˜

in Figure 2.2. The ¬gure shows the weights both under the assumption that χ = 0 (as in our

¬rst approach), and for the positive value χ = .02. The latter is likely to be an over-estimate

of the actual size of real-balance e¬ects on the marginal utility of income, but is considered to

show that even under the most generous assumptions real-balance e¬ects should not matter

greatly. (The value ·i = 28 quarters is assumed in both cases.)

While the real-balance e¬ect 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

ψ

j

1

χj=0

0.8

χj=0.02

0.6

0.4

0.2

0 2 4 6 8 10 12

j

˜

ψj

1.6

1.4 χ =0

j

χj=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 •j with and without real-balance e¬ects.

extremely misleading conclusions, either. Perhaps the most important qualitative di¬erence

is that setting χ = 0.02 results in a slightly larger relative weight on the current period™s

intercept ¯t as opposed to expected future intercepts. Especially in the (realistic) case that

±

both the equilibrium real rate rt and the monetary policy disturbance ¯t exhibit substantial

˜ ±

positive serial correlation, so that it is only smoothed versions of the coe¬cients •j and •j

˜

that matter in practice, the predictions for in¬‚ation under the two assumptions will be quite

similar. For example, let us suppose that rt follows a stationary AR(1) process with serial

˜

correlation coe¬cient 0 < ρr < 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

∞

•j ρj .

∆ 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 ∆, where ∆ is the length of a period in years.

˜

3. PRICE-LEVEL DETERMINATION WITH MONETARY FRICTIONS 69

Quarterly model

2.5

χ=0

2

χ=0.02

1.5

1

0.5

0

0 50 100 150 200 250 300 350 400

Daily model

2.5

χ=0

2 χ=0.02

1.5

1

0.5

0

0 50 100 150 200 250 300 350 400

» (percent per year)

Figure 2.3: In¬‚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, » ≡ ’(log ρr )/∆, where ∆ is the length of a period

in years.54 The results are plotted both under the assumption that χ = 0 and for the upper-

bound case χ = .02. We observe that the error involved in neglecting the real-balance e¬ects

is quite small.

One case in which real balance e¬ects would matter a great deal, however, is that of a

rule (2.43) in which the interest rate depends only upon contemporaneous in¬‚ation, in the

case that the “periods” are very short. Let us consider the behavior of the solution to our

equations as the length of a “period” (∆ > 0 units of calendar time) is made progressively

shorter. We shall ¬x numerical values for the rate of time preference δ ≡ (’ log β)/∆ > 0,

54

This measure is used, instead of ρr itself, in order to allow comparability across models with di¬erent

period lengths. In panel (a), ∆ = .25 years.

70 CHAPTER 2. PRICE-LEVEL DETERMINATION

the intertemporal elasticity of substitution σ > 0, steady-state share of expenditure on

liquidity services sm , the interest-rate semi-elasticity of money demand in units of calendar

time ·i ≡ ∆·i > 0, and the income-elasticity of money demand ·y that are independent of

˜

the assumed size of ∆. Then the same reasoning used to derive (3.23) implies that as ∆ is

made small, the value of χ approaches a well-de¬ned limiting value

·y

’ σ ’1 .

χ = sm

¯ (3.24)

δ ·i

˜

We then obtain the following result.

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.

¯

The proof is in the appendix.55 This result clearly implies that the solution for equilibrium

in¬‚ation in the short-period limit cannot be a continuous function of χ for values of χ near

zero. Thus the value of χ matters in this case, even if it is very small.

However, this failure of continuity in χ 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 period™s in¬‚ation. As periods are made shorter, the central bank

is assumed to respond to a higher-frequency measure of in¬‚ation, and in the limit policy

is assumed to respond solely to an instantaneous rate of in¬‚ation. This is plainly a case

55

Our result agrees with the continuous-time analysis of Benhabib et al. (1998a), who ¬nd that the range

of values of φπ 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 ρ = 0, one has determinacy

in the case of φπ > 1 for all values of χ 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

·i increases as ∆’1 . Hence for any sequence of economies in Proposition 2.14 for which χ < 0, (3.16) is

¯

violated for all small enough values of ∆, 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 in¬‚ation

measure (2.47), and let the rate of decay (in calendar time) of the exponential weights on

past in¬‚ation be ¬xed as we make “periods” 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 in¬‚ation, if we

hold ¬xed the rate of adjustment ψ ≡ ’ log ρ/∆ > 0 as we make ∆ smaller. (We must also

assume that φπ is reduced along with ∆, so as to hold ¬xed the long-run response coe¬cient

¦π ≡ (1 ’ ρ)’1 φπ .) In this case we obtain the following.

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

period lengths ∆, and suppose that

1

χ>’

¯ . (3.25)

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) ± (3.26)

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 (3.25), including values near

¯ ¯

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 χ (of either sign) make no important di¬erence 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 χ 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 “period” is only a day. Thus as long as we assume either a modest degree

of time-averaging in the in¬‚ation measure to which the central bank responds, or a modest

degree of inertia in the central bank™s adjustment of its interest-rate operating target in

response to in¬‚ation variations ” both of which are always characteristic of actual central-

bank policies ” we continue to ¬nd that the cashless analysis gives a good approximation to

the results obtained under a realistic non-zero value of χ, regardless of the assumed period

length.

