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w o r k i n g
p
a
p
e
r
00 12
Implementing the Friedman Rule
by Peter N. Ireland
FEDERAL RESERVE BANK
OF CLEVELAND
Working Paper 00-12
Implementing the Friedman Rule
by Peter N. Ireland
Peter N. Ireland is at Boston College, Chestnut Hill, MA, and is a visiting research
associate at the Federal Reserve Bank of Cleveland.
Working papers of the Federal Reserve Bank of Cleveland are preliminary materials
circulated to stimulate discussion and critical comment on research in progress. They
may not have been subject to the formal editorial review accorded official Federal Reserve
Bank of Cleveland publications. The views stated herein are those of the authors and are
not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of
Governors of the Federal Reserve System
Working papers are now available electronically through the Cleveland Fed’s home page
on the World Wide Web: http://www.clev.frb.org.
August 2000
Implementing the Friedman Rule
by Peter N. Ireland
In cash-in-advance models, necessary and sufficient conditions for the existence of an equilibrium
with zero nominal interest rates and Pareto optimal allocations place restrictions only on the very
long-run, or asymptotic, behavior of the money supply. When these asymptotic conditions are
satisfied, they leave the central bank with a great deal of flexibility to manage the money supply
over any finite horizon. But what happens when these asymptotic conditions fail to hold? This
paper shows that the central bank can still implement the Friedman rule if its actions are
appropriately constrained in the short run.
JEL: E52
Implementing the Friedman Rule
Peter N. Ireland∗
Boston College and NBER
August 2000
Abstract
In cash-in-advance models, necessary and sufficient conditions for the existence of
an equilibrium with zero nominal interest rates and Pareto optimal allocations place
restrictions only on the very long-run, or asymptotic, behavior of the money supply.
When these asymptotic conditions are satisfied, they leave the central bank with a
great deal of flexibility to manage the money supply over any finite horizon. But what
happens when these asymptotic conditions fail to hold? This paper shows that the
central bank can still implement the Friedman rule if its actions are appropriately
constrained in the short run.
JEL: E52.
∗
Please address correspondence to: Peter N. Ireland, Boston College, Department of Economics, 140
Commonwealth Avenue, Chestnut Hill, MA 02467-3806. Tel: (617) 552-3687. Fax: (617) 552-2308. Email:
irelandp@bc.edu. Some of this work was completed while I was visiting the Research Department at the
Federal Reserve Bank of Cleveland; I would like to thank the Bank and its staff for their hospitality and
support. The opinions expressed herein are my own and not those of the Federal Reserve Bank of Cleveland
or the Federal Reserve System.
1
Introduction
Milton Friedman (1969) presents his famous rule for optimal monetary policymaking. ”Our
final rule for the optimum quantity of money,” he writes (p.34), ”is that it will be attained
by a rate of price deflation that makes the nominal rate of interest equal to zero.” Friedman
also suggests that this rule can be implemented by steadily contracting the money supply at
the representative household’s rate of time preference.
Wilson (1979) and Cole and Kocherlakota (1998) assess Friedman’s proposals using fullyspecified, general equilibrium models in which money is introduced through the imposition
of a cash-in-advance constraint. These authors confirm the relevance of the Friedman rule
by demonstrating that equilibrium allocations are efficient if and only if the nominal interest
rate equals zero. But they also find that the Friedman rule can be implemented through any
one from a broad class of monetary policies. Some of these policies call for the money supply
to expand over an arbitrarily long, but finite, horizon; others call for the money supply to
contract, but at a rate that is always slower than the representative household’s rate of time
preference. In fact, Wilson and Cole and Kocherlakota show that necessary and sufficient
conditions for the existence of an equilibrium with zero nominal interest rates and Pareto
optimal allocations place restrictions only on the asymptotic behavior of the money supply:
these restrictions simply require the money supply to eventually contract at a rate that is
no faster than the representative household’s rate of time preference.
