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Working Paper Series
Money and the Gain from Enduring
Relationships in the Turnpike Model
WP 94-07
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Peter N. Ireland
Federal Reserve Bank of Richmond
WORKING PAPER
94-7
MONEYAND THE GAINFROMENDURING
RELATIONSHIPS
IN THE TURNPIKE
MODEL
PETER N.
IRELAND*
Research Department
Federal Reserve Bank of Richmond
June 1994
*I would like to thank Mike Dotsey, Ken Judd, Jeff Lacker, Tom Sargent, and
Stacey Schreft for helpful comments and suggestions. The views expressed here
are my own and do not necessarily represent those of the Federal Reserve Bank
of Richmond or the Federal Reserve System.
Abstract
This paper presents a stochastic version of Townsend's turnpike model
in which the aggregate endowment is distributed randomly between two sets
of agents and in which agents of each type are allowed to remain at a
trading post for multiple periods. Agents use money as a means of exchange
when they meet as strangers but use private securities when they remain
paired at the same trading post.
Both welfare and the income velocity of
money increase monotonically with the length of the trading session.
I.
Introduction
With his turnpike model of exchange, Townsend (1980) demonstrates
that noninterest-bearing fiat money is valued in equilibrium when agents'
itineraries between spatially separated trading posts completely rule out
the existence of markets for private debt.
In contrast to Townsend's model
are those featuring a full set of Arrow-Debreu securities, where trade is
sufficiently centralized so as to never require the use of a low-yielding
outside asset as a means of payment. This paper presents a model
intermediate to these two extremes, in which some trades are effected
through the transfer of private securities while others are possible only
when government-issued money is available. Specifically, it modifies the
turnpike environment by randomly distributing the aggregate endowment
between two sets of agents and by allowing agents of each type to remain at
a trading post for multiple periods.
Comparing Townsend's original turnpike model to the Arrow-Debreu
framework suggests a distinction between money, which is used to facilitate
trade between agents who meet as strangers in isolated markets, and private
securities, which are traded among agents who meet repeatedly in a
centralized market.
The modified turnpike model illuminates this
distinction. Agents in the modified turnpike model use money when they
meet as strangers but use private securities when they remain paired at the
same trading post.
The importance of money in the modified turnpike
economy, therefore, depends directly on the frequency with which agents
meet as strangers and inversely on the length of each multi-period trading
session.
In fact, the length N of the trading session indicates exactly where
each turnpike model lies between the two extremes. Those economies with N
close to unity resemble Townsend's original turnpike economy: agents are
frequently meeting as strangers and money plays an important role in
facilitating trade. Those economies with larger N become increasingly
similar to the Arrow-Debreu case: trade becomes more centralized, the scope
for private debt expands, and the role of money diminishes.
There are two other respects in which the modified turnpike economies
are intermediate to Townsend's pure monetary model on the one hand and the
Arrow-Debreu model on the other, with the session length N measuring
distances between the two extremes. First, Gale (1978) notes that the
differences between monetary and Arrow-Debreu models are summarized by the
form of an agent's budget constraints. In a pure monetary environment like
Townsend's, each agent faces a sequence of budget constraints that must be
balanced period-by-period. In an Arrow-Debreu setting, each agent faces a
single lifetime budget constraint. Agents in the modified turnpike model
face a sequence of budget constraints, but each constraint requires only
that expenditures and receipts balance over the entire course of a multiperiod trading session. Thus, as the trading session is lengthened, the
constraints look less like those in a pure monetary environment and more
like those in an Arrow-Debreu environment.
Second, Hahn (19731 shows that competitive equilibria in pure
monetary economies are generally not Pareto optimal, whereas equilibria in
Arrow-Debreu economies typically are.
In the modified turnpike model,
welfare increases monotonically and approaches the level achieved by Pareto
optimal allocations as N grows larger. Again, the modified turnpike model
2
is seen to be intermediate to the two extremes.
While the implications of longer trading periods in the turnpike
model are explored in great detail by Manuelli and Sargent (19921, the
periodic, deterministic endowment specification that they employ gives rise
to peculiar nonmonotonicities as the trading session is lengthened. For
example, when the length of the trading period coincides with the length of
the cycle in the endowment process, the gains from trade can be completely
exhausted without government-issued money. But when the trading session is
extended one period beyond the length of the endowment cycle, money is
again needed.
