Tài liệu Money and Bonds: An Equivalence Theorem doc

Federal Reserve Ba nk o f Minneapolis
Research Department St aff Report 393
July 2007
Mon ey and Bonds: An Equivalen ce Theorem
Narayana R. Kocherlakota
∗
University of Minnesota,
Federal Reserve Bank of Minneapolis,
and NBER
ABSTRACT
This paper considers four models in which immortal agents face idiosyncratic shocks and trade only
a single risk-free a sset over time. The four models specify this single asset to be private bonds,
public bonds, public money, or private money respectively. I pro ve that, given an equilibrium in
one of these economies, it is possible to pick the exogenous elements in the other three economies so
that there is an outcome-equivalent equilibrium in each of them. (The term "exogenous variables"
refers to the limits on private issue of money or bonds, or the supplies of publicly issued bonds or
money.)
∗
I thank Shouyong Shi and Neil Wallace for great conversations about this paper; I thank Ed Nosal, Chris
Phelan, Adam Slawski, Ha kki Yazici and participants in SED 2007 ses sion 44 for their comments. I ac k nowl-
edge the support of NSF 06-06695. The views expressed herein are mine and not necessarily those of the
Federal Reserve B ank of Minneapolis or the Federal Reserve System.
1. Introduction
In this paper, I examine four different models of asset trade. In all of them , immortal
agents face idiosy n cratic shock s to ta ste s an d /or productivities. They can trade a single
risk-free asset over time. Preferences and risks are the same in all four models. The models
differ in t heir specification of w h at this single asset is.
In the first t wo models, a gen ts trade interest-bearing bonds. In the first model,
agents can trade one period risk-free bonds ava ilable in zero net supply, subject to person-
independen t borro wing res trictions. I n the second m odel, a gents can trade one period risk-free
bonds available in positive net supply, but they cannot s hort-sell the asset. A go v ernm en t

pa ys the interest on these bonds, and regulates their supply, by using time-dependent taxes
that are the same for all a gents.
In th e other two models, agents can trade money. Money is an asset t hat lasts forever,
but pays no dividend. It pla ys no special role in transactions. In the third model, money is
in positive supply. A go vernmen t regulates its supply using lump-sum taxes. In the fourth
model, there i s no g ov ernment. Agen ts c an issue and redeem pri vate money, subject t o a
period-by-period constraint on the difference between past issue a nd past redemption.
These models are designed to be closely related to ones already in the literature. The
first model is ess entially t he famous A iyagari-Bewley m odel o f self-insurance. The s econd
model is motivated by Aiy ag ari and McGrattan’s (1998) study of the optimal quantity of
go vernm ent debt. The third model is a version of Lucas’ (1980) pure currency economy.
It is used by Imroh orog lu (1992) in her study of the welfare costs of inflation. The fourth
model is more nov el, a ltho ug h of course many autho rs hav e been interested in comparin g
the consequences of using inside instead of outside money (see, for example, Cavalcanti and
Wallace (1999)).
The basic lesson of this prior literature is that the exact nature of the traded asset
has important e ffects on model out com e s. In A iyagari a n d McGra ttan (1998) (and later Shin
(2006)), publ ic debt issue generates welfare costs that do not occur in models with private
debt. Lucas (1980) argues that agents cannot achieve as much with mon ey as w ith private
debt, sa ying explicitly, "There is a sense in which money is a second-rate asset." Cavalcanti
and Wallace (1999) argue that u s ing insid e (privately issued) m one y allows a gents to ach iev e
more than outside money.
In c ontras t, I prove th e follo win g equivalence the orem. Take an equ ilibriu m in any of
the four econo mies. Then, it is possible to specify the exogenous elements of the other three
econo m ies so tha t there is an outcome-equivalen t equ ilibrium in eac h . Her e, by "exogenous
elements," I mean specifically:
1. borrowing limits in the first m odel
2. bond supplies in the second model
3. money supplies in the third model
4. money issue limits in the fourth model.

In fact, the equiva len c e is actually even stron ge r : in all of these outcom e -e qu ivalent equilib ria,
agents ha ve ident ica l choic e sets a s in th e original equilibriu m.
Why is m y r esult so differen t from the lesson of the prior literature? In the earlier
analyses, the m odels with different assets also impose differen t assumptions on the n ature
of what might be t erm ed the repayment or collection technology. For examp le, in m odels
with priv ate risk-free debt, the borrow er must mak e a repaym en t that is independent of the
borrower’s dec is ions or shocks . In ess en c e, the lender is essentially able to impose a lump-
sum tax on th e borrower at the tim e of repayment. In models with public risk- free d eb t, like
Aiyagari and McGrattan (1998), the government m ak es a repa ymen t that is independent of
any aggregat e shocks. H owever, it is ty p ically assumed that the go vernment must use linear
taxes to collect the resources for its repa yments. This restriction to linear taxes means that
governmen t repaymen t of public debts must d istort agents’ decisions in a wa y that is not true
of private debt repa ym e nt. T he treatment of taxes in models with outside money is often
ev en more drastic; thu s, in their m odels, Lucas (1980), Cavalcanti and Wallace (1999) and
Kocherlakota (2003) all assume that the government can use no taxes other than inflation
taxes.
In this paper, I elimin ate these differences across the models in their specification of
the repay ment technology. In particular, I assume in the models with public debt issue that
the governmen t is able to levy a head tax — that is, a lump-sum tax that is the same for all
agents. ( Note that giv en the potential heterogeneity in the model setting, t he go v ernm en t
2
cannot generally implement a first-best outcome using the (uniform) head tax.) Once I endow
the go vernmen t with this instrument, I can pro ve the equivalence theorem.
The theorem really contains two distinct results. First, I sho w that the public issue
and private issue of bonds/mon ey are equivalent to one another. In this equ ivalence, the
above head tax plays a key r ole. In the m odels with public issue, the gov ernm ent uses a head
tax t hat i s exactly equal to the interest payment made by a borro wer in t he private-issue
economy who holds the maximal level of debt in e ach period. It is in this sense that the
collection powers of the private sector and public sector are the same. Of course, these
collection powe rs may well be limited by enforcemen t problems of various kinds; the crucial

