<< стр. 3(всего 4)СОДЕРЖАНИЕ >>
weights П•j and П•j appearing in the solution (3.20) for various future horizons j are plotted
Лњ
in Figure 2.2. The п¬Ѓgure shows the weights both under the assumption that П‡ = 0 (as in our
п¬Ѓrst approach), and for the positive value П‡ = .02. The latter is likely to be an over-estimate
of the actual size of real-balance eп¬Ђects on the marginal utility of income, but is considered to
show that even under the most generous assumptions real-balance eп¬Ђects should not matter
greatly. (The value О·i = 28 quarters is assumed in both cases.)
While the real-balance eп¬Ђect matters for such a calculation, neglecting it would not lead to
52
See chapter 4 for further discussion of the calibration of this parameter.
68 CHAPTER 2. PRICE-LEVEL DETERMINATION

П€
j
1

П‡j=0
0.8
П‡j=0.02

0.6

0.4

0.2
0 2 4 6 8 10 12
j
˜
П€j
1.6

1.4 П‡ =0
j
П‡j=0.02
1.2

1

0.8

0.6

0.4

0.2
0 2 4 6 8 10 12
j

Figure 2.2: The weights П•j with and without real-balance eп¬Ђects.

extremely misleading conclusions, either. Perhaps the most important qualitative diп¬Ђerence
is that setting П‡ = 0.02 results in a slightly larger relative weight on the current periodвЂ™s
intercept ВЇt as opposed to expected future intercepts. Especially in the (realistic) case that
Д±
both the equilibrium real rate rt and the monetary policy disturbance ВЇt exhibit substantial
Лњ Д±
positive serial correlation, so that it is only smoothed versions of the coeп¬ѓcients П•j and П•j
Лњ
that matter in practice, the predictions for inп¬‚ation under the two assumptions will be quite
similar. For example, let us suppose that rt follows a stationary AR(1) process with serial
Лњ
correlation coeп¬ѓcient 0 < ПЃr < 1. Then the predicted initial-period jump in the price level
in response to a unit positive innovation in the natural rate of interest53 is given by
в€ћ
П•j ПЃj .
в€† r
j=0

53
We understand this to mean a jump of one percentage point per annum in the natural rate, meaning
that rt jumps by в€†, where в€† is the length of a period in years.
Лњ
3. PRICE-LEVEL DETERMINATION WITH MONETARY FRICTIONS 69

Quarterly model
2.5

П‡=0
2
П‡=0.02
1.5

1

0.5

0
0 50 100 150 200 250 300 350 400

Daily model
2.5

П‡=0
2 П‡=0.02

1.5

1

0.5

0
0 50 100 150 200 250 300 350 400
О» (percent per year)

Figure 2.3: Inп¬‚ation response as a function of shock persistence.

This quantity is plotted, as a function of the degree of persistence of the shocks, in Figure
2.3(a). Here the measure of persistence on the horizontal axis is the rate of decay of the
disturbances per unit of calendar time, О» в‰Ў в€’(log ПЃr )/в€†, where в€† is the length of a period
in years.54 The results are plotted both under the assumption that П‡ = 0 and for the upper-
bound case П‡ = .02. We observe that the error involved in neglecting the real-balance eп¬Ђects
is quite small.
One case in which real balance eп¬Ђects would matter a great deal, however, is that of a
rule (2.43) in which the interest rate depends only upon contemporaneous inп¬‚ation, in the
case that the вЂњperiodsвЂќ are very short. Let us consider the behavior of the solution to our
equations as the length of a вЂњperiodвЂќ (в€† > 0 units of calendar time) is made progressively
shorter. We shall п¬Ѓx numerical values for the rate of time preference Оґ в‰Ў (в€’ log ОІ)/в€† > 0,
54
This measure is used, instead of ПЃr itself, in order to allow comparability across models with diп¬Ђerent
period lengths. In panel (a), в€† = .25 years.
70 CHAPTER 2. PRICE-LEVEL DETERMINATION

the intertemporal elasticity of substitution Пѓ > 0, steady-state share of expenditure on
liquidity services sm , the interest-rate semi-elasticity of money demand in units of calendar
time О·i в‰Ў в€†О·i > 0, and the income-elasticity of money demand О·y that are independent of
Лњ
the assumed size of в€†. Then the same reasoning used to derive (3.23) implies that as в€† is
made small, the value of П‡ approaches a well-deп¬Ѓned limiting value

О·y
в€’ Пѓ в€’1 .
П‡ = sm
ВЇ (3.24)
Оґ О·i
Лњ

We then obtain the following result.

Proposition 2.14. Consider a sequence of economies with progressively smaller period
lengths в€†, calibrated so that П‡ = 0. Assume in each case that monetary policy is speciп¬Ѓed by
ВЇ
a contemporaneous Taylor rule (2.43), with a positive inп¬‚ation-response coeп¬ѓcient П†ПЂ = 1
that is independent of в€†. Assume also that zero interest is paid on money. Then equilibrium
is determinate for all small enough values of в€† if П†ПЂ > 1 and П‡ > 0, or if 0 < П†ПЂ < 1 and
ВЇ
П‡ < 0, but not otherwise.
ВЇ

The proof is in the appendix.55 This result clearly implies that the solution for equilibrium
inп¬‚ation in the short-period limit cannot be a continuous function of П‡ for values of П‡ near
zero. Thus the value of П‡ matters in this case, even if it is very small.
However, this failure of continuity in П‡ in the short-period limit occurs only in the case
of a policy rule that makes the interest-rate operating target a purely contemporaneous
function of the current periodвЂ™s inп¬‚ation. As periods are made shorter, the central bank
is assumed to respond to a higher-frequency measure of inп¬‚ation, and in the limit policy
is assumed to respond solely to an instantaneous rate of inп¬‚ation. This is plainly a case
55
Our result agrees with the continuous-time analysis of Benhabib et al. (1998a), who п¬Ѓnd that the range
of values of П†ПЂ that result in determinacy depends on the sign of ucm . Note that the result does not contradict
Proposition 2.13. That proposition asserts that even in the case of a rule with ПЃ = 0, one has determinacy
in the case of П†ПЂ > 1 for all values of П‡ greater than the negative lower bound (3.16). However, in the
sequence of economies with progressively shorter period lengths considered in Proposition 2.14, the value of
О·i increases as в€†в€’1 . Hence for any sequence of economies in Proposition 2.14 for which П‡ < 0, (3.16) is
ВЇ
violated for all small enough values of в€†, and Proposition 2.13 ceases to apply.
3. PRICE-LEVEL DETERMINATION WITH MONETARY FRICTIONS 71

of no practical interest. If we assume instead that policy responds to a smoothed inп¬‚ation
measure (2.47), and let the rate of decay (in calendar time) of the exponential weights on
past inп¬‚ation be п¬Ѓxed as we make вЂњperiodsвЂќ shorter, no such problem arises. The same is
true in the equivalent case of a policy rule (2.51) with partial adjustment of the interest
rate toward a desired level that depends on the current instantaneous rate of inп¬‚ation, if we
hold п¬Ѓxed the rate of adjustment П€ в‰Ў в€’ log ПЃ/в€† > 0 as we make в€† smaller. (We must also
assume that П†ПЂ is reduced along with в€†, so as to hold п¬Ѓxed the long-run response coeп¬ѓcient
О¦ПЂ в‰Ў (1 в€’ ПЃ)в€’1 П†ПЂ .) In this case we obtain the following.

Proposition 2.15. Again consider a sequence of economies with progressively smaller
period lengths в€†, and suppose that
1
П‡>в€’
ВЇ . (3.25)
П€О¦ПЂ О·i
Лњ
Let monetary policy instead be speciп¬Ѓed by an inertial Taylor rule (2.51), with a long-run
inп¬‚ation-response coeп¬ѓcient О¦ПЂ в‰Ў П†ПЂ /(1 в€’ ПЃ) and a rate of adjustment П€ в‰Ў в€’ log ПЃ/в€† > 0
that are independent of в€†. 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¬Ѓed.
The unique bounded solution for the path of nominal interest rates in the determinate
case is of the form
в€ћ в€ћ
Лњ
j
Оі j EtВЇt+j ,
Л†t = О›ВЇt + О“(1 в€’ Оі)
Д± Д± Оі Et rt+j
Лњ + О“(1 в€’ Оі) Д± (3.26)
j=0 j=0

with the solution for {ПЂt } then obtained by inverting (2.51). In this solution, the coeп¬ѓcients
Лњ
О›, О“, О“ approach well-deп¬Ѓned limiting values as в€† is made arbitrarily small, while the rate
of decay of the weights on expected disturbances farther in the future,

Оѕ в‰Ў в€’ log Оі/в€† > 0,

also approaches a well-deп¬Ѓned limiting value. Furthermore, these limiting values are all
continuous functions of П‡ for values of П‡ in the range satisfying (3.25), including values near
ВЇ ВЇ
zero.
72 CHAPTER 2. PRICE-LEVEL DETERMINATION

Again the proof is in the appendix. Proposition 2.15 implies that in the case of this
kind of rule, small non-zero values of П‡ (of either sign) make no important diп¬Ђerence for our
conclusions regarding price-level determination, even in the limiting case of arbitrarily short
periods.56 For example, our conclusions above about the small consequences of allowing for
a realistic positive value for П‡ continue to hold in the case of periods shorter than a quarter.
As an illustration, Figure 2.3(b) shows the same calculations as in Figure 2.3(a), but for a
model in which the вЂњperiodвЂќ is only a day. Thus as long as we assume either a modest degree
of time-averaging in the inп¬‚ation measure to which the central bank responds, or a modest
degree of inertia in the central bankвЂ™s adjustment of its interest-rate operating target in
response to inп¬‚ation variations вЂ” both of which are always characteristic of actual central-
bank policies вЂ” we continue to п¬Ѓnd that the cashless analysis gives a good approximation to
the results obtained under a realistic non-zero value of П‡, regardless of the assumed period
length.

