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Interest and Prices

Michael Woodford
Princeton University

April 1999
Revised September 2002




Preliminary and Incomplete

c Michael Woodford 2002
Contents

2 Price-Level Determination 1
1 Price-Level Determination in a Cashless Economy . . . . . . . . . . . . . . . 3
1.1 An Asset-Pricing Model with Nominal Assets . . . . . . . . . . . . . 4
1.2 A Wicksellian Policy Regime . . . . . . . . . . . . . . . . . . . . . . . 16
2 Alternative Interest-Rate Rules . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1 Exogenous Interest-Rate Targets . . . . . . . . . . . . . . . . . . . . 30
2.2 The “Taylor Principle” and Determinacy . . . . . . . . . . . . . . . . 35
2.3 Inertial Responses to In¬‚ation Variation . . . . . . . . . . . . . . . . 40
3 Price-Level Determination with Monetary Frictions . . . . . . . . . . . . . . 48
3.1 A Model with Transactions Frictions . . . . . . . . . . . . . . . . . . 49
3.2 Interest-Rate Rules Reconsidered . . . . . . . . . . . . . . . . . . . . 52
3.3 A Comparison with Money-Growth Targeting . . . . . . . . . . . . . 54
3.4 Consequences of Non-Separable Utility . . . . . . . . . . . . . . . . . 60
4 Self-Ful¬lling In¬‚ations and De¬‚ations . . . . . . . . . . . . . . . . . . . . . 72
4.1 Global Multiplicity Despite Local Determinacy . . . . . . . . . . . . . 73
4.2 Policies to Prevent a De¬‚ationary Trap . . . . . . . . . . . . . . . . . 82
4.3 Policies to Prevent an In¬‚ationary Panic . . . . . . . . . . . . . . . . 87
1 APPENDIX TO CHAPTER 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 90
1.2 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 91
1.3 Determinacy of Rational-Expectations Equilibrium . . . . . . . . . . 93
1.4 Proof of Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 93
1.5 Proof of Proposition 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 95
1.6 Proof of Proposition 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . 97
1.7 Proof of Proposition 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . 98
1.8 Proof of Proposition 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . 99
1.9 Proof of Proposition 2.9. . . . . . . . . . . . . . . . . . . . . . . . . . 100
1.10 Proof of Proposition 2.10. . . . . . . . . . . . . . . . . . . . . . . . . 102
1.11 Proof of Proposition 2.11. . . . . . . . . . . . . . . . . . . . . . . . . 103
1.12 Proof of Proposition 2.12. . . . . . . . . . . . . . . . . . . . . . . . . 104
1.13 Proof of Proposition 2.13. . . . . . . . . . . . . . . . . . . . . . . . . 105
1.14 Proof of Proposition 2.14. . . . . . . . . . . . . . . . . . . . . . . . . 106
4 CONTENTS

1.15 Proof of Proposition 2.15. . . . . . . . . . . . . . . . . . . . . . . . . 106
Chapter 2

Price-Level Determination Under
Interest-Rate Rules

While virtually all central banks use a short-term nominal interest rate (typically an overnight
rate, such as the federal funds rate in the U.S.) as their instrument, and an extensive empirical
literature characterizes actual monetary policy in terms of estimated central bank “reaction
functions” for setting such interest rates, the theoretical literature in monetary economics
has almost entirely concerned itself with the analysis of policies that are described by alter-
native (possibly state-contingent) paths for the money supply. The aim of this chapter is to
remedy this oversight by presenting a theory of price-level determination under interest-rate
rules of the sort that are often taken to describe actual central bank policies.
We shall argue that it is not necessary, in order to understand the consequences of such
rules, to ¬rst determine their consequences for the evolution of the money supply, and then
analyze the equivalent money-supply rule. Instead, it is possible to analyze price-level deter-
mination under such rules in terms of an explanatory framework that gives no importance
to either the evolution of the money supply or the determinants of money demand. In
this neo-Wicksellian framework, the fundamental determinants of the equilibrium price level
are instead the real factors that determine the equilibrium real rate of interest, on the one
hand, and the systematic relation between interest rates and prices established by the central
bank™s policy rule, on the other.
We ¬rst expound this approach in the context of a purely cashless economy ” one in

1
2 CHAPTER 2. PRICE-LEVEL DETERMINATION

which there are assumed to be no transactions frictions that can be reduced through the
use of money balances, and that accordingly provide a reason for holding such balances
even when they earn a rate of return that is dominated by that available on other assets.
Such a setting ” one that is commonly assumed in ¬nancial economics and in purely real
models of economic ¬‚uctuations alike ” allows us to display the relations that are of central
importance in the neo-Wicksellian theory in their simplest form.

At the same time, neither the usefulness nor the validity of the approach proposed here
depends on a claim that monetary frictions do not exist in actual present-day economies.
After expounding the theory for the cashless case, we show how the framework can easily be
generalized to allow for monetary frictions, modeled in one or another of the ways that are
common in monetarist models of in¬‚ation determination (by including real balances in the
utility function, or assuming a cash-in-advance constraint). We show in this case that equi-
librium relations continue to be obtained that are direct generalizations of the ones obtained
for the cashless economy, and that need not even imply results that are too di¬erent as a
quantitative matter, if the monetary frictions are parameterized in an empirically plausible
way. Hence the cashless analysis can be viewed as a useful approximation even in the case
of an economy where money balances do facilitate transactions to some extent.

In the case of an economy with transactions frictions, one can also analyze price-level
determination along traditional monetarist lines: one may view the equilibrium price level as
being determined by the expected path of the money supply, although the latter quantity is
endogenous, in the case of an interest-rate rule such as the Taylor rule, so that money, prices,
and interest rates must be simultaneously determined. In the models considered here, this
approach would not yield di¬erent ultimate conclusions than the neo-Wicksellian analysis,
for the system of equilibrium conditions to be solved is actually the same despite the di¬ering
direction of approach. Nonetheless, we shall argue that the neo-Wicksellian interpretation
of these equilibrium conditions is a particularly fruitful one, not least because it continues
to be possible in the limiting case of a cashless economy.

In this chapter, we expound the basic outlines of the neo-Wicksellian theory in the context
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 3

of a model with ¬‚exible prices and an exogenous supply of goods. This allows us to address
a number of basic issues in a particularly simple context, and also allows direct comparison
of this theory with the standard quantity-theoretic approach, which, when derived from
optimizing models, is also most often expounded in a model with ¬‚exible prices. A more
complete development of the theory is possible only after the introduction of nominal price
rigidities in the following chapter.


1 Price-Level Determination in a Cashless Economy
We begin by considering price-level determination in an economy in which both goods mar-
kets and ¬nancial markets are completely frictionless: markets are perfectly competitive,
prices adjust continuously to clear markets, and there exist markets in which state-contingent
securities of any kind may be traded. Under the assumption of frictionless ¬nancial mar-
kets, it is natural to suppose that no “monetary” assets are needed in order to facilitate
transactions.
We shall suppose, however, that there exists a monetary unit of account in terms of
which prices (of both goods and ¬nancial assets) are quoted. This unit of account will be
de¬ned in terms of a claim to a certain quantity of a liability of the central bank, which may
or may not have any physical existence.1 This liability is not a claim to future payment of
anything except future units of the central-bank liability. As argued in section xx of chapter
1, the special situation of the central bank, as issuer of liabilities that promise to pay only
additional units of its own liabilities, allows the central bank to ¬x both the nominal interest
yield on its liabilities and the quantity of them in existence.
1
Under current U.S. arrangements, which are fairly typical, Federal Reserve notes (U.S. currency) and
Federal Reserve balances (credits in an account at the Fed, that can be used for clearing purposes and to
satisfy reserve requirements) are freely convertible into one another, and a promise to pay “a dollar” may
be discharged by transfer to the creditor (or its bank) of either of these types of ¬nancial claim, in the
amount of one dollar. In a cashless economy of the kind that some envision for the future, currency need no
longer exist; in such a world, the “dollar” would be de¬ned by a claim to a one-dollar balance at the Fed.
The fact that in such a world there would be no physical dollars (i.e., dollar bills) would not prevent the
use of dollar accounts in making payments; after all, even now, the dollar is not a claim to anything else,
and is accepted in payment only because of the expectation that it can be transferred to someone else in a
subsequent transaction.
4 CHAPTER 2. PRICE-LEVEL DETERMINATION

While we assume that there is no reason why private parties need to hold this particular
asset, or receive any bene¬t from doing so that would not be obtained by holding any
other similarly riskless ¬nancial claim denominated in terms of the same unit of account, we
assume that they choose to hold ¬nancial claims on the government along with privately-
issued ¬nancial claims. The conditions under which the private sector is willing to hold
the liabilities of the central bank, along with other government liabilities, are described by
arbitrage relations of the kind that are familiar from ¬nancial economics. In an equilibrium,
where these relations are satis¬ed, there then exists a well-de¬ned exchange ratio between
money and real goods and services.
In a frictionless world of this kind, base money ” the monetary liabilities of the central
bank ” is a perfect substitute for other riskless nominal assets of similarly short maturity,
whether these are private obligations or other (non-monetary) government obligations. As
a result, variations in the nominal size of the monetary base, due to for example to open-
market purchases of other sorts of government obligations by the central bank need have no
e¬ect on the prices or interest rates that represent a market equilibrium. Yet this does not
mean that in such a world, the central bank has no control over the equilibrium prices of
goods in terms of money. As we shall see, the central bank™s policy rule is one of the key
determinants of the equilibrium price level even in a cashless economy; and it is possible,
at least in principle, for the central bank to stabilize the price level around a desired level
(or deterministic trend path) through skillful use of the tools at its disposal. But in such a
world, the crucial tool available to the central bank will not be open-market operations, but
the possibility of adjusting the interest rate paid on central-bank balances.