assumption is that the enforcemen t problems are the same in the private and public sectors.
Second, the theorem sho ws that risk-free bonds and money are equiv alen t to one
another. The k ey to this demonstration i s that money can have a positive real rate of r eturn
even th ou gh it does not pay dividends. This p ric e rise can occur in equilibrium in the th ird
model i f the g overnmen t shrinks the supply o f money using t he head tax. The size of t he
needed head tax is exactly the same as in the econom y with public debt issue. It can occur
in th e fourth model if the limits on net money issue are sh rin kin g ov e r time.
Th e theorem is related t o Wallace’s (19 81) famo u s Modigliani-Miller theorem for open
market operations. Wallace p r oves tha t the m o ney/bond c omposition of a government’s debt
portfolio does not affec t equilibrium outcomes. Like my theorem, Wallace ’s relies on two
crucial assumptions. First, a s noted above, the go vernmen t must ha ve access to lump-sum
taxes. Second, money cannot have a transactions a dvantage over bonds. This assumption
is not satisfied by cash-in-advance , money-in -th e -u tility function , or transaction cost models.
Lik e Wallace’s paper, mine is also closely related to Barro’s (1974) analysis of governm ent
debt.
Taub (1994) poses the que stion, "Are currency and credit equivalent mech anisms?"
that motivates this paper. As I do, he answers this question affirmativ ely. Ho wever, he
confine s his analysis to a rather special examp le (linear utilit y ) . Le v ine (19 91) and Green
and Zhou (2005) use linear utility exam ples to demonstrate ho w a go vernment using public
money issue can ac hieve a first-best outcome in a world in whic h a gents experience shocks to
their need for consumption. In their examples, the go vernmen t ac hiev es this good outcome by
3
using an inflationary m onetary policy. My theorem demonstrates that t he gov ernm ent could
instead use an appropriate debt po licy, o r that private agents could achiev e this desirable
outcome with appropriately set borrowing limits.
2. Setup
Consider an infinite horizon e nvironm e nt with a u n it m e asu re o f agents in wh ich time
is indexed by t he na tural numbers. At t he beginning o f period 1, for each agent, Nature
dra ws an infinite sequence (θ
t

)
∞
t=1
from the set Θ
∞
, where Θ is finite. The draws are i.i.d.
acr o ss agents, with m easure μ. Hence, there is no aggregate risk. At the beginning of period
t, a g iven agent ob ser ves his own realization of θ
t
; his information at date t consists of the
history θ
t
=(θ
1
, , θ
t
).
The shocks affect i n d ivid u als as fo llows. Th e typical agent has pre fe ren ces of the f orm
∞
X
t=1
β
t−1
u(c
t
,y
t
,θ
t
),

where c
t
is the agent’s consum ption in period t, y
t
is th e agent’s output in period t, and
0 <β<1. The agent’s utility function u is assumed to be strictly increasin g in c
t
,strictly
decreasing in y
t
, a n d is a function of the realization o f θ
t
.
I t he n consider four d ifferent (possibly inc omplete markets) trading s tructures embed-
ded in this setting.
A. Private-Bond Econom y
The first market structure is a private-bond economy. It is c om pletely characterized b y
a borrowing limit seque nce B
priv
=(B
priv
t+1
)
∞
t=1
, where B
priv
t+1
∈ R
+

. (Note that the borrowing
limits are the s am e for all agents in all periods.) At each date, the agents trade one-period
risk-free real bonds in zero net supply for consumption. They are initially endowed with zero
units of bonds each. Eac h agent’s bond-holdings in period t must be no smaller than −B
priv
t+1
(as measured in terms of consumption in that period).
In this economy, in d ivid ua ls take intere st rates r =(r
t
)
∞
t=1
,r
t
∈ R, as giv en and then
choose consumption, output, and bond-holdings (c, y, b)=(c
t
,y
t
,b
t+1
)
∞
t=1
, (c
t
,y
t
,b
t+1

):Θ
t
→
4
R
2
+
× R. Hence, the agen t’s problem is
max
(c,y,b)
E
∞
X
t=1
β
t−1
u(c
t
,y
t
,θ
t
)
s.t. c
t
(θ
t
)+b
t+1
(θ

t
)
≤ y
t
(θ
t
)+b
t
(θ
t−1
)(1 + r
t−1
) ∀(θ
t
,t≥ 1)
{b
t+1
(θ
t
)+B
priv
t+1
},c
t
(θ
t
),y
t
(θ
t

) ≥ 0 ∀(θ
t
,t≥ 1)
b
1
=0
An equilib rium i n a private-bond economy B
priv
is a specification of (c, y, b, r) such that
(c, y, b) solves the agen t ’s problem given r and m arkets clear for all t:
Z
c
t
dμ =
Z
y
t
dμ
Z
b
t+1
dμ =0
B. Public-Bond Econom y
The second is a pub lic-bond economy. A t eac h date, there is a go vernmen t that sells
one-period risk-free real bonds. The economy is completely characterized by an exogenously
specified bond supply sequence B
pub
=(B
pub
t

)
∞
t=1
,whereB
pub
t+1
∈ R
+
and an in itial period
return r
0
. The government raises B
pub
t+1
units of consumption in period t b y selling one-period
risk-free bonds. I t collects τ
b
t
units of consump tion fro m each agent; th e tax is the same for all
agents, and is d etermined endogenously in equilibrium.
1
Each agen t is in itially endowed with
bonds t hat pay off B
pub
1
(1+r
0
) units of consumption. At eac h date, agen ts trade co nsumption
and the gov ernmen t-issued bonds. Agents are not allow ed to short-sell these bonds.
In this econom y, the individuals take interest rates r =(r

t
)
∞
t=1
as given and then choose
consumption, output, and bond-holdings. Hence, the individual’s problem is
max
(c,y,b)
E
∞
X
t=1
β
t−1
u(c
t
,y
t
,θ
t
)
1
Taxes are endogenously determined in this public-bond economy and in the public-money economy dis-
cussed in the next section. It is important to note that the main equivalence theorem is valid even if taxes
are exogenously specified. I treat taxes as endogenous so as to ensure that the government flow budget
constraint is satisfied for off-equilibrium interest rate/price sequences, as well as in equilibrium.
5
s.t. c
t
(θ

t
)+b
t+1
(θ
t
)
≤ y
t
(θ
t
)+b
t
(θ
t
)(1 + r
t−1
) − τ
bond
t
∀(θ
t
,t≥ 2)
c
1
(θ
1
)+b
2
(θ
1

