawn from the work of Rotemberg and Woodford (1997). The value used for σ is not the
same as in that study, instead, since as discussed earlier, the parameter σ of the baseline
model (and similarly of the model of Rotemberg and Woodford) should not be interpreted
The existence of a unique bounded solution of this form depends as usual upon the roots of a characteristic
equation satisfying certain conditions, that we do not here examine further. We note however that in the
numerical work presented here, we ¬nd that the relevant condition is satis¬ed in the case of what we judge
to be empirically realistic parameter values.
This follows from “solving forward” the corresponding in¬‚ation equation (xx) of chapter 3.


∈ =1.5
∈ =3
0.04 ψ
∈ =12
∈ =100
∈ =500




0 5 10 15 20 25 30 35 40 45 50

Figure 4.19: The coe¬cients Ψj in in¬‚ation equation (5.22), for alternative sizes of investment
adjustment costs.

as the intertemporal elasticity of substitution of non-durable consumption expenditure ”
it instead indicates the substitutability of private expenditure as a whole. In the extended
model, instead, σ does refer solely to the substitutability of consumption; so we now calibrate
this parameter to equal 1, which is roughly the degree of substitutability typically assumed
in the real business cycle literature (see, e.g., Kydland and Prescott, 1982; or King, Plosser
and Rebelo, 1988).82
We must also assign values to two new parameters, δ and ψ, relating to the dynamics of
the capital stock. We note that our model implies that the steady-state capital-output ratio
k ≡ K/Y must satisfy
β ’1 = + (1 ’ δ).
θ φk
Strictly speaking, our calibration here is not identical to the standard RBC choice. For as noted above,
our σ is actually the consumption share in output times the intertemporal elasticity of substitution of
consumption, rather than the elasticity itself; thus a value of 0.7 would be closer to the standard RBC
assumption. But we have no ground for choosing a precise value, and so choose 1 as a round number.

Given the values just assumed for β, θ and φ in our quarterly model, it follows that the
model will predict an average capital-output ratio of 10 quarters (roughly correct for the
U.S.) if and only if we assume a quarterly depreciation rate of δ = .012 (about ¬ve percent
per year). Finally, the ¬gure compares the consequences of a range of di¬erent possible
positive values for ψ. Here the value = 3 (indicated by the solid line in the ¬gure) is the

one that we regard as most empirically plausible; this results in a degree of responsiveness
of overall private expenditure to monetary policy shocks that is similar to that estimated by
Rotemberg and Woodford, as we shall see. But we also consider the consequences of smaller
and larger values for this crucial new parameter.
In the limit of an extremely large value for ψ, the coe¬cients reduce to those implied by
the constant-capital model. One observes from the form given for the polynomial Q(L) in
is made large, the two roots approach limiting values µ1 ’ 1, µ2 ’ β ’1 . It
(5.14) that as ψ

then follows that the coe¬cients in (5.18) approach limiting values a ’ (1 + ωθ)/(1 ’ ±β),
b ’ 0, and hence that the coe¬cients in (5.20) approach limiting values

1 ’ ± 1 ’ ±β
ξ0 ’ ξ ≡ > 0, ξ1 ’ βξ > 0,
± 1 + ωθ

ψ2 ’ β 2 .
ψ1 ’ 2β,

Thus in the limit, (5.20) takes the form

Et [(1 ’ βL’1 )πt ] = ξEt [(1 ’ βL’1 )ˆt ] + βEt [(1 ’ βL’1 )πt+1 ].

This relation has the same bounded solutions as the simpler relation

πt = ξˆt + βEt πt+1

derived for the baseline model in chapter 3, and in particular it implies that (5.22) holds
with coe¬cients Ψj = ξβ j . (These are the coe¬cients indicated by the upper dashed curve
in the ¬gure.)
If is ¬nite but still quite large, the coe¬cients {ψj } again decay only relatively grad-

ually as j increases, though more rapidly than would be predicted by the baseline model.

If instead takes a more moderate value (anything in the range that we could consider

empirically plausible), the coe¬cients decline more sharply with j, and indeed become neg-
ative if horizons as long as ¬ve or six years in the future are considered. Intuitively, the
expectation of a high average level of real marginal cost several years in the future is no
longer a motive to increase prices now, if ¬rms can instead plan to build up their capital
stocks in the meantime. Nonetheless, higher expected future real marginal costs continue to
increase in¬‚ation, as long as the expectations relate to horizons three years in the future or
less. And if the expectations relate to the coming year (i.e., the next four quarters), then the
coe¬cients are not just positive but of roughly the same magnitude as in the baseline model.
And it is these coe¬cients for low j that mainly matter, given that shocks will typically have
a relatively transient e¬ect on average real marginal costs. (With ¬‚exible prices, average
real marginal costs would never vary at all; even with a realistic degree of price stickiness,
price adjustment is rapid enough to make mean-reversion in the level of real marginal costs
relatively rapid.)