4 Self-Fulп¬Ѓlling Inп¬‚ations and Deп¬‚ations

Thus far we have considered only the problem of local determinacy of equilibrium. But it is
appropriate also to consider, at least brieп¬‚y, the question whether rational expectations equi-
librium is globally unique under one policy rule or another. Certainly we may have greater
conп¬Ѓdence that a particular policy regime is desirable if a desirable outcome represents not
merely a locally unique equilibrium, but the unique rational expectations equilibrium, pe-
riod. And insofar as regimes may diп¬Ђer in the matter of global uniqueness, even when they
are equally consistent with the same desired equilibrium, and equally serve to make it locally
determinate, considerations of global uniqueness provide a reasonable further criterion for
reп¬Ѓning oneвЂ™s policy prescription.
The question of global uniqueness requires that we return to a consideration of the exact,
nonlinear equilibrium conditions, as our log-linear approximations can be relied upon to be
56
Wicksellian policy rules of the form (1.30) can similarly be shown to be well-behaved in the continuous-
time limit.
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 73

accurate only in the case of equilibria in which the variables remain within a suп¬ѓciently small
neighborhood of the values at which the log-linearization is done. This makes a complete
treatment of the issue rather complex, and beyond the scope of the present study. However,
a simple example will serve to illustrate how global multiplicity of equilibrium is possible,
despite local determinacy. We shall also give examples of policy regimes that would resolve
this problem.

4.1 Global Multiplicity Despite Local Determinacy

Our example illustrates the potential problem of multiplicity of equilibria under a вЂњTaylor
ruleвЂќ as discussed by Schmitt-GrohВґ and Uribe (1998) and by Benhabib et al. (1998b).
e
Consider a deterministic interest-rate feedback rule of the form

it = П†(О t ), (4.1)

where П† is again an increasing continuous function, satisfying П†(О ) в‰Ґ 0 for all О  > 0.
We suppose once again that this rule incorporates an implicit target inп¬‚ation rate О в€— > ОІ
satisfying П†(О в€— ) = ОІ в€’1 О в€— в€’ 1. The stipulated lower bound вЂ“ which is of some importance
for the present discussion вЂ“ is necessary because it will be impossible for the central bank
to force nominal interest rates to be negative, no matter how much it may increase the
monetary base.
Following Benhabib et al. (1998b), let п¬Ѓscal policy now be speciп¬Ѓed by a rule of the form
it
Tt = О±Wt в€’ Mt (4.2)
1 + it
for determination of net tax collections at each date, for some constant 0 < О± в‰¤ 1. Using
the п¬‚ow government budget constraint (1.7), we see that this rule implies that

Et [Qt,t+1 Wt+1 ] = (1 в€’ О±)Wt .

Thus this п¬Ѓscal policy has the вЂњRicardianвЂќ property that the transversality condition (1.18)
necessarily holds, regardless of the evolution of the endogenous variables.57 This means that
57
Here our terminology follows Benhabib et al. (1998a). See chapter 5 for further discussion.
74 CHAPTER 2. PRICE-LEVEL DETERMINATION

we may omit the transversality condition from our list of requirements for an equilibrium.
Note also that under such a п¬Ѓscal policy, debt management policy is irrelevant for equilibrium
determination.
Under such a regime, a rational expectations equilibrium is a pair of processes {Pt , it }
satisfying (4.1) and
в€’1
О»(yt+1 , it+1 ; Оѕt+1 ) Pt
в€’1
1 + it = ОІ Et (4.3)
О»(yt , it ; Оѕt ) Pt+1
at all dates, together with the bound
в€ћ
iT
ОІ T Et О»(yT , iT ; ОѕT ) yT + L(yT , iT ; ОѕT ) < в€ћ. (4.4)
1 + iT
T =t

Here (4.3) rewrites (1.21) using the function О»( y, i; Оѕ) that gives the value of Uc as a function
of those arguments (by substituting equilibrium real balances for the second argument of
Uc ), and (4.4) similarly rewrites (1.23), also using (3.2) to substitute for Um .
The general existence of multiple solutions can be shown by considering the set of perfect
foresight equilibria (i.e., deterministic solutions) in the absence of shocks (yt = y , Оѕt = 0 for
ВЇ
all t). Then, substituting (4.1) into (4.3), we obtain a nonlinear diп¬Ђerence equation for the
inп¬‚ation rate,
О t+1 О»(П†(О t+1 ))в€’1 = ОІ(1 + П†(О t ))О»(П†(О t ))в€’1 , (4.5)

now writing simply О»(i) for О»(ВЇ, i; 0). In the cashless limit (or the case of additive separa-
y
bility), this reduces to
О t+1 = ОІ(1 + П†(О t )). (4.6)

It is clear in this last case that there exists a solution for О t+1 > 0 in the case of any given
О t > 0. Hence starting from any arbitrarily chosen initial inп¬‚ation rate О 0 > 0, we can
construct a sequence {О t } that satisп¬Ѓes (4.6) at all dates. Associated with this is a sequence
of non-negative interest rates, given by (4.1). As long as these sequences satisfy the bound
(4.4), they represent a perfect foresight equilibrium. In the case that desired real money
balances L(ВЇ, i; 0) are bounded above as i approaches zero,58 because there is satiation at
y
58
In fact, it suп¬ѓces that iL(ВЇ, i; 0) be bounded. Thus even in the case of the log-log money demand function
y
preferred by Lucas (1999), in which desired real balances decline as iв€’1/2 , condition (4.4) is satisп¬Ѓed by all
interest-rate sequences.
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 75

ОІ(1+П†(О ))
О t+1

45Г»

ОІ

О t
0 О в€—в€—= ОІ О 2 О 1 О 0 О в€—
Figure 2.4: A self-fulп¬Ѓlling deп¬‚ation under a Taylor rule.

a п¬Ѓnite level of real money balances, this holds for any sequence {it }. In such a case, it is
clear that there exists a continuum of perfect foresight equilibria, one corresponding to each
possible initial inп¬‚ation rate О 0 .
This result obtains even if the rule (4.1) satisп¬Ѓes П†ПЂ > 1 near the вЂњtargetвЂќ inп¬‚ation rate
О в€— , so that the вЂњTaylor principleвЂќ is satisп¬Ѓed, at least locally. In such a case, one has a large
multiplicity of equilibria globally, despite local determinacy. This is illustrated in Figure
2.4, where the solid curve plots the locus of pairs (О t , О t+1 ) that satisfy (4.6). Note that
this locus crosses the diagonal at the вЂњtargetвЂќ inп¬‚ation rate О в€— , indicating that О t = О в€—
forever is one solution. The вЂњTaylor principleвЂќ implies that the curve cuts the diagonal from
below at this point. In such a case, the fact that П†(О ) в‰Ґ 0 for all О  implies that there also
76 CHAPTER 2. PRICE-LEVEL DETERMINATION

must be another steady state (constant inп¬‚ation rate satisfying (4.6)), at some lower rate of
inп¬‚ation. At this lower steady state, the curve must cut the diagonal from above; thus, as
Benhabib et al. stress, the вЂњTaylor principleвЂќ cannot be globally valid. In the case shown
in the п¬Ѓgure, the вЂњTaylor principleвЂќ is adhered to to the extent possible, which means that
the lower steady state corresponds to a zero nominal interest rate. In this case, the lower
steady-state inп¬‚ation rate is О в€—в€— = ОІ, corresponding to deп¬‚ation at the Friedman rate, the
rate of time preference of the representative household.

The sequence of inп¬‚ation rates corresponding to any given initial inп¬‚ation rate О 0 may
be constructed geometrically as indicated in the п¬Ѓgure; for each value of О t , one п¬Ѓnds the
associated value of О t+1 using the curve, then reп¬‚ects this value down to the horizontal axis
using the diagonal, and repeats the construction. In the п¬Ѓgure, a value О 0 < О в€— is considered.
This is consistent with perfect foresight equilibrium only if О 1 < О 0 , which in turn requires
О 2 < О 1 , and so on. One is able to continue the construction forever, and in the case shown
in the п¬Ѓgure (where П†(О ) < ОІ в€’1 О  в€’ 1 for all О  > ОІ, while П†(О ) = 0 for all О  в‰¤ ОІ), one п¬Ѓnds
that the inп¬‚ation rate must decline monotonically over time, approaching the value О в€—в€— = ОІ
asymptotically. This indicates the possibility of a self-fulп¬Ѓlling deп¬‚ation under such a regime
вЂ“ inп¬‚ation that is perpetually lower than the target rate, and eventually, actual deп¬‚ation,
that represents an equilibrium only because even lower inп¬‚ation is expected in the future.
Along such a path, interest rates are constantly being lowered in response to the decline in
inп¬‚ation, but because expected future inп¬‚ation falls at the same time, real interest rates are
not reduced, and continue to be high enough to restrain demand despite the falling prices.