1.1 An Asset-Pricing Model with Nominal Assets

Consider an economy made up of a large number of identical households. The representative
household seeks to maximize the expected value of a discounted sum of period contributions
to utility of the form

β t u(Ct ; ξt ) .
E0 (1.1)
t=0
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 5

Here 0 < β < 1 is a discount factor, and the period contribution to utility u depends upon the
level of consumption Ct of the economy™s single good. We also allow for exogenous stochastic
disturbances ξt to the period utility function, which we may think of as representing variation
in households™ impatience to consume. The presence of this term represents a ¬rst simple
example of something of considerable importance for our general conception of the problem
facing central banks, namely, the existence of real disturbances that should be expected to
change the equilibrium real rate of return, and hence the level of nominal interest rates
required for price stability. For any given realization of ξt , we assume that the period utility
function u(C; ξt ) is concave and strictly increasing in C.
As noted in the introduction, we shall assume complete ¬nancial markets, i.e., that avail-
able ¬nancial assets completely span the relevant uncertainty faced by households about
future income, prices, taste shocks, and so on, so that each household faces a single in-
tertemporal budget constraint. Under the assumption of complete markets, a household™s
¬‚ow budget constraint each period can be written in the form


Mt + Bt ¤ Wt + Pt Yt ’ Tt ’ Pt Ct . (1.2)


Here Mt denotes the household™s nominal end-of-period balances in the distinguished ¬nan-
cial asset (the monetary base) which represents the economy™s unit of account, Bt represents
the nominal value (in terms of this unit of account) of the household™s end-of-period portfo-
lio of all other ¬nancial assets (whether privately issued or claims on the government), Wt
represents beginning-of-period ¬nancial wealth (now counting the monetary base along with
other assets), Yt is an exogenous (possibly stochastic) endowment of the single good, Pt is
the price of the good in terms of the monetary unit, and Tt represents net (nominal) tax
collections by the government. The constraint says that total end-of-period ¬nancial assets
(money plus bonds) can be worth no more than the value of ¬nancial wealth brought into
the period, plus non-¬nancial income during the period net of taxes and the value of con-
sumption spending. Interest income is not written explicitly in (1.2), because it is assumed
to accrue between the discrete dates at which decisions are made; thus Wt already includes
6 CHAPTER 2. PRICE-LEVEL DETERMINATION

the interest earned on bonds held at the end of period t ’ 1.2
It is important to note that in (1.2), Bt does not refer to the quantity held of some
single type of bond; as we assume complete markets, households must be able, at least
in principle, to hold any of a wide selection of instruments with di¬erent state-contingent
returns. We need not, however, introduce any notation for the particular types of ¬nancial
instruments that are traded. (This is one of the conveniences of the assumption of complete
markets.) Since any pattern of state-contingent payo¬s in the future that a household may
desire can be arranged (for the appropriate price), we can write the household™s consumption
planning and wealth-accumulation problems without any explicit reference to the quantities
that it holds of particular assets; and if there are redundant assets, there will not actually
be determinate demands for individual assets (our assumption in the case of the monetary
base). We distinguish the household™s holdings of the monetary base from the rest of its
end-of-period portfolio, however, in order to allow us to explicitly discuss the central bank™s
supply of this asset and the interest paid on it.
In the proposed notation, we may simply represent the household™s portfolio choice as a
choice of the state-contingent value At+1 of its non-monetary portfolio at the beginning of
the next period. Total beginning-of-period wealth in the following period is then given by

Wt+1 = (1 + im )Mt + At+1 , (1.3)
t


where im is the nominal interest rate paid on money balances held at the end of period t.
t

Note that this implies that Wt+1 , as a function of the state of the world realized in period
t + 1, is determined by decisions made in period t; thus Wt is a predetermined state variable
in (1.2).
At the time of the portfolio decision, At+1 is a random variable, whose value will de-
pend upon the state of the world in period t + 1. But the household chooses the complete
2
See equation (1.6) below. Though we often refer to a succession of “periods”, as is common in the
macroeconomic literature, our models are formally ones in which trading occurs at a sequence of discrete
points in time. References to “beginning-of-period” and “end-of-period” portfolios are simply notation to
keep track of the e¬ects of trades, not references to di¬erent points in time, between which interest may
accrue.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 7

speci¬cation of this random variable, its value in every possible state. The absence of arbi-
trage opportunities (a necessary requirement for equilibrium) then requires that there exist
a (unique) stochastic discount factor (or asset pricing kernel) Qt,t+1 with the property that
the price in period t of any bond portfolio with random value At+1 in the following period
is given by
Bt = Et [Qt,t+1 At+1 ]. (1.4)

(As of date t, Qt,t+1 remains a random variable; and Et refers to the expectation conditional
upon the state of the world at date t.) In terms of this discount factor, the riskless short-term
(one-period) nominal interest rate it corresponds to the solution to the equation

1
= Et [Qt,t+1 ]. (1.5)
1 + it

Note that if it happens that the representative household chooses to hold a purely riskless
portfolio (in nominal terms), so that At+1 is perfectly forecastable at date t, (1.4) states
simply that At+1 = (1 + it )Bt . Substituting this into (1.3), and the resulting expression for
Wt into (1.2), which holds with equality in equilibrium, we obtain the familiar di¬erence
equation
Mt + Bt = (1 + im )Mt’1 + (1 + it’1 )Bt’1 + Pt Yt ’ Tt ’ Pt Ct (1.6)
t


for the evolution of Bt . This will actually be an equilibrium condition in the case that the
government issues only riskless one-period debt; but it is still important, even in that case, to
recognize that an individual household™s budget constraint allows it the possibility of shifting
wealth across states of the world in other ways.3
More generally, then, equations (1.2), (1.3) and (1.4) together give a complete description
of the household™s ¬‚ow budget constraint. Using (1.3) and (1.4) to eliminate Bt from (1.2),
3
Condition (1.6) actually represents the correct ¬‚ow budget constraint if we assume such radically incom-
plete markets that households can neither borrow nor lend except in terms of the single instrument assumed
to be issued by the government. This case of a single traded asset is often considered in the literature on
consumption theory (see, e.g., Obstfeld and Rogo¬, 1996, sec. 2.3). And in the present context, with identi-
cal households, it makes no real di¬erence what we assume about the number of ¬nancial markets that are
open. However, the characterization of optimal household plans is simplest in the case of complete markets,
and the introduction of market valuations for arbitrary random income streams will prove useful, in the next
chapter, when we need to consider the optimal pricing decisions of ¬rms.
8 CHAPTER 2. PRICE-LEVEL DETERMINATION

the constraint can alternatively be written

(1 ’ Et Qt,t+1 (1 + im ))Mt + Et [Qt,t+1 Wt+1 ] ¤ Wt + [Pt Yt ’ Tt ’ Pt Ct ].
t


Using (1.5), this becomes

Pt Ct + ∆t Mt + Et [Qt,t+1 Wt+1 ] ¤ Wt + [Pt Yt ’ Tt ]. (1.7)

where
it ’ im
t
∆t ≡ . (1.8)
1 + it
It is clear from this version that the interest-rate di¬erential ∆t between non-monetary
and monetary assets represents the opportunity cost of holding wealth in monetary form.
Given its planned state-contingent wealth Wt+1 at the beginning of the following period, the
household can choose any values Ct , Mt ≥ 0 that satisfy (1.7).4
A complete description of the household™s budget constraints requires that we also specify
a limit on borrowing, to prevent “Ponzi schemes” of the kind that would otherwise be
consistent with the in¬nite sequence of ¬‚ow budget constraints in an in¬nite-horizon model.
In the spirit of our assumption of perfectly frictionless ¬nancial markets, it is natural to
suppose that there is no obstacle to borrowing against after-tax endowment income that
may be anticipated (even if in only some states of the world) at any future date. The
implied constraint is then that the household must hold a net portfolio at the end of period
t (possibly including issuance of some securities, in order to borrow against future income)
such that the wealth Wt+1 transferred into the next period satis¬es the bound