The remaining relation in the “AS block” of the extended model is the real marginal cost
ˆ ˆ
relation (5.21. This relation reduces to the same one as in the baseline model if the It and Kt
terms are omitted. The relation between real marginal costs and output is no longer as simple
in the extended model, owing to the presence of those additional terms. However, insofar
as cyclical variation in investment is highly correlated with cyclical variation in output, and
cyclical variation in the capital stock is not too great, the implied cyclical variation in real
marginal costs in the extended model is not too di¬erent. (In this case, the cyclical variation
ˆˆ ˆ
in Yt ’ It is highly correlated with, but smaller in amplitude than, the cyclical variation in Yt
itself. One corrects for the di¬erence in amplitude by using a substantially larger value for
σ ’1 when calibrating the constant-capital model.) We illustrate this below when we report
the impulse response of real marginal costs to a monetary policy shock, in Figure 4.21. Of
ˆ ˆ
course, in the limiting case of large ψ, the equilibrium ¬‚uctuations in both It and Kt are
negligible, and the entire “AS block” reduces to an AS relation like that of the baseline

The “IS block” of the extended model also retains broad similarities to that of the baseline
model. It is ¬rst useful to consider the implied long-run average values for capital, output
and investment as a function of the long-run average rate of in¬‚ation π∞ implied by a given
monetary policy. Equations (5.8) “ (5.9) imply that the long-run average values of the various
state variables must satisfy

ˆ ˆ ˆ
»∞ = ρy Y∞ ’ ρk K∞ ,
ˆ ˆ
I∞ = δk K∞ ,
ˆ ˆ ˆ
»∞ = ’σ ’1 (Y∞ ’ I∞ ).

ˆˆ ˆ
These relations can be solved for Y∞ , I∞ and K∞ as multiples of »∞ ; this generalizes the
relation between Y∞ and »∞ obtained for the baseline model. Equation (5.11) similarly
implies that the long-run average level of real marginal cost must satisfy

ˆ ˆ
s∞ = ω Y∞ ’ (ω ’ ν)K∞ ’ »∞ ;

substituting the above solutions, we obtain s∞ as a multiple of »∞ as well. Finally, (5.20)
implies that
ξ0 ’ ξ1
π∞ = s∞ .
1 ’ ψ1 + ψ2
ˆ ˆˆ
Using this together with the previous solution allows us to solve for »∞ , and hence for Y∞ , I∞
and K∞ as well, as multiples of π∞ .
We turn next to the characterization of transitory ¬‚uctuations around these long-run
average values. Using (5.8) “ (5.9) to eliminate Yt+1 from (5.7), we obtain a relation of the
Et [A(L)Kt+2 ] = Et [B(L)»t+1 ] + zt , (5.23)

where A(L) is a quadratic lag polynomial, B(L) is linear, and zt is a linear combination of
the disturbances gt and qt . For empirically realistic parameter values, the polynomial A(L)
can be factored as (1 ’ µ1 L)(1 ’ µ2 L), where the two real roots satisfy 0 < µ1 < 1 < µ2 .
˜ ˜ ˜ ˜
ˆ ˆ
It follows that there is a unique bounded solution for Kt+1 as a linear function of Kt , the

expectations Et »t+j for j ≥ 0, and the expectations Et zt+j for j ≥ 0. Then solving (5.10)
forward to obtain

ˆ ˆ
»t = »∞ + Et (ˆt+j ’ πt+j+1 ),
± (5.24)

and using this to eliminate the expectations Et »t+j , we ¬nally obtain a solution of the form

ˆ ˜ˆ ˜ˆ χj Et (ˆt+j ’ πt+j+1 ) + ek ,
Kt+1 = (1 ’ µ1 )K∞ + µ1 Kt ’ ˜ ± (5.25)

where the {χj } are constant coe¬cients and ek is an exogenous disturbance term (a linear
˜ t

combination of the {Et zt+j }). This can be solved iteratively for the dynamics of the cap-
ital stock, starting from an initial capital stock and given the evolution of the exogenous
disturbances and of real interest-rate expectations.
Equation (5.24) can also be substituted into (5.9) to yield

