<< стр. 4(всего 4)СОДЕРЖАНИЕ

is easily seen to be monotonically increasing in ПҒ, so that the right-hand side of (A.13) is
monotonically decreasing in ПҒ, for any given value of ПҶПҖ .

1.10 Proof of Proposition 2.10.

Proposition 2.10. Consider a policy rule of the form (2.51), where ПҒ > 1 and {ВҜt } is a
Д±
bounded process, to be adopted beginning at some date t0 . Then any bounded processes
{ПҖt , ЛҶt } that satisfy (2.51) for all t вүҘ t0 must be such that the predicted path of inп¬‚ation,
Д±
looking forward from any date t вүҘ t0 , satisп¬Ғes (2.56). Conversely, any bounded processes
satisfying (2.56) for all t вүҘ t0 also satisfy (2.51) for all t вүҘ t0 .

Proof: Note that (2.51) can equivalently be written

ЛҶtвҲ’1 вҲ’ ВҜtвҲ’1 = вҲ’ПҒвҲ’1 ПҶПҖ ПҖt + ПҒвҲ’1 Et [ЛҶt вҲ’ ВҜt ].
Д± Д± Д± Д±

This is yet another stochastic diп¬Җerence equation of the form (A.4), where now zt вүЎ ЛҶtвҲ’1 вҲ’ВҜtвҲ’1
Д± Д±
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 ПҒ > 1, since in this case 0 < ПҒвҲ’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 satisп¬Ғes (A.4). This was
implicit in our previous characterization of (A.6) as a вҖңsolutionвҖқ of equation (A.4). In the
present application, it does not make sense to call condition (2.56) a вҖңsolutionвҖқ for the the
variable ЛҶtвҲ’1 вҲ’ ВҜtвҲ’1 , for this variable is determined at date t вҲ’ 1, while the right-hand side
Д± Д±
of (2.56) depends on information at date t. Instead, (2.56) indicates the way in which the
central bankвҖ™s inп¬‚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 speciп¬Ғed in terms of exogenous
paths {Mts , im }, together with a п¬Ғscal policy speciп¬Ғed 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 п¬‚uctuations in the exogenous processes {Yt , Оҫt , Mts /MtвҲ’1 , im , Dt /DtвҲ’1 }, there
s
t

exists a unique rational-expectations equilibrium in which Pt /Mts вҲҲ P and it вҲҲ 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(||Оҫ||2 ), takes the form
вҲһ
П•j Et [log Mt+j вҲ’ О·i log(1 + im ) вҲ’ ut+j ] вҲ’ log m,
s
log Pt = ВҜ (A.16)
t
j=0

where the weights
j
О·i
П•j вүЎ >0
(1 + О·i )j+1
sum to one, and ut is a composite exogenous disturbance

ЛҶ m
вҲ’ О·i log(1 + ВҜm ).
ut вүЎ О·y Yt вҲ’ О·i rt +
ЛҶ Д±
t

Proof: Once again, we may ignore conditions (1.24) and (3.4), as these will be satisп¬Ғed
by any processes {Pt /Mts , it , im , Mts /MtвҲ’1 , Dt /DtвҲ’1 , Yt , Оҫt } that satisfy tight enough bounds.
s
t

It then suп¬ғces 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 = mtвҲ’1 + Вµt вҲ’ ПҖt ,
ЛҶ ЛҶ (A.17)

where Вµt вүЎ log(Mts /MtвҲ’1 is the exogenous rate of growth in the monetary base. This
s

comprises a system of three expectational diп¬Җerence equations per period to determine the
three endogenous variables mt , ПҖt , and ЛҶt .
ЛҶ Д±
104 CHAPTER 2. PRICE-LEVEL DETERMINATION

Using (1.32) to eliminate ЛҶt in (3.6), we obtain a discrete-time rational-expectations ver-
Д±
sion of the Cagan model of inп¬‚ation determination,

mt = вҲ’О·i Et ПҖt+1 + [ut + О·i log(1 + im )],
ЛҶ (A.18)
t

where ut is the composite exogenous disturbance deп¬Ғned in the statement of the proposition.
Then using (A.17) to substitute for ПҖt+1 , (A.18) implies that

mt = О±Et mt+1 + (1 вҲ’ О±)[ut + О·i log(1 + im ) вҲ’ О·i Et Вµt+1 ],
ЛҶ ЛҶ (A.19)
t

where О± вүЎ О·i /(1 + О·i ).
This is once again an expectational diп¬Җerence equation of the form (A.4). Because О·i > 0
implies that 0 < О± < 1, (A.19) can be solved forward to obtain a unique bounded solution
for {mt }, given by
ЛҶ
вҲһ
О±j Et [ut+j + О·i log(1 + im ) вҲ’ О·i Вµt+j+1 ].
mt = (1 вҲ’ О±)
ЛҶ (A.20)
t
j=0