) ≤ y
1
(θ
1
)+B
pub
1
(1 + r
0
) − τ
bond
1
∀θ
1
b
t+1
(θ
t
),c
t
(θ
t
),y
t
(θ
t
) ≥ 0 ∀θ
t
,t≥ 1
An equilibrium in a public-bond economy (B

pub
,r
0
) is a s pecification of (c, y, b, r, τ
bond
) suc h
that (c, y, b) solves the i n div id ual’s problem given (r, τ
bond
) and mar kets clear for all t:
Z
c
t
dμ =
Z
y
t
dμ for all t
Z
b
t+1
dμ = B
pub
t+1
for all t
Together, these imply t hat a gov ernm en t budget constraint holds at eac h date:
τ
b
t
= −B
pub

t+1
+ B
pub
t
(1 + r
t−1
)
C. Public-Money Economy
The third economy is a public-m oney economy. By mo ney, I mean an infinite ly -lived
asset that pays no dividends. Each a gen t is initially endowed with M
pub
1
units of money.
Then, the economy is completely characterized by an exogenously specified money supply
sequence M
pub
=(M
pub
t+1
)
∞
t=1
,whereM
pub
t+1
∈ R
+
.Inperiodt,thegovernmentcollectsτ
mon
t

units of consumption from eac h agent; again, the taxes are the same for all agents, and are
determ in ed endogenou sly in equilibrium . At eac h date, agen ts t rad e money a nd consumption ;
the go vernment trades so as to ensure that there are M
pub
t+1
units of mon ey outstandin g.
In this econom y, the i ndividuals tak e money prices p as g iv en and then c hoose con-
sumption, output, and m oney-holdings. Hence, the individual’s problem is
max
(c,y,M )
E
∞
X
t=1
β
t−1
u(c
t
,y
t
,θ
t
)
s.t. c
t
(θ
t
)+M
t+1
(θ

t
)p
t
≤ y
t
(θ
t
)+M
t
(θ
t−1
)p
t
− τ
mon
t
∀θ
t
,t≥ 2
c
1
(θ
1
)+M
2
(θ
1
)p
1
≤ y

1
(θ
1
)+M
pub
1
p
1
− τ
mon
1
∀θ
1
M
t+1
(θ
t
),c
t
(θ
t
),y
t
(θ
t
) ≥ 0 ∀(θ
t
,t≥ 1)
6
An equilibrium in a public -money ec on omy (M

pub
) is a specification of (c, y, M, p, τ
mon
) such
that (c, y, M) s olves the individual’s problem given (p, τ
mon
) and markets clear for all t:
Z
c
t
dμ =
Z
y
t
dμ
Z
M
t+1
dμ = M
pub
t+1
Again, a go vernment budget constrain t is implied at eac h date b y market-clearing:
τ
mon
t
= M
pub
t
p
t

− M
pub
t+1
p
t
There is no cash-in-advance constrain t or any transaction cost advantage associated
with mon ey in this se tting.
D. Private Money Econom y
The fourth and final econo my is a private-money economy. In this economy, there
is no gov ernm en t. A gen ts are able to issue their money in exchange for consumption, and
redeem others’ monies in exc hange for consumption. Howev er, in eac h history, they face
an ex ogen ous upper bound on the net am ou nt of money issue that they have done in their
lifetimes. The econom y is completely characterized b y the exogenous upper bound process
M
priv
=(M
priv
t+1
)
∞
t=1
, where M
priv
t+1
∈ R
+
.
In this econom y, the individuals take money prices p as given and then c hoose con-
sumption, how m u ch money to issue and ho w muc h money to redeem. (I assume that all
monies are traded at the same price p; there m ay be other equilibria in which this restriction

is n ot satisfied.) H enc e, the individual’s pro ble m is
max
(c,y,M
iss
,M
red
)
E
∞
X
t=1
β
t−1
u(c
t
,y
t
,θ
t
)
s.t. c
t
(θ
t
)+M
red
t+1
(θ
t
)p

t
≤ y
t
(θ
t
)+M
iss
t+1
(θ
t
)p
t
∀θ
t
,t≥ 1
M
iss
t+1
(θ
t
),M
red
t+1
(θ
t
),c
t
(θ
t
),y

t
(θ
t
) ≥ 0 for all θ
t
,t
t
X
s=1
[M
iss
s+1
(θ
s
) − M
red
s+1
(θ
s
)] ≤ M
priv
t+1
∀θ
t
,t≥ 1
An equ ilib rium in a private-money e conomy (M
priv
) is a specification of (c, y, M
red
,M

iss
,p)
7
suc h that (c, y, M
red
,M
iss
,p) solves the individual’s problem and markets clear for all t:
Z
c
t
dμ =
Z
y
t
dμ
Z
M
red
t+1
dμ =
Z
M
iss
t+1
dμ
Again, there is no cash-in-advance constrain t or transaction cost advantages associated
with mon ey in this se tting.
3. Example Economy
In this section, I w o rk through an example of the abo ve general structure that il-