ˆ ˆ ˆ ˆ
Yt = (Y∞ ’ I∞ ) + It + gt ’ σ Et (ˆt+j ’ πt+j+1 ),

a direct generalization of (1.4), which now however takes account of investment spending.
Using (5.8) and (5.25) to substitute for It , this expression takes the form

ˆ ˆ ˆ ˆ χj Et (ˆt+j ’ πt+j+1 ) + ey ,
Yt = (Y∞ ’ ΣK∞ ) + ΣKt ’ ± (5.26)

where Σ ≡ k[˜1 ’ (1 ’ δ)], {χj } is another set of constant coe¬cients, and ey is another
µ t

exogenous disturbance term (a linear combination of gt and of the {Et zt+j }). The joint
evolution of output and of the capital stock are then determined by the pair of equations
(5.25) “ (5.26), starting from an initial capital stock and given the evolution of the exogenous
disturbances and of real interest-rate expectations.
Except for the need to jointly model the evolution of output and of the capital stock,
this system of equations has implications rather similar to those of the “IS relation” (1.4)
of the baseline model. In particular, for typical parameter values, the coe¬cients {χj } in
(5.26) are all positive, and even of roughly similar magnitude for all j. For example, these
coe¬cients are plotted in Figure 4.20 for a model that is calibrated in the same way as in


12 ∈ψ=0.5





0 2 4 6 8 10 12 14 16 18 20

Figure 4.20: The coe¬cients χj in aggregate demand relation (5.26), for alternative sizes of
investment adjustment costs.

Figure 4.19, again allowing for a range of di¬erent possible values of ψ. In the limit of very
large ψ, the coe¬cients all approach the constant value σ (here assigned the value 1), as
in (1.4). For most lower values of ψ, the coe¬cients are not exactly equal in magnitude,
and each coe¬cient is larger the smaller are the adjustment costs associated with investment
spending. However, the coe¬cients all remain positive, and quite similar in magnitude to
one another, especially for values of near our baseline value of 3.

We can show analytically that χj takes the same value for all j (though a value greater
than σ) if it happens that B(L) in (5.23) is of the form ’h(1 ’ µ2 L), where h > 0 and µ2 is
the root greater than one in the factorization of A(L). In this case (5.23) is equivalent to

(1 ’ µ1 L)Kt+1 = ’h»t ,

and substitution of (5.24) yields a solution for aggregate demand of the form (5.26) with
χj = σ+h for all j ≥ 0. For our calibrated parameter values, the root of B(L) coincides with a

root of A(L) in this way if and only if happens to take a speci¬c value, equal approximately

to 3.23. This is in fact not an unrealistic value to assume. Perhaps more interesting, however,
is the fact that the coe¬cients {χj } are all reasonably similar in magnitude even when is

larger or smaller than the critical value.

Thus it continues to be true, as in the baseline model, that changes in interest-rate
expectations (due, for example, to a shift in monetary policy) a¬ect aggregate demand
through their e¬ect upon a very long real rate; the existence of endogenous variation in
investment spending simply makes the degree of sensitivity of aggregate demand to the level
of the very long real rate greater. For example, we see from the ¬gure that when = 3,

the degree of interest-sensitivity of aggregate demand is about four times as large as if ψ

were extremely large; the response to interest-rate changes is thus roughly the same as in
a constant-capital model with a value of σ near 4, rather than equal to 1 as assumed here.
This justi¬es our use of a value of σ much larger than 1 when calibrating the baseline model.

However, even if we adjust the value assumed for σ in this way, the predictions of the
constant-capital model as to the e¬ects of real interest rate changes are not exactly the same
as those of the model with variable capital. This is because lower investment spending as
a result of high long real rates of interest soon results in a lower capital stock, and once
this occurs aggregate demand is a¬ected through the change in the size of the ΣKt term in
(5.26). In the case of su¬ciently moderate adjustment costs (the empirically realistic case),
the value of Σ is negative; for given real interest-rate expectations, a higher existing capital
stock depresses investment demand (because returns to existing capital are low).83 Thus a
sustained increase in long real rates of interest will initially depress aggregate demand, in
the variable-capital model, by more than it does later on; once the capital stock has fallen
this fact helps investment demand to recover, despite the continued high real rates.