In this case, an increase in session length yields an
increase in the demand for money, obscuring the connection between the
value of money and the prevalence of trade among strangers. The additional
stochastic endowment feature introduced here restores this connection, so
that the turnpike economies clearly represent cases intermediate to pure
monetary and Arrow-Debreu environments, with the session length N measuring
distances between the two extremes.
The modified turnpike model is specified in the next section.
Section III describes both nonmonetary and monetary competitive equilibria
in the modified turnpike environment; agents' consumption patterns in these
equilibria are compared to the consumption patterns provided by Pareto
optimal resource allocations. Section IV explores how competitive
equilibria change as the trading session is lengthened. Section V
concludes.
3
II. The Modified Turnpike Model
Following Townsend (1980) and Manuelli and Sargent (19921, the
economy is assumed to consist of a large number of infinitely-lived agents
who are distributed uniformly into a countable number of trading posts
located at discrete intervals along a turnpike of infinite length. These
trading posts are spatially isolated, meaning that agents at one post can
neither trade nor communicate with agents at other posts.
The agents are of two types, labelled i=A and i=B. At any given time
t=1,2,3..., there are equal numbers of each type at each trading post.
At
the end of each period t=N,2N,3N,..., where Nz-1,each agent of type A
leaves his trading post, arriving at the next post to the east along the
turnpike at the beginning of the following period. Simultaneously, each
type B agent moves one market to the west.
The length N of a trading
session, therefore, measures the frequency with which agents meet as
strangers along the turnpike.
In each period t=1,2,3,..., an endowment shock ste{a,b) is revealed
to the agents.
The timing of this revelation is such that when an agent is
moving between two posts he leaves the first before, but arrives at the
second after, st becomes known.
If st=a, then each type A agent in the
economy receives an endowment of one unit of a perishable consumption good
in period t, while type B agents receive no endowment. If st=b, then each
type B agent receives a unit of the good in period t, while type A agents
get nothing.
The shock st is iid over time and has a binomial distribution
with equal probability of either outcome.
Preferences over state-contingent consumption plans are common to all
4
agents and are represented by the additively time-separable expected
utility function
where @(O,l)
is the discount factor, ct is consumption at date t, and u(e)
is strictly increasing, strictly concave, and twice continuously
differentiable with lim
u'(c)=~.
C+O
III. Competitive Equilibrium: Nonmonetary and ,Monetary
As in the earlier studies by Townsend (1980) and Manuelli and Sargent
(1992), attention is confined here to symmetric equilibria, meaning those
in which agents with identical endowments and (in the monetary case)
identical money holdings receive identical consumption allocations. Also,
just as the earlier studies focus on equilibria with periodic outcomes
reflecting periodic endowment specifications, the analysis here is focused
on equilibria in which outcomes are described as time-invariant functions
of a minimal number of state variables, reflecting the simple structure of
the stochastic endowment process. Under these additional assumptions,
competitive equilibria for the modified turnpike economy may be completely
characterized by describing the optimizing behavior of two representative
agents, one of each type, and the market clearing process at a single
representative trading post.
5
A.
Nonmonetary Equilibrium
Itineraries are such that two agents of different types will never
meet more than once along the turnpike. Moreover, there will never be a
third agent who meets the first two at different times. Consequently,
private debt contracts cannot be extended beyond a single N-period trading
session in this environment. Each agent's expenditures and receipts must
balance over the course of the N periods. This constraint on trade, as
well as the time-separability of preferences, the iid structure of shocks,
and the absence of storage possibilities implies that the infinite horizon
economy without money may be studied as a sequence of N-period economies in
which agents trade securities at each date teT={l,N+1,2N+l,...) for statecontingent consumption over the following N-l periods, taking the realized
history of shocks st={sI,sz,...,st)as given (since st is revealed before
agents arrive at their new posts at the beginning of the time t trading
session).
Thus, for any t=1,2,3,..., let y'(st) and c'(st) denote the endowment
and consumption of a representative type i agent in period t after the
history st has been realized. Since the endowment process is iid,
yi(st)=y'(st),with yA(a)=yB(b)=l and yA(b)=yB(a)=O.
For any t=1,2,3,..., let ht denote the set of all possible histories
st through time t.