Wt+1 ≥ ’ Et+1 [Qt+1,T (PT YT ’ TT )] (1.9)
T =t+1

with certainty, i.e., in each state of the world that may be reached in period t + 1. Here the
general stochastic discount factor Qt,T for discounting (nominal) income in period T back to
4
We assume that money balances must be non-negative because this asset is de¬ned as a liability of the
central bank, that accordingly cannot be issued by any other parties, even though (under our assumption
of complete markets) private securities are issued that are equivalent in terms of their state-contingent
payouts. This non-negativity constraint is another reason to single out this asset from the others in writing
the household™s budget constraints; for we assume no short-sale constraints in the case of any other securities
in our model with frictionless ¬nancial markets.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 9

an earlier period t is de¬ned by
T
Qt,T ≡ Qs’1,s .
s=t+1

(We also use the notation Qt,t ≡ 1.) Condition (1.9) then says that a household cannot plan
to be indebted in any state in an amount greater than the present value of all subsequent
after-tax non-¬nancial income.
The entire in¬nite sequence of ¬‚ow budget constraints (1.7) and borrowing limits (1.9)
are equivalent to a single intertemporal (or lifetime) budget constraint for the household. We
note ¬rst of all that unless the present value on the right-hand side of (1.9) is well-de¬ned
(i.e., the in¬nite sum converges), the household has no budget constraint: Ponzi schemes
are possible, hence unlimited consumption is a¬ordable. Furthermore, if the present value is
in¬nite looking forward from any state of the world, at any date, unbounded consumption
is possible not only at that date and in all other states (including along histories under
which the state in question never occurs); for with complete markets, it is possible to borrow
against that state to ¬nance unbounded consumption in any other state.
We may thus restrict attention to the case in which Ponzi schemes are not possible,
because

Et [Qt,T (PT yT ’ TT )] < ∞ (1.10)
T =t

at all times.5 The budget constraint is also unde¬ned unless interest rates satisfy the lower
bound

it ≥ im (1.11)
t


at all times. For otherwise, an arbitrage opportunity exists; a household can ¬nance unlim-
ited consumption by shorting riskless one-period bonds (i.e., borrowing at the short riskless
rate, assumed to be negative) and using the proceeds partly to hold cash su¬cient to repur-
chase the bonds (repay its debt) a period later and partly to ¬nance additional consumption.
Because utility is assumed to be strictly increasing in consumption, such an operation con-
5
Throughout, it should be understood that when we say that such a relation holds “at all times,” this
also means in all possible states of the world at each date.
10 CHAPTER 2. PRICE-LEVEL DETERMINATION

tinues to increase utility no much matter how much it may be engaged in. Hence we may
also restrict attention to the case in which (1.11) holds at all times.
We are then able to establish that the in¬nite sequence of ¬‚ow budget constraints (1.7)
are equivalent to a single intertemporal budget constraint.


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 ]. (1.12)
t=0 t=0




The proof is given in the appendix. Note that the intertemporal budget constraint states
simply that the present value of the household™s planned consumption over the entire indef-
inite future, plus the cost to it of its planned money holdings, must not exceed its initial
¬nancial wealth plus the present value of its expected after-tax income from sources other
than ¬nancial assets. One can also show (see the proof in the appendix) that the household™s
continuation plan, looking forward from any date t (i.e., its plan for dates T ≥ t, in all of
the states that remain possible given the state of the world at date t), must satisfy the
corresponding intertemporal budget constraint
∞ ∞
Et Qt,s [Ps cs + ∆s Ms ] ¤ Wt + Et Qt,s [Ps ys ’ Ts ]. (1.13)
s=t s=t


The household™s optimization problem is then to choose processes Ct , Mt ≥ 0 for all
dates t ≥ 0, satisfying (1.12) given its initial wealth W0 and the goods prices and asset
prices (indicated by the stochastic discount factors Qt,t+1 ) that it expects to face, so as to
maximize (1.1). Given an optimal choice of these processes, an optimal path for Wt may
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 11

be constructed as in the proof of Proposition 2.1. Given a stochastic process for Wt+1 , the
implied processes for At+1 and for Bt are given by (1.3) and (1.4).
Because this is essentially a standard concave optimization problem subject to a single
budget constraint, necessary and su¬cient conditions for household optimization are easily
given. First of all, (1.10) and (1.11) must hold at all times, since otherwise no optimal plan
exists (as more consumption is always possible). Second, since in the cashless economy there
is no non-pecuniary bene¬t to holding money balances, household optimization requires that
either
Mt = 0 (1.14)

or
it = im (1.15)
t


at each date and in each possible state (though which condition obtains may di¬er across
dates and across states).
Third, by equating marginal rates of substitution to relative prices, we obtain the ¬rst-
order conditions
uc (Ct ; ξt ) β Pt
= . (1.16)
uc (Ct+1 ; ξt+1 ) Qt,t+1 Pt+1
Here Uc is the partial derivative of U with respect to the level of consumption. This condition
must hold for each possible state at each date t ≥ 0, and for each possible state that may
occur at date t + 1, given the state that has occurred at date t. (Qt,t+1 indicates the value of
the discount factor in a particular state at date t + 1.) Using (1.5), condition (1.16) implies
that the short-term nominal interest rate must satisfy
’1
uc (Ct+1 ; ξt+1 ) Pt
’1
1 + it = β Et (1.17)
uc (Ct , ; ξt ) Pt+1

at each date.
Finally, optimization requires that the household exhaust its intertemporal budget con-
straint; that is, (1.13) must hold as an equality at each date. Equivalently, the ¬‚ow budget
constraint (1.2) must hold as an equality at each date, and in addition, the household™s
12 CHAPTER 2. PRICE-LEVEL DETERMINATION

wealth accumulation must satisfy the transversality condition

lim Et [Qt,T WT ] = 0. (1.18)
T ’∞


(Condition (1.13), stated as a strict equality, implies both that (1.2) must hold as a strict
equality at each date T ≥ t, and that (1.18) must hold. Conversely, the latter set of conditions
imply that (1.13) holds with strict equality, looking forward from date t.) Finally, given that
(1.13) must hold with strict equality, condition (1.10) may equivalently be written

Et Qt,T [PT CT + ∆T MT ] < ∞. (1.19)
T =t

We thus obtain a set of conditions ” (1.2) as a strict equality; the requirement that either
(1.14) or (1.15) hold with equality, in addition to the inequality conditions (1.11) and Mt ≥ 0;
(1.16); (1.18); and (1.19) ” that must hold at all times in order for the representative
household™s actions to be optimal. At the same time, one can show that this set of conditions
su¬ces for optimality as well.
We may now state the complete set of conditions for a rational expectations (or intertem-
poral) equilibrium in this model. In addition to the conditions just stated for household
optimization, markets must clear at all dates. This means that household demands must
satisfy
Mt = Mts , At+1 = As
Ct = Yt , t+1


at all dates. Here Mts refers to the supply of base money by the central bank, which we
assume to be positive at all dates. As refers to the aggregate value at the beginning of
t+1

period t + 1 of government bonds in the hands of the public at the end of period t . (In
s s
general, it would not su¬ce for bond-market clearing to require that Bt = Bt , where Bt
denotes the market value of government bonds outstanding at the end of period t, as this
could allow households to demand a portfolio with di¬erent state-contingent payo¬s than
the aggregate supply of government bonds.) If we specify the supply of government bonds
s
in more primitive terms by specifying the variables {Bt,t+j }, where for each date t and each
s
j ≥ 1, Bt,t+j denotes the total (nominal) coupons that the government promises to pay at
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 13

date t + j on bonds that are outstanding at the end of period t, then

As s
≡ Et+1 [Qt+1,t+j Bt,t+j ]
t+1
j=1

in each possible state that may be reached at date t + 1. Finally, note that we abstract here
from government purchases of real goods and services (though the model is extended below
to allow for them).
Given that Mts > 0, market-clearing implies that (1.14) cannot hold, and hence that
(1.15) must hold at all times. Substituting the market-clearing conditions into conditions
(1.16) “ (1.17) for household optimization, we obtain equilibrium conditions