For central bankers who wish to implement the Friedman rule, these asymptotic conditions are a double-edged sword. For when the conditions are satisfied, they leave the policymaker with considerable leeway in managing the money supply over any finite horizon.
1
But what should the central banker do when, for some reason, these asymptotic conditions
fail to hold? Must the Friedman rule be abandoned altogether? Or is there still a way to
manage the money supply so that nominal interest rates are zero and equilibrium allocations
are efficient, at least in the short run?
To answer these questions, section 2 sets up a cash-in-advance model like those used
by Wilson (1979) and Cole and Kocherlakota (1998) and, for the sake of completeness,
restates the asymptotic conditions that are both necessary and sufficient for implementing
the Friedman rule over the infinite horizon. Section 3 then assumes that these asymptotic
conditions do not hold and characterizes optimal monetary policies in this alternative case.
Section 4 concludes by reinterpreting and extending the results of section 3 using a version
of the model in which private agents are boundedly rational in a very special way.
2
A Cash-in-Advance Model
An infinitely-lived representative household is endowed with one unit of productive time
during each period t = 0, 1, 2, .... Its preferences are described by
∞
X
β t u(ct , 1 − nt ),
t=0
where ct denotes its consumption and 1 − nt its leisure during period t. The discount
factor satisfies 1 > β > 0. The single-period utility function u is strictly increasing in both
arguments, strictly concave, and twice continuously differentiable. Let ui and uij , i, j = 1, 2,
denote the first and second derivatives of u, and for y ∈ (0, 1), define
V (y) =
u1 (y, 1 − y)
.
u2 (y, 1 − y)
2
It will be useful in what follows to assume that V is strictly decreasing with limy→0 V (y) = ∞
and limy→1 V (y) = 0. Since u is strictly increasing and concave, a sufficient condition for
V 0 (y) < 0 is u12 ≥ 0.
The household enters period t with money Mt and bonds Bt . The goods market opens
first; here, the description of production and trade draws on Lucas’ (1980) interpretation
of the cash-in-advance model. Suppose that the representative household consists of two
members: a shopper and a worker. The shopper purchases consumption from workers from
other households at the nominal price Pt , subject to the cash-in-advance constraint
Mt
≥ ct .
Pt
Meanwhile, the worker produces output according to the linear technology yt = nt and sells
this output to shoppers from other households for Pt nt units of money.
The asset market opens last. In this end-of-period asset market, the representative
household receives a lump-sum nominal transfer Ht from the central bank and the household’s
bonds mature, providing Bt additional units of money. The household spends Bt+1 /(1 + rt )
on new bonds, where rt is the net nominal interest rate, and carries Mt+1 units of money
into period t + 1. The household’s budget constraint is therefore
Mt + Ht + Bt
Bt+1 /(1 + rt ) + Mt+1
+ nt ≥ ct +
.
Pt
Pt
In addition to the cash-in-advance and budget constraints, the household’s choices must
satisfy the nonnegativity constraints
ct ≥ 0, 1 ≥ nt ≥ 0, Mt+1 ≥ 0.
3
And while the household is allowed to borrow by choosing negative values of Bt+1 , it is not
permitted to borrow more than it can ever repay. Let Qt denote the present discounted value
in the period-0 asset market of one unit of money received in the period-t asset market, so
that Q0 = 1 and
Qt =
t−1
Y
s=0
1
1 + rs
for t = 1, 2, 3, .... Then the no-Ponzi-game constraint can be formalized as
Wt+1 = Qt
Bt+1
Mt+1 +
1 + rt
∞
X
+
Qs (Hs + Ps ns ) ≥ 0.
s=t+1
Thus, the representative household chooses {ct , nt , Mt+1 , Bt+1 }∞
t=0 to maximize its utility
subject to the cash-in-advance constraint, the budget constraint, the nonnegativity constraints, and the no-Ponzi-game constraint, each of which must hold for all t = 0, 1, 2, ....