lustrates the gene ral equivalence theore m th at follows in the next section. I sta r t with an
equilibrium in a private-bond econom y. I then construct a public-bond economy, a public-
money econom y, and a private-money economy. I show that in eac h of these economies,
there is an equilibrium with the same consu mption allocation as the original, p rivate-bond,
equilibrium. Even more stron gly, agents have exactly the same budget set s in each of these
equilibria.
In t he example, output is inelastically su p plied. Half of the agents rece ive an endow -
ment stream of the f orm (1 + h, 1, 1 , ) and the ot her half get an endowm ent stream of the
form (1 − h, 1, 1, ), where 1 >h>0. I will call the first half "rich" and the second half
"poor." The agents have identical preferences of the form
∞
X
t=1
β
t−1
ln(c
t
),
where 1 >β>0.
A. Private-Bond Econom y
Consider first a priva te-bond ec on omy in which th e borrow in g limit B
priv
t+1
is con stant
at βλ(1 − β)
−1
, where (1 − β) <λ<1. We can construct an equilibriu m in th is eco n omy as
follows. Set the interest rate r
t
to be constant at 1/β − 1. Ric h agents consume a constant

amount c
r
,where
c
r
=(1+h)(1 − β)+β =1+h(1 − β)
8
Ric h agents’ bond-holdings b
r
t+1
equal
βh
for period t ≥ 1. Poor agents consu me a constant amount c
p
=2− c
r
. Poor agents’ bond-
holdin gs b
p
t+1
equal
−hβ
for period t ≥ 1. Note that the borrowing limit has been chosen so that it nev er binds in
equilib rium.
It is readily checked tha t the abo ve specification forms an equilibrium. Markets clear.
The agents’ flo w budget constrain ts are satisfied because
c
r
+ b
r

2
− 1 − h =0
c
r
+ b
r
t+1
− 1 − b
r
t
β
−1
=0for all t ≥ 2
and similarly for poor agents. Because the borrow in g limit d oes not bind, the agents’ Euler
equations are satisfied. We need only chec k the agents’ transv ersality conditions, whic h are
satisfied because the two limits
lim
t→∞
β
t−1
u
0
(c
r
)(b
r
t+1
− 2β)
lim
t→∞

β
t−1
u
0
(c
p
)(b
p
t+1
− 2β)
are both equal to zero.
B. Public-Bond Econom y
I n ow want to design a public-bond ec on o my with an outcome-equ ivalent equilib rium.
In a public-bond econom y, agen ts are not allowed to borrow. Hence, to get a non-autarkic
equilib rium, th ere must be a positiv e amou nt of d e b t outstan d in g. I set B
pub
t
= βλ(1 − β)
−1
(the private economy borrowing limit) for all t and r
0
=1/β − 1. As above, w e can c onstruct
an equilibriu m in this economy in which the equ ilibriu m intere st rate r
t
is constant at 1/β −1.
Rich agents consume c
r
(as defined above) in eac h period, and poor agents consume c
p
in

9
eac h period. Each agent pa y s a lum p-sum tax τ
b
t
= λ at ever y date. Then, in period t ≥ 1,
ric h agen ts’ bond-holdings b
r
t+1
equal
βh + βλ(1 − β)
−1
and poor agents’ bond-holdings b
p
t+1
equal
−βh + βλ(1 − β)
−1
Note that the agents’ bond-holdin gs are a lways positive .
Again, it is sim p le to ve rify that these in te r es t rates and quantities form an equ ilibrium.
Markets clear. T he rich agen t’s flow budget constrain t in period 1 is satisfied because
c
r
+ b
r
2
− 1 − h − B
pub
1
β
−1

+ τ
bond
=1+h(1 − β)+hβ + βλ(1 − β)
−1
− 1 − h − λ(1 − β)
−1
+ λ
=0.
The rich agen t’s flow budget constrain t in period t>1 is satisfied because
c
r
+ b
r
t+1
− 1 − b
r
t
β
−1
+ τ
bond
=1+h(1 − β)+hβ + βλ(1 − β)
−1
− 1 − h − λ(1 − β)
−1
+ λ
=0.
We can check t he poor agents’ flo w constraints in a similar fashion.
The agents’ Eu ler equations are clearly satisfied, because their no-short-sales constraint
never binds. Finally, we n ee d to ve r ify the transvers ality conditions:

lim
t→∞
β
t
u
0
(c
r
)b
r
t+1
=0
lim
t→∞
β
t
u
0
(c
p
)b
p
t+1
=0
Hence, th ere is an equilibrium in this public-bond econom y with the same consumption
allocation as the original, p r ivate-bond, equilibrium .
10
C. Public-Money Economy
I now w ant to design a public-money economy with an outcome-equivalent equilibrium.
Clearly, the gr oss rate of return on m oney m ust be β

−1
, in orde r to satisfy the agen ts’ first-
order conditions. Since money pays no dividend, this rate of return implies that the price
of money m ust rise at rate β
−1
. A t the same time, we need the rich agen t s’ transv ersality
condition:
lim
t→∞
β
t
u
0
(c
r
)M
r
t+1
p
t
=0
to be satisfied. This requires that the money stock must converge to zero ov er time.
Giv en these considerations, consider a public-money economy in which M
pub
t
= β
t
λ(1−
β)
−1

. I claim that there is an e quilib riu m in this eco no my in whic h p
t
= β
−t
for all t. Rich
agents consume c
r
in eac h period, and poor agents consume c
p
in eac h period. As in the
public-bond econom y, each agen t pays a lump-sum tax τ
mon
t
= λ at every da te. Then, in
period t ≥ 1, rich agents’ money-h o ld ings M
r
t+1
equal
β
t+1
h + β
t+1
λ(1 − β)
−1
and poor agents’ bond-holdings M
p
t+1
equal
−β
t+1

h + β
t+1
λ(1 − β)
−1
To v erify the claim that these prices and quan titie s form an equilibrium, note that
mark ets clear and that individual Euler equations are satisfied. Clearly, the transvers a lity
conditions are also satisfied, because the money supply converges to zero. F inally, we can ver-
ify that the flow budget constraints are satisfied. In period 1, the rich agen ts’ flow constraints
are:
c
r
+ p
1
M
r
2
− 1 − h − p
1
M
r
1
+ τ
mon
=1+h(1 − β)+β
−1
(β
2
h + β
2
λ(1 − β)