Such a unique solution for {mt } then implies a unique solution for {ЛҶt }, using (3.6), and
ЛҶ Д±
for {ПҖt }, 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 ЛҶt . (Since Pt /Mts = 1/mt , this allows us to ensure that
ЛҶ Д±
Pt /Mts вҲҲ 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 speciп¬Ғed by a Wicksellian rule (1.30) for the central bankвҖ™s interest-
rate operating target. Suppose that im = ВҜ at all times, for some 0 вү¤ ВҜ < ОІ вҲ’1 вҲ’ 1; and let
Д± Д±
t

п¬Ғscal policy again be speciп¬Ғed by an exogenous process {Dt }. Finally, suppose that

1
ПҮ>вҲ’ . (A.21)
2О·i
1. APPENDIX TO CHAPTER 2 105

Then equilibrium is determinate in the case of any policy rule with ПҶp > 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) вҖ“ (3.19).

Proof: Substituting (1.34) for ЛҶt in (3.12), and setting ЛҶm = 0, we obtain the equilibrium
Д± Д±t
relation

ЛҶ ЛҶ вҲ—
(1 + (1 + О·i ПҮ)ПҶp )Pt = (1 + О·i ПҮПҶp )Et Pt+1 + [Et ПҖt+1 + (Лңt вҲ’ ОҪt ) + О·i ПҮ(Et ОҪt+1 вҲ’ ОҪt )] (A.22)
r

as a generalization of (1.36). This is once again a stochastic diп¬Җerence equation of the form
(A.4). It then follows from our discussion in section xx above that (A.22) can be solved
ЛҶ
forward to yield a unique bounded solution for {Pt }, if and only if

|1 + О·i ПҮПҶp | < |1 + ПҶp (1 + О·i ПҮ)|.

We observe that as long as (A.21) holds, this determinacy condition is satisп¬Ғed for all ПҶp > 0,
just as was concluded in section xx for the case ПҮ = 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 speciп¬Ғed by an interest-rate rule of
the form (2.51), with coeп¬ғcients ПҶПҖ , ПҒ вүҘ 0, and again suppose that im = ВҜm at all times.
Д±t
t

Finally, suppose that ПҮ satisп¬Ғes (A.21).Then equilibrium is determinate if and only if ПҶПҖ > 0
and (A.11) holds. When these conditions are satisп¬Ғed, a log-linear approximation to the
equilibrium evolution of inп¬‚ation is given by (3.20), where the weights are given by (3.21) вҖ“
(3.22).

Proof: Using (2.51) to eliminate ПҖt+1 in (3.12), and again setting ЛҶm = 0, one obtains
Д±t
a stochastic diп¬Җerence equation for the interest rate of the form

[(1 + О·i ПҮ)ПҶПҖ + ПҒ]ЛҶt = [1 + О·i ПҮПҶПҖ ]EtЛҶt+1 + ПҶПҖ rt + ПҒВҜt вҲ’ EtВҜt+1 .
Д± Д± Лң Д± Д± (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 Оі0 ) is larger in absolute
value than the term in square brackets on the right-hand side (call it Оі1 ). Condition (A.21)
suп¬ғces to guarantee that Оі0 + Оі1 > 0. It then follows that |Оі0 | > |Оі1 |, so that determinacy
obtains, if and only if Оі0 > Оі1 . 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 вҲҶ, calibrated so that ПҮ = 0. Assume in each case that monetary policy is speciп¬Ғed
ВҜ
by a contemporaneous Taylor rule (2.43), with a positive inп¬‚ation-response coeп¬ғcient ПҶПҖ = 1
that is independent of вҲҶ. Assume also that zero interest is paid on money. Then equilibrium
is determinate for all small enough values of вҲҶ if ПҶПҖ > 1 and ПҮ > 0, or if 0 < ПҶПҖ < 1 and
ВҜ
ПҮ < 0, but not otherwise.
ВҜ

Proof: As discussed in the proof of Proposition 2.13, determinacy obtains if and only
if |Оі0 | > |Оі1 |. In the limit as вҲҶ вҶ’ 0, both Оі0 and Оі1 become unboundedly large, while

вҲҶОі0 , вҲҶОі1 вҶ’ О·i ПҮПҶПҖ .
ЛңВҜ (A.24)

Thus both Оі0 and Оі1 have the same sign as ПҮ, for all small enough values of вҲҶ. We note
ВҜ
furthermore that
Оі0 вҲ’ Оі1 = ПҶПҖ вҲ’ 1,

so that Оі0 > Оі1 if and only if ПҶПҖ > 1. It then follows that |Оі0 | > |Оі1 |, and determinacy
obtains, for all small enough values of вҲҶ, if and only if either ПҶПҖ > 1 and ПҮ > 0 (so that
ВҜ
Оі0 > Оі1 > 0) or 0 < ПҶПҖ < 1 and ПҮ < 0 (so that Оі0 < Оі1 < 0).
ВҜ

