is easily seen to be monotonically increasing in ρ, so that the right-hand side of (A.13) is

monotonically decreasing in ρ, for any given value of φπ .

1.10 Proof of Proposition 2.10.

Proposition 2.10. Consider a policy rule of the form (2.51), where ρ > 1 and {¯t } is a

±

bounded process, to be adopted beginning at some date t0 . Then any bounded processes

{πt , ˆt } that satisfy (2.51) for all t ≥ t0 must be such that the predicted path of in¬‚ation,

±

looking forward from any date t ≥ t0 , satis¬es (2.56). Conversely, any bounded processes

satisfying (2.56) for all t ≥ t0 also satisfy (2.51) for all t ≥ t0 .

Proof: Note that (2.51) can equivalently be written

ˆt’1 ’ ¯t’1 = ’ρ’1 φπ πt + ρ’1 Et [ˆt ’ ¯t ].

± ± ± ±

This is yet another stochastic di¬erence equation of the form (A.4), where now zt ≡ ˆt’1 ’¯t’1

± ±

happens to be a variable that is predetermined at date t. It follows from the discussion in

the proof of Proposition 2.3 that if |a| < 1, any bounded solution to (A.4) must satisfy (A.6).

In the present case, this result applies if ρ > 1, since in this case 0 < ρ’1 < 1, and (2.56) is

just the condition corresponding to (A.6).

The converse is established by noting that (A.6) implies that zt satis¬es (A.4). This was

implicit in our previous characterization of (A.6) as a “solution” of equation (A.4). In the

present application, it does not make sense to call condition (2.56) a “solution” for the the

variable ˆt’1 ’ ¯t’1 , for this variable is determined at date t ’ 1, while the right-hand side

± ±

of (2.56) depends on information at date t. Instead, (2.56) indicates the way in which the

central bank™s in¬‚ation target at date t varies depending on past conditions.

1. APPENDIX TO CHAPTER 2 103

1.11 Proof of Proposition 2.11.

Proposition 2.11. In the context of a Sidrauski-Brock model with additively separable

preferences, consider the consequences of a monetary policy speci¬ed in terms of exogenous

paths {Mts , im }, together with a ¬scal policy speci¬ed by an exogenous path {Dt }. Under such

t

a regime, the rational-expectations equilibrium paths of prices and interest rates are (locally)

determinate; that is, there exist open sets P and I such that in the case of any tight enough

bounds on the ¬‚uctuations in the exogenous processes {Yt , ξt , Mts /Mt’1 , im , Dt /Dt’1 }, there

s

t

exists a unique rational-expectations equilibrium in which Pt /Mts ∈ P and it ∈ I at all

times. Furthermore, a log-linear approximation to the equilibrium path of the price level,

accurate up to a residual of order O(||ξ||2 ), takes the form

∞

•j Et [log Mt+j ’ ·i log(1 + im ) ’ ut+j ] ’ log m,

s

log Pt = ¯ (A.16)

t

j=0

where the weights

j

·i

•j ≡ >0

(1 + ·i )j+1

sum to one, and ut is a composite exogenous disturbance

ˆ m

’ ·i log(1 + ¯m ).

ut ≡ ·y Yt ’ ·i rt +

ˆ ±

t

Proof: Once again, we may ignore conditions (1.24) and (3.4), as these will be satis¬ed

by any processes {Pt /Mts , it , im , Mts /Mt’1 , Dt /Dt’1 , Yt , ξt } that satisfy tight enough bounds.

s

t

It then su¬ces that we consider the existence of bounded solutions to the system of log-linear

relations consisting of (1.32) and (3.6), augmented by the identity

mt = mt’1 + µt ’ πt ,

ˆ ˆ (A.17)

where µt ≡ log(Mts /Mt’1 is the exogenous rate of growth in the monetary base. This

s

comprises a system of three expectational di¬erence equations per period to determine the

three endogenous variables mt , πt , and ˆt .

ˆ ±

104 CHAPTER 2. PRICE-LEVEL DETERMINATION

Using (1.32) to eliminate ˆt in (3.6), we obtain a discrete-time rational-expectations ver-

±

sion of the Cagan model of in¬‚ation determination,

mt = ’·i Et πt+1 + [ut + ·i log(1 + im )],

ˆ (A.18)

t

where ut is the composite exogenous disturbance de¬ned in the statement of the proposition.

Then using (A.17) to substitute for πt+1 , (A.18) implies that

mt = ±Et mt+1 + (1 ’ ±)[ut + ·i log(1 + im ) ’ ·i Et µt+1 ],

ˆ ˆ (A.19)

t

where ± ≡ ·i /(1 + ·i ).