uc (Yt ; ξt ) β Pt
= , (1.20)
uc (Yt+1 ; ξt+1 ) Qt,t+1 Pt+1
’1
uc (Yt+1 ; ξt+1 ) Pt
1 + it = β ’1 Et (1.21)
uc (Yt ; ξt ) Pt+1
for each date. Note that the latter relation takes the form of a “Fisher equation” for the nom-
inal interest rate, where the intertemporal marginal rate of substitution of the representative
household plays the role of the real-interest factor.
Finally, substituting our solution (1.20) for the stochastic discount factor into (1.18) and
(1.19), the latter conditions take the form

lim β T Et [uc (Yt ; ξT )WT /PT ] = 0,
s
(1.22)
T ’∞


β T Et [uc (Yt ; ξT )YT ] < ∞. (1.23)
T =t
s
(Here we have also used the market-clearing conditions to equate WT with WT ≡ (1 +
im’1 )MT ’1 + As , the total supply of nominal claims on the government at the beginning
s
T T

of period T , and (1.15) to substitute for the factor ∆T in (1.19).) A rational-expectations
equilibrium is then a collection of processes that satisfy (1.15), (1.21), (1.22) and (1.23) at
all dates t ≥ 0.
The transversality condition (1.22) can equivalently be written in a possibly more familiar
form, in terms of the end-of-period value of total government liabilities, Dt ≡ Mts + Bt .
s
14 CHAPTER 2. PRICE-LEVEL DETERMINATION

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
t

(1.23) 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 ; ξT )DT /PT ] = 0. (1.24)
T ’∞



The proof is given in the appendix. It follows that we can equivalently de¬ne equilibrium as
follows.


Definition. A rational-expectations equilibrium of the cashless economy is a pair of
processes {Pt , it } that satisfy (1.15), (1.21), (1.23), and (1.24) at all dates t ≥ 0, given the
exogenous processes {Yt , ξt }, and evolution of the variables {im , Mts , Dt } consistent with the
t

monetary-¬scal policy regime.


This latter formulation is especially useful in that it allows us to specify ¬scal policy in
terms of restrictions on the evolution of the total government liabilities, or alternatively,
restrictions on the path of the conventional government budget de¬cit.
Note that we need not include the additional equilibrium condition (1.20) in our de¬-
nition of rational-expectations equilibrium, if we are interested only in the determination
of equilibrium prices and interest rates. (The additional condition must be appended to
our system, of course, if we are interested in other equilibrium asset prices.) Nor is there
any additional equilibrium condition corresponding to the requirement that (1.2) hold with
equality; this condition is necessarily satis¬ed (when we substitute the market-clearing con-
ditions) as long as the supplies of government liabilities evolve in accordance with the ¬‚ow
government budget constraint

Et [Qt,t+1 Wt+1 ] = Wts ’ Tt ’ ∆t Mts .
s
(1.25)

We shall assume that the monetary-¬scal policy regime satis¬es this constraint at all times.
We then have a system of two equalities at each date, (1.15) and (1.21), to determine the two
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 15

endogenous variables Pt and it , together with the bounds (1.23) and (1.24) that the solution
must satisfy.
Our notation thus far allows only for ¬scal policies consisting of taxes or transfers. But the
framework above is easily extended to allow for government purchases of goods and services
as well, without any material change being required in the above equilibrium conditions. Let
government purchases of the single good in period t be denoted Gt , and suppose that {Gt }
is an exogenous process, such that Gt < Yt at all dates.6 Market clearing then requires that
Ct + Gt = Yt at all dates. Substitution of this relation into the conditions for optimization
by the representative household then leads to equilibrium conditions such as
’1
uc (Yt+1 ’ Gt+1 ; ξt+1 ) Pt
1 + it = β ’1 Et , (1.26)
uc (Yt ’ Gt ; ξt ) Pt+1
generalizing (1.21).
We note that (1.26) is obtained from the previous equation (1.21) by replacing uc (Yt ; ξt )
by uc (Yt ’ Gt ; ξt ) each time it occurs. The same is true for the other equilibrium conditions
(1.20), (1.23) and (1.24) as well. Alternatively, we obtain the equilibrium conditions for the
general case by replacing the “direct” utility function u(Ct ; ξt ) throughout our calculations
by the “indirect” utility
˜
u(Yt ; ξt ) ≡ u(Yt ’ Gt ; ξt ), )
˜ (1.27)

indicating the utility ¬‚ow to the representative household as a function of its “total demand”
for resources Yt , where total demand adds the resources consumed by the government on the
household™s behalf (its per-capita share of government purchases) to the household™s private
˜
consumption.7 In this indirect utility function, ξt indicates a vector of disturbances that
includes both Gt and the taste shock ξt .
6
One might, of course, also consider ¬scal policies under which Gt is endogenously determined, for example
as the solution to some welfare-maximization problem of the government™s. In the present study, however,
we shall assume that government purchases are given exogenously. We allow for endogeneity of the level
of net tax collections, as for example in the next section, and this is some importance for our theory of
price-level determination. See section xx of chapter 4 for further discussion.
7
Here we use the same notation Yt for a choice variable of the household as has previously been used
for the exogenous supply of goods. In fact, in the model with endogenous output presented in chapter 4,
equilibrium conditions such as (1.27) continue to apply, but with Yt referring to aggregate demand, and not
to any exogenously given supply of goods.
16 CHAPTER 2. PRICE-LEVEL DETERMINATION

The household™s problem can then be written as one of choosing the state-contingent
evolution of total demand to maximize its expected discounted ¬‚ow of indirect utility subject
to an intertemporal budget constraint of the form (1.12), if Ct in this constraint is taken to
refer to total demand, and Tt to the primary government budget surplus (tax collections in
excess of government spending).8 In this case, we can derive exactly the same equilibrium
conditions as were obtained earlier, except that the function u is everywhere replaced by u.
˜
Hence variations in the level of government purchases Gt have exactly the same e¬ect as the
˜
taste shock ξt ; they are simply another source of exogenous variations in the relation uc (Yt ; ξt )
˜
between the marginal utility of income to the representative household and aggregate output,
and hence of variations in the equilibrium real rate of interest.


1.2 A Wicksellian Policy Regime

We now o¬er a simple example of a complete speci¬cation of monetary and ¬scal policy rules
for a cashless economy, and consider the determinants of the equilibrium path of the money
price of goods under such a regime. Note that, as a consequence of the forward-looking
character of households™ asset accumulation problems, the determination of equilibrium at
any point in time requires that we specify how policy is expected to be conducted into the
inde¬nite future, and in all possible future states. This is one reason for our speci¬cation
of government policy in terms of systematic rules for the determination of both the central
bank™s actions and the government™s budget.
Our speci¬cation of monetary policy will be in the spirit of Wicksell™s (1898, 1907) pro-
posed rule. As discussed in the previous chapter, this rule can be expressed in terms of a
formula for the central bank™s interest-rate operating target. We shall also explicitly specify
the way in which the central bank adjusts the two instruments at its disposal ” the nominal
value of the monetary base Mts , on the one hand, and the interest rate im paid on base money
t

8
The de¬nition of equilibrium that results from this formulation of the household™s problem is equivalent
to the standard one, subject to the proviso that the processes {Gt , Yt , ξt } are such that the present value of
government purchases is ¬nite. Technically, one can imagine an equilibrium in which the present value of per
capita output is not ¬nite, though the present value of the resources left for the private sector to consume is
¬nite. But this special case is of little interest and is ignored here.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 17

on the other ” in order to achieve its operating target. In a world with monetary frictions
(discussed below in section xx), it is possible to use either of these instruments to a¬ect
the level of short-term nominal interest rates, and as discussed in chapter 1, actual central
banks di¬er in the extent to which they use these two means to implement policy (though
almost all central banks formulate policy in terms of an interest-rate operating target). In
the cashless economy described above, however, changes in the quantity of base money (for
example, through open-market purchases of government securities) have no consequences for
the equilibrium determination of interest rates or other variables. (Note that Mts does not
appear in any of the equilibrium conditions obtained at the end of the previous section.)
Hence policy targets must be implemented exclusively through adjustment of the interest
paid on base money.
Speci¬cally, we assume a regime under which the interest paid on base money is equal
at all times to the central bank™s current interest-rate target, determined in response to
the bank™s assessment of current aggregate conditions. Such a system would resemble the
“channel systems” described in chapter 1, under which the interest paid on central-bank
balances is equal to the target rate minus a ¬xed spread; here the spread is assumed to be
zero, since in equilibrium, the market interest rate it will actually equal im , and not im plus
t t