Such an equilibrium exists for each possible choice of О 0 in the interval ОІ < О 0 < О в€— . At
the same time, for any inп¬‚ation rate higher than the target rate, there exists an equilibrium
in which the equilibrium inп¬‚ation rate rises over time, eventually growing unboundedly large.
Thus self-fulп¬Ѓlling inп¬‚ation is equally possible under such a regime. Furthermore, because
(4.3) need only hold in expectation for an inп¬‚ation process {О t } to constitute a rational
expectations equilibrium, there is also an even larger set of equilibria in which the rate of
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 77

inп¬‚ation or deп¬‚ation depends upon вЂњsunspotвЂќ variables.59
Note that this global multiplicity of solutions does not contradict our previous results with
regard to local determinacy. One observes from Figure 2.4 that any deterministic equilibrium
other than the one with О t = О в€— forever involves an inп¬‚ation rate that diverges farther and
farther from the target inп¬‚ation rate as time passes. Thus every other equilibrium eventually
leaves a neighborhood of О в€— , even if the initial inп¬‚ation rate is very close to it. The same
can be shown to be true of all of the stochastic equilibria as well,60 so that the desired
equilibrium is indeed locally unique in the sense discussed above. Note also that equilibrium
is indeterminate even locally, near the deп¬‚ationary steady state; for any neighborhood of
О в€—в€— , there exist a continuum of distinct equilibria in which inп¬‚ation remains forever within
this range. But this too is consistent with our previous results, since the вЂњTaylor principleвЂќ
is violated near this steady state.
These conclusions are largely unchanged when we take account of real balance eп¬Ђects.
As long as О /О»(П†(О )) is still a monotonically increasing function of О , we can solve (4.5)
in the same manner as (4.6). In the case that real balances are complementary with private
expenditure (Ucm > 0), as was suggested above to be reasonable, О»(i) is a decreasing function,
and this condition is necessarily satisп¬Ѓed. And even if Ucm < 0, the monotonicity condition
may still hold вЂ“ it suп¬ѓces that О» not be too strongly increasing in i. In particular, as long as
О»(i) has a п¬Ѓnite limiting value for i = 0 вЂ“ which makes sense, as there should be a limit to
the value of expenditure, even when it is completely unimpeded by transactions frictions вЂ“
then the assumptions above about the form of П†(О ) suп¬ѓce to imply that the curve in Figure
2.4 cuts the diagonal from above at the Friedman rate of deп¬‚ation. This suп¬ѓces to imply
the existence of a continuum of solutions to (4.5) involving self-fulп¬Ѓlling deп¬‚ation. These
solutions will also satisfy (4.4), and hence represent perfect foresight equilibria, as long as
desired real balances are bounded, or indeed, as long iL(ВЇ, i; 0) has a п¬Ѓnite bound for i near
y
zero.
59
The analysis of this possibility may be conducted along lines like those followed in Woodford (1994a),
in the analysis of multiple equilibria under a money growth rule.
60
Again, see the related analysis in Woodford (1994a).
78 CHAPTER 2. PRICE-LEVEL DETERMINATION

These results may make it seem that a Taylor rule is not a very reliable way of ensuring
a determinate equilibrium price level after all, even if the вЂњTaylor principleвЂќ is adhered to
except when interest rates become very low (in which case it cannot be). Several responses
may be made to this criticism. One is to note that the equilibrium in which inп¬‚ation is
stabilized at the вЂњtargetвЂќ level is nonetheless locally unique, which may be enough to allow
expectations to coordinate upon that equilibrium rather than one of the others. Here it
might seem that the existence of other equilibria with initial inп¬‚ation rates arbitrarily close
to the target rate should make it easy for the economy to вЂњslipвЂќ into one of those other
equilibria. Indeed, it is often said that in the case of perfect foresight dynamics like those
shown in Figure 2.4, the steady state with inп¬‚ation rate О в€— is вЂњunstableвЂќ, implying that an
economy should be expected almost inevitably to experience either a self-fulп¬Ѓlling inп¬‚ation
or a self-fulп¬Ѓlling deп¬‚ation under such a regime.

Such reasoning involves a serious misunderstanding of the causal logic of diп¬Ђerence equa-
tion (4.5). The equation does not indicate how the equilibrium inп¬‚ation rate in period t + 1
is determined by the inп¬‚ation that happens to have occurred in the previous period. If it did,
it would be correct to call О в€— an unstable п¬Ѓxed point of the dynamics вЂ“ even if that point
were fortuitously reached, any small perturbation would result in divergence from it. But
instead, the equation indicates how the equilibrium inп¬‚ation rate in period t is determined
by expectations regarding inп¬‚ation in the following period. These expectations determine
the real interest rate, and hence the incentive for spending, associated with the nominal rate
that the central bank sets in response to any given current inп¬‚ation rate. The equilibria
that involve initial inп¬‚ation rates near (but not equal to) О в€— can only occur as a result of
expectations of future inп¬‚ation rates (at least in some states) that are even farther from the
target inп¬‚ation rate. Thus the economy can only move to one of these alternative paths if
expectations about the future change signiп¬Ѓcantly, something that one may suppose should
not easily occur.

Indeed, many analyses of convergence to rational expectations equilibrium as a result
of adaptive learning dynamics п¬Ѓnd that equilibria are stable under the learning dynamics
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 79

exactly in the case that they are вЂњstable under the backward perfect foresight dynamics,вЂќ
which is exactly the case of the steady state О в€— in Figure 2.4.61 The key to such results
is that any deviation in expected future inп¬‚ation from the target rate results in an actual
inп¬‚ation rate that is closer to the target rate than is the expected rate. If expectations
evolve relatively slowly (as an average of experience over a period of time), then one will
persistently observe inп¬‚ation closer to the target rate than one is expecting, as a result of
which expectations eventually adjust toward a value closer to the target rate themselves.
But this makes actual inп¬‚ation even closer to the target rate, and so on, until the process
eventually converges to an equilibrium in which both expected and actual inп¬‚ation equal the
target rate forever.

Nonetheless, other types of learning processes, that allow extrapolation of paths diverging
from the target steady state, can result in convergence to one of the other equilibria.62 And
even if one regards the target steady state as locally stable, one must worry that a large
shock could nonetheless perturb the economy enough that expectations settle upon another
equilibrium; thus the problem of self-fulп¬Ѓlling inп¬‚ations and deп¬‚ations should probably not
be dismissed out of hand. But it is also important to note that this problem is in no way
special to the formulation of monetary policy in terms of an interest-rate feedback rule. In
particular, exactly the same sort of problems may arise in the case of monetary targeting.

Let us recall the equations that deп¬Ѓned equilibrium real balances in the case of a mon-
etary targeting regime, and once again restrict our attention to the case in which there are
no exogenous shocks (Вµt = Вµ > ОІ, yt = y , Оѕt = 0 for all t). Solving (3.3) for it , and substitut-
ВЇ ВЇ
ing this into the Fisher equation (1.21), we obtain a stochastic diп¬Ђerence equation for real
balances of the form

F (mt ) = ОІ/ВЇEt G(mt+1 ),
Вµ (4.7)

61
For examples of results of this kind, see Grandmont (1985), Grandmont and Laroque (1986), Marcet
and Sargent (1989), and Lettau and Van Zandt (1998). Lucas (1986) uses a result of this kind to argue that
self-fulп¬Ѓlling inп¬‚ations should not be expected to occur, in the case of a monetary targeting regime.
62
See Grandmont and Laroque (19xx) and Lettau and Van Zandt (1998).
80 CHAPTER 2. PRICE-LEVEL DETERMINATION

в€’
G -1 ( Вµ / ОІ F ( m ) )
mt+1

45Г»

mt
0 mв€—
m2 m1 m0
Figure 2.5: A self-fulп¬Ѓlling inп¬‚ation under monetary targeting.

where
F (m) в‰Ў [Uc (ВЇ, m; 0) в€’ Um (ВЇ, m; 0)]m,
y y G(m) в‰Ў Uc (ВЇ, m; 0)m.
y

This is the exact equation to which (A.19) represents a log-linear approximation, except that
we have here suppressed the dependence upon exogenous disturbances.
If we again consider perfect foresight solutions, we look for sequences {mt } that satisfy

G(mt+1 ) = Вµ/ОІF (mt ).
ВЇ (4.8)

In the additively separable case, G(m) is linearly increasing in m, and so can be inverted,
while the function F (m) equals m times an increasing function of m. If this latter function
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 81

(Uc в€’ Um ) is positive for all m > 0 вЂ“ which is to say, if desired real balances can be made
arbitrarily small by making the interest rate high enough вЂ“ then F (m) is positive, increasing,
and convex, and takes the limiting value F (0) = 0. In such a case, (4.8) can be uniquely solved
for mt+1 > 0 given any value mt > 0, and the graph of the solution is of the form shown in
Figure 2.5. Once again, the steady-state level of real money balances mв€— = L(ВЇ, (ВЇ в€’ОІ)/ОІ; 0)
yВµ
is the point at which this graph intersects the diagonal; we note that at this point the curve
necessarily cuts the diagonal from below. And once again there is a second steady state, this
time at zero real balances.