For the particular parameter values discussed in the text, our baseline value ψ = 3 implies that Σ =
’1.246 in our quarterly model. Note that our model does not require Σ to be negative; one can show
that Σ > 0 (because µ1 > 1 ’ δ) if and only if ψ exceeds the critical value ρk (1 ’ δ)/δ > 0. For our
calibrated parameter values, this critical value is approximately equal to 114.5, and thus would imply a
level of adjustment costs in investment that would be inconsistent with the observed degree of volatility of
investment spending.

nominal interest inflation
1 1


’2 ’3
0 5 10 15 20 0 5 10 15 20
real interest output


0 5 10 15 20 0 5 10 15 20
capital stock real marginal cost
var. capital
’1 constant capital
0 5 10 15 20 0 5 10 15 20

Figure 4.21: Impulse responses to an unexpected monetary tightening: the constant-capital
and variable-capital models compared.

The degree to which endogenous variation in the capital stock is likely to matter in
practice can be illustrated by considering the predicted e¬ects of a monetary policy shock,
in the case of a systematic monetary policy rule again given by (2.22) with coe¬cients
φπ = 2, φx = 1, and ρ = 0.8. Impulses responses to an unexpected monetary tightening
(again increasing nominal interest rates by one percent per year) are plotted in Figure 4.21,
which also reproduces the predictions of the baseline model corresponding to the case ρ = 0.8
in Figure 4.7. We observe that when we assume = 3 (and all other parameter values as

in Figures 4.19 and 4.20), the predicted output response in the extended model is essentially
the same as in the baseline model. (It is for this reason that we choose = 3 as our baseline

calibration of the extended model.)
However, this does not mean that the extended model with = 3 and σ = 1 makes

predictions that in all respects identical to those of a constant-capital model with σ = 6.37.

For example, we see from Figure 4.20 that when = 3, the coe¬cients {χj } of the IS relation

are approximately constant, but the constant value is a bit less than 4, rather than being
greater than 6 as in the baseline model. We could instead arrange for the extended model to
predict the same degree of interest-sensitivity of aggregate demand as in the baseline model
if we were to assume a value near = 1.5. But this would then result in overprediction of

the output contraction that should result from a monetary policy tightening. The reason
has to do with the e¬ects of endogenous variation in the capital stock, abstracted from
in the baseline model. The ΣKt term in (5.26) contributes a positive stimulus to output
(i.e., reduces the size of the output decline) several quarters after the shock, as the low
capital stock induces greater investment spending than would otherwise be chosen given the
higher-than-average real interest rates.84 This means that an output response that decays
at the same rate as in the constant-capital model (and as in the estimates of Rotemberg and
Woodford, 1997, discussed below) requires a more persistent increase in real interest rates
in the case of the variable-capital model. (One can see from the ¬gure that the simulation
does indeed have this property.) On the other hand, because of the very forward-looking
character of the model ” real interest-rate expectations several years in the future a¬ect
aggregate demand to essentially the same extent as the current short-term real rate ” a
more persistent increase in the short real rate will cause a larger immmediate contraction
of aggregate demand, unless the size of the coe¬cients {χj } is reduced. This is achieved in
the simulation shown in Figure 4.21 by assuming a large enough value of to reduce the

interest-rate sensitivity of aggregate demand by a factor of about 40 percent, relative to the
baseline model.

The fact that the two alternative models can be calibrated so as to predict very similar
output and in¬‚ation dynamics in response to the same kind of monetary policy shock indi-
cates that the predictions of the models need not be too di¬erent; indeed, their respective

The e¬ect is quite signi¬cant. For example, in the simulation shown in Figure 4.21, by the eighth quarter
following the shock the positive e¬ect of the ΣKt term is 65 percent of the size of the cumulative negative
e¬ect of all of the real-interest-rate terms, so that the contraction in aggregate demand is only a bit more
than a third of the size it would otherwise have.

abilities to ¬t empirical evidence as to the response to a particular kind of (historically typ-
ical) disturbance might well be quite similar. On the other hand, the mechanisms within
the models that produce these predictions are not too closely parallel, owing to the signif-
icant e¬ects of endogenous capital accumulation in the extended model. This means that
even if the models are calibrated so as to predict similar responses to a particular kind of
policy (as in Figure 4.21), it will not follow that the models calibrated in this way would
also predict similar responses to all other policies that might be contemplated. Thus while
a number of general lessons that may be drawn from the baseline model (e.g., the degree
to which aggregate demand should depend upon expected real rates years in the future) are
found to be robust to an explicit consideration of endogenous capital accumulation, accurate
quantitative conclusions about the nature of optimal monetary policy are likely to require
explicit allowance for the dynamics of the capital stock.