For any teT, steht, and j=O,l,...,N-1, let hJ(st)
t+J) that can possibly
denote the set of all histories st+'={si,...,st,...,s
follow St. For any st+'ehJ(st),let qt(st+'ldenote the price, in the time
t securities market following the realization of st, of a claim to one unit
of consumption in time t+j upon the realization of st+J , in terms of
consumption. Note that by definition, qt(st)=l.
6
time
t
During each trading session, each agent chooses a state-contingent
consumption plan that maximizes his utility subject to the requirement that
his expenditures and receipts balance over the course of the N periods.
That is, at each date teT, the representative type i agent solves
Problem 1: Given stsht and given Y'(s~+~)and qt(stsJ)for all
j=O,l,...,N-1 and st+'ehJ(st),choose ci(st+J) for all j=O,l,...,N-1 and
S t+JehJ(st)to
maximize
"z'
c
J=O $+JEh' (St)
(B/2~Jutc'(st+J~1
subject to
7i
J=O
qt(st+J)yW+Jl
c
st+‘,h’
=
tSt 1
qt(st+JIcW+J~.
Y
c
J=o st+J,hJ~st~
The objective function in problem 1 indicates that the probability that any
history st+'ehJ(st) will follow st is just (l/2)'. To repeat, the budget
constraint indicates that because the spatial organization of trade rules
out the use of private credit agreements across trading sessions,
expenditures and receipts must balance over the course of the N periods.
As N approaches infinity, the constraint becomes increasingly similar to
the lifetime budget constraint faced by agents in an Arrow-Debreu economy.
The interpretation of the infinite horizon economy as a sequence of
N-period economies is also exploited in defining a nonmonetary competitive
equilibrium.
In a nonmonetary competitive equilibrium, each agent solves
problem 1 at each date teT, and markets clear at every date and state:
Definition 1:
A nonmonetary competitive equilibrium consists of prices
7
qt(st+')and quantities yi(st*')and c'(st+')for i=A and i=B, all teT,
steht, j=O,l,...,N-1, and st+'EhJ(st)such that
(i) For i=A and i=B, the c'(st+')solve problem 1 given the qt(st+')
and yi(st+').
(ii) The yi(st+')and c'(st+')satisfy
cAG+J1
=
+ cB&+')
yW+J)
+ yB(St+J)
=
1.
The market clearing condition listed as (ii) in definition 1 reflects the
fact that there are equal numbers of agents of each type at each trading
post, so that the aggregate endowment at the representative trading post
can be normalized to unity.
The representative type i agent's first order conditions for problem
1 are given by
(fm’ll
(11
tc’b3t+JH
=
qtkt+Jw
[CWH,
for all j=O,l,...,N-1 and st+'shJ(st).
Equation (l), along with the market clearing condition from definition 1,
implies that qt(st+')=(B/2)'and
(2)
cw+'l
=
CWI,
for all j=O,l,...,N-1 and st+'shJ(st).
Thus, for each agent, consumption is constant across all date-state
combinations within a given trading session. The budget constraint from
problem 1 pins down the constant level of consumption as
(3)
CWI
=
~(l-Ply'(st)+(~/2)(1-BN-1)l/(1-BN).
It is useful to compare the equilibrium allocation described by (2)
and (3) to the Pareto optimal allocation under which all agents are treated
alike.
This allocation would be supported, for example, in the competitive
8
equilibrium of an economy having the same endowment specification as the
turnpike model but in which the spatial constraints on exchange are relaxed
to permit agents to trade in a centralized Arrow-Debreu securities market.
The Pareto optimal allocation maximizes the sum of the two representative
agents' expected utilities subject to an aggregate resource constraint for
each date-state combination. Thus, it may be characterized by solving
Problem 2:
Choose cA(st) and cB(stl for all t=1,2,3,... and stsht to
maximize
w
00
c
t=1
c
(P/21tu[cAktll
+
1
steht
1
Q3/2Pu[cB(stIl
t=l stfht
subject to
1
1
CAbA
+ CB(2)
for all t=1,2,3,... and stsht.
The solution to problem 2 sets cA(st)=cB(st1=l/2for all t=1,2,3,...
and steht. In contrast to the optimal allocation, which equates each
agent's marginal utilities across all date-state combinations, equilibrium
consumptions depend on the realization of the time t shock s
and hence
t'
marginal utilities are equated only during those periods in which groups of
agents remain together at a trading post.