4 Self-Ful¬lling In¬‚ations and De¬‚ations

Thus far we have considered only the problem of local determinacy of equilibrium. But it is

appropriate also to consider, at least brie¬‚y, the question whether rational expectations equi-

librium is globally unique under one policy rule or another. Certainly we may have greater

con¬dence 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 di¬er 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

re¬ning one™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 su¬ciently 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 “Taylor

rule” as discussed by Schmitt-Groh´ and Uribe (1998) and by Benhabib et al. (1998b).

e

Consider a deterministic interest-rate feedback rule of the form

it = φ(Πt ), (4.1)

where φ is again an increasing continuous function, satisfying φ(Π) ≥ 0 for all Π > 0.

We suppose once again that this rule incorporates an implicit target in¬‚ation rate Π— > β

satisfying φ(Π— ) = β ’1 Π— ’ 1. The stipulated lower bound “ which is of some importance

for the present discussion “ 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 ¬scal policy now be speci¬ed by a rule of the form

it

Tt = ±Wt ’ Mt (4.2)

1 + it

for determination of net tax collections at each date, for some constant 0 < ± ¤ 1. Using

the ¬‚ow government budget constraint (1.7), we see that this rule implies that

Et [Qt,t+1 Wt+1 ] = (1 ’ ±)Wt .

Thus this ¬scal policy has the “Ricardian” 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 ¬scal 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

’1

»(yt+1 , it+1 ; ξt+1 ) Pt

’1

1 + it = β Et (4.3)

»(yt , it ; ξt ) Pt+1

at all dates, together with the bound

∞

iT

β T Et »(yT , iT ; ξT ) yT + L(yT , iT ; ξT ) < ∞. (4.4)

1 + iT

T =t

Here (4.3) rewrites (1.21) using the function »( y, i; ξ) 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 , ξt = 0 for

¯

all t). Then, substituting (4.1) into (4.3), we obtain a nonlinear di¬erence equation for the

in¬‚ation rate,

Πt+1 »(φ(Πt+1 ))’1 = β(1 + φ(Πt ))»(φ(Πt ))’1 , (4.5)

now writing simply »(i) for »(¯, i; 0). In the cashless limit (or the case of additive separa-

y

bility), this reduces to

Πt+1 = β(1 + φ(Πt )). (4.6)

It is clear in this last case that there exists a solution for Πt+1 > 0 in the case of any given

Πt > 0. Hence starting from any arbitrarily chosen initial in¬‚ation rate Π0 > 0, we can

construct a sequence {Πt } that satis¬es (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(¯, i; 0) are bounded above as i approaches zero,58 because there is satiation at

y

58

In fact, it su¬ces that iL(¯, 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 i’1/2 , condition (4.4) is satis¬ed by all

interest-rate sequences.

4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 75

β(1+φ(Π))

Πt+1

45»

β

Πt

0 Π——= β Π2 Π1 Π0 Π—

Figure 2.4: A self-ful¬lling de¬‚ation under a Taylor rule.

a ¬nite 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 in¬‚ation rate Π0 .

This result obtains even if the rule (4.1) satis¬es φπ > 1 near the “target” in¬‚ation rate

Π— , so that the “Taylor principle” is satis¬ed, 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 (Πt , Πt+1 ) that satisfy (4.6). Note that

this locus crosses the diagonal at the “target” in¬‚ation rate Π— , indicating that Πt = Π—

forever is one solution. The “Taylor principle” implies that the curve cuts the diagonal from

below at this point. In such a case, the fact that φ(Π) ≥ 0 for all Π implies that there also

76 CHAPTER 2. PRICE-LEVEL DETERMINATION

must be another steady state (constant in¬‚ation rate satisfying (4.6)), at some lower rate of

in¬‚ation. At this lower steady state, the curve must cut the diagonal from above; thus, as

Benhabib et al. stress, the “Taylor principle” cannot be globally valid. In the case shown

in the ¬gure, the “Taylor principle” 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 in¬‚ation rate is Π—— = β, corresponding to de¬‚ation at the Friedman rate, the

rate of time preference of the representative household.

The sequence of in¬‚ation rates corresponding to any given initial in¬‚ation rate Π0 may

be constructed geometrically as indicated in the ¬gure; for each value of Πt , one ¬nds the

associated value of Πt+1 using the curve, then re¬‚ects this value down to the horizontal axis

using the diagonal, and repeats the construction. In the ¬gure, a value Π0 < Π— is considered.

This is consistent with perfect foresight equilibrium only if Π1 < Π0 , which in turn requires

Π2 < Π1 , and so on. One is able to continue the construction forever, and in the case shown

in the ¬gure (where φ(Π) < β ’1 Π ’ 1 for all Π > β, while φ(Π) = 0 for all Π ¤ β), one ¬nds

that the in¬‚ation rate must decline monotonically over time, approaching the value Π—— = β

asymptotically. This indicates the possibility of a self-ful¬lling de¬‚ation under such a regime

“ in¬‚ation that is perpetually lower than the target rate, and eventually, actual de¬‚ation,

that represents an equilibrium only because even lower in¬‚ation is expected in the future.