any positive spread. It is true that current channel systems pay interest only on central-bank
clearing balances, and not on currency. But we can interpret the regime analyzed here to
have this property as well; in a cashless world, this would simply mean that currency would
not be held in equilibrium (any initially existing currency would be promptly deposited with
the central bank in an interest-earning account), so that “base money” Mts would correspond
to the supply of clearing balances.
Under a Wicksellian rule for the interest-rate target, the interest rate paid on central-bank
balances equals
im = φ(Pt /Pt— ; νt ) (1.28)
t


where Pt— > 0 de¬nes a target path for the price level, νt is an additional possible exogenous
random disturbance to the policy rule (or to its implementation), and φ(·; ν) is a non-
18 CHAPTER 2. PRICE-LEVEL DETERMINATION

negative-valued, non-decreasing function for each possible value of the disturbance ν.9
Here the function φ indicates the rule used by the central bank to sets its operating target,
while equation (1.28) indicates the way in which the rule is implemented. The inclusion of a
time-varying price-level target Pt— allows us to treat the case of a rule that seeks to stabilize
the price level around a modestly growing trend path ” say, a rule that provides for one
or two percent in¬‚ation per year, perhaps to compensate for bias in the price index that
is targeted ” rather than necessarily assuming a constant price level target, as Wicksell
did. The inclusion of the random disturbance νt allows us to consider the e¬ects of random
variations in policy, or in its implementation, that we may not wish to model as changes in
the target price level itself. This includes the possibility that the central bank may respond
to output variations as well as the path of prices, as called for by the Taylor (1993) rule;
or that the central bank may respond to perceived variation in the equilibrium real rate of
return. (In the present model, both output and the equilibrium real interest rate are purely
exogenous; hence systematic responses to these variables can be modeled by the inclusion of
an exogenous disturbance term in the policy rule.)
We also need to specify the rule by which the evolution of the monetary base is deter-
mined. Here we assume simply that {Mts } is an exogenous, positive-valued sequence. The
logic of the Wicksellian regime requires no variation over time in the supply of base money
at all; however, we allow for possible variation over time in the monetary base, in order to
analyze the equilibrium consequences of this kind of policy action.
Finally, ¬scal policy is speci¬ed by a rule for the evolution of the total supply of gov-
ernment liabilities {Dt }, and by a speci¬cation of the composition of government liabilities
(debt-management policy) at each point in time. For simplicity, we let {Dt } be an exogenous

9
The function is assumed to be non-negative on the ground that it is not possible for the central bank
to drive nominal interest rates to negative levels. We assume that, as under typical current arrangements,
the holders of central-bank balances have the right to ask for currency in exchange for such balances at any
time, and that it is infeasible to pay negative interest on currency. Hence an attempt to pay negative interest
on central-bank balances would lead to zero demand for such balances, and a market overnight interest rate
of zero (the rate available on currency), rather than a negative overnight interest rate. The assumed non-
negativity of the function requires that φ(P/P — ; ν) not be an exactly linear function of log(P/P — ), though
we make use a local log-linear approximation to the function below.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 19

process. One simple example of such a ¬scal rule would be a balanced-budget rule of the
kind analyzed by Schmitt-Groh´ and Uribe (2000), where ∆Dt = 0 each period; another
e
would be a policy under which no government bonds are ever issued, so that Dt = Mts each
period. We also simplify by assuming that all government debt consists entirely of riskless
s
one-period nominal bonds. The variable Bt then indicates the supply of such bonds at the
end of period t, in terms of their nominal value at the time of issuance (rather than maturity).
The implied rule for net tax collections Tt is then given by

s
Tt = (1 + it’1 )(Dt’1 ’ ∆t’1 Mt’1 ) ’ Dt , (1.29)

using the fact that As = (1 + it’1 )Bt’1 ) = (1 + it’1 )(Dt’1 ’ Mt’1 ).
s s
t

A rational-expectations equilibrium is then a set of processes {Pt , it , im } that satisfy
t

(1.15), (1.21), (1.23), (1.24), and (1.28) at all dates t ≥ 0, given the exogenous processes
{Yt , ξt , Mts , Dt }.10 Using (1.15) to eliminate im in (1.28), we obtain
t


it = φ(Pt /Pt— ; νt ), (1.30)

as an equilibrium condition linking the paths of interest rates and prices. (Note that this
equation directly expresses the interest-rate rule that the central bank implements through
its adjustment of the interest rate paid on base money.) We note furthermore that condition
(1.23) does not involve any endogenous variables, and thus plays no role in equilibrium
determination. We assume processes {Yt , ξt } that satisfy this condition; having done so, we
can drop (1.23) from our list of requirements for equilibrium. We can thus identify rational-
expectations equilibrium with a set of processes {Pt , it } that satisfy (1.21), (1.24) and (1.30)
each period.
We are interested not only in whether a solution to this system of equilibrium conditions
exists, but in whether these relations su¬ce to uniquely determine the equilibrium paths of
interest rates and prices. The question of the determinacy of equilibrium is a preliminary,
more basic issue, before we can hope to address the question of what factors a¬ect the
10
In the case that we allow for government purchases, one should replace u by u in each equation, and ξt
˜
˜t .
by ξ
20 CHAPTER 2. PRICE-LEVEL DETERMINATION

equilibrium price level and how they a¬ect it. And there are obvious reasons to worry about
determinacy under the kind of regime just described. In the celebrated analysis of Sargent
and Wallace (1975), interest-rate rules as such are to be avoided, on the ground that they
result in indeterminacy of the equilibrium price level (and hence, in their model, of the
equilibrium paths of real variables as well). And it is also often worried that in a cashless
economy, there should be nothing to pin down the equilibrium price level, given that there
is in such an environment no determinate demand for the monetary base; this is sometimes
argued to be an important reason to head o¬ ¬nancial innovations that could lead to this
kind of world (e.g., Friedman, 2000).

A general analysis of the existence and uniqueness of solutions to our system of equa-
tions, for arbitrary monetary and ¬scal policy rules, is beyond the scope of this study. We
shall instead address here, and in most of this study, a more limited question, namely the
local determinacy of equilibrium, in the case of policies that involve only small ¬‚uctuations
over time in the monetary policy rule, and assuming that other exogenous disturbances are
similarly small. By local determinacy we mean the question of whether there is a unique
equilibrium within a su¬ciently small neighborhood of certain paths for the endogenous
variables. If so, this equilibrium is at least locally unique, and such local uniqueness makes
possible a well-de¬ned “comparative statics” analysis of the e¬ects of small disturbances or
parameter changes.11 In fact, we shall analyze the e¬ects of small ¬‚uctuations in the price-
level target, and other small disturbances, through exactly such a consideration of how the
steady-state equilibrium associated with steady trend growth of the price-level target and
zero disturbances is perturbed by small stochastic variations in the exogenous variables.

An advantage of restricting our attention to this question is that it can be addressed using
purely linear methods; we analyze a log-linear approximation to the structural equations
derived above, and characterize the (bounded) log-linear solutions to these equations. This
way of characterizing both the important structural relations implied by our model, and the

11
See section xx of the appendix for further general discussion of the issue of local determinacy and of the
methods used in this section to analyze it.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 21

predicted equilibrium evolution of economic time series under alternative policies, is useful
not only because of its tractability, but because of the ubiquity of linear time series models
in empirical studies. Our approach will thus allow a direct mapping between the predictions
of optimizing models of economic behavior and the kinds of structural models and data
characterizations already used in quantitative monetary policy analysis. This will facilitate
both evaluation of the empirical adequacy of the optimizing models, and productive dialogue
between optimization-based and more traditional approaches to policy evaluation.
Furthermore, such an analysis corresponds reasonably closely to the way that the deter-
minacy of rational expectations equilibrium is considered by Sargent and Wallace (1975).
That paper assumes a log-linear model, and considers the uniqueness of non-explosive solu-
tions to the log-linear structural equations. Our method will be essentially the same, except
that the log-linear structural equations can be justi¬ed as a log-linear approximation to exact
relations derived from an explicit intertemporal general equilibrium model, and that we shall
be more explicit about the class of solutions to be considered. Thus the analysis here su¬ces
to address the particular issue relating to determinacy of equilibrium under interest-rate
rules raised by Sargent and Wallace.12
We begin by characterizing the steady state near which we shall look for other solutions.
¯
Consider an environment in which Yt = Y > 0 and ξt = 0 at all times. Let us assume a
policy regime under which the price-level target grows at the constant rate