5.4 Capital and the Natural Rate of Interest

We now consider the extent to which the concept of the “natural rate of interest”, introduced
in section 2 in connection with our baseline model, can be extended to a model which allows
for endogenous variation in the capital stock. The most important di¬erence in the case
of our extended model is that the equilibrium real rate of return under ¬‚exible prices is no
longer a function solely of current and expected future exogenous disturbances; it depends
upon the capital stock as well, which is now an endogenous state variable (and so a function
of past monetary policy, among other things, when prices are sticky). Hence if we continue
to de¬ne the natural rate of interest in this way, it ceases to refer to an exogenous process.
An alternative possibility (pursued in Neiss and Nelson, 2000) would be to de¬ne the
“natural rate of interest” as what the equilibrium real rate of return would be if prices not
only were currently ¬‚exible and were expected always to be in the future, but also had
always been in the past ” so that what matters for the computation is not the capital stock
that actually exists, but the one that would exist under this counterfactual, given the actual
history of exogenous real disturbances. Under that de¬nition the natural rate of interest

would be exogenous, but at the cost of less connection with equilibrium determination in
the actual (sticky-price) economy. It seems odd to de¬ne the economy™s “natural” level
of activity, and correspondingly the associated “natural” level of interest rates, in a way
that makes irrelevant the capital stock that actually exists and the e¬ects of this upon the
economy™s productive capacity. And a clear cost of the alternative de¬nition would be less
connection between this concept of “natural” output and the e¬cient level of output, which
clearly depends on the actual capital stock. For this reason, we instead continue to de¬ne
the “natural rates” of output and interest as those that would result from price ¬‚exibility
now and in the future, given all exogenous and predetermined state variables, including the
economy™s capital stock. (The connection between this de¬nition and a welfare-based policy
objective is taken up in chapter 6.)
Since the equilibrium with ¬‚exible prices at any date t depends only on the capital stock
at that date85 and current and expected future exogenous real disturbances, we can write a
log-linear approximation to the solution in the form

ˆ ˆ ˆ
Ytn = Ytncc + ·y Kt ,

ˆn ˆncc
rt = rt + ·r Kt ,

and so on, where the terms Ytncc and rt refer to exogenous processes (functions solely of
the exogenous real disturbances). These “intercept” terms in each expression indicate what
the level of real output (or the real interest rate, and so on) would be, given current and
expected future real disturbances, if prices were ¬‚exible and the capital stock did not di¬er
from its steady-state level; we shall call this the constant-capital natural rate of output (or
of interest, and so on). We shall also ¬nd it useful to de¬ne a “natural rate” of investment
Itn and of the marginal utility of income »n in a similar way. We can even de¬ne a “natural”
capital stock Kt+1 , as what the capital stock in period t + 1 would be if it had been chosen
in a ¬‚exible-price equilibrium in period t, as a function of the actually existing capital stock
This is true up to the log-linear approximation that we use here to characterize equilibrium. More
precisely, it would depend on the capital stock in place in each of the ¬rms producing di¬erentiated goods.

Kt and the exogenous disturbances at that time; thus we similarly write

ˆn ˆ ncc ˆ
Kt+1 = Kt=1 + ·k Kt .

Finally, we shall use tildes to indicate the “gaps” between the actual and “natural” values
˜ ˆ ˆ˜
of these several variables: Yt ≡ Yt ’ Ytn , rt ≡ rt ’ rt , and so on.
ˆ ˆn
Once again, in a ¬‚exible-price equilibrium, real marginal cost must at all times be equal
to a constant, (θ ’ 1)/θ. It then follows from (5.21) that ¬‚uctuations in the natural rate of
output satisfy
’1 ’1
ˆ n = ω ’ ν Kt + σ ˆt + σ ω
Yt I gt + qt .
ω + σ ’1 ω + σ ’1 ω + σ ’1 ω + σ ’1

This relation generalizes equation (2.2) for the baseline model. Note that this does not
allow us to solve for the natural rate of output as a function of the capital stock and the
real disturbances without also simultaneously solving for the natural rate of investment.
However, comparison with (5.21) allows us to derive an expression for real marginal cost in
terms of the “gaps”,
˜ ˜
st = (ω + σ ’1 )Yt ’ σ ’1 It ,
ˆ (5.27)

generalizing equation (xx) of chapter 3.
Because condition (5.7) must also hold in a ¬‚exible-price equilibrium, we observe that

ˆ ˆ
ˆn ˆ ˜
»n + ψ (Kt+1 ’ Kt ) = β(1 ’ δ)[Et »t+1 ’ ·» Kt+1 ] +

ˆn ˜ ˆn ˆn ˜ ˆn
[1 ’ β(1 ’ δ)][ρy (Et Yt+1 ’ ·y Kt+1 ) ’ ρk Kt+1 ’ ωqt ] + β ψ [(Et Kt+2 ’ ·k Kt+1 ) ’ Kt+1 ].