Along such a path, interest rates are constantly being lowered in response to the decline in

in¬‚ation, but because expected future in¬‚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 Π0 in the interval β < Π0 < Π— . At

the same time, for any in¬‚ation rate higher than the target rate, there exists an equilibrium

in which the equilibrium in¬‚ation rate rises over time, eventually growing unboundedly large.

Thus self-ful¬lling in¬‚ation is equally possible under such a regime. Furthermore, because

(4.3) need only hold in expectation for an in¬‚ation process {Πt } 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

in¬‚ation or de¬‚ation depends upon “sunspot” 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 Πt = Π— forever involves an in¬‚ation rate that diverges farther and

farther from the target in¬‚ation rate as time passes. Thus every other equilibrium eventually

leaves a neighborhood of Π— , even if the initial in¬‚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 de¬‚ationary steady state; for any neighborhood of

Π—— , there exist a continuum of distinct equilibria in which in¬‚ation remains forever within

this range. But this too is consistent with our previous results, since the “Taylor principle”

is violated near this steady state.

These conclusions are largely unchanged when we take account of real balance e¬ects.

As long as Π/»(φ(Π)) is still a monotonically increasing function of Π, 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, »(i) is a decreasing function,

and this condition is necessarily satis¬ed. And even if Ucm < 0, the monotonicity condition

may still hold “ it su¬ces that » not be too strongly increasing in i. In particular, as long as

»(i) has a ¬nite limiting value for i = 0 “ which makes sense, as there should be a limit to

the value of expenditure, even when it is completely unimpeded by transactions frictions “

then the assumptions above about the form of φ(Π) su¬ce to imply that the curve in Figure

2.4 cuts the diagonal from above at the Friedman rate of de¬‚ation. This su¬ces to imply

the existence of a continuum of solutions to (4.5) involving self-ful¬lling de¬‚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(¯, i; 0) has a ¬nite 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 “Taylor principle” 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 in¬‚ation is

stabilized at the “target” 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 in¬‚ation rates arbitrarily close

to the target rate should make it easy for the economy to “slip” 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 in¬‚ation rate Π— is “unstable”, implying that an

economy should be expected almost inevitably to experience either a self-ful¬lling in¬‚ation

or a self-ful¬lling de¬‚ation under such a regime.

Such reasoning involves a serious misunderstanding of the causal logic of di¬erence equa-

tion (4.5). The equation does not indicate how the equilibrium in¬‚ation rate in period t + 1

is determined by the in¬‚ation that happens to have occurred in the previous period. If it did,

it would be correct to call Π— an unstable ¬xed point of the dynamics “ even if that point

were fortuitously reached, any small perturbation would result in divergence from it. But

instead, the equation indicates how the equilibrium in¬‚ation rate in period t is determined

by expectations regarding in¬‚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 in¬‚ation rate. The equilibria

that involve initial in¬‚ation rates near (but not equal to) Π— can only occur as a result of

expectations of future in¬‚ation rates (at least in some states) that are even farther from the

target in¬‚ation rate. Thus the economy can only move to one of these alternative paths if

expectations about the future change signi¬cantly, 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 ¬nd that equilibria are stable under the learning dynamics

4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 79

exactly in the case that they are “stable under the backward perfect foresight dynamics,”

which is exactly the case of the steady state Π— in Figure 2.4.61 The key to such results

is that any deviation in expected future in¬‚ation from the target rate results in an actual

in¬‚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 in¬‚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 in¬‚ation even closer to the target rate, and so on, until the process

eventually converges to an equilibrium in which both expected and actual in¬‚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-ful¬lling in¬‚ations and de¬‚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 de¬ned 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 (µt = µ > β, yt = y , ξt = 0 for all t). Solving (3.3) for it , and substitut-

¯ ¯

ing this into the Fisher equation (1.21), we obtain a stochastic di¬erence equation for real

balances of the form

F (mt ) = β/¯Et G(mt+1 ),

µ (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-ful¬lling in¬‚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

’

G -1 ( µ / β F ( m ) )

mt+1

45»

mt

0 m—

m2 m1 m0

Figure 2.5: A self-ful¬lling in¬‚ation under monetary targeting.

where

F (m) ≡ [Uc (¯, m; 0) ’ Um (¯, m; 0)]m,

y y G(m) ≡ Uc (¯, 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 ) = µ/βF (mt ).

¯ (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 ’ Um ) is positive for all m > 0 “ which is to say, if desired real balances can be made

arbitrarily small by making the interest rate high enough “ 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 m— = L(¯, (¯ ’β)/β; 0)

yµ

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 < m— , real balances

must be expected to decrease over time (as in the equilibrium illustrated in the ¬gure),

converging to zero asymptotically. This corresponds to a self-ful¬lling in¬‚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 in¬‚ation; asymptotically,

in¬‚ation approaches the rate (that may or may not be ¬nite) that causes complete demoneti-

zation of the economy. If instead m0 > m— , real balances must increase over time, eventually

increasing without bound. If desired real balances are ¬nite at all positive interest rates,

then such equilibria involve interest rates falling to zero, at least asymptotically, and so

eventually involve de¬‚ation at a rate approaching the Friedman rate. They thus correspond

to the self-ful¬lling de¬‚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 di¬erence 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 De¬‚ationary Trap

The result that self-ful¬lling de¬‚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 ¬nd that such equilibria are impossible.63 But the reason for this is that standard

analyses do not specify a Ricardian ¬scal policy, as we have above. As noted in section 1.2,

analyses of monetary targeting typically assume a ¬scal policy under which there is zero

government debt at all times. Under this speci¬cation, the transversality condition (1.24) is

no longer redundant; it holds only if the path of real balances satis¬es

lim β T Et [uc (YT , mT ; ξT )mT ] = 0. (4.9)

T ’∞

As a result, self-ful¬lling de¬‚ations generally cannot occur in a rational expectations

equilibrium, if the money growth rate satis¬es µ ≥ 1, i.e., the money supply is non-decreasing.