−1
) − 1 − h − λ(1 − β)
−1
+ λ
=0.
In period t>1, the rich agents’ flow constraints are
c
r
+ p
t
M
r
t+1
− 1 − p
t
M
r
t
+ τ
mon
11
=1+h(1 − β)+β
−t
(β
t+1
h + β
t+1
λ(1 − β)
−1
) − 1 − (h + λ(1 − β)

−1
)+λ
=0.
The poor agents’ flow constraints are similar.
Mo n e y p ays n o dividend an d has no liquidity benefits. N onetheless, money has a
positive valu e in this equilibrium. This positiv e value se ems to create t he possibilit y for an
arbitrage, in which a given agen t permanen tly reduces his money-h olding s by a small amount
ε. Howev er, this arbitrage is infeasible, because the supply of money ev entually falls to zero.
D. Private-Money Economy
Finally, I desig n a private-money economy that induces the same equilibrium con-
sumption allocation. Consider a private-money economy in which M
priv
t+1
= β
t+1
λ(1 − β)
−1
.
Then , I claim tha t th ere is an eq u ilibriu m in t his econ omy in which p
t
= β
−t
for all t. Rich
agents consume c
r
in each period, and poor agents consume c
p
in each period. In period 1,
poor agents issue M
iss,p

2
= β
2
h and rich agents redeem M
red,r
2
= β
2
h. In period t>1, poor
agents redeem M
red,p
t+1
= h(1 − β)β
t
andrichagentsissueM
iss,r
t+1
= h(1 − β)β
t
. Note that in
period t,
t
X
s=1
(M
iss,p
s+1
− M
red,p
s+1

)=β
2
h − β
2
h(1 − β) − β
3
h(1 − β) − − β
t
h(1 − β)=β
t+1
h
which is less than β
t+1
λ(1 − β)
−1
. In this private-money econom y, neither rich nor poor
agen ts ev er have net m oney issue equal to β
t
. Hence, the bound on money issue nev er binds
in e qu ilib riu m .
We can verify the validity of th is putativ e e qu ilib rium as follo ws . Markets c lea r, and
the agen t s’ Euler equations are clearly satisfied. In period 1,therichagents’flow budget
constraints are
c
r
+ p
1
M
red,r
2

− 1 − h − p
1
M
iss,r
2
=1+h(1 − β)+βh − 1 − h
=0.
12
In period t>1, the rich agents’ flo w budget constraints are
c
r
+ p
t
M
red,r
t+1
− 1 − p
t
M
iss,r
t+1
=1+h(1 − β) − h(1 − β) − 1
=0.
The poor agents’ flow constraints can be check ed in a similar fa shion.
Finally, we need to verify the agen ts’ transversality conditions. To do so, note that
lim
t→∞
β
t
u

0
(c
r
)p
t
{−
t
X
s=1
[M
iss
s+1
− M
red
s+1
]+M
priv
t+1
}
= lim
t→∞
β
t
u
0
(c
r
)p
t
{β

t+1
h}[λ(1 − β)
−1
− 1]
=0
In this economy, money has a positive value and pa ys no dividends. Wh y is it not
optimal for an agent to issue ε units more m oney in period 1? Issuing ε more units of m oney
in period 1 means that the agent’s net money issue would be ε + β
t+1
h in period t. But for
large t, this amoun t will exceed the upper bound on money issue. Put another way, ev en
though it never binds, the money issue constrain t is structured so that agents must eventually
redeem whatev er currency they hav e issued.
E. Budget Set Equivalen ce
The argu ments abo ve estab lish th at t h e equ ilibriu m outcome s are identical across the
four econom ies. But, with som e algebra, it is possible to prove an even stronger equivalence:
the equilibrium budget sets of (c, y) are the same across the four economies. Agents are
confronted with exactly the same sets of possible choices in the four economies. As w e shall
see, this deeper isomorphism ca n be generalized.
4. An Equivalence Theorem
In this section, I pro ve the main theorem in the paper. The theorem starts with an
equilib rium (c, y, b, r) in a private-bond economy defined by B
priv
. It then sho w s how , by
translating the bond-holdings upward b y B
priv
and craftin g taxes in the right way, we c an
13
get an ou tc o me-equivalent equilibrium in the pu blic-bond economy. Th e k e y to the th eorem
is that these taxes equal the net interest paymen ts o n the bond-holding limits.

Then, the theorem goes on to show tha t, by setting monetary policy in the righ t way,
we can design an outco me-equivalen t equ ilibriu m in th e pu blic-money economy. T h e key here
is that monetary policy must be design ed so that t h e price level falls at the rate of int er es t,
and the real value of aggregate money-holdings always equals a ggregate bond-holdings. As
we will see later, this will typically mean that aggregate money-holdings will be sh rin king
(but not necessarily at the Friedman Rule 1/β − 1).
Finally, the theorem turns to the private-money econom y. Here, I set the upper bounds
on net real money issue e qual to the borrow ing limit in the private-bond economy. By doing
so, I can induc e an outcom e -e qu ivalent equilibrium in this setting.
2
Theorem 1. Suppose (r
t−1
,p
t
,B
priv
t+1
,B
pub
t
,τ
bond
t
,M
pub
t
,τ
mon
t
,M

priv
t+1
)
∞
t=1
are sequences such that
for all t:
p
t
=(1+r
t−1
)p
t−1
; p
1
= M
pub
1
(1 + r
0
)
−1
/B
pub
1
B
priv
t+1
= B
pub

t+1
= M
pub
t+1
p
t
= M
priv
t+1
p
t
τ
bond
t
= τ
mon
t
= B
pub
t
(1 + r
t−1
) − B
pub
t+1
Then the following four statements are equ ivalent.
1. (c, y, b, r) is an equilibrium in a private-bond economy defined by B
priv
.
2. (c, y, b + B

priv
,r,τ
bond
) is an equilibrium in a public-bond economy de fined by B
pub
and
r
0
.
3. (c, y, (b + B
priv
)/p,p,τ
mon
) is an equilibrium in a public-money economy defined by
M
pub
.
4. (c, y, M
red
,M
iss
,p) is an equilibrium in a private-m on ey economy defined by M
priv
,
where for all t:
t
X
s=1
[M
red

s+1
(θ
s
) − M
iss
s+1
(θ
s
)] = b
t+1
/p
t
for all θ
s
.
2
The proof of this theorem relies on techniques similar to those that I use in Kocherlakota (forthcoming).
14
Proof. I first prove the equivalence of statements 1 and 2. Pick an elem ent (
b
c,
b
b) of an agent’s
budget set in the private-bond economy. Then, define
b
b
0
t+1
(θ
t