Just as in the case of the Taylor rule, it will be observed that there is a distinct perfect
foresight equilibrium corresponding to each possible value of m0 . If m0 < mв€— , real balances
must be expected to decrease over time (as in the equilibrium illustrated in the п¬Ѓgure),
converging to zero asymptotically. This corresponds to a self-fulп¬Ѓlling inп¬‚ation. For the
progressively lower levels of real balances are associated with progressively higher nominal
interest rates, which in turn require progressively higher rates of inп¬‚ation; asymptotically,
inп¬‚ation approaches the rate (that may or may not be п¬Ѓnite) that causes complete demoneti-
zation of the economy. If instead m0 > mв€— , real balances must increase over time, eventually
increasing without bound. If desired real balances are п¬Ѓnite at all positive interest rates,
then such equilibria involve interest rates falling to zero, at least asymptotically, and so
eventually involve deп¬‚ation at a rate approaching the Friedman rate. They thus correspond
to the self-fulп¬Ѓlling deп¬‚ations discovered to be possible in the case of a Taylor rule. Thus, as
a general rule, monetary targeting as such avoids neither of these types of potential problem.

In fact, as we shall see, it is possible to choose a policy regime under which equilibrium
is globally unique. But here again, there is little diп¬Ђerence between the degree to which this
is possible when monetary policy takes the form of an interest-rate feedback rule, such as a
Taylor rule, and when it instead takes the form of monetary targeting.
82 CHAPTER 2. PRICE-LEVEL DETERMINATION

4.2 Policies to Prevent a Deп¬‚ationary Trap

The result that self-fulп¬Ѓlling deп¬‚ations are possible in the case of monetary targeting may
seem surprising, as many papers that consider equilibrium in the case of a constant money
supply п¬Ѓnd that such equilibria are impossible.63 But the reason for this is that standard
analyses do not specify a Ricardian п¬Ѓscal policy, as we have above. As noted in section 1.2,
analyses of monetary targeting typically assume a п¬Ѓscal policy under which there is zero
government debt at all times. Under this speciп¬Ѓcation, the transversality condition (1.24) is
no longer redundant; it holds only if the path of real balances satisп¬Ѓes

lim ОІ T Et [uc (YT , mT ; ОѕT )mT ] = 0. (4.9)
T в†’в€ћ

As a result, self-fulп¬Ѓlling deп¬‚ations generally cannot occur in a rational expectations
equilibrium, if the money growth rate satisп¬Ѓes Вµ в‰Ґ 1, i.e., the money supply is non-decreasing.
ВЇ
This is most easily shown in the case that there is satiation in real balances at some п¬Ѓnite
level. Then for all values of m above this level, F (m) = G(m), and (4.8) requires that

mt+1 = Вµ/ОІmt
ВЇ

each period, after real balances exceed the satiation level. At the same time, Uc (ВЇ, mt ; 0)
y
remains constant, even if preferences are not additively separable. It follows that in the case
of any deп¬‚ationary solution to (4.8),

ОІ T Uc (ВЇ, mT ; 0)mT = ОІ t ВµT в€’t Uc (ВЇ, mt ; 0)mt в‰Ґ ОІ t Uc (ВЇ, mt ; 0)mt > 0
y ВЇ y y

at all dates T в‰Ґ t, where t is some date at which real balances have already reached the
satiation level. Thus the transversality condition is violated in the case of any such path,
and it does not represent a perfect foresight equilibrium. A generalization of this argument
can be used to exclude stochastic equilibria in which real balances eventually exceed the
satiation level as well; as a result one can show that one must have mt в‰¤ mв€— at all times in
any rational expectations equilibrium.64
63
See, for example, Brock (1975), Obstfeld and Rogoп¬Ђ (1986), and Woodford (1994a).
64
See Woodford (1994a) for details.
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 83

But this result has nothing to do with the fact that monetary policy is speciп¬Ѓed in terms
of a target path for the money supply. Instead it depends upon the fact that п¬Ѓscal policy
is assumed, under such a regime, not to be Ricardian. Government policy implies that not
just the money supply, but also the nominal value of government liabilities Dt , grows at the
rate Вµ, and it is the latter fact that rules out self-fulп¬Ѓlling deп¬‚ations. But we could equally
ВЇ
well combine this kind of п¬Ѓscal policy with an interest rate rule, by specifying п¬Ѓscal policy
in terms of a target path for Dt , as in section 2.1 above. We would then obtain exactly the
same result.
Let total nominal government liabilities Dt be speciп¬Ѓed to grow at a constant rate Вµ в‰Ґ 1,
ВЇ
starting from an initial size D0 > 0, while monetary policy is described by the Taylor rule
(4.1). (Note that a balanced-budget policy, in the sense of Schmitt-GrohВґ and Uribe (1998),
e
is one example of such a п¬Ѓscal policy.) Let us also suppose, to simplify the analysis, that
there exists an inп¬‚ation rate О  > ОІ such that П†(О ) = 0 for all О  в‰¤ О .65
Then any deп¬‚ationary solution to (4.5) involves a zero nominal interest rate in each
period after some п¬Ѓnite date t, which in turn implies that О T = ОІ for all T в‰Ґ t + 1. It follows
that
ОІ T О»(iT )DT /PT = ОІ t ВµT в€’t О»(0)Dt /Pt в‰Ґ ОІ t О»(0)Dt /Pt > 0
ВЇ

at all dates T > t, so that the transversality condition (1.24) is violated in the case of
any such path.66 In essence, the speciп¬Ѓed п¬Ѓscal policy is too stimulative, in the case of a
deп¬‚ationary path, to be consistent with market clearing; if the private sector consumes only
as much as the economy produces, it п¬Ѓnds that its real wealth grows explosively, as a result
of which it wishes to consume more.67
The result just mentioned depends upon interest rates being driven to zero suп¬ѓciently
quickly in the event that inп¬‚ation falls. However, even if П†(О ) > 0 for all О  > ОІ, as long
65
One interpretation of this assumption is that the policy rule conforms to the вЂњTaylor principleвЂќ except
when the zero bound prevents any further decreases in the nominal interest rate.
66
This is how self-fulп¬Ѓlling deп¬‚ations are excluded under a Taylor rule in Schmitt-GrohВґ and Uribe (1998),
e
in the context of a related model.
67
The eп¬Ђects of п¬Ѓscal policy upon aggregate demand under a non-Ricardian regime are discussed further
in chapter 5.
84 CHAPTER 2. PRICE-LEVEL DETERMINATION

as П†(О ) < (О  в€’ ОІ)/ОІ for all ОІ < О  < О в€— , we can still exclude self-fulп¬Ѓlling deп¬‚ations in the
case of any п¬Ѓscal policy that ensures a growth rate Вµ > 1 for government liabilities. For this
ВЇ
ensures that every deп¬‚ationary solution involves an inп¬‚ation rate converging to О в€—в€— = ОІ, as
shown in Figure 2.4. Hence there must exist a date t such that 1 + П†(О T ) в‰Ґ Вµ for all T в‰Ґ t.
ВЇ
It follows that
T в€’1
T t
(1 + is )в€’1 ВµT в€’t Dt /Pt в‰Ґ ОІ t О»(it )Dt /Pt > 0,
ОІ О»(iT )DT /PT = ОІ О»(it ) ВЇ
s=t

so that again the transversality condition is violated. Finally, even if interest rates level oп¬Ђ
sooner, so that there exists a lower steady-state inп¬‚ation rate О в€—в€— > ОІ, the same argument
works as long as Вµ > ОІ в€’1 О в€—в€— .
ВЇ
Thus, in the case of an appropriate п¬Ѓscal policy rule, a deп¬‚ationary trap is not a possible
rational expectations equilibrium, even under an interest-rate rule. Furthermore, the type
of п¬Ѓscal commitment that is required to exclude such a possibility is essentially the same
in the case of an interest-rate rule as in the case of monetary targeting: п¬Ѓscal policy must
ensure that the nominal value of total government liabilities Dt will not decline, even in the
case of sustained deп¬‚ation.
Indeed, an interest-rate rule has an advantage over the classic formulation of a monetary-
targeting regime, in at least one regard. Self-fulп¬Ѓlling deп¬‚ations may be excluded in the case
of an interest-rate rule, even when the target inп¬‚ation rate О в€— associated with the Taylor
rule (or with a Wicksellian regime with a non-constant target path for the price level) implies
deп¬‚ation at a rate less than the rate of time preference (ОІ < О в€— < 1). Under the monetary
targeting regime (with п¬Ѓscal policy maintaining zero government debt at all times), instead,
the argument given above to exclude self-fulп¬Ѓlling deп¬‚ations fails when Вµ < 1. In this case,
ВЇ
the transversality condition is satisп¬Ѓed by all of the solutions to (4.8) that involve inп¬‚ation
rates eventually falling to the level О t = ОІ, and so a continuum of perfect foresight equilibria
(and similarly a large set of вЂњsunspotвЂќ equilibria) exist.68 Of course, this result could be
avoided if we assume a diп¬Ђerent п¬Ѓscal policy; it is simply necessary that the nominal value of
68
See Woodford (1994a) for demonstration of this for a closely related model.
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 85

total liabilities be non-decreasing, even as the money supply contracts. Thus Dupor (1999)
shows that self-fulп¬Ѓlling deп¬‚ations can be avoided if the money supply is contracted through
open-market operations (that replace the money with corresponding increases in government
debt), rather than through the lump-sum tax collections envisioned by Friedman (1969). But
this emphasizes that monetary targeting as such is not the key to excluding the possibility
of a deп¬‚ationary trap.