¯

This is most easily shown in the case that there is satiation in real balances at some ¬nite

level. Then for all values of m above this level, F (m) = G(m), and (4.8) requires that

mt+1 = µ/βmt

¯

each period, after real balances exceed the satiation level. At the same time, Uc (¯, mt ; 0)

y

remains constant, even if preferences are not additively separable. It follows that in the case

of any de¬‚ationary solution to (4.8),

β T Uc (¯, mT ; 0)mT = β t µT ’t Uc (¯, mt ; 0)mt ≥ β t Uc (¯, mt ; 0)mt > 0

y ¯ y y

at all dates T ≥ 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 ¤ m— at all times in

any rational expectations equilibrium.64

63

See, for example, Brock (1975), Obstfeld and Rogo¬ (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 speci¬ed in terms

of a target path for the money supply. Instead it depends upon the fact that ¬scal 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 µ, and it is the latter fact that rules out self-ful¬lling de¬‚ations. But we could equally

¯

well combine this kind of ¬scal policy with an interest rate rule, by specifying ¬scal 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 speci¬ed to grow at a constant rate µ ≥ 1,

¯

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-Groh´ and Uribe (1998),

e

is one example of such a ¬scal policy.) Let us also suppose, to simplify the analysis, that

there exists an in¬‚ation rate Π > β such that φ(Π) = 0 for all Π ¤ Π.65

Then any de¬‚ationary solution to (4.5) involves a zero nominal interest rate in each

period after some ¬nite date t, which in turn implies that ΠT = β for all T ≥ t + 1. It follows

that

β T »(iT )DT /PT = β t µT ’t »(0)Dt /Pt ≥ β t »(0)Dt /Pt > 0

¯

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 speci¬ed ¬scal policy is too stimulative, in the case of a

de¬‚ationary path, to be consistent with market clearing; if the private sector consumes only

as much as the economy produces, it ¬nds 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 su¬ciently

quickly in the event that in¬‚ation falls. However, even if φ(Π) > 0 for all Π > β, as long

65

One interpretation of this assumption is that the policy rule conforms to the “Taylor principle” except

when the zero bound prevents any further decreases in the nominal interest rate.

66

This is how self-ful¬lling de¬‚ations are excluded under a Taylor rule in Schmitt-Groh´ and Uribe (1998),

e

in the context of a related model.

67

The e¬ects of ¬scal policy upon aggregate demand under a non-Ricardian regime are discussed further

in chapter 5.

84 CHAPTER 2. PRICE-LEVEL DETERMINATION

as φ(Π) < (Π ’ β)/β for all β < Π < Π— , we can still exclude self-ful¬lling de¬‚ations in the

case of any ¬scal policy that ensures a growth rate µ > 1 for government liabilities. For this

¯

ensures that every de¬‚ationary solution involves an in¬‚ation rate converging to Π—— = β, as

shown in Figure 2.4. Hence there must exist a date t such that 1 + φ(ΠT ) ≥ µ for all T ≥ t.

¯

It follows that

T ’1

T t

(1 + is )’1 µT ’t Dt /Pt ≥ β t »(it )Dt /Pt > 0,

β »(iT )DT /PT = β »(it ) ¯

s=t

so that again the transversality condition is violated. Finally, even if interest rates level o¬

sooner, so that there exists a lower steady-state in¬‚ation rate Π—— > β, the same argument

works as long as µ > β ’1 Π—— .

¯

Thus, in the case of an appropriate ¬scal policy rule, a de¬‚ationary trap is not a possible

rational expectations equilibrium, even under an interest-rate rule. Furthermore, the type

of ¬scal 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: ¬scal policy must

ensure that the nominal value of total government liabilities Dt will not decline, even in the

case of sustained de¬‚ation.