In equilibrium when N=l,
equation (31 reveals that agents must simply consume their own endowments
in every period.
N
The exchange of private securities supports more trade as
grows larger, but even when N=w the optimum is not achieved since agents
cannot trade before the realization of s
1'
9
B.
Monetary Equilibrium
Noninterest-bearing government-issued money may be introduced into
this economy via lump-sum transfers of rn:to each type i agent just prior
to the initial period.
The constant size of the aggregate money supply is
normalized so that mt+mf=l. The potential value of money in this economy
rests on its status as the only available outside asset.
Unlike private
securities that must be redeemed before agents move on, money can be
carried across trading sessions and thereby permits otherwise infeasible
transactions to occur between strangers.
While the infinite horizon monetary economy may still be studied as a
sequence of N-period economies with state-contingent securities markets
open at each date tsT, now each member of the sequence is linked to its
predecessor and its follower by the distribution of cash balances across
agent types at the beginning and end of each trading session.
In the time
t securities market, agents trade claims for state-contingent consumption
over the following N-l periods and claims for state-contingent delivery in
period t+N-1 of money to be carried into period t+N.
Extending the notational conventions established above, let m'(st-'1
denote the nominal balances held at the end of period t-l, and hence
carried into period t, by the representative type i agent after the history
S t-1 has
been realized. For teT, stcht, and j=O,l,...,N-1, let pt(st+j)be
the price, in the time t securities market following the realization of st,
of a claim to one unit of money in time t+j upon the realization of
S t+J,hJ
kt),
in terms of time t consumption. Note that pt(st) is just the
inverse of the price level at time t.
As sources of funds during the time t trading session, an agent has
10
the money that he carries into the trading session as well as the receipts
from the sale of his endowment. As uses of funds, an agent has the money
that he carries out of the trading session as well as his expenditures on
consumption. During each trading session, an agent chooses a statecontingent consumption and money-holdings plan that maximizes his utility
subject to the constraint that his sources and uses of funds balance over
the course of the N periods. Thus, the representative type i agent solves
Problem 3:
Given rn:and given y'(st+'),qt(s'+'),and pt(st+')for all
teT, steht, j=O,l,...,N-1, and st+'ehJ(st),choose c'(st+')and rn'(~~+~-l)
for all teT, stsht, j=O,l,...,N-1, st+'ehJ(st),and st+N-lehN-'(st)
to
maximize
c INil c
tET steht J=O
(p/2)t+Ju[c’
(St+‘)
I
st+'ehJ(st)
subject to
pt(st)ml(st-l)+
qp
Y
c
j=O st+JshJ(st)
t+Jly’
Q’J)
q@+J)cW+J)
t+N-l~mi~St+N-lI~
+
J=o st+JEhJ(st)
c
t+N-1
S
Pt(s
EhN-’(st)
for all teT and stsht
and
t+N-1
m'(s
1 z
0
for all tcT, st.sht,and st+N-iehN-l(st).
The objective function in problem 3 indicates that the probability that any
history st+' will be realized through time t+j is just (1/2)t+J. The
11
budget constraints indicate that, while expenditures and receipts must
still balance over the course of the N-period trading session, the
availability of valued money now permits agents to carry purchasing power
across trading sessions. In the modified turnpike environment with N=l, as
in the pure monetary economies studied by Gale (1978) and Townsend (1980),
money is used as a means of exchange in every period. As N gets larger,
the constraints reveal that money will change hands less frequently.
A monetary equilibrium is defined as a set of prices and quantities
such that each agent is solving problem 3 and such that all markets clear:
Definition 2:
A monetary competitive equilibrium consists of the initial
conditions rn:and rn:,the prices qt(st+')and pt(st+'),and the quantities
yi(st+'),c'(st+'),and mi(st+"-'
) for i=A and i=B, all teT, stsht,
j=O,l,...,N-1, st+'ehJ(st),and st+N-lehN-'(st)
such that
(i) For i=A and i=B, the ci(st+')and mi(st+N-l)solve problem 3 given
the m:, qt(st*'),pt(st+'),and yi(st+').