Here we use the fact that in a ¬‚exible-price equilibrium, the conditional expectation at t of
period t + 1 output does not correspond to the value of Et Yt+1 at date t in the sticky-price
equilibrium, for the latter depends upon the value of Kt+1 in the sticky-price equilibrium,
which is generally not the same as what the period t + 1 capital stock would be in a ¬‚exible-
price equilibrium. Thus the conditional expectation at t of period t + 1 output in a ¬‚exible-
price equilibrium (beginning at t and conditional upon the actual period t capital stock) is
ˆn ˜
actually equal to Et Yt+1 ’ ·y Kt+1 ; and similarly for the expectation at t of other variables

determined at t + 1. Comparing this with (5.7), we see that in the sticky-price equilibrium,
the “gap” variables must satisfy

˜ ˜
˜ ˜
»t + ψ Kt+1 = β(1 ’ δ)[Et »t+1 + ·» Kt+1 ] +

˜ ˜ ˜ ˜ ˜ ˜
[1 ’ β(1 ’ δ)][ρy (Et Yt+1 + ·y Kt+1 ) ’ ρk Kt+1 ] + β ψ [(Et Kt+2 + ·k Kt+1 ) ’ Kt+1 ]. (5.28)

This equation is similar in form to (5.7), except that it is purely forward-looking: it de-
termines the equilibrium size of the gap Kt+1 without any reference to predetermined state
variables such as Kt .
Equations (5.8) “ (5.10) similarly must hold in a ¬‚exible-price equilibrium, implying that
the “gaps” must also satisfy equations

˜ ˜
It = k Kt+1 , (5.29)

˜ ˜ ˜
»t = ’σ ’1 (Yt ’ It ), (5.30)
˜ ˜ ˜
rt = »t ’ (Et »t+1 + ·» Kt+1 ).
˜ (5.31)
˜ ˜
Using equations (5.29) and (5.30) to eliminate »t and Kt+1 from (5.28) and (5.31), we are
left with a system of two equations that can be written in the form

Et zt+1 = Azt + a˜t ,
r (5.32)

for a certain matrix A and vector a of coe¬cients, where now
zt ≡ .
This pair of coupled di¬erence equations generalizes the “gap” version (1.8) of the IS relation
of the baseline model.86
Let us now close our system by specifying monetary policy in terms of an interest-rate
feedback rule of the form

ˆt = ¯t + φπ (πt ’ π ) + φx (xt ’ x)/4,
± ± ¯ ¯ (5.33)
Note, however, that it is typically not possible to solve this system “forward” to obtain a solution for
the output gap as a function of current and expected future interest-rate gaps, as in equation (2.19) for the
baseline model. [ADD MORE]

where we re-introduce the notation xt ≡ Yt for the output gap. (Once again, x is the steady-
state output gap corresponding to the steady-state in¬‚ation rate π .) With policy speci¬ed
by a “Taylor rule” of this kind, the interest-rate gap will be given by

ˆn ¯
rt = (¯t ’ rt ’ π ) ’ Et (πt+1 ’ π ) + φπ (πt ’ π ) + φx (xt ’ x)/4.
˜ ± ¯ ¯ ¯ (5.34)

Note that in this last relation, the only endogenous variables are “gap” variables if we make
the further assumption that
¯t = ¯cc + ·r Kt ,
± ±t (5.35)

where ¯cc } is an exogenous process. This implies that in addition to systematic responses to
endogenous variation in in¬‚ation and in the output gap, the policy rule (5.33) also involves
a systematic response to endogenous variation in the capital stock, of a speci¬c sort: the
central bank™s interest rate operating target is adjusted to exactly the same extent as the
natural rate of interest is changed by the variation in the capital stock. This is obviously a
special case, but not an entirely implausible one, if we posit a desire to stabilize in¬‚ation, and
hence to arrange for interest rates to vary one-for-one with variation in the natural rate of
interest. For of all the possible sources of variation in the natural rate of interest, variations
due to changes in the economy™s aggregate capital ought to be the easiest for a central bank
to track with some accuracy (owing to the slowness of movements in the capital stock).
A complete system of equilibrium conditions for the determination of the variables
˜ ˜˜ˆ
{Yt , It , rt , st , πt } is then given by (5.20), (5.27), (5.32), and (5.34). The system of equa-
tions may furthermore be written in the form