Indeed, an interest-rate rule has an advantage over the classic formulation of a monetary-

targeting regime, in at least one regard. Self-ful¬lling de¬‚ations may be excluded in the case

of an interest-rate rule, even when the target in¬‚ation rate Π— associated with the Taylor

rule (or with a Wicksellian regime with a non-constant target path for the price level) implies

de¬‚ation at a rate less than the rate of time preference (β < Π— < 1). Under the monetary

targeting regime (with ¬scal policy maintaining zero government debt at all times), instead,

the argument given above to exclude self-ful¬lling de¬‚ations fails when µ < 1. In this case,

¯

the transversality condition is satis¬ed by all of the solutions to (4.8) that involve in¬‚ation

rates eventually falling to the level Πt = β, and so a continuum of perfect foresight equilibria

(and similarly a large set of “sunspot” equilibria) exist.68 Of course, this result could be

avoided if we assume a di¬erent ¬scal 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-ful¬lling de¬‚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 de¬‚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 de¬‚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 e¬ect 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 ¬‚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 ; ξt ) de-

scribing the marginal utility of additional expenditure by »(yt , it ; ξt ). Thus additional real

balances a¬ect 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 e¬ect upon aggregate demand, through either a real-

interest-rate e¬ect or a direct real-balance e¬ect. 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 ine¬ective, due to the economy™s being in a “liquidity trap” (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 e¬cacy of a commitment to monetary

targets in this regard. Commitment to a high rate of money growth does nothing to exclude

a de¬‚ationary trap, when coupled with a Ricardian ¬scal policy; the de¬‚ationary equilibrium

simply involves a higher level of excess money balances in that case. Only an anti-de¬‚ationary

¬scal 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-ful¬lling de¬‚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 de¬‚ation when the disturbance occurs may be a regime

that creates expectations of subsequent in¬‚ation. Let us suppose that the problem of self-

ful¬lling de¬‚ations may be put aside (perhaps because of an appropriate ¬scal 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 de¬‚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 de¬‚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 In¬‚ationary Panic

We next consider the opposite sort of instability due to self-ful¬lling expectations, the pos-

sibility of spontaneous ¬‚ight from a country™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-ful¬lling in¬‚ations have generally been considered a more trou-

bling possibility than self-ful¬lling de¬‚ations in the literature on monetary targeting regimes.

However, conditions have been identi¬ed under which such equilibria would not exist in the

case of a constant money growth rate.

In particular, Obstfeld and Rogo¬ (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)

m’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 ¤ m. But then it follows from (4.8) that no perfect foresight equilibrium can ever have

mt ¤ m1 ≡ F ’1 (β/¯G(m)),

µ

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

mt ¤ m2 ≡ F ’1 (β/¯G(m1 )),

µ

and so on. One shows in this way that no equilibrium is possible with mt < m— 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 su¬ciently increasing the interest rate; for it

implies that L(¯, 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 Rogo¬ point out that the conditions required for this to be true are somewhat

88 CHAPTER 2. PRICE-LEVEL DETERMINATION

implausible, though theoretically possible. Observed hyperin¬‚ations in several countries have

also shown that real balances do indeed fall to a small fraction of their normal level when

in¬‚ation becomes su¬ciently 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 in¬‚ation. Hence it is not clear that one can rely upon

this mechanism to prevent self-ful¬lling in¬‚ations in an actual economy.

What if monetary policy is instead speci¬ed by a Taylor rule of the form (4.1)? A

similar argument excluding self-ful¬lling in¬‚ations would be possible only if the graph of

¯

(4.5) becomes vertical at some ¬nite in¬‚ation rate Π, so that (4.5) has no solution for Πt+1

¯

in the case of Πt > Π. In the additively separable case, this might seem to be impossible, as

there could fail to be a solution for Πt+1 in (4.6) only if φ(Πt ) is itself not de¬ned. However,

it is not clear that a function φ that becomes unboundedly large at a ¬nite in¬‚ation rate

¯

Π must be excluded as a possible policy. After all, in the case of monetary targeting when

(4.10) holds, the policy that excludes self-ful¬lling in¬‚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

φ becomes unboundedly large at a ¬nite level of Pt /Pt— . 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

¯

makes φ unde¬ned for Πt ≥ Π 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-

ful¬lling in¬‚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-ful¬lling in¬‚ations

through commitment to increase interest rates su¬ciently sharply at high rates of in¬‚ation,

even when φ(Π) is well-de¬ned for any ¬nite in¬‚ation rate.If Ucm < 0 at low levels of real

balances, the function »(i) is increasing in i, at high levels of i. Suppose that in fact the

4. SELF-FULFILLING INFLATIONS AND DEFLATIONS 89

elasticity of »(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 Πt , for high in¬‚ation rates, eventually

growing without bound. However, if φ(Π) increases su¬ciently rapidly for high values of Π,

the function Π/»(φ(Π)) may be bounded above. In this case, (4.5) has no solution for Πt+1

¯

in the case of values of Πt above some ¬nite bound Π.69

But this then allows us to exclude self-ful¬lling in¬‚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 ¬scal policy of the kind discussed

above, that excludes self-ful¬lling de¬‚ations, one can show that the equilibrium with in¬‚a-

tion forever at the target in¬‚ation rate Π— 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 su¬ciently small random variations in the exogenous variables {yt , ξt , νt , Π— }, 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-ful¬lling in¬‚ations may be excluded

through the addition of policy provisos that apply only in the case of hyperin¬‚ation. For

example, Obstfeld and Rogo¬ (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-ful¬lling in¬‚ation, the commitment has no e¬ect

except to exclude such paths as possible equilibria. Obstfeld and Rogo¬ propose this as a

solution to the problem of self-ful¬lling in¬‚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 hyperin¬‚ation proviso in a regime that otherwise follows a Taylor rule.