The appendix shows that when the market-clearing conditions
yt = ct = nt , Mt+1 = Mt + Ht , Bt+1 = 0
are imposed, necessary and sufficient conditions for a solution to the household’s problem
can be written as
u1 (yt , 1 − yt ) = λt + µt ,
(1)
u2 (yt , 1 − yt ) = λt ,
(2)
λt
β(λt+1 + µt+1 )
=
,
Pt
Pt+1
(3)
λt
βλt+1
=
,
(1 + rt )Pt
Pt+1
(4)
and
Mt
µt ≥ 0,
≥ y t , µt
Pt
4
Mt
− yt
Pt
=0
(5)
for all t = 0, 1, 2, ... and
β t λt Mt+1
= 0,
t→∞
Pt
lim
(6)
where λt and µt are Lagrange multipliers on the period-t budget and cash-in-advance constraints. Accordingly, an equilibrium can be defined as a set of sequences {yt , λt , µt , rt , Pt , Mt+1 }∞
t=0
that satisfy (1)-(6), with the initial condition M0 pinned down by a choice of nominal units.
Under the maintained assumptions on the household’s utility function, there is a unique
symmetric Pareto optimal allocation for this economy. This allocation has yt = y ∗ for all
t = 0, 1, 2, ..., where y ∗ is the unique value that satisfies the efficiency condition V (y ∗ ) = 1:
the marginal rate of substitution between leisure and consumption equals the corresponding
marginal rate of transformation. What monetary policies, defined as sequences {Mt+1 }∞
t=0 ,
allow for the existence of an equilibrium in which allocations are Pareto optimal? To answer this question, Wilson (1979) and Cole and Kocherlakota (1998) obtain results like the
following.
Proposition 1 An equilibrium with yt = y ∗ for all t = 0, 1, 2, ... exists if and only if
inf β −t Mt > 0
(7)
lim Mt+1 = 0.
(8)
t
and
t→∞
Proof To begin, suppose that (7) and (8) are satisfied, and set yt = y ∗ , λt = u1 (y ∗ , 1 − y ∗ ) =
u2 (y ∗ , 1 − y ∗ ), µt = 0, rt = 0, and Pt = β t P0 for all t = 0, 1, 2, ..., where P0 > 0 is
chosen below. Clearly, these values satisfy (1)-(4). Since µt = 0, (5) requires that
β −t Mt ≥ P0 y ∗
5
for all t = 0, 1, 2, .... But (7) guarantees the existence of an ε > 0 such that β −t Mt ≥ ε
for all t = 0, 1, 2, ..., and thereby allows this last condition to be satisfied for any choice
of P0 ≤ ε/y ∗ . Meanwhile, (8) guarantees that (6) will hold. Thus, (7) and (8) are
sufficient conditions for the existence of an optimal equilibrium.
Next, suppose that an equilibrium with yt = y ∗ for all t = 0, 1, 2, ... exists. By (1)-(4),
λt = u1 (y ∗ , 1 − y ∗ ) = u2 (y ∗ , 1 − y ∗ ), µt = 0, rt = 0, and Pt = β t P0 > 0 for all
t = 0, 1, 2, ... in any such equilibrium. Thus, (5) requires that
β −t Mt ≥ P0 y ∗ > 0
for all t = 0, 1, 2, ..., which implies that (7) must be satisfied. Meanwhile (6) implies
that (8) must hold. This establishes that (7) and (8) are also necessary conditions for
the existence of an optimal equilibrium, completing the proof.
Proposition 1 and its proof support Friedman’s (1969) assertion that Pareto optimal
allocations are associated with price deflation and zero nominal interest rates. Friedman also
suggests that his zero-nominal-interest-rate rule can be implemented by steadily contracting
the money supply at the representative household’s rate of time preference and, indeed, the
policy that sets Mt = β t M0 for all t = 0, 1, 2, ... satisfies both (7) and (8). As emphasized by
Wilson (1979) and Cole and Kocherlakota (1998), however, many other monetary policies
also satisfy (7) and (8), including ones that call for positive rates of money growth over
arbitrarily long, but finite, horizons, and ones that set Mt = γ t M0 , with 1 > γ ≥ β, for all
t = 0, 1, 2, ....