ˆz ˆ rn ±
Et zt+1 = Aˆt + a(ˆt ’ ¯t + π ),
ˆ ¯ (5.36)

where now ® 
Yt ’ x¯
 
 
It ’ I
zt ≡  ,
ˆ  
Et πt+1 ’ π
° »
πt ’ π¯
¯ ˜ ˆ
I is the steady-state value of It corresponding to steady in¬‚ation at the rate π , and A and a
¯ ˆ
are again a matrix and vector of coe¬cients. We obtain this system as follows. The ¬rst two

rows are obtained by substituting for rt in (5.32) using (5.34).87 The third row is obtained
by solving (5.20) for Et πt+2 , and then substituting for st and Et st+1 using (5.27); one ¬nally
ˆ ˆ
˜ ˜
substitutes for Et Yt+1 and Et It+1 using the ¬rst two rows of (5.36), just derived. The fourth
row is simply an identity.
Because the system (5.36) is purely forward-looking (i.e., there are no predetermined
endogenous state variables), a policy rule of the kind de¬ned by (5.33) and (5.35) then
results in determinate equilibrium dynamics for in¬‚ation and the output gap (among other
variables) if and only if the matrix A has all four eigenvalues outside the unit circle. When
this is true, the system can be “solved forward” in the usual way to obtain a unique bounded
solution. The solutions for in¬‚ation and the output gap will once again be of the form (2.18)
“ (2.19), and the implied solution for the nominal interest rate will correspondingly again
be of the form (2.20), just as in the baseline model, though the numerical values of the
π x i
coe¬cients {ψj , ψj , ψj } in these expressions will be di¬erent. Figure 4.22 plots coe¬cients
π x
{ψj , ψj } for j = 0 through 10, in the case of model parameters chosen as in the earlier
¬gures, and a policy rule of the form de¬ned by (5.33) and (5.35), with feedback coe¬cients
φπ = 2, φx = 1. Here the solid line indicates the coe¬cients in the case of the variable-capital
model, and the dashed line the coe¬cients in the case of the baseline model. (The dashed
line here corresponds to the baseline case shown in Figures 4 and 5 earlier.)
Once again one ¬nds that, at least in the case of changes in the expected gap Et (ˆt+j ’¯t+j )
only a few quarters in the future, increases in the expected gap increase both in¬‚ation and
the output gap; and once again, this is true even many quarters in the future in the case
of in¬‚ation, whereas expected gaps Et (ˆt+j ’ ¯t+j ) more than a few quarters in the future
have little e¬ect upon the output gap. Thus ¬‚uctuations in in¬‚ation and the output gap
can still be explained in essentially the same way as in the constant-capital model. Once
again, in¬‚ation and positive output gaps result from increases in the natural rate of output
that are not fully matched by a tightening of monetary policy, or by loosenings of monetary
Note that all of these equations continue to be valid when we replace variables by the di¬erence of those
variables from their steady-state values. We choose to express the equations in this form in (5.36) because
the policy rule (5.33) has already been expressed in this form.






0 1 2 3 4 5 6 7 8 9 10


var. capital
const. capital




0 1 2 3 4 5 6 7 8 9 10

Figure 4.22: Coe¬cients of the forward-looking solution for in¬‚ation and the output gap.

policy not justi¬ed by any decline in the natural rate of interest.
An immediate consequence is that once again a possible approach to the goal of in¬‚ation
stabilization is to commit to a policy rule of the form (5.33) such that (i) the coe¬cients
φπ , φx are chosen so as to imply a determinate equilibrium, and (ii) the intercept adjustments
track variations in the natural rate of interest as well as possible, i.e., the central bank seeks
ˆn ¯
to set ¯t = rt + π at all times. (Note that this is an example of a rule of the form (5.35), with
¯cc = rt .) If it is possible to satisfy this condition with su¬cient accuracy, then in¬‚ation can
in principle be completely stabilized with ¬nite response coe¬cients. Thus the requirement
of tracking variations in the natural rate of interest continues to be as important to the
pursuit of price stability as in our analysis of the baseline model.
Aiyagari, S. Rao, and R. Anton Braun, “Some Models to Guide Monetary Policymakers,”
Carnegie-Rochester Conference Series on Public Policy 48: 1-42 (1998).