69

This is essentially the way in which self-ful¬lling in¬‚ations are excluded under a Taylor rule in the

analysis of Schmitt-Groh´ 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

speci¬cation of the household™s portfolio plan at each date satisfying both the ¬‚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

∞ ∞

E0 Q0,t [Pt Ct + ∆t Mt ] ¤ W0 + E0 Q0,t [Pt Yt ’ 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 ’ 1, we obtain

T ’1 T ’1

Et Qt,s [Ps Cs + ∆s Ms ] + Et [Qt,T WT ] ¤ Wt + Et Qt,s [Ps Ys ’ Ts ]

s=t s=t

for any date T ≥ t + 1. Combining this with the bounds (1.9) on the portfolios that may be

chosen in the various possible states at date T ’ 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 ’1 ∞

Et Qt,s [Ps Cs + ∆s Ms ] ¤ Wt + Et Qt,s [Ps Ys ’ 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 ¬nite 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 ¬nite 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 in¬nite streams {Ct , Mt } from the initial date t = 0 onward must

satisfy this constraint for date zero, given the household™s initial ¬nancial wealth W0 . This

establishes the necessity of (A.1).

It remains to show that this constraint is also su¬cient for processes {Ct , Mt } to be

attainable. One easily shows that processes that satisfy (A.1) can be achieved, by letting

the household™s choice of Wt at each date t ≥ 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 speci¬es the

household™s portfolio plan at each date t ≥ 0. The resulting plan obviously satis¬es (1.9) at

each date, and it is easily veri¬ed that it satis¬es (1.7), and hence (1.2), at each date as well.

Thus the entire sequence of ¬‚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 , ξt }. Then these processes satisfy (1.22) as well

if and only if they satisfy

lim β T Et [uc (YT ; ξT )DT /PT ] = 0. (A.2)

T ’∞

In this proposition, note that the path of {Dt } can be inferred from the processes that are

speci¬ed using the identity

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

s

t

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

βEt [uc (YT +1 ; ξT +1 )(1 + im )MT /PT +1 ] = βEt [uc (YT +1 ; ξT +1 )(1 + iT )MT /PT +1 ]

s s

T

92 CHAPTER 2. PRICE-LEVEL DETERMINATION

s

= Et [uc (YT ; ξT )MT /PT ].

Adding this to the relation

βEt [uc (YT +1 ; ξT +1 )As +1 /PT +1 ] = Et [uc (YT ; ξT )BT /PT ]

s

T

that follows from (1.4) and (1.20), we ¬nd that

s

βEt [uc (YT +1 ; ξT +1 )WT +1 /PT +1 ] = Et [uc (YT ; ξT )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 , ξt }. Then these processes satisfy (1.22) as

well if and only if they satisfy

lim β T Et [uc (YT , MT /PT ; ξT )DT /PT ] = 0.

s

(A.3)

T ’∞

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

βEt [uc (YT +1 , MT +1 /PT +1 ; ξT +1 )(1 + im )MT /PT +1 ] = Et [uc (YT , MT /PT ; ξT )(1 ’ ∆T )MT /PT ]

s s s s

T

s s s

= Et [(uc (YT , MT /PT ; ξT ) ’ um (YT , MT /PT ; ξT ))MT /P

from which it follows as above that

s s s s s

βEt [uc (YT +1 , MT +1 /PT +1 ; ξT +1 )WT +1 /PT +1 ] = Et [uc (YT , MT /PT ; ξT )DT /PT ]’Et [um (YT , MT /PT ; ξT )MT /PT

Furthermore, (3.4) implies that

lim β T Et [um (YT , MT /PT ; ξT )MT /PT ] = 0.

s s

T ’∞

Hence (1.22) holds if and only if (A.3) holds.

1. APPENDIX TO CHAPTER 2 93

1.3 Determinacy of Rational-Expectations Equilibrium

[TO BE ADDED][See Woodford (1986)]

1.4 Proof of Proposition 2.3

Proposition 2.3. Under a Wicksellian policy rule (1.30) with φp > 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 ¬‚uc-

r—

tuations in the exogenous processes {ˆt , πt , νt }, there exists a unique rational-expectations

equilibrium in which Pt /Pt— ∈ P and it ∈ I at all times. Furthermore, equations (1.37) and

(1.38) give a log-linear (¬rst-order Taylor series) approximation to that solution, accurate up

to a residual of order O(||ξ||2 ), where ||ξ|| 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 (¬rst-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 ¬rst-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 φp > 0. As shown in

the text, these equations imply (1.36). This is a form of expectational di¬erence 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

ˆ

application, zt = Pt , a = (1 + φp )’1 , and

ut = (1 + φp )’1 (ˆt + Et πt+1 ’ νt ).

—

r

Any expectational di¬erence 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 ≥ 0. Multiplying this equation by aj and summing from j = 0 through k ’ 1,

we obtain

k’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.

k’∞

Then since the left-hand side of (A.5) is independent of k, it follows that the ¬nal term on

the right must converge in value as k is made unboundedly large, and speci¬cally to the

value of the left-hand side. Thus we must have

∞

aj Et ut+j .

zt = (A.6)

j=0

(This solution is sometimes said to be obtained by “solving (A.4) forward.”)