In fact, (7) and (8) impose restrictions only on the very long-run behavior of the money
6
supply. Condition (7) places a lower bound on the asymptotic money growth rate: since
the gross inflation rate equals β under the Friedman rule, the money stock must eventually
grow at a rate that is at least as large as β if the cash-in-advance constraint is not to
become binding. Condition (8) places an upper bound on the asymptotic money growth
rate: evidently, the money supply must eventually contract if the nominal interest rate is
to remain at zero. Together, therefore, (7) and (8) simply require the money supply to
asymptotically contract at a rate no faster than the household’s rate of time preference.
3
Implementing the Friedman Rule in the Short Run
When (7) and (8) hold, they leave the central bank with a great deal of flexibility; in fact,
they allow the central bank to choose any time path for the money supply over any finite
horizon while still implementing the Friedman rule. But what is a central banker to do when
(7) or (8) fails to hold?
When (7) fails to hold, the money supply contracts asymptotically at a rate that exceeds
the representative household’s rate of time preference. A second result, also adapted from
Cole and Kocherlakota (1998), is useful in considering this case.
Proposition 2 Let the single-period utility function take the form
u(c, 1 − n) = ln(c) + v(1 − n),
where v is strictly increasing, strictly concave, and twice continuously differentiable
with limn→1 v 0 (1 − n) = ∞. If Mt+1 /Mt < β for all t = 0, 1, 2, ..., then no equilibrium
exists.
7
Proof When Mt+1 /Mt < β for all t = 0, 1, 2, ..., (7) fails to hold. Hence, by proposition
1, there is no equilibrium with yt = y ∗ for all t = 0, 1, 2, .... So suppose there is an
equilibrium with yt = ỹ 6= y ∗ for some t. Equations (1), (2), and (5) then imply
1
= λt + µt ≥ λt = v 0 (1 − ỹ),
ỹ
a condition that, along with the concavity of v, rules out the possibility that ỹ > y ∗ .
Hence, any such equilibrium must have yt = ỹ < y ∗ and µt > 0. But in this case, (5)
also implies that Mt = Pt ỹ. Using this result, together with (1), (3), and (5) again,
1
β
β
1
=
≥
≥
Mt
Pt ỹ
Pt+1 yt+1
Mt+1
or, more simply, Mt+1 /Mt ≥ β. But this contradicts the original assumption that
Mt+1 /Mt < β for all t = 0, 1, 2, ...; evidently, no equilibrium exists.
Proposition 2 suggests that when (7) fails to hold, the problem has to do with the
possible nonexistence of an equilibrium, rather than merely the suboptimality of equilibrium
allocations. What happens when a central bank adopts a policy that is inconsistent with
the existence of an equilibrium? Exploring the subtleties of this issue is left for future
research; instead, the remainder of this paper will focus on the case in which the conditions
of proposition 1 are violated because (8) fails to hold.
Suppose, for example, that a central banker is appointed at the beginning of period 0 and
−1
granted the authority to choose {Ht }Tt=0
, the monetary transfers for the first T periods. With
−1
the initial condition M0 taken as given, this central banker’s control over {Ht }Tt=0
provides
−1
him or her with control over {Mt+1 }Tt=0
, the time path for the money supply through the
beginning of period T .
8
This central banker’s term lasts only T periods, however: during period T , a new central
banker takes over and arbitrarily decides that the money supply will grow at the constant
gross rate π ≥ 1, so that MT +j = π j MT for all j = 0, 1, 2, .... Under the maintained
assumptions on the household’s utility function, there is a unique steady-state equilibrium
under the constant money growth rate π, in which output yt and real balances mt = Mt /Pt
are both constant and equal to ȳ, where ȳ < y ∗ is the unique value that satisfies V (ȳ) = π/β.