Amato, Je¬ery D., and Thomas Laubach, “Estimation and Control of an Optimization-
Based Model with Sticky Prices and Wages,” Journal of Economic Dynamics and
Control, forthcoming 2002.

Bernanke, Ben S., and Jean Boivin, “Monetary Policy in a Data-Rich Environment,”
unpublished, Princeton University, September 2000.

Bernanke, Ben S., and Michael Woodford, “In¬‚ation Forecasts and Monetary Policy,”
Journal of Money, Credit, and Banking, 24: 653-84 (1997).

Black, Richard, Vincenzo Cassino, Aaron Drew, Eric Hansen, Benjamin Hunt, David
Rose, and Alasdair Scott, “The Forecasting and Policy System: The Core Model,”
Research Paper no. 43, Reserve Bank of New Zealand, August 1997.

Blinder, Alan S., Central Banking in Theory and Practice, Cambridge: M.I.T. Press,

Boivin, Jean, and Marc Giannoni, “Has Monetary Policy Become Less Powerful?” Fed-
eral Reserve Bank of New York, Sta¬ Report no. 144, March 2002.

Brayton, Flint, Andrew Levin, Ralph Tryon, and John Williams, “The Evolution of
Macro Models at the Federal Reserve Board,” Carnegie-Rochester Conference Se-
ries on Public Policy 47: 43-81 (1997).

Calvo, Guillermo, “Staggered Prices in a Utility-Maximizing Framework,” Journal of
Monetary Economics, 12: 383-98 (1983).

Chari, V.V., Patrick J. Kehoe, and Ellen R. McGrattan, “Sticky Price Models of the Busi-
ness Cycle: Can the Contract Multiplier Solve the Persistence Problem?” Econo-
metrica 68: 1151-1179 (2000).

Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans, “Monetary Policy
Shocks: What Have We Learned and to What End?” in J.B. Taylor and M. Wood-
ford, eds., Handbook of Macroeconomics, vol. 1A, Amsterdam: North Holland,

Clarida, Richard, Jordi Gali and Mark Gertler, “The Science of Monetary Policy: A New
Keynesian Perspective,” Journal of Economic Literature 37: 1661-1707 (1999).


Clarida, Richard, Jordi Gali and Mark Gertler, “Monetary Policy Rules and Macroeco-
nomic Stability: Evidence and Some Theory,” Quarterly Journal of Economics 115:
147-180 (2000).

Coletti, Donald, Benjamin Hunt, David Rose and Robert Tetlow, “The Dynamic Model:
QPM, The Bank of Canada™s New Quarterly Projection Model, Part 3,” Technical
Report no. 75, Bank of Canada, May 1996.

Edge, Rochelle M., “Time-to-Build, Time-to-Plan, Habit-Persistence, and the Liquidity
E¬ect,” International Finance Discussion Paper no. 673, Federal Reserve Board,
July 2000.

Fisher, Mark, and Christian Gilles, “Modeling the State-Price De¬‚ator and the Term
Structure of Interest Rates,” unpublished, Research Department, Federal Reserve
Bank of Atlanta, February 2000.

Friedman, Milton,“The Role of Monetary Policy,” American Economic Review 58: 1-17

Fuhrer, Je¬rey C., “Comment,” NBER Macroeconomics Annual 12: 346-355 (1997).

Fuhrer, Je¬rey C., and Geo¬rey R. Moore, “Monetary Policy Trade-o¬s and the Cor-
relation between Nominal Interest Rates and Real Output,” American Economic
Review 85: 219-239 (1995a).

Fuhrer, Je¬rey C., and Geo¬rey R. Moore, “Forward-Looking Behavior and the Stability
of a Conventional Monetary Policy Rule,” Journal of Money, Credit and Banking
27: 1060-1070 (1995b).

Giannoni, Marc P., “Optimal Interest-Rate Rules in a Forward-Looking Model, and In¬‚a-
tion Stabilization versus Price-Level Stabilization,” unpublished, Federal Reserve
Bank of New York, September 2000.

Hairault, Jean-Olivier, and Franck Portier, “Money, New Keynesian Macroeconomics,
and the Business Cycle,” European Economic Review 37: 1533-1568 (1993).

Humphrey, T.M., “Price-Level Stabilization Rules in a Wicksellian Model of the Cumu-
lative Process,” Scandinavian Journal of Economics 94: 509-518 (1992).


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