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 вҲҶ, and suppose that
1
ПҮ>вҲ’
ВҜ . (A.25)
ПҲО¦ПҖ О·i
Лң
Let monetary policy instead be speciп¬Ғed by an inertial Taylor rule (2.51), with a long-run
inп¬‚ation-response coeп¬ғcient О¦ПҖ вүЎ ПҶПҖ /(1 вҲ’ ПҒ) and a rate of adjustment ПҲ вүЎ вҲ’ log ПҒ/вҲҶ > 0
that are independent of вҲҶ. Assume again that zero interest is paid on money. Then rational-
expectations equilibrium is determinate if and only if О¦p i > 1, i.e., if and only if the Taylor
Principle is satisп¬Ғed.
The unique bounded solution for the path of nominal interest rates in the determinate
case is of the form
вҲһ вҲһ
Лң
j
Оі j EtВҜt+j ,
ЛҶt = ОӣВҜt + О“(1 вҲ’ Оі)
Д± Д± Оі Et rt+j + О“(1 вҲ’ Оі)
Лң Д± (A.26)
j=0 j=0

with the solution for {ПҖt } then obtained by inverting (2.51). In this solution, the coeп¬ғcients
Лң
Оӣ, О“, О“ approach well-deп¬Ғned limiting values as вҲҶ is made arbitrarily small, while the rate
of decay of the weights on expected disturbances farther in the future,

Оҫ вүЎ вҲ’ log Оі/вҲҶ > 0,

also approaches a well-deп¬Ғned limiting value. Furthermore, these limiting values are all
continuous functions of ПҮ for values of ПҮ in the range satisfying (A.25), including values near
ВҜ ВҜ
zero.

Proof: In this case, because ПҶПҖ approaches zero at the same rate as вҲҶ (though we
again maintain a long-run inп¬‚ation-response coeп¬ғcient independent of вҲҶ), Оі0 and Оі1 do not
become unboundedly large for small вҲҶ. Instead of (A.24), we obtain

Оі0 , Оі1 вҶ’ 1 + О·i ПҮО¦ПҖ ПҲ,
ЛңВҜ

and this limiting value is positive given (A.25). Hence Оі0 , Оі1 > 0 in the case of any small
enough value of вҲҶ, even if ПҮ is (modestly) negative. We furthermore observe that in this
ВҜ
case,
Оі0 вҲ’ Оі1 = (О¦p i вҲ’ 1)(1 вҲ’ ПҒ).
108 CHAPTER 2. PRICE-LEVEL DETERMINATION

Hence we п¬Ғnd once again that Оі0 > Оі1 if and only if О¦ > 1. Thus |Оі0 | > |Оі1 |, and equilibrium
is determinate, if and only if О¦ > 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
вҲһ вҲһ
ЛҶt = ВҜt +
Д± Д± ПҶПҖ П•j Et rt+j вҲ’
Лң ПҶПҖ П•j EtВҜt+j ,
ЛңД±
j=0 j=0

where the coeп¬ғcients {П•j , П•j } are deп¬Ғned in (3.21) вҖ“ (3.22). This is observed to be of the
Лң
Лң
form (A.26), and allows us to identify the coeп¬ғcients Оӣ, О“, О“, and Оі in that representation
of the solution.
We then observe that as вҲҶ is made arbitrarily small,

(1 + О·i ПҮ)ПҶПҖ О·i ПҮО¦ПҖ ПҲ
ЛңВҜ
Оӣ + О“(1 вҲ’ Оі) вҲ’ 1 = вҶ’ ,
(1 + О·i ПҮ)ПҶПҖ + ПҒ 1 + О·i ПҮО¦ПҖ ПҲ
ЛңВҜ

ПҶПҖ О¦ПҖ
О“= = > 0,
ПҶПҖ + ПҒ вҲ’ 1 О¦ПҖ вҲ’ 1
Лң 1 + (1 вҲ’ ПҒ)О·i ПҮ О“ вҶ’ 1 + О·i ПҮПҲ
ЛңВҜ О¦ПҖ
О“= > 0,
1 + О·i ПҮПҶПҖ 1 + О·i ПҮО¦ПҖ ПҲ О¦ПҖ вҲ’ 1
ЛңВҜ
and that Оҫ has the same limiting value as

1вҲ’Оі 1 ПҶПҖ + ПҒ вҲ’ 1 (О¦ПҖ вҲ’ 1)ПҲ
= вҶ’ > 0.
вҲҶ вҲҶ (1 + О·i ПҮ)ПҶПҖ + ПҒ 1 + О·i ПҮО¦ПҖ ПҲ
ЛңВҜ

Since Оі вҶ’ 1, we also observe that

О·i ПҮО¦ПҖ ПҲ
ЛңВҜ
ОӣвҶ’1+ > 0.
1 + О·i ПҮО¦ПҖ ПҲ
ЛңВҜ
Лң
Hence Оӣ, О“, О“, and Оҫ all have well-deп¬Ғned limiting values, and each is a continuous function
of ПҮ.
ВҜ

 << стр. 4(всего 4)СОДЕРЖАНИЕ