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

’θ

pt (i)

yt (i) = Yt , (5.2)

Pt

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

ψ

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

98 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

ψ

costs.

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

ρ

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

5. MONETARY POLICY AND INVESTMENT DYNAMICS 99

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)

At

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)

»t

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)

φ’1

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

ˆ

ˆ

product, the marginal utility of income »t , and the exogenous disturbance qt .

75

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.

100 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

φ

ρy ≡ νφ + ωp > ρk ≡ ρy ’ ν > 0.

φ’1

The implied dynamics of investment spending are then given by

ˆ ˆ ˆ

It = k[Kt+1 ’ (1 ’ δ)Kt ], (5.8)

ˆ

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)

76

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 101

The system of equations (5.7) “ (5.10) then comprise the “IS block” of our model. These

ˆˆˆ

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

below.

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

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

102 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

Letting st without the index i denote the average level of real marginal cost in the

ˆ

economy as a whole, we note that (5.11) implies that

ˆ

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

ˆ

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

k=0

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)

k=0

We can as before express the entire sequence of values {ˆt+k (i)} as a linear function of the

p

relative price p— chosen at date t and aggregate variables (namely, the overall rate of price

ˆt

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

ˆt

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

˜ ˜ ˜ ˜ ˜

ˆ

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)).

y

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 103

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

’1

+ L2 ,

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

and

’1

Ξ ≡ (1 ’ β(1 ’ δ))ρy θ > 0.

ψ

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

p

˜

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).

’1

zt (i) ≡ β Ξ ˆ

2

j=1

˜

∞ 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)

78

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.

104 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

for all k ≥ 0. Integrating this law of motion we then ¬nd that

k’1

˜ ˜ µk’1’j Et zt+j (i),

µk kt (i)

Et kt+k (i) = ’ 1

1

j=0

from which it follows that

∞ ∞

1 ±β

˜ ˜

k

(±β)j Et zt+j (i).

(±β) Et kt+k (i) = kt (i) ’ (5.16)

1 ’ ±βµ1 1 ’ ±βµ1 j=0

k=0

The ¬nal term in this last relation can furthermore be expressed in terms of expected relative

prices, yielding

®

∞ ∞ ∞

Ξ

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

ˆ

as

k

p—

ˆt ’ Et πt+j ,

j=1

where p— denotes the average relative price (average value of log pt (i)/Pt ) for the ¬rms that

ˆt

choose new prices at date t. With these substitutions, (5.13) yields an equation for p— of the

ˆt

form

∞ ∞ ∞

b)ˆ— k k

µ’k Et πt+k ,

(a ’ pt = (±β) Et st+k + a

ˆ (±β) Et πt+k ’ b (5.18)

2

k=0 k=1 k=1

where

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 )

2

5. MONETARY POLICY AND INVESTMENT DYNAMICS 105

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— ] =

pt

2

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)

s

2 2 2

We then recall that in the Calvo pricing model the overall rate of price in¬‚ation will be given

by

1’± —

πt = p.

ˆ

±t

Using this to substitute for p— in (5.19), we obtain an in¬‚ation equation of the form

ˆt

πt = ξ0 st ’ ξ1 Et st+1 + ψ1 Et πt+1 ’ ψ2 Et πt+2 ,

ˆ ˆ (5.20)

where

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

a’b

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

ˆ

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

106 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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)

model.

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

79

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 107

ˆ

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

ˆn

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)

j=0

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

80

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.

81

This follows from “solving forward” the corresponding in¬‚ation equation (xx) of chapter 3.

108 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

0.05

∈ =1.5

ψ

∈ =3

0.04 ψ

∈ψ=6

∈ =12

ψ

∈ψ=30

0.03

∈ =100

ψ

∈ =500

ψ

∈ψ=∞

0.02

0.01

0

’0.01

’0.02

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

θ’1φ’11

β ’1 = + (1 ’ δ).

θ φk

82

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 109

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

ψ

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 ].

s

This relation has the same bounded solutions as the simpler relation

πt = ξˆt + βEt πt+1

s

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-

ψ

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

110 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

ψ

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

model.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 111

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∞ ).

ˆ

ˆˆ ˆ

These relations can be solved for Y∞ , I∞ and K∞ as multiples of »∞ ; this generalizes the

ˆ

ˆ

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 ˆ

s∞ = ω Y∞ ’ (ω ’ ν)K∞ ’ »∞ ;

ˆ

ˆ

substituting the above solutions, we obtain s∞ as a multiple of »∞ as well. Finally, (5.20)

ˆ

implies that

ξ0 ’ ξ1

π∞ = s∞ .

ˆ

1 ’ ψ1 + ψ2

ˆ ˆˆ

Using this together with the previous solution allows us to solve for »∞ , and hence for Y∞ , I∞

ˆ

and K∞ as well, as multiples of π∞ .

We turn next to the characterization of transitory ¬‚uctuations around these long-run

ˆ

average values. Using (5.8) “ (5.9) to eliminate Yt+1 from (5.7), we obtain a relation of the

form

ˆ

ˆ

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

112 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

ˆ

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)

j=0

ˆ

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)

t

j=0

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 ),

±

j=0

a direct generalization of (1.4), which now however takes account of investment spending.

ˆ

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)

t

j=0

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

5. MONETARY POLICY AND INVESTMENT DYNAMICS 113

14

12 ∈ψ=0.5

∈ψ=1.5

∈ψ=3

∈ψ=6

10

∈ψ=12

∈ψ=∞

8

6

4

2

0

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.

ψ

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

ˆ

ˆ

(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

114 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

ψ

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

ψ

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,

ψ

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

ˆ

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.

83

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 115

nominal interest inflation

1 1

0

0

’1

’1

’2

’2 ’3

0 5 10 15 20 0 5 10 15 20

real interest output

1

0

’2

0.5

’4

0

’6

0 5 10 15 20 0 5 10 15 20

capital stock real marginal cost

1

0

0

’2

var. capital

’1 constant capital

’4

’2

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

ψ

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

ψ

calibration of the extended model.)

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

ψ

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

116 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

ψ

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

ψ

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

ˆ

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

ψ

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

84

The e¬ect is quite signi¬cant. For example, in the simulation shown in Figure 4.21, by the eighth quarter

ˆ

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 117

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

118 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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 ,

ˆ

ˆn ˆncc

rt = rt + ·r Kt ,

ˆ

and so on, where the terms Ytncc and rt refer to exogenous processes (functions solely of

ˆncc

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

ˆ

ˆ

Itn and of the marginal utility of income »n in a similar way. We can even de¬ne a “natural”

t

ˆn

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

85

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 119

ˆ

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

ˆ

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 ] +

t

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

n

period t + 1 output does not correspond to the value of Et Yt+1 at date t in the sticky-price

ˆ

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

120 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

˜

Yt

zt ≡ .

˜

It

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)

86

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]

5. MONETARY POLICY AND INVESTMENT DYNAMICS 121

˜

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

ˆ

¯t = ¯cc + ·r Kt ,

± ±t (5.35)

where ¯cc } is an exogenous process. This implies that in addition to systematic responses to

±t

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

122 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

ˆ

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.)

rn

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

rn

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

87

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.

5. MONETARY POLICY AND INVESTMENT DYNAMICS 123

ψπ

j

0.1

0.08

0.06

0.04

0.02

0

0 1 2 3 4 5 6 7 8 9 10

ψx

j

2.5

var. capital

2

const. capital

1.5

1

0.5

0

’0.5

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

ˆncc

±t

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.

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