This is once again an expectational di¬erence equation of the form (A.4). Because ·i > 0

implies that 0 < ± < 1, (A.19) can be solved forward to obtain a unique bounded solution

for {mt }, given by

ˆ

∞

±j Et [ut+j + ·i log(1 + im ) ’ ·i µt+j+1 ].

mt = (1 ’ ±)

ˆ (A.20)

t

j=0

Such a unique solution for {mt } then implies a unique solution for {ˆt }, using (3.6), and

ˆ ±

for {πt }, using (A.17). It then follows from the discussion in section xx above that there

will also be a locally unique solution to the exact equilibrium relations (1.21) and (3.5) in

the case of tight enough bounds on the exogenous processes. Furthermore, this solution will

satisfy any desired bounds on mt and ˆt . (Since Pt /Mts = 1/mt , this allows us to ensure that

ˆ ±

Pt /Mts ∈ P at all times.) Finally, we observe that (A.19) can be rewritten as (A.16).

1.12 Proof of Proposition 2.12.

Proposition 2.12. In a Sidrauski-Brock model where utility is not necessarily separable,

let monetary policy be speci¬ed by a Wicksellian rule (1.30) for the central bank™s interest-

rate operating target. Suppose that im = ¯ at all times, for some 0 ¤ ¯ < β ’1 ’ 1; and let

± ±

t

¬scal policy again be speci¬ed by an exogenous process {Dt }. Finally, suppose that

1

χ>’ . (A.21)

2·i

1. APPENDIX TO CHAPTER 2 105

Then equilibrium is determinate in the case of any policy rule with φp > 0. A log-linear

approximation to the locally unique equilibrium price process is given by (3.17), where the

weights are given by (3.18) “ (3.19).

Proof: Substituting (1.34) for ˆt in (3.12), and setting ˆm = 0, we obtain the equilibrium

± ±t

relation

ˆ ˆ —

(1 + (1 + ·i χ)φp )Pt = (1 + ·i χφp )Et Pt+1 + [Et πt+1 + (˜t ’ νt ) + ·i χ(Et νt+1 ’ νt )] (A.22)

r

as a generalization of (1.36). This is once again a stochastic di¬erence equation of the form

(A.4). It then follows from our discussion in section xx above that (A.22) can be solved

ˆ

forward to yield a unique bounded solution for {Pt }, if and only if

|1 + ·i χφp | < |1 + φp (1 + ·i χ)|.

We observe that as long as (A.21) holds, this determinacy condition is satis¬ed for all φp > 0,

just as was concluded in section xx for the case χ = 0. Furthermore, in this case, the unique

bounded solution is given by (A.6). Applying this result and rearranging terms, one obtains

(3.17).

1.13 Proof of Proposition 2.13.

Proposition 2.13. Let monetary policy instead be speci¬ed by an interest-rate rule of

the form (2.51), with coe¬cients φπ , ρ ≥ 0, and again suppose that im = ¯m at all times.

±t

t

Finally, suppose that χ satis¬es (A.21).Then equilibrium is determinate if and only if φπ > 0

and (A.11) holds. When these conditions are satis¬ed, a log-linear approximation to the

equilibrium evolution of in¬‚ation is given by (3.20), where the weights are given by (3.21) “

(3.22).

Proof: Using (2.51) to eliminate πt+1 in (3.12), and again setting ˆm = 0, one obtains

±t

a stochastic di¬erence equation for the interest rate of the form

[(1 + ·i χ)φπ + ρ]ˆt = [1 + ·i χφπ ]Etˆt+1 + φπ rt + ρ¯t ’ Et¯t+1 .

± ± ˜ ± ± (A.23)

106 CHAPTER 2. PRICE-LEVEL DETERMINATION

This is again an equation of the form (A.4). It follows that equilibrium is locally determinate

if and only if the term in square brackets on the left-hand side (call it γ0 ) is larger in absolute

value than the term in square brackets on the right-hand side (call it γ1 ). Condition (A.21)

su¬ces to guarantee that γ0 + γ1 > 0. It then follows that |γ0 | > |γ1 |, so that determinacy

obtains, if and only if γ0 > γ1 . This last inequality is in turn seen to be equivalent to (A.11).

In the case that equilibrium is determinate, (A.6) can again be applied, yielding (3.20).

1.14 Proof of Proposition 2.14.

Proposition 2.14. Consider a sequence of economies with progressively smaller period

lengths ∆, calibrated so that χ = 0. Assume in each case that monetary policy is speci¬ed

¯

by a contemporaneous Taylor rule (2.43), with a positive in¬‚ation-response coe¬cient φπ = 1

that is independent of ∆. Assume also that zero interest is paid on money. Then equilibrium

is determinate for all small enough values of ∆ if φπ > 1 and χ > 0, or if 0 < φπ < 1 and

¯

χ < 0, but not otherwise.