(ii) The yi(st+'),ci(st+'),and mi(st+N-l)satisfy
CA(S t+J ) + CB(st+J)
=
yA(st+J)
+ yB($+J)
=
1
and
t+N-1
mA(s
) + mB(st+N-l) =
1.
As before, the market clearing conditions in definition 3 reflect the fact
that the aggregate endowment and money supply are normalized to unity.
The type i representative agent's first order conditions for problem
3
are
(41
(Is/2IJu’
[cw+J~l
=
q$?+%Y
[c’(Al,
for all teT, steht, j=O,l,...,N-1, and st+'ehJ(st)
12
and
(5)
Pt(st+N-')~'[~i(~t)]h
Et+N-I (/3N/2N-1)pt+N(st+N1~'
[c'(s~+~)]
with equality if m'(st+N-l)>O,for all teT, stEht, and st+N-lshN-l(st),
where E
t+N-1
(0) indicates that the expectation is conditional on time t+N-1
information, that is, conditional on the realization of s~+~-'.
As in the nonmonetary case, equation (4) along with the market
clearing conditions for goods implies that qt(stcJ)=(~/21Jand
(6)
ci(st+J)
= C’(2),
for all tcT, steht, j=O,l,...,N-1, and st+'ehJ(st).
The absence of arbitrage opportunities in the money market guarantees that
c
pt(21
=
(7)
S
t+N-1
t+N-1
Pt(s
)
chN-‘(st)
will hold for all teT and stsht. Otherwise, any agent could buy (sell)
nominal balances at time t after st has been realized, simultaneously sell
(buy) claims to equal quantities of nominal balances for each
S t+N-lehN-'(st),
and
St+N-l ehNW1(st)and
thereby make a certain profit. Summing (5) over all
using (7) yields a stochastic Euler equation of the
form
(81
p$h’
[ci(st)l =
Et ,!3Npt+N(~t+N)~'
[c’~~+~H
I
which holds with equality whenever m'(st+N-lI>Ofor all st+N-lshN-l(st).
Equations (2) and (6) reveal that in both nonmonetary and monetary
equilibria, marginal utilities are equated across periods, and hence the
gains from trade are exhausted, within each trading session. Equation (8)
summarizes the extent to which the availability of outside money permits
13
additional welfare-improving exchange across sessions as trading partners
change. The nonnegativity constraint on money holdings makes this intersession Euler equation an inequality. As emphasized by Zeldes (19891, even
if the nonnegativity constraint does not bind in the current period, the
possibility that it will bind in some future period is enough to lower
consumption today. That is, even when it holds with equality, equation (8)
indicates that the gains from trade are only partially exhausted across
trading sessions. Thus, government-issued money is a valuable, but
imperfect, means of exchange between agents who are meeting as strangers.
No arbitrage conditions may be used to deduce what within-session
paths for prices and money holdings are if trading in money, goods, and
one-period-ahead contingent claims takes place at each date t=1,2,3,....
For teT and j=1,2,...,N-1, let p,+'(~~"1 be the spot price of money in
terms of goods in period t+j when history st+' has been realized. Since
agents have no reason to prefer money to privately-issued securities within
a given trading session, cash must yield the same return as is available
from any other asset between t and t+j; the Euler equation
pt(21u’
&211
=
/3’pt+)(st+‘h~
&2”~1
must hold in equilibrium. Equation (6) then implies that
pt(st)=gJpt+j(~t+JI;
deflation prevails within each session.
Also because private securities can substitute perfectly for money
within a trading session, it may be assumed without loss of generality that
m' (st+J-l
)=mi(st-')for all tcT, all j=O,l,...,N-1 and all st+"l that
follow S?
That is, money only changes hands at the end of each trading
session. Money in the modified turnpike model is used exclusively to
facilitate trade across trading sessions; within trading sessions, private
14
securities do all the work.
IV.
The Effects of Lengthening the Trading Session
It is easy, using equations (2) and (31, to numerically compute a
nonmonetary equilibrium for the model economy with trading sessions of
length N.
Once a realization for {st>yZIis drawn, the implied values for
the yi(st) can be plugged into (3) to determine the ci(st), which by (2)
completely describe consumptions at each date-state combination within each
trading session. Finding monetary equilibria is a more difficult task,
however, since the ci(st), rn'(~~+~-'
1, and pt(st) must be constructed to
satisfy the stochastic Euler equation given by (81 for both i=A and i=B.