Pt+1 ¯
=Π>β

Pt


at all dates (with some initial P0 > 0), νt = 0 at all dates, and suppose that the function φ
satis¬es
¯
Π
1 + φ(1; 0) = ,
β

so that the policy rule is consistent with the assumed target path for prices. Let us also
12
In chapter 4, we consider an extension of the model in this chapter which is essentially a rational-
expectations IS-LM model, though not identical in structure to the one analyzed by Sargent and Wallace
(1975). The determinacy of equilibrium under alternative policy rules is taken up again in that chapter.
22 CHAPTER 2. PRICE-LEVEL DETERMINATION

assume a ¬scal rule under which
¯
Dt+1 Π
= γD <
Dt β
at all dates (given initial government liabilities D0 > 0), and an open-market policy under
which {Mts } is an arbitrary sequence, satisfying

0 < Mts < Dt (1.31)

at all times. Under such a policy regime, we easily observe that the paths
¯
Π’β
Pt— ,
Pt = it = ¯ ≡
± >0
β

for all t represent a rational-expectations equilibrium.
Next, let us consider an environment in which there are only small ¬‚uctuations in the
exogenous variables Yt , Pt+1 /Pt— , νt , and Dt+1 /Dt around the constant values speci¬ed in the



previous paragraph. (Speci¬cally, we suppose that each of these variables remains forever
within a bounded interval containing a neighborhood of the steady-state value.) We also
continue to assume that the process {Mts } satis¬es the bounds (1.31). We wish to look for
rational-expectations equilibria in which the endogenous variables it and Pt /Pt— similarly
remain forever within certain neighborhoods of their steady-state values. In this case, condi-
tion (1.24) is necessarily satis¬ed (for any tight enough bounds on the allowable variation in
the variables listed above); thus we need only consider the (local) existence and uniqueness
of solutions to the system of equations consisting of (1.21) and (1.30).
In the case of tight enough bounds on the variations that we consider in these variables,
it su¬ces that we consider the bounded solutions to a system consisting of log-linear approx-
imations to conditions (1.21) and (1.30).13 For the sake of simplicity (and continuity with
the assumptions made in more complex examples in the next two chapters), we shall specify
¯
that the steady state around which we log-linearize is one in which Π = 1, so that there is
zero in¬‚ation. (This does not require that we only analyze policies under which there is no
13
As is further explained in the appendix, this amounts to using the inverse function theorem to demon-
strate local uniqueness of the solution to our system of equilibrium conditions, and using the implicit function
theorem to give a log-linear local approximation to the solution.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 23

trend growth in the price-level target: it only requires that the target in¬‚ation rate is never
very large.)
A log-linear approximation to (1.21) is then given by14

ˆt = rt + Et πt+1 ,
± ˆ (1.32)

where
1 + it Pt
ˆt ≡ log
± , πt ≡ log .
1 +¯± Pt’1
Here the (percentage deviation in the) ex-ante short-term equilibrium real rate of return rt
ˆ
is an exogenous process given by

ˆ ˆ
rt = σ ’1 [Et (Yt+1 ’ gt+1 ) ’ (Yt ’ gt )],
ˆ (1.33)

where the constant coe¬cient
uc
σ≡’ ¯ >0
ucc Y
measures the intertemporal elasticity of substitution in private spending, and the disturbance
term
ucξ
gt ≡ ’ ¯ ξt
Y ucc
indicates the percentage increase in output required to keep the marginal utility of income
constant, given the change that has occurred in the impatience to consume.
In the case of the model extended to allow for government purchases, equations (1.32)
¯¯
“ (1.33) still apply, under the alternative de¬nitions σ ≡ sC σC , where sC ≡ C/Y is the
steady-state share of private expenditure in total demand and
uc
˜
σC ≡ ’ >0
˜¯
ucc C
ˆ ¯ ˆ ¯¯
is the intertemporal elasticity of substitution; and gt = Gt + sC Ct , where Gt ≡ (Gt ’ G)/Y
indicates ¬‚uctuations in government purchases, measured in units of steady-state GDP, and
ucξ
˜
¯
Ct ≡ ’ ¯ ξt
C ucc
˜
14
Note that the appearance in (1.32) of the in¬‚ation rate rather than of the in¬‚ation rate relative to
steady-state in¬‚ation depends on our assumption that the steady-state in¬‚ation rate is zero.
24 CHAPTER 2. PRICE-LEVEL DETERMINATION

indicates the percentage change in private expenditure required to keep marginal utility
constant. (In fact, the notation “gt ” is used for the disturbance in (1.33) because variations
in government purchases are one of the most obvious reasons for the existence of shifts in
this factor.)
A corresponding log-linear approximation to the Wicksellian policy rule (1.30) is given
by
ˆ
ˆt = φp Pt + νt ,
± (1.34)

ˆ
where Pt ≡ log(Pt /Pt— ), and φp ≥ 0 represents the elasticity of φ with respect to P/P — ,
evaluated at the steady-state values of its arguments.15 To the system of log-linear equations
(1.32) and (1.34) we must also adjoin the identity

ˆ ˆ —
πt = Pt ’ Pt’1 + πt , (1.35)


where πt ≡ log(Pt— /Pt’1 ) indicates the exogenous ¬‚uctuations, if any, in the target in¬‚ation
— —


rate. We then wish to examine the bounded solutions to the system of log-linear equations
(1.32), (1.34), and (1.35).
Using (1.34) and (1.35) to substitute for ˆt and πt+1 in the Fisher equation (1.32), we
±
ˆ
obtain an expectational di¬erence equation in the variable Pt alone, given by

ˆ ˆ —
(1 + φp )Pt = Et Pt+1 + (ˆt + Et πt+1 ’ νt ).
r (1.36)


Given a policy rule for which φp > 0, as called for by Wicksell, so that 0 < (1 + φp )’1 < 1,
this equation can be solved forward (as discussed further in the appendix), to obtain a unique
bounded solution

ˆ (1 + φp )’(j+1) Et [ˆt+j + πt+j+1 ’ νt+j ],

Pt = r (1.37)
j=0

r—
in the case of any bounded exogenous processes {ˆt , πt , νt }. Substitution of this solution
into (1.34) then yields an associated unique bounded solution for the nominal interest-rate
15
In writing the disturbance term simply as νt , we adopt the normalization under which ‚φ/‚ν = 1 + ¯ =
±
’1
β , when also evaluated at the steady-state values of the arguments.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 25

dynamics as well, namely

φp (1 + φp )’(j+1) Et [ˆt+j + πt+j+1 ’ νt+j ] + νt ,

ˆt =
± r (1.38)
j=0

while substitution into (1.35) yields a corresponding solution for the equilibrium rate of
in¬‚ation.
We thus obtain the main result of this section.16


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.


See further discussion in the appendix.
Thus there may be a well-de¬ned rational-expectations equilibrium path for the price
level, even in a purely cashless economy, and even under a policy rule that is formulated
in terms of an interest-rate rule ” i.e., a rule for setting a short-term nominal interest
rate that is independent of the evolution of any monetary aggregate. It is true that the
regime described above assumes an exogenously given path for the monetary base Mts . But
it should not therefore be assumed that it is the existence of a “money growth target” that
is responsible for the existence of a determinate price level; for the equilibrium price level
(1.37) is independent of the assumed path of the monetary base.
Indeed, in a cashless economy, a money growth target will not succeed in determining
16
Kerr and King (1996) provide an early discussion of the determinacy of equilibrium under a rule of this
kind, in a log-linear framework similar to the one derived here as an approximation to our exact equilibrium
conditions. Woodford (1998a) demonstrates determinacy of equilibrium under a Wicksellian regime in the
context of a complete intertemporal equilibrium model, along lines similar to those followed here, although
the means by which the central bank is assumed to implement its interest-rate operating target is di¬erent
there.
26 CHAPTER 2. PRICE-LEVEL DETERMINATION

an equilibrium price level ” at least if such a policy is understood to involve a constant
(typically zero) rate of interest on the monetary base. In the case that no interest is paid on
money, the following result implies that no equilibrium price level is possible at all.


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, per-
s
haps the monetary base is non-decreasing over time, starting from an initial level M0 > 0.)
Suppose 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,
t

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


The proof is in the appendix. Results of this kind are often taken (see, e.g., Sargent, 1987,
sec. 4.1) to imply that it is not possible to model the determinants of the exchange value of
money without introducing some kind of frictions that create a demand for money despite
its low rate of return. Hence monetary frictions are thought to be an essential element of
any theory of the e¬ects of monetary policy. But we have seen instead that an equilibrium in
which money exchanges for goods is possible even in a cashless economy, as long as the rate
of interest paid on money is high enough relative to the growth rate of the money supply.