It is sometimes supposed that the conduct of monetary policy through interest-rate con-
trol leaves monetary policy impotent in the case of a deп¬‚ationary trap, because the nominal
interest rate instrument cannot be lowered below zero, whereas it is actually still possible to
stimulate aggregate demand by increasing the money supply. But monetary control has no
such advantage. It is important to remember that there is no real-balance eп¬Ђect once the
short nominal interest rate falls to zero, even if it is still possible to increase the size of the
excess real money balances (i.e., balances in excess of the satiation level) held by the public.
This is because higher real balances increase desired spending, for any given expected path of
real interest rates, only insofar as they are able to increase the marginal utility of additional
expenditure associated with a given level of real expenditure. Once the satiation level of real
balances is reached, additional money balances no longer lead to any further relaxation of
constraints upon transactional п¬‚exibility, and so they cannot stimulate aggregate demand.

Recall that in section 3.3 above, we were able to replace the function Uc (yt , mt ; Оѕt ) de-
scribing the marginal utility of additional expenditure by О»(yt , it ; Оѕt ). Thus additional real
balances aп¬Ђect the marginal utility of expenditure only insofar as they can reduce the short-
term nominal interest rate it , which indicates the value of further relaxation of transaction
constraints. Once the nominal interest rate cannot be further lowered, an increase in real
money balances as such can have no eп¬Ђect upon aggregate demand, through either a real-
interest-rate eп¬Ђect or a direct real-balance eп¬Ђect. Thus monetary policy is impotent under
such circumstances; but this is not a limitation of the use of an interest-rate instrument.

Alternatively, it is sometimes argued that even when increases in the current money
supply are ineп¬Ђective, due to the economyвЂ™s being in a вЂњliquidity trapвЂќ (i.e., money balances
86 CHAPTER 2. PRICE-LEVEL DETERMINATION

already in excess of the satiation level), a commitment to future money supply increases can
nonetheless stimulate aggregate demand. The idea is that increasing the expected future
price level can lower real interest rates even when nominal rates cannot be further reduced
(Krugman, 1999). This is right, but there is no special eп¬ѓcacy of a commitment to monetary
targets in this regard. Commitment to a high rate of money growth does nothing to exclude
a deп¬‚ationary trap, when coupled with a Ricardian п¬Ѓscal policy; the deп¬‚ationary equilibrium
simply involves a higher level of excess money balances in that case. Only an anti-deп¬‚ationary
п¬Ѓscal commitment can solve this problem, and it can do so whether or not there are monetary
targets.

On the other hand, the problem of a country like Japan at present may not be so much
that it has fallen into a self-fulп¬Ѓlling deп¬‚ationary trap, despite the existence of an equilibrium
with stable prices if only expectations were to coordinate upon it, as that a temporary
reduction in the equilibrium real rate of return has made stable prices incompatible with
the zero bound on nominal interest rates, as suggested by Krugman. In such a case, the
only way to avoid a period of sharp deп¬‚ation when the disturbance occurs may be a regime
that creates expectations of subsequent inп¬‚ation. Let us suppose that the problem of self-
fulп¬Ѓlling deп¬‚ations may be put aside (perhaps because of an appropriate п¬Ѓscal commitment),
and that the economy is expected to converge to a near-steady-state equilibrium after the
real disturbance subsides (and the equilibrium real rate returns to its normal positive level).
Then whether the path of this sort that is followed involves a sharp initial deп¬‚ation or
not depends upon expected monetary policy after the shock subsides, although, for some
expectations regarding future policy, there may be nothing current monetary policy can
do to prevent deп¬‚ation. In such a case, as Krugman argues, the future monetary policy
commitment and its credibility are crucial. But the kind of policy commitment that can
imply expectations of a suitably high future price level may well be a Wicksellian regime or
a Taylor rule, which would determine the expected future price level in the way explained
in section 2 above.
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 87

4.3 Policies to Prevent an Inп¬‚ationary Panic

We next consider the opposite sort of instability due to self-fulп¬Ѓlling expectations, the pos-
sibility of spontaneous п¬‚ight from a countryвЂ™s currency, with its loss in value over each time
period resulting from expectations of even further declines in value, at an even faster rate,
in the near future. Such self-fulп¬Ѓlling inп¬‚ations have generally been considered a more trou-
bling possibility than self-fulп¬Ѓlling deп¬‚ations in the literature on monetary targeting regimes.
However, conditions have been identiп¬Ѓed under which such equilibria would not exist in the
case of a constant money growth rate.
In particular, Obstfeld and Rogoп¬Ђ (1986) show that in the model considered here, if pref-
erences are of the additively separable form considered in section xx above, and in addition

lim mum (m; 0) > 0, (4.10)
mв†’0

then no such perfect foresight equilibria exist under monetary targeting. For in this case,
the function F (m) is negative for all levels of real balances below a critical level m > 0, as a
result of which the graph of (4.8) cuts the horizontal axis at the point mt = m > 0, rather
than passing through the origin as shown in Figure 2.5. Thus no equilibrium can ever have
mt в‰¤ m. But then it follows from (4.8) that no perfect foresight equilibrium can ever have

mt в‰¤ m1 в‰Ў F в€’1 (ОІ/ВЇG(m)),
Вµ

as a result of which no perfect foresight equilibrium can ever have

mt в‰¤ m2 в‰Ў F в€’1 (ОІ/ВЇG(m1 )),
Вµ

and so on. One shows in this way that no equilibrium is possible with mt < mв€— at any time.
Condition (4.10) was excluded in our analysis above by the assumption that desired real
balances can be made arbitrarily small by suп¬ѓciently increasing the interest rate; for it
implies that L(ВЇ, i; 0) is bounded below by m for all i. (The existence of such a bound is
y
the key to the above construction, for which additive separability is actually not important.)
Obstfeld and Rogoп¬Ђ point out that the conditions required for this to be true are somewhat
88 CHAPTER 2. PRICE-LEVEL DETERMINATION

implausible, though theoretically possible. Observed hyperinп¬‚ations in several countries have
also shown that real balances do indeed fall to a small fraction of their normal level when
inп¬‚ation becomes suп¬ѓciently severe, and the money demand functions estimated from such
data (as in the classic study by Cagan, 1956) imply that real balances should approach zero
in the case of high enough expected inп¬‚ation. Hence it is not clear that one can rely upon
this mechanism to prevent self-fulп¬Ѓlling inп¬‚ations in an actual economy.

What if monetary policy is instead speciп¬Ѓed by a Taylor rule of the form (4.1)? A
similar argument excluding self-fulп¬Ѓlling inп¬‚ations would be possible only if the graph of
ВЇ
(4.5) becomes vertical at some п¬Ѓnite inп¬‚ation rate О , so that (4.5) has no solution for О t+1
ВЇ
in the case of О t > О . In the additively separable case, this might seem to be impossible, as
there could fail to be a solution for О t+1 in (4.6) only if П†(О t ) is itself not deп¬Ѓned. However,
it is not clear that a function П† that becomes unboundedly large at a п¬Ѓnite inп¬‚ation rate
ВЇ
О  must be excluded as a possible policy. After all, in the case of monetary targeting when
(4.10) holds, the policy that excludes self-fulп¬Ѓlling inп¬‚ations is one of commitment never to
supply more than a certain quantity of money, no matter how high this may require interest
rates to be driven. This is equivalent to a Wicksellian rule (1.30) in which the function
П† becomes unboundedly large at a п¬Ѓnite level of Pt /Ptв€— . If such a policy is considered to
be feasible, despite the fact that it commits the central bank to something that is simply
impossible in the case of too high a price level, then it is not clear why a Taylor rule that
ВЇ
makes П† undeп¬Ѓned for О t в‰Ґ О  is not an equally feasible policy. But if we consider a Taylor
rule of this kind to represent a credible commitment, then such a strategy to exclude self-
fulп¬Ѓlling inп¬‚ations should actually be superior to monetary targeting. For its applicability
would not depend upon the implausible assumption that desired real balances are bounded
away from zero.

Furthermore, under some circumstances it is possible to exclude self-fulп¬Ѓlling inп¬‚ations
through commitment to increase interest rates suп¬ѓciently sharply at high rates of inп¬‚ation,
even when П†(О ) is well-deп¬Ѓned for any п¬Ѓnite inп¬‚ation rate.If Ucm < 0 at low levels of real
balances, the function О»(i) is increasing in i, at high levels of i. Suppose that in fact the
4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 89

elasticity of О»(i) with respect to 1 + i is positive, but bounded below one for all high enough
i. Then the right-hand side of (4.5) increases with О t , for high inп¬‚ation rates, eventually
growing without bound. However, if П†(О ) increases suп¬ѓciently rapidly for high values of О ,
the function О /О»(П†(О )) may be bounded above. In this case, (4.5) has no solution for О t+1
ВЇ
in the case of values of О t above some п¬Ѓnite bound О .69
But this then allows us to exclude self-fulп¬Ѓlling inп¬‚ations altogether among the set of
perfect foresight equilibria, using an iterative argument like that made above in the case
of the lower bound on real balances. Then, assuming a п¬Ѓscal policy of the kind discussed
above, that excludes self-fulп¬Ѓlling deп¬‚ations, one can show that the equilibrium with inп¬‚a-
tion forever at the target inп¬‚ation rate О в€— is the unique perfect foresight equilibrium. In
fact, using similar arguments, one can show that it is the unique rational expectations equi-
librium, even allowing for stochastic equilibria of arbitrary form. And, at least in the case
of suп¬ѓciently small random variations in the exogenous variables {yt , Оѕt , ОЅt , О в€— }, the locally
t

unique equilibrium that was approximately characterized in section 2.3 above can similarly
be shown to be globally unique.
Even if solutions of these kinds are unavailable, self-fulп¬Ѓlling inп¬‚ations may be excluded
through the addition of policy provisos that apply only in the case of hyperinп¬‚ation. For
example, Obstfeld and Rogoп¬Ђ (1986) propose that the central bank commit itself to peg the
value of the monetary unit in terms of some real commodity, by standing ready to exchange
the commodity for money, in the event that the real value of the total money supply ever
shrinks to a certain very low level. Assuming that this level of real balances is one that would
never be reached except in the case of a self-fulп¬Ѓlling inп¬‚ation, the commitment has no eп¬Ђect
except to exclude such paths as possible equilibria. Obstfeld and Rogoп¬Ђ propose this as a
solution to the problem of self-fulп¬Ѓlling inп¬‚ations under a regime that otherwise targets the
money supply; but it has no intrinsic connection to monetary targeting, and could equally
well be added as a hyperinп¬‚ation proviso in a regime that otherwise follows a Taylor rule.