For teT, define
(9)
The symmetry and stationarity assumptions made in the previous section
justify the conjecture that because the time t division of the money supply
and the time t realization of the endowment shock are the only variables
that distinguish the time t trading session from any other, equilibrium
values for ci(stI, mi(st+N-lI,and pt(stl may be expressed as timeinvariant functions of mt=mA(st-')=l-mB(st-l)
and <
CA($)
(10)
m
t+N
Pt(21
=
1 - CB(SC) =
t+N-1
=
mA<s
=
I =
t'
so that
c$,E,l
1 - mB(st+N-') =
p(mt,<,)
pht.Et)
The functions c(m,<), p(rn,<),and p(m,c) that satisfy equations (8)-
15
(10) may be solved for using the numerical procedures outlined in the
appendix. Once these functions have been found, the initial condition
mi=m: and a sample realization for (st)yZ1may be fed into equations (9)
and (10) to generate the equilibrium values for ci(st), mi(st+N-l),and
ptdl.
In particular, the function p(m,<) governs how the distribution of
money balances across agent types evolves over time. For teT, let At(m)
denote the probability, prior to the initial period, that type A agents
will each carry m units of money into the time t trading session.
In each
of the examples considered below, the function p(m,<) is such that At(m)
converges to a limiting distribution as t approaches infinity, regardless
of the initial conditions rn:,just as in Foley and Hellwig (1975). That
is, in each case the economy reaches a stochastic steady state in which
prices and quantities are strictly stationary random variables.
The numerical procedures are implemented to construct nonmonetary and
monetary equilibria for values of N ranging from 1 through 6.
The utility
function is specialized to the CES form,
u(c)
=
(c'-c-1)/(1-c),
cr>o.
In order to focus on the effects of changing N, the preference parameters
are held fixed, with p=O.95 and (r=2.
Figures l-6 display the functions c(m,<) and p(m,<) for the monetary
equilibria.
In each case, c and p are increasing in both of their
arguments (except for minor nonmonotonicities that are artifacts of the
numerical approximation procedure). An agent with larger cash holdings
consumes more than an agent with less money. An agent who receives an
endowment at the beginning of a trading session both consumes more and
16
accumulates money relative to an agent who receives no endowment. Since
additional trade becomes possible as N increases, the function c(m,c) is
closer to being constant at the Pareto optimal level of l/2 in the
economies with larger N than in those with smaller N.
Also, the
constraints 15~ and ~20 are more likely to be binding as N increases; as
the scope for private debt expands, agents no longer need to constantly
hold precautionary balances.
With both the aggregate endowment and the money supply fixed at
unity, the equation of exchange MV=PY reduces to V=P, indicating that the
income velocity of money is exactly equal to the price level in the
modified turnpike economies. Since the price level at time tsT in state
steht is just l/pt(stl, velocity is described by the time-invariant
function v(m,~l=l/p(m,<l. Table 1 reports selected moments for the price
level/velocity process and a representative agent's endowment and
consumption process in each example economy's stochastic steady state.
An agent's endowment is either 0 or 1 with equal probability, so the
variance of this process is 0.25 in every case.
In both nonmonetary and
monetary equilibria, the variance of consumption decreases monotonically as
N increases, reflecting the additional trade that becomes possible with
longer trading sessions. The gains are largest moving from N=l to N=2 and
from N=2 to N=3.
As N increases, equilibrium consumption patterns approach
those provided by Pareto optimal allocations, which eliminate all
variability in individual agents' consumption streams.
Comparing the variance of consumption in nonmonetary and monetary
equilibria for any fixed N reveals the extent to which the availability of
an outside asset expands opportunities for exchange. Money is especially
17
important in economies with low values of N, where agents are frequently
meeting as strangers. Hence, the price level and the velocity of money are
low in these equilibria and increase monotonically with N.
The additional welfare-improving exchange made possible by
lengthening the trading session and introducing money is also illustrated
in table 2, which reports a representative agent's expected utility in the
stochastic steady state of each example economy. The welfare gain is most
dramatic when money is introduced into the economy with N=l, since the
nonmonetary equilibrium there is autarkic. As suggested by the comparison
of consumption patterns, expected utility in nonmonetary equilibria,
monetary equilibria, and under Pareto optimal allocations converge as N
increases.