So suppose in addition that, independent of the first central banker’s decisions, yT +j =
mT +j = ȳ for all j = 0, 1, 2, ....
The assumption that π ≥ 1 implies that (8) will not hold when the first central banker
takes office at the beginning of period 0. The question now becomes: can this first central
T −1
banker, through an appropriate choice of {Mt+1 }t=0
, nevertheless guarantee the existence of
an equilibrium in which nominal interest rates are zero and allocations are efficient, at least
in the short run?
As a first step in answering this question, note that with MT +j = π j MT and yT +j =
mT +j = ȳ for all j = 0, 1, 2, ..., (1)-(5) are satisfied for all t = T, T +1, T +2, ... and (6) is satis−1
fied as well. Hence, the values of concern to the first central banker, {yt , λt , µt , rt , Pt , Mt+1 }Tt=0
,
need only satisfy (1), (2), and (5) for all t = 0, 1, ..., T −1, (3) and (4) for all t = 0, 1, ..., T −2,
βu1 (ȳ, 1 − ȳ)
λT −1
=
,
PT −1
MT /ȳ
(9)
λT −1
βu2 (ȳ, 1 − ȳ)
=
,
(1 + rT −1 )PT −1
MT /ȳ
(10)
and
where these last two conditions correspond to (3) and (4) for t = T − 1 and make use of the
fact that in the inefficient steady state, λT = u2 (ȳ, 1 − ȳ), µT = u1 (ȳ, 1 − ȳ) − u2 (ȳ, 1 − ȳ) > 0,
9
and mT = MT /PT = ȳ. These observations are useful in establishing the following.
Proposition 3 Suppose that MT +j = π j MT , with π ≥ 1, for all j = 0, 1, 2, ... and that from
period T forward, the economy is in its unique steady state, with yT +j = mT +j = ȳ for
all j = 0, 1, 2, .... Then an equilibrium with yt = y ∗ for all t = 0, 1, ..., T − 1 exists if
and only if
MT > 0
(11)
and
Mt ≥ β
t
u1 (y ∗ , 1 − y ∗ )y ∗
MT
β T u1 (ȳ, 1 − ȳ)ȳ
(12)
for all t = 0, 1, ..., T − 1.
Proof To begin, suppose that (11) and (12) are satisfied, and set yt = y ∗ , λt = u1 (y ∗ , 1 −
y ∗ ) = u2 (y ∗ , 1 − y ∗ ), µt = 0, and
Pt = β
t
u1 (y ∗ , 1 − y ∗ )
MT
β T u1 (ȳ, 1 − ȳ)ȳ
for all t = 0, 1, ..., T − 1. In addition, set rt = 0 for all t = 0, 1, ..., T − 2, and set
rT −1 = V (ȳ) − 1. Equation (11) guarantees that Pt > 0 for all t = 0, 1, ..., T − 1,
as required for the existence of this equilibrium. Clearly, (1) and (2) hold for all
t = 0, 1, ..., T − 1 and, since Pt+1 = βPt , (3) and (4) hold for all t = 0, 1, ..., T − 2.
Equations (9) and (10) hold as well. Since µt = 0, (5) requires that
Mt ≥ β
t
u1 (y ∗ , 1 − y ∗ )y ∗
MT
β T u1 (ȳ, 1 − ȳ)ȳ
for all t = 0, 1, ..., T − 1, but this last condition coincides with (12) and is therefore
guaranteed to hold. Thus, (11) and (12) are sufficient conditions for the existence of
an equilibrium with yt = y ∗ for all t = 0, 1, ..., T − 1.