69
This is essentially the way in which self-fulп¬Ѓlling inп¬‚ations are excluded under a Taylor rule in the
analysis of Schmitt-GrohВґ and Uribe (1998), though their model is not precisely consistent with ours.
e
90 CHAPTER 2. PRICE-LEVEL DETERMINATION

1 APPENDIX TO CHAPTER 2
1.1 Proof of Proposition 2.1

Proposition 2.1. Consider positive-valued stochastic processes {Pt , Qt,T } satisfying (1.10)
and (1.11) at all dates, and let {Ct , Mt } be non-negative-valued processes representing a
possible consumption and money-accumulation plan for the household. Then there exists a
speciп¬Ѓcation of the householdвЂ™s portfolio plan at each date satisfying both the п¬‚ow budget
constraint (1.7) and the borrowing limit (1.9) at each date, if and only if the plans {Ct , Mt }
satisfy the constraint
в€ћ в€ћ
E0 Q0,t [Pt Ct + в€†t Mt ] в‰¤ W0 + E0 Q0,t [Pt Yt в€’ Tt ]. (A.1)
t=0 t=0

Proof: Substituting s for the time index t in (1.7), taking the present value of both
sides of the inequality at an earlier (or no later) date t, and summing over dates s from t
through T в€’ 1, we obtain
T в€’1 T в€’1
Et Qt,s [Ps Cs + в€†s Ms ] + Et [Qt,T WT ] в‰¤ Wt + Et Qt,s [Ps Ys в€’ Ts ]
s=t s=t

for any date T в‰Ґ t + 1. Combining this with the bounds (1.9) on the portfolios that may be
chosen in the various possible states at date T в€’ 1 (which are lower bounds upon the values
of those portfolios in possible states at date T ), one sees that a feasible plan must satisfy
T в€’1 в€ћ
Et Qt,s [Ps Cs + в€†s Ms ] в‰¤ Wt + Et Qt,s [Ps Ys в€’ Ts ].
s=t s=t

Note that the right-hand side is now independent of the terminal date T . The left-hand
side is instead a non-decreasing series in T, given positive goods prices at all dates (necessary
for any п¬Ѓnite level of consumption to be optimal), interest rates satisfying (1.11), and non-
negative levels of consumption and money balances at all times. The right-hand side, which
is п¬Ѓnite by (1.10), provides an upper bound for this series, which accordingly must converge
as T grows. Furthermore, the limiting value of the series must itself satisfy the upper bound.
Thus any feasible plan involves a sequence of state-dependent consumption levels and money
1. APPENDIX TO CHAPTER 2 91

balances satisfying the intertemporal budget constraint (1.13). In particular, under any
feasible plan, the entire inп¬Ѓnite streams {Ct , Mt } from the initial date t = 0 onward must
satisfy this constraint for date zero, given the householdвЂ™s initial п¬Ѓnancial wealth W0 . This
establishes the necessity of (A.1).
It remains to show that this constraint is also suп¬ѓcient for processes {Ct , Mt } to be
attainable. One easily shows that processes that satisfy (A.1) can be achieved, by letting
the householdвЂ™s choice of Wt at each date t в‰Ґ 1 (in each possible state) be given by the value
that makes (1.13) hold at that date with equality. Given the hypothesized process for Mt ,
this then implies a value for At+1 in each possible state, and thus completely speciп¬Ѓes the
householdвЂ™s portfolio plan at each date t в‰Ґ 0. The resulting plan obviously satisп¬Ѓes (1.9) at
each date, and it is easily veriп¬Ѓed that it satisп¬Ѓes (1.7), and hence (1.2), at each date as well.
Thus the entire sequence of п¬‚ow budget constraints is equivalent to the single intertemporal
constraint (A.1).

1.2 Proof of Proposition 2.2

Proposition 2.2. Let assets be priced by a system of stochastic discount factors that satisfy
(1.20), and consider processes {Pt , it , im , Mts , Wts } that satisfy (1.15), (1.21), and (1.23) at
t

all dates, given the exogenous processes {Yt , Оѕt }. Then these processes satisfy (1.22) as well
if and only if they satisfy

lim ОІ T Et [uc (YT ; ОѕT )DT /PT ] = 0. (A.2)
T в†’в€ћ

In this proposition, note that the path of {Dt } can be inferred from the processes that are
speciп¬Ѓed using the identity

Dt = Mts + Et [Qt,t+1 (Wt+1 в€’ (1 + im )Mts )].
s
t

Proof: Note that (1.15) and (1.21) imply that

ОІEt [uc (YT +1 ; ОѕT +1 )(1 + im )MT /PT +1 ] = ОІEt [uc (YT +1 ; ОѕT +1 )(1 + iT )MT /PT +1 ]
s s
T
92 CHAPTER 2. PRICE-LEVEL DETERMINATION

s
= Et [uc (YT ; ОѕT )MT /PT ].

ОІEt [uc (YT +1 ; ОѕT +1 )As +1 /PT +1 ] = Et [uc (YT ; ОѕT )BT /PT ]
s
T

that follows from (1.4) and (1.20), we п¬Ѓnd that

s
ОІEt [uc (YT +1 ; ОѕT +1 )WT +1 /PT +1 ] = Et [uc (YT ; ОѕT )DT /PT ].

It then follows that (1.22) holds if and only if (A.2) does.
In the case of the model with transactions frictions introduced in section xx, a similar
proposition continues to hold. A precise statement can be given as follows.

Proposition 2.2 . Let assets be priced by a system of stochastic discount factors that
satisfy (1.20), and consider processes {Pt , it , im , Mts , Wts } that satisfy (1.21), (3.3), and (3.4)
t

at all dates, given the exogenous processes {Yt , Оѕt }. Then these processes satisfy (1.22) as
well if and only if they satisfy

lim ОІ T Et [uc (YT , MT /PT ; ОѕT )DT /PT ] = 0.
s
(A.3)
T в†’в€ћ

In this more general case, (1.21) and (3.3) can be used to show that

ОІEt [uc (YT +1 , MT +1 /PT +1 ; ОѕT +1 )(1 + im )MT /PT +1 ] = Et [uc (YT , MT /PT ; ОѕT )(1 в€’ в€†T )MT /PT ]
s s s s
T

s s s
= Et [(uc (YT , MT /PT ; ОѕT ) в€’ um (YT , MT /PT ; ОѕT ))MT /P

from which it follows as above that

s s s s s
ОІEt [uc (YT +1 , MT +1 /PT +1 ; ОѕT +1 )WT +1 /PT +1 ] = Et [uc (YT , MT /PT ; ОѕT )DT /PT ]в€’Et [um (YT , MT /PT ; ОѕT )MT /PT

Furthermore, (3.4) implies that

lim ОІ T Et [um (YT , MT /PT ; ОѕT )MT /PT ] = 0.
s s
T в†’в€ћ

Hence (1.22) holds if and only if (A.3) holds.
1. APPENDIX TO CHAPTER 2 93

1.3 Determinacy of Rational-Expectations Equilibrium

1.4 Proof of Proposition 2.3

Proposition 2.3. Under a Wicksellian policy rule (1.30) with П†p > 0, 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¬‚uc-
rв€—
tuations in the exogenous processes {Л†t , ПЂt , ОЅt }, there exists a unique rational-expectations
equilibrium in which Pt /Ptв€— в€€ P and it в€€ I at all times. Furthermore, equations (1.37) and
(1.38) give a log-linear (п¬Ѓrst-order Taylor series) approximation to that solution, accurate up
to a residual of order O(||Оѕ||2 ), where ||Оѕ|| indexes the bounds on the disturbance processes.