)=
b
b
t+1
(θ
t
)+B
priv
t+1
,t≥ 1, all θ
t
My claim is that (
b
c,
b
y,
b
b
0
) is in the agen t’s budget set in the public-bond economy. Ob viously,
b
b
0
t+1
(θ
t
) ≥ 0, because
b
b
t+1

(θ
t
) ≥−B
priv
t+1
. Note that for t ≥ 2
b
c
t
(θ
t
)+
b
b
0
t+1
(θ
t
) −
b
y
t
(θ
t
) − (1 + r
t−1
)
b
b
0

t
(θ
t−1
)+τ
bond
t
=
b
c
t
(θ
t
)+
b
b
t+1
(θ
t
)+B
priv
t+1
−
b
y
t
(θ
t
) − (1 + r
t−1
)

b
b
t
(θ
t−1
) − (1 + r
t−1
)B
priv
t
+ τ
bond
t
=
b
c
t
(θ
t
)+
b
b
t+1
(θ
t
) −
b
y
t
(θ

t
) − (1 + r
t−1
)
b
b
t
(θ
t−1
)
≤ 0 for all θ
t
We can use similar logic to check the flow budget constraint for t =1:
b
c
1
(θ
1
)+
b
b
0
2
(θ
1
) −
b
y
1
(θ

1
) − B
pub
1
(1 + r
0
)+τ
bond
1
=
b
c
1
(θ
1
)+
b
b
2
(θ
1
)+B
priv
2
−
b
y
1
(θ
1

) − B
pub
1
(1 + r
0
)+τ
bond
1
=
b
c
1
(θ
1
)+
b
b
2
(θ
1
) −
b
y
1
(θ
1
) ≤ 0
for all (θ
1
). Running the steps in reverse establishes the converse. Hence, the agent’s budget

sets are (c, y)-equivalen t in th e two econo m ie s. But it is then straightf o rwar d to see that
statemen ts 1 and 2 are equivalen t.
I now prove the equivalence of statemen ts 2 and 3. Pic k an elemen t (
b
c,
b
y,
b
b
0
) of an
agent’s budget set in the public-bond econom y. Define
c
M
t+1
=
b
b
0
t+1
/p
t
. I claim that (
b
c,
b
y,
c
M)
is in the agent’s budget set in the p ub lic -money economy. Then for all θ

t
,t≥ 2:
b
c
t
(θ
t
)+
c
M
t+1
(θ
t
)p
t
−
b
y
t
(θ
t
) −
c
M
t
(θ
t−1
)p
t
+ τ

mon
t
=
b
c
t
(θ
t
)+
b
b
0
t+1
(θ
t
) −
b
y
t
(θ
t
) − (1 + r
t−1
)
c
M
t
(θ
t
)p

t−1
+ τ
bond
t
=
b
c
t
(θ
t
)+
b
b
0
t+1
(θ
t
) −
b
y
t
(θ
t
) − (1 + r
t−1
)
b
b
0
t

(θ
t
)+τ
bond
t
≤ 0
15
In period 1, w e can verify
b
c
1
(θ
1
)+
c
M
2
(θ
1
)p
1
−
b
y
1
(θ
1
) − M
pub
1

p
1
+ τ
mon
1
=
b
c
1
(θ
1
)+
b
b
0
2
(θ
1
) −
b
y
1
(θ
1
) − B
pub
1
+ τ
bond
1

≤ 0
for all (θ
1
). We can run the logic in reverse to c hec k the con verse. Thus, the agent’s budget
sets are (c, y)-equiva lent; th is in turn establishes that statem ents 2 and 3 are equivalent.
Finally, I pro ve the equivalence of statements 1 and 4. Pick a n element (
b
c,
b
y,
b
b) of an
agent’s budget set in the private-bond econom y. Then, define
M
red
t+1
(θ
t
)=max(
b
b
t+1
(θ
t
) − (1 + r
t−1
)
b
b
t

(θ
t−1
), 0)/p
t
,t≥ 2
M
iss
t+1
(θ
t
) = max((1 + r
t−1
)
b
b
t
(θ
t−1
) −
b
b
t+1
(θ
t
), 0)/p
t
,t≥ 2
M
red
2

(θ
1
)=max(
b
b
2
(θ
1
), 0)/p
1
M
iss
2
(θ
1
)=max(−
b
b
2
(θ
1
), 0)/p
1
I claim that (
b
c,
b
y, M
red
,M

iss
) is in the agent’s budget set in the private-money econom y. In
period t>1:
b
c
t
(θ
t
)+M
red
t+1
(θ
t
)p
t
−
b
y
t
(θ
t
) − M
iss
t+1
(θ
t
)p
t
=
b

c
t
(θ
t
)+
b
b
t+1
(θ
t
) − (1 + r
t−1
)
b
b
t
(θ
t
) −
b
y
t
(θ
t
)
≤ 0
for all (θ
t
) and in period 1, for all (θ
1

):
b
c
1
(θ
1
)+M
red
2
(θ
1
)p
1
−
b
y
1
(θ
1
) − M
iss
2
(θ
1
)p
1
=
b
c
1

+
b
b
2
(θ
1
) −
b
y
1
(θ
1
)
≤ 0
16
Note too tha t:
t
X
s=1
(M
red
s+1
(θ
s
) − M
iss
s+1
(θ
s
))

=
t
X
s=1
[
b
b
s+1
(θ
s
) − (1 + r
s−1
)
b
b
s
(θ
s−1
)]/p
s
=
t
X
s=1
[
b
b
s+1
(θ
s