10
Next, suppose that an equilibrium with yt = y ∗ for all t = 0, 1, ..., T − 1 exists. By (1)-(3)
and (9), λt = u1 (y ∗ , 1 − y ∗ ) = u2 (y ∗ , 1 − y ∗ ), µt = 0, and
Pt = β
t
u1 (y ∗ , 1 − y ∗ )
MT > 0
β T u1 (ȳ, 1 − ȳ)ȳ
for all t = 0, 1, ..., T − 1 in any such equilibrium; this last condition implies that (11)
must hold. In addition, (5) requires that
Mt ≥ β
t
u1 (y ∗ , 1 − y ∗ )y ∗
MT
β T u1 (ȳ, 1 − ȳ)ȳ
for all t = 0, 1, ..., T − 1, but this condition simply says that (12) must hold. This
establishes that (11) and (12) are also necessary conditions for the existence of an
equilibrium with yt = y ∗ for all t = 0, 1, ..., T − 1, completing the proof.
Before going on to interpret conditions (11) and (12), it is useful to note that proposition
3 holds much more generally. In particular, the assumption that the economy is in a steady
state from period T forward is not essential. All that is required is that the monetary policy
adopted from period T forward give rise to an equilibrium in which the cash-in-advance
constraint binds in period T , so that MT /PT = yT for some yT < y ∗ . In the more general
case, the proof goes through unchanged, with yT in place of ȳ. As stated, however, the
proposition makes clear that optimal allocations can be achieved in periods t = 0, 1, ..., T − 1
even when the rate of money growth is positive for all t = T, T + 1, T + 2, ..., even when the
cash-in-advance constraint binds for all t = T, T + 1, T + 2, ..., and even when allocations are
suboptimal for all t = T, T + 1, T + 2, ....
Proposition 3 implies that the Friedman rule need not be abandoned when (8) fails to
−1
hold: the central bank can still select {Mt+1 }Tt=0
in a way that guarantees the existence of an
11
equilibrium in which the nominal interest rate is zero for all t = 0, 1, ..., T − 2 and allocations
are efficient for all t = 0, 1, ..., T − 1. Condition (11) simply insures that money is always in
positive supply, given that (12) must hold for all t = 0, 1, ..., T − 1 and that Mt+1 = πMt for
all t = T, T + 1, T + 2, .... Condition (12), in turn, places upper and lower bounds on the
growth rate of the money supply and thereby provides finite-horizon analogs to (7) and (8).
Consider (12) for t = 0. Since the initial condition M0 is given, this constraint places an
upper bound on MT :
β T u1 (ȳ, 1 − ȳ)ȳ
M0 ≥ MT .
u1 (y ∗ , 1 − y ∗ )y ∗
(13)
Thus, like (8), (12) implies that money growth must be sufficiently slow if the nominal
interest rate is to remain at zero. Given a choice of MT that satisfies (13), (12) also places
lower bounds on Mt , t = 1, 2, ..., T − 1. Thus, like (7), (12) implies that money growth must
be sufficiently large if the cash-in-advance constraint is not to become binding.
Conditions (11) and (12) still leave the central bank with a great deal of flexibility in
choosing its policy: the money supply can expand for the first T − 1 periods, for instance,
so long as it eventually contracts so that (13) holds. Unlike (7) and (8), however, (11) and
(12) do constrain the money supply over a finite horizon. Thus, proposition 3 implies that
the central bank must act in the short run in order to implement the Friedman rule in the
short run.
4
Interpretation as Limited Forecast Equilibria
Extending the example from the previous section, suppose that the central banker at period
−1
0 announces a policy {Mt+1 }Tt=0
that satisfies the conditions of proposition 3 and thereby
12
succeeds in giving rise to an equilibrium with r0 = 0 and y0 = y ∗ . But now, suppose that
at the beginning of period 1, the representative household discovers that this first central
banker will also be permitted to choose MT +1 and thereby delay by one period the economy’s
convergence to its inefficient steady state. And suppose further that this scenario repeats
itself over the infinite horizon: at the beginning of each period t = 0, 1, 2, ..., the household
believes that the money supply {Mt+j }Tj=1 over the next T periods will be chosen by the
benevolent central banker and that the money supply will grow at the constant gross rate
π ≥ 1 thereafter. In this case, the representative household behaves like the boundedly
rational players in Jehiel’s (1998) game-theoretic framework, having perfect foresight over
the first T periods but having what might be considered vague, and in this case incorrect,
beliefs about what will happen beyond this limited forecast horizon.