V.
Conclusions and Directions for Future Research
Among the economies studied here, the stochastic turnpike model with
N=l most closely resembles Townsend's (1980) original turnpike environment.
Agents in this economy continually meet as strangers at isolated trading
posts. Markets for private securities are ruled out completely.
Government-issued money is critical in facilitating trade. Monetary
equilibria, though far superior to nonmonetary equilibria, are generally
not Pareto optimal.
In stark contrast to the N=l economy is one in which the spatial
constraints on exchange are relaxed to permit agents to trade a full set of
Arrow-Debreu securities in a single centralized market.
18
There are no
strangers in this alternative environment. The gains from trade are
completely exhausted through the transfer of privately-issued financial
assets. There is no need for money. Competitive equilibria are Pareto
optimal.
The results presented in section IV show how the stochastic turnpike
economies with Nr2 represent cases intermediate to the pure monetary
environment on the one hand and the Arrow-Debreu environment on the other.
Agents encounter strangers less frequently as N increases. More trade
becomes possible through the exchange of private securities. Money becomes
less valuable, so that the price level and velocity both rise. Welfare
increases, approaching the level achieved by Pareto optimal allocations.
All of these changes proceed smoothly as the trading session is lengthened,
making N an exact index measuring distances between the two extreme cases.
In actual economies, payments systems usually -featureboth privatelyissued and government-issued assets as means of exchange. The model
economies presented here capture this reality; the original turnpike model
and the Arrow-Debreu framework do not.
The results from section IV imply
that asset velocities ought to vary to the extent that decentralized trade
among strangers becomes more or less common, both across countries at any
given point in time and over time within any given country. Evidence
consistent with this implication is found, in fact, by Townsend (1983) and
Bordo and Jonung (1987).
Since the analysis in this paper takes the nominal money supply as
fixed, two issues are left for future research. The first concerns the
effects of monetary injections, either once-and-for-all or ongoing.
If,
following Scheinkman and Weiss (19861, preferences are respecified to
19
include leisure in the utility function and a technology is specified so
that st summarizes idiosyncratic productivity shocks instead of endowment
shocks, then monetary policies that change the cross-sectional distribution
of money holdings will have real aggregate effects.
It will be more
/
difficult, however, for monetary policy to generate significant
distributive effects when real balances represent a smaller fraction of
each agent’s total asset holdings. Thus, the magnitude and duration of the
nonneutralities--the efficacy of monetary policy--is likely to diminish as
N increases.
Once the effects of arbitrary monetary policies have been analyzed,
the next question concerns what constitutes the optimal monetary policy in
this environment. The answer is likely to depend sensitively on the
informational constraints that the government is assumed to face: whether
it can observe the realization of the endowment shock, whether it can keep
track of histories of shocks, and whether it can observe individual agents’
asset holdings. Work by Levine (19911, which also uses a model in which
agents have a demand for precautionary balances, suggests that if the
government is constrained‘to treat all agents alike, the traditional result
of Friedman (19691 calling for a steady contraction of the money supply may
be overturned in favor of an inflationary policy that helps to prevent
unlucky agents from running out of cash.
20
Appendix: Numerical Methods
Monetary equilibria for the stochastic turnpike economy are
constructed numerically as solutions to a system of functional
eqUatiOnS.
Substitute the functions defined by equation (10) into the representative
type A agent's budget constraint, which will always hold with equality, to
obtain
(l-pN)c(mt,‘t)
BW3N-1)
(A.11
p(mtXt)mt + C, +
=
+ P(m,.E,)cl(m,.S,),
2(1-B)
l-8
have
where equations (61, (71, and (9) and the result that qt(st+'I=(/3/2)'
all been used.
Substitute the functions into the stochastic Euler equation
given by (8) to obtain
(A.21
p(mt,Et)u'[c(mt,Ft)l
2
Et
BNP(~(m,,E,l.S,+,lu~
[c(Ccht,St)94t+N11
I
I
and
(A.31
p(mt,St)u'[l-c(mt,E,)l
2
Et
BNp(Ccht,Etl
,Ct+Nlu’ [l-c(phttEt),Et+N)l
.