Of course, in actual economies like that of the U.S., interest is not paid on the monetary
base, so that the policy regime in place would seem to be one to which Proposition 2.4
would apply, were the economy cashless. One might then conclude that monetary frictions
are essential to understanding in¬‚ation determination in an economy like that of the U.S.
But in fact, monetary frictions are essential only to understand one aspect of current U.S.
policy ” the fact that it is possible for the Fed to implement its operating targets for the
federal funds rate without paying interest on Fed balances. As we shall see, the details of
how the Fed is able to implement its interest-rate targets are of relatively little importance
for the e¬ects of its interest-rate policy on price-level determination; and a cashless model
may give a good account of the latter question.
1. PRICE-LEVEL DETERMINATION IN A CASHLESS ECONOMY 27

Even when interest is paid on the monetary base at a su¬cient rate to allow a monetary
equilibrium to exist, if the interest rate on money is constant, the equilibrium price level
will be indeterminate, as a consequence of Proposition 2.5 in the next section, whether
or not the monetary base is kept on a precise deterministic growth path. Determinacy of
equilibrium requires a regime under which interest rates respond systematically ” in the
right way, of course ” to variations in the price level. In standard models with monetary
frictions, money-growth targeting (with a constant rate of interest on money) is an example
of a policy with this property: price-level increases automatically result in increases in short-
term nominal interest rates. But in a cashless economy, money-growth targeting has no such
consequence for interest rates, and so fails to determine an equilibrium price level. It is
actually the presence of a systematic relation between prices and interest rates, of the kind
called for Wicksell, that is essential for determinacy of equilibrium, and not control of the
money supply as such.
Proposition 2.3 does not simply establish conditions under which a monetary equilibrium
is possible: it gives a precise account of the factors that determine the equilibrium price level
under such a regime. Equation (1.37) may equivalently be written

•j Et [log Pt+j + φ’1 (ˆt+j ’ νt+j )],

log Pt = pr (1.39)
j=0


where the weights •j ≡ φp (1+φp )’(j+1) are all positive and sum to one. Thus the equilibrium
log price level is equal (up to terms of order O(||ξ||2 )) to a weighted average of the current
and expected future log price level targets, plus a deviation term that is itself a weighted
average of current and expected future disturbances to the equilibrium real rate of interest
and disturbances to the monetary policy rule other than those represented by variations in
the “target” price level.17 The determinants of variations in the equilibrium real interest rate
17
Note that insofar as we regard the target price level as being implicit in the central bank™s reaction
function, rather than an explicit target, the distinction between variations in the target price level and
variations in νt is an arbitrary one; and as we should expect, (1.39) indicates that only the sum log Pt— ’
(φ’1 + 1)νt actually matters, in our log-linear approximation, for determination of the equilibrium price
p
level. But for some purposes, it remains useful to distinguish the two components; for example, we allow
log Pt— to possess a trend, but assume that νt is stationary.
28 CHAPTER 2. PRICE-LEVEL DETERMINATION

are in turn given by (1.33); these depend solely upon exogenous real factors, independent
of monetary policy. Note that the associated evolution of the monetary base {Mts } is not
among the relevant factors.
The theory of price-level determination under such a regime that we obtain has, in fact, a
distinctly Wicksellian ¬‚avor. Equation (1.39) indicates that increases in the equilibrium price
level result either from exogenous increases in the equilibrium real rate of return (Wicksell™s
“natural” rate of interest),18 that are not su¬ciently o¬set by an adjustment of the central
bank™s operating target, or from a loosening of monetary policy (corresponding either to
an increase in Pt— or a decrease in νt ), that is not justi¬ed by any real disturbance. This
emphasis upon the interplay between variations in the equilibrium real rate and the stance of
monetary policy (and speci¬cally upon the gap between the current level of the “natural rate”
of interest and the interest rate controlled by the central bank) as the source of in¬‚ationary
or de¬‚ationary pressures recalls Wicksell™s theory (Wicksell, 1898, 1915).
Of course, our rational-expectations equilibrium version of Wicksellian theory di¬ers in
important ways from the original. For example, our solution (1.39) for the equilibrium
price level is forward-looking, in much the same way as the rational-expectations monetarist
analysis presented in section xx below. It is not simply the current equilibrium real rate of
return that matters, but a weighted average of current and expected future rates, and it is
not simply the current stance of monetary policy that matters, but a weighted average of
the current and expected future shifts in the central bank™s feedback rule. Perhaps more
crucially, our theory does not determine an equilibrium price level as a function of the
path (even including expectations about the future path) of the central bank™s interest-rate
instrument. Rather, the price level depends upon the current and expected future feedback
rules for determination of the interest rate as a function of the evolution of the price level.
(For an exogenously speci¬ed interest-rate process leaves the price level indeterminate, as
shown above.)
18
In the simple analysis here, there is no distinction between the actual and “natural” real rates of interest.
This is introduced in chapter 4, where we ¬nd that it continues to be possible to understand price-level
determination along similar lines.
2. ALTERNATIVE INTEREST-RATE RULES 29

Nonetheless, the implications of this theory for the conduct of monetary policy with a
view to price stability are reminiscent of Wicksell™s prescriptions. First of all, supposing that
the target price level Pt— is constant,19 and letting φp > 0 be ¬xed, then monetary policy
achieves the constant price level Pt = P — if and only if νt = rt at all times. Failure of policy
ˆ
to track such variations in the “natural rate” with su¬cient accuracy was, in Wicksell™s
account, the primary explanation for price-level instability.20
Second, our results imply that for any given degree that the shift factor νt fails to track
the exogenous variation in rt , the price-level instability that results can be reduced by a
ˆ
sharper automatic positive response of the central bank™s operating target to price-level
increases. The e¬ects upon log Pt of variations in the gap rt ’ νt are smaller the larger is φp ,
ˆ
and in fact they can be made arbitrarily small, in principle, by choosing φp large enough.21
Thus, a positive automatic response to price-level deviations from target is desirable, not
only because it is necessary for determinacy (this would be achieved by even a very small
φp > 0), but because it reduces the degree to which accurate direct observation of the current
equilibrium real rate is necessary in order for price-level variability to be kept at a given level.
This too is a theme in Wicksell™s discussions of desirable policy.


2 Alternative Interest-Rate Rules

While most central banks organize their monetary-policy deliberations around the choice of
an operating target for a short-term nominal interest rate, and pay a great deal of attention
19
Similar conclusions are reached in the case of a target price level that grows at a constant rate; one must
simply add a constant to the value of νt required each period to achieve the target.
20
See, e.g., Wicksell (1915, sec. IV.9). He found evidence for this in the often-remarked tendency of the
price level to covary positively with the level of nominal interest rates during the period of the classical gold
standard (the so-called “Gibson paradox”). Note that if νt = 0, (1.39) implies that log Pt should vary with
exogenous variations in the real rate, while (1.34) then implies that the endogenous variations in the short
nominal rate should perfectly coincide with these ¬‚uctuations in the price level. A similar conclusion is
obtained if νt covaries positively, but less than one-for-one, with the variations in rt , so that the in¬‚ationary
and de¬‚ationary impulses are not completely eliminated.
21
It is unlikely to be desirable, however, to seek to completely eliminate price-level variation without any
need to directly respond to variations in the equilibrium real rate, by choosing a rule with an extremely large
elasticity φp . For in this case, errors in the measurement of the price level by the central bank would have
very large e¬ects upon policy, and hence upon the economy, as discussed in Bernanke and Woodford (1997).
30 CHAPTER 2. PRICE-LEVEL DETERMINATION

to in¬‚ation measures in those deliberations, it would be hard to argue that the Wicksellian
rule (1.30) represents even a rough description of the current behavior of any central banks.
(In particular, central banks clearly accept “base drift” in the price level, rather than seeking
to stabilize a price index around an exogenously given target path.) Nor shall we argue that
this is actually an optimal rule, though such a rule can be shown to have desirable properties,
relative to other rules of equal simplicity (Giannoni, 2000). It is therefore useful to extend
our analysis to other types of interest-rate rules. While a complete treatment of the topic
would be beyond the scope of this study, we here take up a few additional cases suggested
by the discussion of empirical central-bank reaction functions in chapter 1.