Proof: This is a direct application of the implicit function theorem, as discussed in the
previous section. As discussed in the text, (1.32) and (1.34) represent log-linear (п¬Ѓrst-order
Taylor-series) approximations to the equilibrium relations (1.21) and (1.30). The existence
of a unique bounded solution to the log-linearized relations implies the existence of a locally
unique solution to the exact relations as well, in the case of any tight enough bound on
the exogenous disturbances, using the inverse function theorem; and that solution to the
log-linearized relations provides a п¬Ѓrst-order Taylor-series approximation to the solution to
the exact relations, using the implicit function theorem. If the neighborhoods P and I are
small enough, any solution to the exact relations restricted to these sets must satisfy the
transversality condition (1.24) as well, and so represents a rational-expectations equilibrium.
It thus remains only to demonstrate that the system consisting of (1.32) and (1.34),
together with the identity (1.35), has a unique bounded solution when П†p > 0. As shown in
the text, these equations imply (1.36). This is a form of expectational diп¬Ђerence equation
that occurs repeatedly in this chapter, that may be written in the form

zt = aEt zt+1 + ut , (A.4)
94 CHAPTER 2. PRICE-LEVEL DETERMINATION

where zt is an endogenous variable and ut is an exogenous disturbance process. In the present
Л†
application, zt = Pt , a = (1 + П†p )в€’1 , and

ut = (1 + П†p )в€’1 (Л†t + Et ПЂt+1 в€’ ОЅt ).
в€—
r

Any expectational diп¬Ђerence equation of the form (A.4) has a unique bounded solution
{zt } in the case of an arbitrary bounded disturbance process {ut } in the case that |a| < 1.
Note that (A.4) implies that

Et zt+j = aEt zt+j+1 + Et ut+j

for arbitrary j в‰Ґ 0. Multiplying this equation by aj and summing from j = 0 through k в€’ 1,
we obtain
kв€’1
k
aj Et ut+j .
zt = a Et zt+k + (A.5)
j=0

Note that this equation must hold for arbitrary k. If {zt } is a bounded process and |a| < 1,
it follows that
lim ak Et zt+k = 0.
kв†’в€ћ

Then since the left-hand side of (A.5) is independent of k, it follows that the п¬Ѓnal term on
the right must converge in value as k is made unboundedly large, and speciп¬Ѓcally to the
value of the left-hand side. Thus we must have
в€ћ
aj Et ut+j .
zt = (A.6)
j=0

(This solution is sometimes said to be obtained by вЂњsolving (A.4) forward.вЂќ)
Equation (A.5) represents not just one possible solution to (A.4), but the unique bounded
solution. In the present application, (A.6) yields equation (1.37) in the text. Substitution
of this into (1.34) then yields (1.38) as well.
For future reference, it is also useful to consider the case in which |a| в‰Ґ 1. In this case,
the process {zt } recursively deп¬Ѓned by

zt = aв€’1 (ztв€’1 в€’ utв€’1 ) + ОЅt (A.7)
1. APPENDIX TO CHAPTER 2 95

for all t в‰Ґ 1, starting from an arbitrary initial condition z0 , represents a solution to (A.4)
in the case of any process {ОЅt } such that Et ОЅt+1 = 0 for all t. If |a| > 1, (A.7) represents
a bounded solution for {zt } in the case of any bounded process {ОЅt }, assuming that {ut } is
bounded as well. Hence there is an extremely large set of bounded solutions {zt } to equation
(A.4).
In the case that |a| = 1 exactly, not all solutions of the form (A.7) are bounded, even if
both {ut } and {ОЅt } are bounded processes. Nonetheless, if (A.4) has any bounded solution,
it must have an uncountably inп¬Ѓnite number of them. Let {ВЇt } be one bounded solution.
z
(For example, in the case that
в€ћ
vt в‰Ў Et ut+j
j=0

is well-deп¬Ѓned and bounded, as is true for any stationary ARMA process {ut } with bounded
innovations, then one bounded solution to (A.4) is given by zt = vt , the solution obtained
ВЇ
by solving forward.) Then another bounded solution is recursively deп¬Ѓned by

zt = zt + (ztв€’1 в€’ ztв€’1 + ОЅt
ВЇ ВЇ

for all t в‰Ґ 1, starting from an arbitrary initial condition z0 , where {ОЅt } is any stochastic
T
process such that Et ОЅt+1 = 0 for all t, and such that ОЅt remains bounded for arbitrarily
t=1

large T . This last stipulation can obviously be satisп¬Ѓed by a large number of mean-zero,
unforecastable processes; for example, it suп¬ѓces that ОЅt be bounded for each t, and equal
to zero with probability one for all t greater than some date T . Hence in this case as well,
there are clearly an uncountably inп¬Ѓnite number of bounded solutions. Thus the condition
that |a| < 1 is both necessary and suп¬ѓcient for the existence of a unique bounded solution
to (A.4).

1.5 Proof of Proposition 2.4

Proposition 2.4. Consider a monetary policy under which the monetary base is bounded
below by a positive quantity: Mts в‰Ґ M > 0 at all times. (For example, perhaps the
s
monetary base is non-decreasing over time, starting from an initial level M0 > 0.) Suppose
96 CHAPTER 2. PRICE-LEVEL DETERMINATION

furthermore that government debt is non-negative at all times, so that Dt в‰Ґ Mts . Finally,
suppose that im = 0 at all times. Then in the cashless economy described above, there exists
t

no rational-expectations equilibrium path for the price level {Pt }.

Proof: If im = 0 at all times, (1.15) requires that in any equilibrium, it = 0 at all times.
t

Then (1.21) requires that

ОІEt [uc (Yt+1 ; Оѕt+1 /Pt+1 ] = uc (Yt ; Оѕt )/Pt

at all times, and hence, by iteration, that

ОІ T Et [uc (YT ; ОѕT /PT ] = ОІ t uc (Yt ; Оѕt )/Pt

for all T в‰Ґ t. But this, together with the lower bound Dt в‰Ґ M > 0, implies that

ОІ T Et [uc (YT ; ОѕT )DT /PT ] в‰Ґ M ОІ T Et [uc (YT ; ОѕT /PT ]

= M ОІ t Et [uc (Yt ; Оѕt /Pt ] > 0,

which contradicts (A.2). Hence no equilibrium is possible.
In fact, equilibrium values could be deп¬Ѓned under such a regime for real rates of return and
asset prices; if we write prices in terms of some real numeraire rather than monetary units,
a well-deп¬Ѓned equilibrium would exist, but would involve zero exchange value for money.
In essence, under the regime described, money is a pure вЂњbubbleвЂќ вЂ” an asset the exchange
value of which would have to be sustained purely by the expectation of a future exchange
value, and not any dividends ever yielded by the asset вЂ” and cannot have an exchange value
in a rational-expectations equilibrium, at least not in a representative-household model of
the kind assumed here. (In fact, a similar result can be obtained in much more general
environments, as shown by Santos and Woodford, 1997.) Instead, under the regime to which
Proposition 2.3 applies, interest is paid on money, and вЂ” the crucial point вЂ” this interest
is not simply additional money that remains forever in circulation. Because private-sector
nominal claims on the government Dt are assumed to grow at a rate less than the rate at
1. APPENDIX TO CHAPTER 2 97

ВЇ
which interest is paid on money вЂ” recall that ОіD < О /ОІ = 1 + П†(1; 0) = 1 + ВЇm вЂ” at least
Д±
some of the money received as interest payments is eventually redeemed by the government
(accepted as payment for taxes), so that money ceases to be a pure вЂњbubbleвЂќ.

1.6 Proof of Proposition 2.5.

Proposition 2.5. Let monetary policy be speciп¬Ѓed by an exogenous sequence of interest-
rate targets, assumed to remain forever within a neighborhood of the interest rate ВЇ > 0
Д±
associated with the zero-inп¬‚ation steady state; and let these be implemented by setting im
t

equal to the interest-rate target each period. Let {Mts , Dt } be exogenous sequences of the
kind assumed in Proposition 2.3. Finally, let P be any neighborhoods of the real number zero.
Then for any tight enough bounds on the exogenous processes {Yt , Оѕt Dt /Dtв€’1 } and on the
interest-rate target process, there exists an uncountably inп¬Ѓnite set of rational-expectations
equilibrium paths for the price level, in each of which the inп¬‚ation rate satisп¬Ѓes ПЂt в€€ P for
all t. These include equilibria in which the inп¬‚ation rate is aп¬Ђected to an arbitrary extent
by вЂњfundamentalвЂќ disturbances (unexpected changes in Yt or Оѕt ), by pure вЂњsunspotвЂќ states
(exogenous randomness unrelated to the вЂњfundamentalвЂќ variables), or both.

As discussed in the text, this can be established using the local method discussed in section
xx above. However, in the present case, the equilibrium relations are simply enough to
analyze without any resort to linear approximation.

Proof: Let {ОЅt } be any unforecastable mean-zero random variable (or martingale dif-
ference) such that ОЅt < 1 at all times. Then the inп¬‚ation process given by
1 + im
Pt tв€’1
=ОІ Etв€’1 [uc (Yt ; Оѕt )(1 в€’ ОЅt )] f rac11 в€’ ОЅt
Ptв€’1 uc (Ytв€’1 ; Оѕtв€’1 )
satisп¬Ѓes (1.21) at all times. (Note that the solutions (2.42) presented in the text are log-
linear approximations to these processes.) In the case of tight enough bounds on both the
exogenous variables and the п¬‚uctuations in {ОЅt }, this yields an inп¬‚ation process such that
ПЂt в€€ P at all times, and that satisп¬Ѓes (A.2) as well. Hence any such solution represents a
98 CHAPTER 2. PRICE-LEVEL DETERMINATION

rational-expectations equilibrium. The solutions corresponding to diп¬Ђerent choices of {ОЅt }
represent distinct equilibria, since in each case the surprise component of inп¬‚ation is given
by
ПЂt в€’ Etв€’1 ПЂt = в€’ log(1 в€’ ОЅt ).

Finally, the variable {ОЅt } may be correlated in an arbitrary way with any of the вЂњfundamen-
talвЂќ variables, or it may be completely independent of them.