¯

Proof: As discussed in the proof of Proposition 2.13, determinacy obtains if and only

if |γ0 | > |γ1 |. In the limit as ∆ ’ 0, both γ0 and γ1 become unboundedly large, while

∆γ0 , ∆γ1 ’ ·i χφπ .

˜¯ (A.24)

Thus both γ0 and γ1 have the same sign as χ, for all small enough values of ∆. We note

¯

furthermore that

γ0 ’ γ1 = φπ ’ 1,

so that γ0 > γ1 if and only if φπ > 1. It then follows that |γ0 | > |γ1 |, and determinacy

obtains, for all small enough values of ∆, if and only if either φπ > 1 and χ > 0 (so that

¯

γ0 > γ1 > 0) or 0 < φπ < 1 and χ < 0 (so that γ0 < γ1 < 0).

¯

1.15 Proof of Proposition 2.15.

Proposition 2.15. Again consider a sequence of economies with progressively smaller

1. APPENDIX TO CHAPTER 2 107

period lengths ∆, and suppose that

1

χ>’

¯ . (A.25)

ψ¦π ·i

˜

Let monetary policy instead be speci¬ed by an inertial Taylor rule (2.51), with a long-run

in¬‚ation-response coe¬cient ¦π ≡ φπ /(1 ’ ρ) and a rate of adjustment ψ ≡ ’ log ρ/∆ > 0

that are independent of ∆. Assume again that zero interest is paid on money. Then rational-

expectations equilibrium is determinate if and only if ¦p i > 1, i.e., if and only if the Taylor

Principle is satis¬ed.

The unique bounded solution for the path of nominal interest rates in the determinate

case is of the form

∞ ∞

˜

j

γ j Et¯t+j ,

ˆt = Λ¯t + “(1 ’ γ)

± ± γ Et rt+j + “(1 ’ γ)

˜ ± (A.26)

j=0 j=0

with the solution for {πt } then obtained by inverting (2.51). In this solution, the coe¬cients

˜

Λ, “, “ approach well-de¬ned limiting values as ∆ is made arbitrarily small, while the rate

of decay of the weights on expected disturbances farther in the future,

ξ ≡ ’ log γ/∆ > 0,

also approaches a well-de¬ned limiting value. Furthermore, these limiting values are all

continuous functions of χ for values of χ in the range satisfying (A.25), including values near

¯ ¯

zero.

Proof: In this case, because φπ approaches zero at the same rate as ∆ (though we

again maintain a long-run in¬‚ation-response coe¬cient independent of ∆), γ0 and γ1 do not

become unboundedly large for small ∆. Instead of (A.24), we obtain

γ0 , γ1 ’ 1 + ·i χ¦π ψ,

˜¯

and this limiting value is positive given (A.25). Hence γ0 , γ1 > 0 in the case of any small

enough value of ∆, even if χ is (modestly) negative. We furthermore observe that in this

¯

case,

γ0 ’ γ1 = (¦p i ’ 1)(1 ’ ρ).

108 CHAPTER 2. PRICE-LEVEL DETERMINATION

Hence we ¬nd once again that γ0 > γ1 if and only if ¦ > 1. Thus |γ0 | > |γ1 |, and equilibrium

is determinate, if and only if ¦ > 1.

In the case of determinacy, the equilibrium solution for the nominal interest rate, derived

in the course of the proof of Proposition 2.13, is given by

∞ ∞

ˆt = ¯t +

± ± φπ •j Et rt+j ’

˜ φπ •j Et¯t+j ,

˜±

j=0 j=0

where the coe¬cients {•j , •j } are de¬ned in (3.21) “ (3.22). This is observed to be of the

˜

˜

form (A.26), and allows us to identify the coe¬cients Λ, “, “, and γ in that representation

of the solution.

We then observe that as ∆ is made arbitrarily small,

(1 + ·i χ)φπ ·i χ¦π ψ

˜¯

Λ + “(1 ’ γ) ’ 1 = ’ ,

(1 + ·i χ)φπ + ρ 1 + ·i χ¦π ψ

˜¯

φπ ¦π

“= = > 0,

φπ + ρ ’ 1 ¦π ’ 1

˜ 1 + (1 ’ ρ)·i χ “ ’ 1 + ·i χψ

˜¯ ¦π

“= > 0,

1 + ·i χφπ 1 + ·i χ¦π ψ ¦π ’ 1

˜¯

and that ξ has the same limiting value as

1’γ 1 φπ + ρ ’ 1 (¦π ’ 1)ψ

= ’ > 0.

∆ ∆ (1 + ·i χ)φπ + ρ 1 + ·i χ¦π ψ

˜¯

Since γ ’ 1, we also observe that

·i χ¦π ψ

˜¯

Λ’1+ > 0.

1 + ·i χ¦π ψ

˜¯

˜

Hence Λ, “, “, and ξ all have well-de¬ned limiting values, and each is a continuous function

of χ.

¯