2.1 Exogenous Interest-Rate Targets

The simplest sort of interest-rate rule, of course, would be one that involves no feedback from
any endogenous variables. We might instead suppose that the central bank™s interest-rate
operating target is given by some exogenous stochastic process {¯t }. This need not imply that
±
the sole objective of policy is interest-rate stabilization (or maintenance of “easy money”). A
central bank concerned with price stability, and believing in the theoretical model of section
1, might reason that a rational-expectations equilibrium with constant prices is possible only
if
’1
uc (Yt+1 ; ξt+1 )
’1
1 + it = 1 + rt ≡ β Et , (2.40)
uc (Yt ; ξt )
as a consequence of (1.21). Supposing that it is possible for the central bank to measure the
exogenous variation in the right-hand side of this equation in time to use this information in
the conduct of policy, would the bank achieve its objective by committing itself to a policy
of always adopting the current value of rt as its operating target? (Such a proposal might
appear to be suggested by a Wicksellian analysis of the determinants of in¬‚ation, although,
as noted in chapter 1 and in the previous section, this is the not the kind of rule actually
proposed by Wicksell.)
While such a policy would be consistent with the desired rational-expectations equi-
librium, it would also be equally consistent with an extremely large class of alternative
2. ALTERNATIVE INTEREST-RATE RULES 31

rational-expectations equilibria, in most of which prices vary randomly. This is true even if,
as above, we restrict attention to alternative equilibria that remain forever near the reference
equilibrium, i.e., the steady state with zero in¬‚ation. In fact, this is a consequence of any
policy commitment that makes the interest-rate operating target purely a function of the
economy™s exogenous state (i.e., the history of disturbances alone), regardless of how sensibly
the exogenous sequence of interest-rate targets may have been chosen.


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 equal to the interest-rate target each period. Let {Mts , Dt } be exogenous sequences of
t

the kind assumed in Proposition 2.3. Finally, let P be any neighborhoods of the real num-
ber 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.


In this case, any process {Pt } that satis¬es both (1.21) and (1.24), given the exogenous
processes {Yt , ξt , im , Mts , Dt }, and with the exogenous target im substituted for it in (1.21),
t t

represents a rational-expectations equilibrium. For any tight enough bounds on the exoge-
nous processes and on the neighborhood P, (1.24) is necessarily satis¬ed, so our question
reduces to an analysis of the local uniqueness of solutions to (1.21) for a given interest-rate
process. As in the previous section, this can be addressed through a consideration of the
uniqueness of bounded solutions to the log-linearized equilibrium condition (1.32). This now
takes the form

¯t = rt + Et πt+1 ,
± ˆ (2.41)
32 CHAPTER 2. PRICE-LEVEL DETERMINATION

where ¯t indicates the exogenous ¬‚uctuations in the interest-rate target and rt the exogenous
± ˆ
¬‚uctuations in rt owing to random “fundamentals”, again given by 1.33). This equation
obviously has a unique solution for Et πt+1 , and if the exogenous terms are bounded, the
implied ¬‚uctuations in the expected in¬‚ation rate will be bounded as well. But this equation
does not have a unique bounded solution for the stochastic process {πt }, for absolutely any
pattern of bounded ¬‚uctuations in the unexpected component of the in¬‚ation rate will be
consistent with it.
Writing this explicitly, we observe that

πt = ¯t’1 ’ rt’1 + νt
± ˆ (2.42)

is a bounded solution to (2.41), where {νt } represents any mean-zero bounded process that
is completely unforecastable a period in advance, i.e., that satis¬es

Et νt+1 = 0

at all dates. It then follows from the discussion in section xx of the appendix that we similarly
have an uncountably in¬nite set of bounded solutions to the exact equilibrium condition.
(1.21).2223
Here the random variable νt may be correlated in an arbitrary way with unforecastable
±ˆ
variations in “fundamental” variables such as ¯t , Yt , and ξt ; but it may also be completely un-
related to economic fundamentals. Thus even when we restrict attention to nearby solutions,
the rational expectations equilibrium price level is quite indeterminate under such a regime.
Note furthermore that even though we consider only alternative solutions in which in¬‚a-
tion is always within a certain neighborhood of zero, this set of solutions includes alternative
22
There is actually no need for linearization to establish this, as shown in the appendix. Here we use the
method of linear approximation in the analysis of determinacy to illustrate the method, that is used again
later in more complicated cases.
23
It is important to note that this conclusion depends on a particular assumption about the character of
¬scal policy. If ¬scal policy were not assumed, as it has been here, to be “locally Ricardian,” then there
might be a locally, or even globally unique equilibrium in the case of an exogenous path for the interest-rate
operating target. This occurs, for example, in the analysis of price-level determination under a bond-price
support regime in Woodford (2001xx). See further discussion in section xx of chapter 4.
2. ALTERNATIVE INTEREST-RATE RULES 33

paths for the price level that wander arbitrarily far from one another, once su¬cient time has
passed. This is the basis for the conclusion of Sargent and Wallace (1975) that interest-rate
rules are ¬‚awed as a general approach to monetary policy, and that policy should instead be
formulated in terms of monetary targets.24

One might ask, is this sort of price-level indeterminacy really a problem? It will be
observed in our discussion here that real quantities are una¬ected by the indeterminacy of
the price level, and the same conclusion is true even in the case of the model with monetary
frictions considered below in section xx, and even in a more elaborate model, in which
output is endogenous and may depend upon the level of real money balances. Thus no
variables that actually a¬ect household utility are a¬ected. However, the indeterminacy is
plainly undesirable if price stability is a concern, as Sargent and Wallace assume in their
analysis of optimal monetary policy. Indeed, since the class of bounded solutions includes
solutions in which the unexpected ¬‚uctuations in in¬‚ation are arbitrarily large, at least
some of the equilibria consistent with the interest-rate targeting policy are worse (assuming
a loss function that penalizes squared deviations of in¬‚ation from target, say) than the
equilibrium associated with any policy that makes equilibrium determinate. Furthermore,
similar conclusions are shown below (in chapter 4) to hold in the case of a model with
nominal price rigidity, in which case the self-ful¬lling expectations also a¬ect real variables,
that matter for household utility. Thus, if one evaluates policy rules according to how bad
is the worst outcome that they might allow, it would be appropriate to assign an absolute
priority to the selection of a rule that would guarantee determinacy of equilibrium.

This argument might seem inconsistent with our use above of a purely local analysis of
determinacy. One response to such a concern would be to refer to the exact analysis in
section xx of the appendix, and show that there are indeed solutions involving arbitrarily
large unexpected changes in the log price level. But in fact the local analysis is also valid,
when correctly interpreted. Let us ¬x neighborhoods of the steady-state values of πt , it ,

24
See Walsh (1998, sec. 10.2.1) for an exposition of their analysis in the context of an IS-LM model with
rational expectations that is closer to the structure of the model actually used in the original paper.
34 CHAPTER 2. PRICE-LEVEL DETERMINATION

and so on, that are small enough that the approximation error in the log-linearized relation
(1.32) is of an acceptable size, for all paths remaining within these neighborhoods; we shall
restrict attention to solutions of this kind. The analysis of the log-linearized equations
shows that in the case of the exogenous interest-rate target, there exist solutions in which
the in¬‚ation rate ¬‚uctuates over the entire admissible neighborhood, no matter how small the
¬‚uctuations in the exogenous disturbances may be. Now let us compare such a policy to one
that results in a determinate equilibrium (and hence a solution in which πt and the other
endogenous variables are linear functions of the exogenous disturbances, with coe¬cients
that are independent of the assumed shock variances). We observe that by making the
exogenous disturbances small enough, we obtain a case in which the in¬‚ation variability
in at least certain equilibria associated with interest-rate targeting is much greater than in
the locally unique equilibrium associated with the other policy. Thus, at least in the case
of small enough exogenous disturbances, the conclusion reached from the analysis of the
log-linearized equations is correct.

Whether one should only care about the worst possible equilibrium might be doubted, if
a particular policy also allows very desirable equilibria, that are better than those associated
with any other policies. But in fact, this is unlikely to be a serious problem, once the class
of policies that are considered is su¬ciently broad; for it is often possible to achieve any
desired equilibrium through a policy rule that makes equilibrium determinate, in addition
to its being consistent with rules that would make equilibrium indeterminate. There are
typically many policy rules consistent with the desired equilibrium; these coincide in what
they prescribe should occur in the desired equilibrium, but di¬er in how policy is speci¬ed
“o¬ the equilibrium path”, and thus may di¬er as to whether they exclude other nearby
equilibria. In such a case, it seems reasonable to accept as a principle of policy design
that one should choose one of the rules that makes the desired equilibrium at least locally
determinate, if not globally unique. We shall take this perspective in the current study; see
chapter 8.

Fortunately, interest-rate rules as such need not imply indeterminacy of equilibrium, as
2. ALTERNATIVE INTEREST-RATE RULES 35

McCallum (1981) ¬rst noted, in the context of the model of Sargent and Wallace. A rule
that involves a commitment to feedback from endogenous state variables such as the price
level to the level of nominal interest rates can result in a determinate equilibrium, as our
analysis of Wicksellian rules in section 1 has shown. We now turn to additional examples
that are better descriptions of current policies.


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