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| ≥ 1. In this case,

the process {zt } recursively de¬ned by

zt = a’1 (zt’1 ’ ut’1 ) + νt (A.7)

1. APPENDIX TO CHAPTER 2 95

for all t ≥ 1, starting from an arbitrary initial condition z0 , represents a solution to (A.4)

in the case of any process {νt } such that Et νt+1 = 0 for all t. If |a| > 1, (A.7) represents

a bounded solution for {zt } in the case of any bounded process {νt }, 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 {νt } are bounded processes. Nonetheless, if (A.4) has any bounded solution,

it must have an uncountably in¬nite number of them. Let {¯t } be one bounded solution.

z

(For example, in the case that

∞

vt ≡ Et ut+j

j=0

is well-de¬ned 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

¯

by solving forward.) Then another bounded solution is recursively de¬ned by

zt = zt + (zt’1 ’ zt’1 + νt

¯ ¯

for all t ≥ 1, starting from an arbitrary initial condition z0 , where {νt } is any stochastic

T

process such that Et νt+1 = 0 for all t, and such that νt remains bounded for arbitrarily

t=1

large T . This last stipulation can obviously be satis¬ed by a large number of mean-zero,

unforecastable processes; for example, it su¬ces that νt 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 in¬nite number of bounded solutions. Thus the condition

that |a| < 1 is both necessary and su¬cient 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 ≥ 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 ≥ 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

βEt [uc (Yt+1 ; ξt+1 /Pt+1 ] = uc (Yt ; ξt )/Pt

at all times, and hence, by iteration, that

β T Et [uc (YT ; ξT /PT ] = β t uc (Yt ; ξt )/Pt

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

β T Et [uc (YT ; ξT )DT /PT ] ≥ M β T Et [uc (YT ; ξT /PT ]

= M β t Et [uc (Yt ; ξt /Pt ] > 0,

which contradicts (A.2). Hence no equilibrium is possible.

In fact, equilibrium values could be de¬ned 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-de¬ned equilibrium would exist, but would involve zero exchange value for money.

In essence, under the regime described, money is a pure “bubble” ” 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 ” 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 ” the crucial point ” 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

¯

which interest is paid on money ” recall that γD < Π/β = 1 + φ(1; 0) = 1 + ¯m ” at least

±

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 “bubble”.

1.6 Proof of Proposition 2.5.

Proposition 2.5. Let monetary policy be speci¬ed by an exogenous sequence of interest-

rate targets, assumed to remain forever within a neighborhood of the interest rate ¯ > 0

±

associated with the zero-in¬‚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 , ξt Dt /Dt’1 } and on the

interest-rate target process, there exists an uncountably in¬nite set of rational-expectations

equilibrium paths for the price level, in each of which the in¬‚ation rate satis¬es πt ∈ P for

all t. These include equilibria in which the in¬‚ation rate is a¬ected to an arbitrary extent

by “fundamental” disturbances (unexpected changes in Yt or ξt ), by pure “sunspot” states

(exogenous randomness unrelated to the “fundamental” 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 {νt } be any unforecastable mean-zero random variable (or martingale dif-

ference) such that νt < 1 at all times. Then the in¬‚ation process given by

1 + im

Pt t’1

=β Et’1 [uc (Yt ; ξt )(1 ’ νt )] f rac11 ’ νt

Pt’1 uc (Yt’1 ; ξt’1 )

satis¬es (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 ¬‚uctuations in {νt }, this yields an in¬‚ation process such that

πt ∈ P at all times, and that satis¬es (A.2) as well. Hence any such solution represents a

98 CHAPTER 2. PRICE-LEVEL DETERMINATION

rational-expectations equilibrium. The solutions corresponding to di¬erent choices of {νt }

represent distinct equilibria, since in each case the surprise component of in¬‚ation is given

by

πt ’ Et’1 πt = ’ log(1 ’ νt ).

Finally, the variable {νt } may be correlated in an arbitrary way with any of the “fundamen-

tal” 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-in¬‚ation steady state, with ¦π ≥ 0. Then equilibrium is determinate

if and only if ¦π > 1. When this condition is satis¬ed, a log-linear approximation to the

equilibrium evolution of the smoothed in¬‚ation process is given by

∞

(δ + (1 ’ δ)¦π )’(j+1) Et [ˆt+j ’ ¯t+j ].

πt = (1 ’ δ)

¯ r ± (A.8)

j=0

A corresponding approximation to the equilibrium evolution of the single-period in¬‚ation

rate πt is then obtained by substituting (A.8) into

πt ’ δ¯t’1

¯ π

πt = . (A.9)

1’δ

Proof: The solution (A.9) for πt given the evolution of πt is obtained by inverting (2.47).

¯

Then substituting (2.48) into (1.32) to eliminate it , we obtain

¦π πt = Et πt+1 + (ˆt ’ ¯t ).

¯ r ±

Substituting (A.9) for πt in this equation, we obtain an expectational di¬erence equation for

the smoothed in¬‚ation measure,

[δ + (1 ’ δ)¦π ]¯t = Et πt+1 + (1 ’ δ)(ˆt ’ ¯t ).