)/p
s
−
b
b
s
(θ
s
)/p
s−1
]
=
b
b
t+1
(θ
s
)/p
t
≥−B
priv
t+1
/p
t
= −M
priv
t+1
This confirms that the budget-feasible consumptions in the private-bond econom y are a subset
of the budget-feasible consu mp tio n set in the private-m oney economy. Conversely, suppose
(

b
c,
b
y, M
red
,M
iss
) is in the agent’s budget set in the private-money economy. Then define:
b
b
t+1
(θ
s
)=p
t
t
X
s=1
(M
red
s+1
(θ
s
) − M
iss
s+1
(θ
s
))
I claim that (

b
c,
b
y,
b
b) is budget-feasible in the private-bond economy. In period t>1, for an y
(θ
t
):
b
c
t
(θ
t
)+
b
b
t+1
(θ
t
) −
b
y
t
(θ
t
) −
b
b
t

(θ
t
)(1 + r
t−1
)
=
b
c
t
(θ
t
)+p
t
t
X
s=1
(M
red
s+1
(θ
s
) − M
iss
s+1
(θ
s
)) −
b
y
t

(θ
t
) − p
t
t−1
X
s=1
(M
red
s+1
(θ
s
,y
s
) − M
iss
s+1
(θ
s
,y
s
))
=
b
c
t
(θ
t
,y
t

)+p
t
(M
red
t+1
(θ
t
) − M
iss
t+1
(θ
t
)) −
b
y
t
(θ
t
)
≤ 0
In period 1,forany(θ
1
):
b
c
1
(θ
1
)+
b

b
2
(θ
1
) −
b
y
1
(θ
1
)
=
b
c
1
(θ
1
,y
1
)+p
1
(M
red
2
(θ
1
) − M
iss
2
(θ

1
)) −
b
y
1
(θ
1
)
≤ 0
Agent j’s bond-holdings
b
b c le arly satisfy the borrowing limit −B
priv
.Hence,(
b
c,
b
y,
b
b) is budget-
17
feasible in the private-bond econom y. T he budget sets are (c, y)-equivalen t. It follows that
Statements 1 and 4 are equivalent. QED
5. Discussion
In this section , I d isc u ss several a s pects of Theorem 1.
A. Equivalences
Theorem 1 establishes two kinds of equivalences. The first is between private issue a nd
public issue (of m oney or bonds). C onsider, for example, a private-bond econom y in whic h
agents ha ve a consta nt borrow ing lim it B
priv

. In this economy, all agents begin with the
same holdings of bonds (z e r o). They can run down their h old in gs to −B
priv
. Now, consider
a public-bond economy in w hich all ag ents begin th eir lives by holding B
priv
units of bonds.
They face taxes with present value equal to B
priv
and can ru n down their h oldings to zero. In
the public-bond econom y, a gen ts’ initial wealths are the sam e as in the private debt economy.
As well, they can run do wn their initial bond-holdings by exactly the same amount (B
priv
)
as in the private debt econom y. Hence, their budget sets are the same in the two kinds of
economies.
3
Th e secon d kind of equivalen c e is betwe en moneta ry e c on omies and bond econ omies.
Con side r a public-bond ec on omy in w hich the equilibriu m rate of return is constant a t r>0,
and the value of outstanding public debt is constant at B
pub
. Theorem 1 designs a public-
mon ey economy in which the equilibrium rate of return is also r, and value of outstanding
public obligations (now in the form of m oney) is B
pub
. In this public-money economy, the
price of money m u st rise at rate r. H ence, the quantit y of money must fall at this same rate.
The go vernmen t sucks out this m oney using t he same taxes that it used to financ e its interest
payme nts in the public-bond economy.
It is worth poin ting out that the proof of Theorem 1 establishes a stronger result than

Theorem 1 itself. The statemen t of Theorem 1 is that the equilibrium outcomes across the
3
It is possible to extend Theorem 1 to include model economies in which a gents ca n trade both public and
private debt. In particular, suppose there is an equilibrium (c, y, b, r) in a private-bond economy defined by
the sequence B
priv
. Suppose too that there i s an economy in whic h the sequence o f supplies of outside debt
is given by B
pub0
and the borrowing limit sequence is given by B
priv0
. Then, ther e is an equilibrium with
allocation (c, y) in this latter private-public bond economy if B
pub0
+ B
priv0
= B
priv
in all periods.
18
four economies are the same in terms of (c, y). Actually, the proof es tab lish e s a muc h strong er
result: the equilibrium budget sets of (c, y) are the same in the four econom ies. Agents face
exactly the s ame choice problem s in the four equilibria o f the f ou r econom ies.
It is possible to extend T h eore m 1 to include model economies in wh ich agents can
trade both public and private debt. In particular, suppose there is an equilibrium (c, y, b, r) in
a private-bond economy de fined by the sequence B
priv
. Suppose too that there is an economy
in which the sequence of supplies of outs id e debt is given by B
pub0

and the borro wing limit
sequence is given b y B
priv0
. Then, there is an equilibrium with allocation (c, y) in this latter
private-public bond economy if B
pub0
+ B
priv0
= B
priv
in all periods. Public and private debt
are perfect substitutes.
B. Lump-Sum Taxes
A key elemen t of the proof of Theorem 1 is that the taxes τ
mon
and τ
bond
don’t depend
on y; hence, they are lump-sum. These lum p -s u m taxes simply give the governme nt similar
collection power s to those of the private sector. In particular, consid er an equilibrium
(c, y, b, r) in a private bond economy defined by B
priv
. In this equilibrium, a borrower may
owe as much as
B
priv
t+1
(1 + r
t
)

in period (t +1). The borro wer can only borrow up to B
priv
t+2
to repay this loan. Hence, a
lende r must be able to collect
B
priv
t+1
(1 + r
t
) − B
priv
t+2
in period (t +1). This collection limit is independent of decision s about (c, y) being made by
the borro wer. In this sense, lenders are able to levy a lump-sum tax on the borrower equal
to B
priv
t+1
(1 + r
t
) − B
priv
t+2
in period (t +1).
Th is collection limit is exactly eq ua l to τ
bond
and τ
mon
in the equ ivalent public bond
and money economies constructed in Theorem 1. H ence, by assuming that the go vernment

can levy taxes equal to τ
bond
and τ
mon
, I am assuming tha t the government can levy the same
lump-sum taxes as can a private lender. This m eans, for e xample, that the private and public
sector must face the same lim its on enforcement across the two k inds of models.
19
In realit y, governments can use a broader range of taxes than is assumed in the above
economies. More generally, suppose that in all four econom ies, the governm e nt can use any
element of a class C of tax sch edules ψ = {ψ
t
}
∞
t=1
, where an agent who has production history
y
t
in period t pays a t ax ψ
t
(y
t
). We can pro ve a v ersion of Theorem 1 if C is closed under the
addition of a sequence of con stants, so that if α = {α
t
}
∞
t=1
∈ R
∞