The logic used to prove proposition 3 now implies that the central bank can guarantee
the existence of an equilibrium with rt = 0 and yt = y ∗ for all t = 0, 1, 2, ... by choosing
−1
{Mt+1 }Tt=0
at the beginning of period 0 to satisfy (11) and (12) and by choosing Mt+T > 0
at the beginning of each period t = 1, 2, 3, ... to satisfy the appropriately generalized version
of (12):
Mt+j ≥ β
j
u1 (y ∗ , 1 − y ∗ )y ∗
Mt+T
β T u1 (ȳ, 1 − ȳ)ȳ
(14)
for all j = 0, 1, ..., T − 1.
Thus, when the representative household is boundedly rational in this very special way,
the Friedman rule can be implemented over the infinite horizon by any policy {Mt+1 }∞
t=0
that satisfies Mt+1 > 0 and (14) for all t = 0, 1, 2, .... Like (12), but unlike (7) and (8), (14)
places constraints on the short-run behavior of the money supply. Once again, therefore, the
13
central bank must act in the short run to implement the Friedman rule.
It should be noted, however, that the boundedly rational household in this example does
not learn: it continues to believe that the economy will eventually move to its inefficient
steady state, even though this steady state is never reached. What monetary policies will
implement the Friedman rule in environments where private agents’ expectations gradually
change in response to the observed actions of the central bank? This is another question for
future research.
5
Appendix
This appendix shows how (1)-(6) in the text can be derived from conditions that are both
necessary and sufficient for a solution to the representative household’s optimization problem. Since the household’s utility function is strictly concave, the necessary conditions for
optimality include the usual first-order conditions, which are given by
u1 (ct , 1 − nt ) = λt + µt ,
(A.1)
u2 (ct , 1 − nt ) = λt ,
(A.2)
λt
β(λt+1 + µt+1 )
=
,
Pt
Pt+1
(A.3)
λt
βλt+1
=
,
(1 + rt )Pt
Pt+1
(A.4)
Mt + Ht + Bt
Bt+1 /(1 + rt ) + Mt+1
+ nt = ct +
,
Pt
Pt
(A.5)
and
Mt
µt ≥ 0,
≥ c t , µt
Pt
14
Mt
− yt
Pt
=0
(A.6)
for all t = 0, 1, 2, ..., where λt and µt are Lagrange multipliers on the period-t budget and
cash-in-advance constraints.
The necessary conditions also include the transversality condition
lim Wt+1 = lim Qt
t→∞
t→∞
Bt+1
Mt+1 +
1 + rt
= 0.
(A.7)
To derive (A.7), note first that the sequence {Wt+1 }∞
t=0 is nonincreasing since, using the
period-t budget constraint,
Wt+1 − Wt = Qt
Bt+1
Mt+1 +
1 + rt
− Qt−1
Bt
Mt +
1 + rt−1
− Qt (Ht + Pt nt )
≤ −Qt Pt ct − (Qt−1 − Qt )Mt .
In any equilibrium, Pt > 0, ct ≥ 0, Mt ≥ 0, and Qt−1 ≥ Qt > 0 must hold; it therefore
follows from the expression above that Wt+1 ≤ Wt must also hold.
Next, note that if {ct , nt , Mt+1 , Bt+1 }∞
t=0 solve the household’s problem, the implied sequence {Wt+1 }∞
t=0 must satisfy inf t Wt+1 = 0. To see this, suppose to the contrary that there
exists an ε > 0 such that Wt+1 ≥ ε for all t = 0, 1, 2, ... and construct alternative sequences
∞
{c̃t , ñt , M̃t+1 , B̃t+1 }∞
t=0 that coincide with {ct , nt , Mt+1 , Bt+1 }t=0 except for
c̃1 = c1 +
ε
,
P1
M̃1 = M1 + ε,
and
B̃t+1 = Bt+1 −
ε
Qt+1
for all t = 0, 1, 2, .... These alternative sequences satisfy all of the cash-in-advance, budget,
nonnegativity, and no-Ponzi-game constraints and provide the household with a higher level
15
of utility than the original sequences, which contradicts the assumption that the original
sequences are optimal. Thus, inf t Wt+1 = 0 must hold.