π ¯ r ± (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

δ + (1 ’ δ)¦π > 1,

which is to say, if and only if ¦π > 1, as required by the “Taylor principle”. 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-in¬‚ation steady state, with φπ , rho ≥ 0. Then equilibrium is determinate

if and only if φπ > 0 and

φπ + ρ > 1 (A.11)

When these conditions are satis¬ed, a log-linear approximation to the equilibrium evolution

of in¬‚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 πt+1 in (1.32), one obtains an expectational di¬erence equation

(φπ + ρ)ˆt = Etˆt+1 + φπ rt + ρ¯t ’ Et¯t+1 ,

± ± ˆ ± ±

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 ¬nds that there is a unique bounded solution

for {ˆt } if and only if

±

|φπ + ρ| > 1 (A.12)

is satis¬ed. 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

{πt }, obtained by inverting (2.51), if and only if φπ = 0. Hence there is a unique bounded

solution for {πt } if φπ > 0 and (A.11) applies. Using the solution obtained for {ˆt }, one

±

obtains the solution (2.53) for the in¬‚ation process.

100 CHAPTER 2. PRICE-LEVEL DETERMINATION

If instead φπ = 0, there is a multiplicity of possible solutions for {πt }, even when the

equilibrium path {ˆt } is uniquely determined. In fact, Proposition 2.5 again applies in this

±

case. If φπ > 0 but 0 < φπ + ρ < 1, there is an uncountably in¬nite number of solutions

for {ˆt }, as one can show using the method discussed following the proof of Proposition

±

2.3. To each of these there corresponds a unique associated in¬‚ation process, but the set of

equilibrium in¬‚ation processes is uncountably in¬nite.

As remarked in the text, determinacy of equilibrium does not require that φπ > 0,

though that is the case of primary practical interest. Our analysis above shows that in

fact all that is required is that φπ = 0 and that (A.12) be satis¬ed. When ρ > 1, the latter

condition is satis¬ed by all non-zero in¬‚ation-response coe¬cients φπ > ’(ρ’1), which would

include moderately negative values. In such a case, (2.53) continues to provide a log-linear

approximation to the equilibrium in¬‚ation process. The conditions for determinacy would

also be satis¬ed by all φπ < ’(1 + ρ). (Note that this means that determinacy results from

su¬ciently large negative values of φπ even in the case that ρ < 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 {ˆt } follows an exogenously given

r

stationary AR(1) process, and let the monetary policy rule be of the form (2.51), with

ρ ≥ 0, φπ > 0 and a constant intercept consistent with the zero-in¬‚ation steady state (i.e.,

¯t = 0). Consider the choice of a policy rule (ρ, φπ ) within this class so as to bring about

±

a certain desired unconditional variance of in¬‚ation var(π) > 0 around the mean in¬‚ation

rate of zero. For any large enough value of ρ, there exists a φπ satisfying (A.11) such that

the unconditional variance of in¬‚ation in the stationary rational-expectations equilibrium

associated with this rule is of the desired magnitude. Furthermore, the larger is ρ, the

smaller is the unconditional variance of interest-rate ¬‚uctuations var(ˆ) in this equilibrium.

±

1. APPENDIX TO CHAPTER 2 101

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

rt = ρr rt’1 +

ˆ ˆ t

for the equilibrium real rate, Et rt+j = ρj rt for all j ≥ 0. Then for any policy rule (ρ, φπ )

ˆ rˆ

that satis¬es (A.11) (2.54) implies that

φπ

ˆt =

± rt .

ˆ

φπ + ρ ’ ρr

Hence {ˆt } is also an AR(1) process, with variance and ¬rst-order autocovariance

±

2

φπ

var(ˆ) =

± var(ˆ),

r (A.13)

φπ + ρ ’ ρr

cov(ˆt , ˆt’1 ) = ρr var(ˆ).

±± ±

Inverting (2.51) implies that πt = φ’1 (ˆt ’ ˆt’1 ), from which it follows that

π± ±

var(π) = φ’2 var(ˆt ’ ρˆt’1 )

± ±

π

= φ’2 [(1 + ρ2 )var(ˆ) ’ 2ρcov(ˆt , ˆt’1 )]

± ±±

π

1 ’ 2ρr ρ + ρ2

= var(ˆ).

r (A.14)

(φπ + ρ ’ ρr )2

Condition (A.14) can be solved for the required in¬‚ation-response coe¬cient in order to

obtain a given degree of variability of in¬‚ation, yielding

1/2

φπ = (1 ’ 2ρr ρ + ρ2 )f racvar(ˆ)var(π)

r + ρr ’ ρ.

(Here we select the positive square root because we know that if a solution exists that satis¬es

(A.11), it must be such that φπ > ρr ’ ρ.) We note that for all large enough ρ > 0, the right-

hand side expression must exceed 1 ’ ρ, in which case there is indeed a solution satisfying

(A.11), as asserted in the Proposition.

This solution can then be substituted for φπ in (A.13), yielding

ρ ’ ρr

σ(ˆ) = σ(ˆ) ’

± r σ(π), (A.15)

(1 ’ 2ρr ρ + ρ2 )1/2

102 CHAPTER 2. PRICE-LEVEL DETERMINATION

where σ(x) ≡ (var(x))1/2 in the case of any stationary random variable x. The expression

ρ ’ ρr ρ ’ ρr

=

(1 ’ 2ρr ρ + ρ2 )1/2 [(1 ’ ρ2 ) + (ρ ’ ρr )2 ]1/2

r