, and ψ
t
(y
t
) − ψ
0
t
(y
t
)=α
t
for all t, y
t
, then
ψ ∈ C =⇒ ψ
0
∈ C
For example, the class C may consist of the affine tax codes discussed b y Werning (forthcom-
ing).
4
C. Multiple Equilib ria
Th e equivalence described in Theorem 1 is limited in the following wa y. Suppose c
is an equilibrium consumption allocation in a public-money economy defined b y M
pub
and
τ. Then, Theorem 1 says that there is an outcome -eq u ivalent equilibrium outcome in the
private deb t economy defined by B
priv
. The theorem does not say that the sets of equilibrium
outcomes are the same in th e two economies.

The follo wing example makes th is point m ore forcefully.
Exam ple 1. Suppose that Θ = {1, 2}, that θ
t
=1for all t with probability 1/2, and that
θ
t
=2for all t with probab ility 1/2. Suppose that u(c, y)=ln(c), so that output is inelastically
supplied and β>1/2. Finally, suppose that for all agen ts, y
t
= θ
1
if t is odd and y
t
=3− θ
1
if t is even.
Consider a public-money economy in which M
pub
t
=1for all t. It is well-known that
there are (at least) two equilibria in this econo my. In the one equilibrium, p
t
=0for all t,
and c
t
= y
t
for all t. In the oth er equilib rium:
p
1

=(4β − 2)/(5 + 2β)
4
To impose lump-sum taxes in the public-money economy, the government must be able to threaten agents
with some kind of penalty if they fail to pay those taxes. Kocherlakota (2003) and Berentsen and Waller
(2006) argue that these penalties could be used to enforce cross-agent transfers of resources, and thereby
eliminate the need for money altogether. How ever, to ensure that there is no need for m oney, the government
must be able to impose an arbitra rily large penalty, and know the realization of θ for each agent. Neither of
these assumptions i s implied by the government’s being able to levy a particular lump-sum tax of size τ
mon
.
20
p
t
= p
∗
≡ (2β − 1)/(2 + 2β),t>1
c
1
(2) = 2 − p
1
; c
1
(1) = 1 + p
1
c
2t+1
(2) = c
2t
(1) = 2 − 2p
∗

,t=1, 2, 3,
c
2t
(2) = c
2t+1
(1) = 1 + 2p
∗
,t=1, 2, 3
(Here, c
t
(y) is consumption in period t if y
1
= y.) In this latter equilibrium, the t wo age nts
sw ap the money stock bac k and forth in exc hange for consumption.
As in Theorem 1, construct a private bond economy by setting
B
priv
t
= p
t
,t>1
for all t. There is an equilibrium in th is economy of the form
r
1
=(1− 2β)/(4 + 4β)
r
t
=0,t>1
c
1

(2) = 2 − p
1
; c
1
(1) = 1 + p
1
c
2t+1
(2) = c
2t
(1) = 2 − 2p
∗
,t=1, 2, 3,
c
2t
(2) = c
2t+1
(1) = 1 + 2p
∗
,t=1, 2, 3
This equilibriu m is consumption - equivalen t to the second equilibrium in the public-m on ey
economy. (In t h is equilibrium, the poor agents (with in c om e 1) borro w as much as possible in
each period.) Ho wev e r, in the private bond economy, there is no autarkic equilibrium. Hen ce,
the set of equilibriu m outc o mes is not the same a s in th e origin al public-m oney economy.
D. Welfar e Im plica tion s
Cole and Koc h erlakota (2 001) con sider a setup in wh ich agents have hidden endow-
men ts and store secretly over time. They pro ve that, under w eak conditions, the symmetric
Pareto optimum is the equilibrium outcome of risk-free borrowing and lending, subject to the
natural borrowing constraint. The analysis in this paper sho ws that this symm e tric Pareto
optimum could also be impleme nted as an eq uilib riu m in an economy in which agents can

only hold money. Note that the rate of retur n on money in this economy is necessarily less
thantherateoftimepreference.
21
Aiyagari (1995 ) s tudies th e properties of optima l capital inc om e tax es in economies of
this kind (in which age nts trade only ca pital and risk-free bonds, and face sh oc ks to their la bor
productivities). Aiyagari (1995) finds that the optimal capital income tax rate is positiv e,
and the equilib riu m interest rate in the eco no my is less than th e rate of time preference.
Using the logic of T h eo rem 1, one could redo Aiyagari’s a nalysis in a p ub lic -money ec on omy.
Aiyaga ri’s result implies th a t in su ch a settin g, the optimal rate of re tu rn on money is less
thantherateoftimepreference.
5
6. Conclusions
In this paper, I make two k ey assumptions. First, I assume that the private and
public sectors have the same collection powers. Second, I assume that money has no distinct
transaction advantage ov er bonds. Under these t wo assumptions, I am able to establish an
isomorphism across a broad class of one-asset incomplete markets economies.
In reality, the collection po wers of the private and pu blic sector may well differ, and
mon ey almos t certain ly does provide liquidity bene fits that bonds do not. A great deal
of attention has been given to modelling and understanding the latter phenomenon. In
light of the theorem in this paper, th e former issue seems an especially importan t one for
understanding the impact of go vernmen t financin g decisions.
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