Together, {Wt+1 }∞
t=0 nonincreasing and inf t Wt+1 = 0 imply that (A.7) must hold and
that, more generally, the first-order and transversality conditions are necessary for optimality.
It is also possible to show that the first-order and transversality conditions are sufficient
∞
for optimality. Suppose that {ct , nt , Mt+1 , Bt+1 }∞
t=0 satisfy (A.1)-(A.7) but that {c̃t , ñt , M̃t+1 , B̃t+1 }t=0
satisfy all of the constraints and yield a higher level of utility. Then
lim
T →∞
<
=
lim
T →∞
lim
T →∞
T
X
t=0
T
X
β t [u(c̃t , 1 − ñt ) − u(ct , 1 − nt )]
β t [u1 (ct , 1 − nt )(c̃t − ct ) − u2 (ct , 1 − nt )(ñt − nt )]
t=0
T
X
t=0
T
X
β t [λt (c̃t − ct ) − λt (ñt − nt ) + µt (c̃t − ct )]
"
M̃t − Mt B̃t − Bt M̃t+1 − Mt+1 B̃t+1 − Bt+1
+
−
−
≤ lim
β t λt
T →∞
P
P
P
(1 + rt )Pt
t
t
t
t=0
!
T
X
M̃t − Mt
+
β t µt
Pt
t=0
#
β T λT (MT +1 − M̃T +1 ) β T λT (BT +1 − B̃T +1 )
+
T →∞
PT
(1 + rT )PT
"
!#
λ0
BT +1
B̃T +1
=
lim QT MT +1 +
− QT M̃T +1 +
P0 T →∞
1 + rT
1 + rT
!
λ0
B̃T +1
= −
lim QT M̃T +1 +
P0 T →∞
1 + rT
=
lim
≤ 0
by concavity, by the first-order conditions for ct and nt , by the budget constraint, the cash-inadvance constraint, and the complementary slackness condition, by the first-order conditions
for Mt+1 and Bt+1 , by the first-order conditions for Mt+1 and Bt+1 again, by the transversality
16
condition, and by the no-Ponzi-game constraint. But all of this contradicts the assumption
∞
that {c̃t , ñt , M̃t+1 , B̃t+1 }∞
t=0 yield a higher level of utility than {ct , nt , Mt+1 , Bt+1 }t=0 .
Thus, (A.1)-(A.7) are both necessary and sufficient conditions for a solution to the household’s problem. After the market-clearing conditions are imposed, these equations can be
rewritten as (1)-(6) in the text.
6
References
Cole, Harold L. and Narayana Kocherlakota. ”Zero Nominal Interest Rates: Why They’re
Good and How to Get Them.” Federal Reserve Bank of Minneapolis Quarterly Review
22 (Spring 1998): 2-10.
Friedman, Milton. ”The Optimum Quantity of Money.” In The Optimum Quantity of
Money and Other Essays. Chicago: Aldine Publishing Company, 1969.
Jehiel, Philippe. ”Repeated Games and Limited Forecasting.” European Economic Review
42 (May 1998): 543-551.
Lucas, Robert E., Jr. ”Equilibrium in a Pure Currency Economy.” Economic Inquiry 18
(April 1980): 203-220.
Wilson, Charles. ”An Infinite Horizon Model with Money.” In Jerry R. Green and Jose
Alexandre Scheinkman, eds. General Equilibrium, Growth, and Trade: Essays in
Honor of Lionel McKenzie. New York: Academic Press, 1979.
17
@FedFRASER