1.7 Proof of Proposition 2.7

Proposition 2.7. Let monetary policy be described by a feedback rule of the form (2.48),
at least near the zero-inп¬‚ation steady state, with О¦ПЂ в‰Ґ 0. Then equilibrium is determinate
if and only if О¦ПЂ > 1. When this condition is satisп¬Ѓed, a log-linear approximation to the
equilibrium evolution of the smoothed inп¬‚ation process is given by
в€ћ
(Оґ + (1 в€’ Оґ)О¦ПЂ )в€’(j+1) Et [Л†t+j в€’ ВЇt+j ].
ПЂt = (1 в€’ Оґ)
ВЇ r Д± (A.8)
j=0

A corresponding approximation to the equilibrium evolution of the single-period inп¬‚ation
rate ПЂt is then obtained by substituting (A.8) into
ПЂt в€’ ОґВЇtв€’1
ВЇ ПЂ
ПЂt = . (A.9)
1в€’Оґ

Proof: The solution (A.9) for ПЂt given the evolution of ПЂt is obtained by inverting (2.47).
ВЇ
Then substituting (2.48) into (1.32) to eliminate it , we obtain

О¦ПЂ ПЂt = Et ПЂt+1 + (Л†t в€’ ВЇt ).
ВЇ r Д±

Substituting (A.9) for ПЂt in this equation, we obtain an expectational diп¬Ђerence equation for
the smoothed inп¬‚ation measure,

[Оґ + (1 в€’ Оґ)О¦ПЂ ]ВЇt = Et ПЂt+1 + (1 в€’ Оґ)(Л†t в€’ ВЇt ).
ПЂ ВЇ r Д± (A.10)

This is again an equation of the form (A.4), allowing us to apply the same method as in the
proof of Proposition 2.3. The equation can be solved forward to obtain a unique bounded
1. APPENDIX TO CHAPTER 2 99

solution if and only if
Оґ + (1 в€’ Оґ)О¦ПЂ > 1,

which is to say, if and only if О¦ПЂ > 1, as required by the вЂњTaylor principleвЂќ. When this
condition holds, the solution (A.5) is given by (A.8).

1.8 Proof of Proposition 2.8

Proposition 2.8. Let monetary policy be described by a feedback rule of the form (2.51), at
least near the zero-inп¬‚ation steady state, with П†ПЂ , rho в‰Ґ 0. Then equilibrium is determinate
if and only if П†ПЂ > 0 and
П†ПЂ + ПЃ > 1 (A.11)

When these conditions are satisп¬Ѓed, a log-linear approximation to the equilibrium evolution
of inп¬‚ation is given by (2.53).

Proof: The proof follows the same lines as in the case of Proposition 2.7. Using (2.51)
to eliminate ПЂt+1 in (1.32), one obtains an expectational diп¬Ђerence equation

(П†ПЂ + ПЃ)Л†t = EtЛ†t+1 + П†ПЂ rt + ПЃВЇt в€’ EtВЇt+1 ,
Д± Д± Л† Д± Д±

corresponding to (A.10) above, and once again this is of the form (A.4). Applying the same
method as in the proof of Proposition 2.3, one п¬Ѓnds that there is a unique bounded solution
for {Л†t } if and only if
Д±
|П†ПЂ + ПЃ| > 1 (A.12)

is satisп¬Ѓed. Under the sign assumptions made in the statement of the Proposition, condition
(A.12) reduces to (A.11). By solving forward, i.e., applying (A.5), one obtains (2.54) as the
solution for the interest-rate process.
Corresponding to this solution for the path of the interest rate is a unique solution for
{ПЂt }, obtained by inverting (2.51), if and only if П†ПЂ = 0. Hence there is a unique bounded
solution for {ПЂt } if П†ПЂ > 0 and (A.11) applies. Using the solution obtained for {Л†t }, one
Д±
obtains the solution (2.53) for the inп¬‚ation process.
100 CHAPTER 2. PRICE-LEVEL DETERMINATION

If instead П†ПЂ = 0, there is a multiplicity of possible solutions for {ПЂt }, even when the
equilibrium path {Л†t } is uniquely determined. In fact, Proposition 2.5 again applies in this
Д±
case. If П†ПЂ > 0 but 0 < П†ПЂ + ПЃ < 1, there is an uncountably inп¬Ѓnite number of solutions
for {Л†t }, as one can show using the method discussed following the proof of Proposition
Д±
2.3. To each of these there corresponds a unique associated inп¬‚ation process, but the set of
equilibrium inп¬‚ation processes is uncountably inп¬Ѓnite.
As remarked in the text, determinacy of equilibrium does not require that П†ПЂ > 0,
though that is the case of primary practical interest. Our analysis above shows that in
fact all that is required is that П†ПЂ = 0 and that (A.12) be satisп¬Ѓed. When ПЃ > 1, the latter
condition is satisп¬Ѓed by all non-zero inп¬‚ation-response coeп¬ѓcients П†ПЂ > в€’(ПЃв€’1), which would
include moderately negative values. In such a case, (2.53) continues to provide a log-linear
approximation to the equilibrium inп¬‚ation process. The conditions for determinacy would
also be satisп¬Ѓed by all П†ПЂ < в€’(1 + ПЃ). (Note that this means that determinacy results from
suп¬ѓciently large negative values of П†ПЂ even in the case that ПЃ < 1.) However, the equilibrium
obtained in this case depends too crucially upon the assumption of a discrete sequence of
dates on which markets are open to be of practical interest. (In the continuous-time limit of
the model, no such equilibria are possible. See the discussion below at xxxx.)

1.9 Proof of Proposition 2.9.

Proposition 2.9. Suppose that the equilibrium real rate {Л†t } follows an exogenously given
r
stationary AR(1) process, and let the monetary policy rule be of the form (2.51), with
ПЃ в‰Ґ 0, П†ПЂ > 0 and a constant intercept consistent with the zero-inп¬‚ation steady state (i.e.,
ВЇt = 0). Consider the choice of a policy rule (ПЃ, П†ПЂ ) within this class so as to bring about
Д±
a certain desired unconditional variance of inп¬‚ation var(ПЂ) > 0 around the mean inп¬‚ation
rate of zero. For any large enough value of ПЃ, there exists a П†ПЂ satisfying (A.11) such that
the unconditional variance of inп¬‚ation in the stationary rational-expectations equilibrium
associated with this rule is of the desired magnitude. Furthermore, the larger is ПЃ, the
smaller is the unconditional variance of interest-rate п¬‚uctuations var(Л†) in this equilibrium.
Д±
1. APPENDIX TO CHAPTER 2 101

Proof: In the case of an AR(1) process

rt = ПЃr rtв€’1 +
Л† Л† t

for the equilibrium real rate, Et rt+j = ПЃj rt for all j в‰Ґ 0. Then for any policy rule (ПЃ, П†ПЂ )
Л† rЛ†

that satisп¬Ѓes (A.11) (2.54) implies that

П†ПЂ
Л†t =
Д± rt .
Л†
П†ПЂ + ПЃ в€’ ПЃr

Hence {Л†t } is also an AR(1) process, with variance and п¬Ѓrst-order autocovariance
Д±
2
П†ПЂ
var(Л†) =
Д± var(Л†),
r (A.13)
П†ПЂ + ПЃ в€’ ПЃr
cov(Л†t , Л†tв€’1 ) = ПЃr var(Л†).
Д±Д± Д±

Inverting (2.51) implies that ПЂt = П†в€’1 (Л†t в€’ Л†tв€’1 ), from which it follows that
ПЂД± Д±

var(ПЂ) = П†в€’2 var(Л†t в€’ ПЃЛ†tв€’1 )
Д± Д±
ПЂ

= П†в€’2 [(1 + ПЃ2 )var(Л†) в€’ 2ПЃcov(Л†t , Л†tв€’1 )]
Д± Д±Д±
ПЂ
1 в€’ 2ПЃr ПЃ + ПЃ2
= var(Л†).
r (A.14)
(П†ПЂ + ПЃ в€’ ПЃr )2

Condition (A.14) can be solved for the required inп¬‚ation-response coeп¬ѓcient in order to
obtain a given degree of variability of inп¬‚ation, yielding

1/2
П†ПЂ = (1 в€’ 2ПЃr ПЃ + ПЃ2 )f racvar(Л†)var(ПЂ)
r + ПЃr в€’ ПЃ.

(Here we select the positive square root because we know that if a solution exists that satisп¬Ѓes
(A.11), it must be such that П†ПЂ > ПЃr в€’ ПЃ.) We note that for all large enough ПЃ > 0, the right-
hand side expression must exceed 1 в€’ ПЃ, in which case there is indeed a solution satisfying
(A.11), as asserted in the Proposition.
This solution can then be substituted for П†ПЂ in (A.13), yielding

ПЃ в€’ ПЃr
Пѓ(Л†) = Пѓ(Л†) в€’
Д± r Пѓ(ПЂ), (A.15)
(1 в€’ 2ПЃr ПЃ + ПЃ2 )1/2
102 CHAPTER 2. PRICE-LEVEL DETERMINATION

where Пѓ(x) в‰Ў (var(x))1/2 in the case of any stationary random variable x. The expression

ПЃ в€’ ПЃr ПЃ в€’ ПЃr
=
(1 в€’ 2ПЃr ПЃ + ПЃ2 )1/2 [(1 в€’ ПЃ2 ) + (ПЃ в€’ ПЃr )2 ]1/2
r

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