I
t
Since (A.l)-(A.31 must hold for all mte[O,ll and Ets{O,l), they may
be rewritten as the functional equations
p(1-8N-31 =
(A.41
p(m,S)m + C +
2(1-B)
(A.51
p(m,c)u' [c(m,<Il
=
IsN
-
2 II
(l-f3NIc(m,~)
+ p(m,C)l-c(m,C),
1-s
- r(rn,S)
+ p(~(rn,~),O)u'tc(~(rn,~),O~l
] ,
p(Cr(m,S),l)u'Ic(CL(m,S),l)l
and
(A.61
p(m,<)u'[l-c(m.E)l - (p(m,C)
21
= -SN
2 C
p(~(m,~),l)u'[l-c(~(m,5),1~1+ p(~(m,~),O)u'[l-c(~(m,S),O)l] ,
where
(A.71
Y(m,S)p(m,<) =
;r(m,EkO,
0,
and
p(m,<)[l-P(m,<)l =
(A.81
0,
(p(m,<)=O.
In addition, the symmetry assumption requires that
c(m,<I =
(A.91
(A.101
p(rn,E) =
(A.111
p(m,Cl
1 - c(l-m,l-cl,
1 - I.r(l-m,l-El,
=
p(l-m,l-cl,
'b'(m,Sl=
p(l-m,l-El.
and
(A.121
Solutions to the system (A.4)-(A.121 are found numerically by
modifying the spline approximation technique discussed in Hildebrand (1974,
Ch.91.
The domain [O,ll is divided into five intervals of equal length.
The functions p(m,O) and p(m,O) are approximated by a cubic polynomial on
each interval:
(A.131
p(m,O)
x
ak + bkm + ckm" + dkm3
p(m,Ol
x
ek + fkm + gkrn'+ hkm3
where k=l if me[0,0.2), k=2 if ms[0.2,0.4), k=3 if me[O.4,0.61, k=4 if
me[O.6,0.8), and k=5 if me[0.8,11. Given these approximations for P(m,O)
p(m,l), and c(m,l) are
and p(m,O), approximations for c(m,O), r.l(m.11,
constructed so that equations (A.41 and (A.9)-(A.111 are satisfied.
The 40 unknown coefficients {ak,b~,c~,d~.e~,f~,g~,h~}~=~
are
determined by requiring that p(m,O) and p(m,O) be continuous at m=0.2,
m=0.4, m=0.6, and m=0.8 (a total of 8 equations that must be satisfied) and
22
by requiring that p(m,O) and p(m,O) satisfy (A.5) and (A.6) at 16
uniformly-spaced points on [O,ll (a total of 32 equations that must be
satisfied). Approximations for r(m,O) and p(m,O) are determined so that
(A.7) and (A.8) also hold at the same 16 uniformly-spaced points.
Finally,
approximations for r(m,l) and (p(m,l)are constructed to satisfy equation
(A.12). Throughout, the constraints l>c(m,S)>O, llp(m,<)rO, and p(m,<)>O
for all me[O,ll and <e{O,l) are imposed.
The limiting distribution of money holdings, h(m), is found by
confining the approximation for p(m,S) to the finite set
M={0,0.01,0.02,...,1) and solving the equations
A(m')
=
i
1
A(m) + i
msMo(m )
C
A(m)
meM1(m')
for all h(m') such that m'44, where
Mo(m') = { m&l : p(m,O)=m' )
and
Ml(m’ 1 = 4 meM : cL(m.l)=m').
This solution procedure specializes the projection methods outlined
by Judd (1992) by choosing the set of piecewise cubic functions in equation
(A.13) as a basis and a collocation method for the projection conditions.
Piecewise cubic approximation allows for considerable nonlinearity in the
functions describing consumptions, money holdings, and prices.
Imposing
continuity on the functions but not (as is usually done in spline
approximation) their first and second derivatives allows for kinks
associated with binding nonnegativity constraints on money holdings.
23
References
Bordo, Michael D. and Lars Jonung. The Long-Run Behavior of the Velocity of
Circulation. Cambridge: Cambridge University Press, 1987.
Foley, Duncan K. and Martin F. Hellwig. "Asset Management with Trading
Uncertainty." Review of Economic Studies 42 (July 1975): 327-346.
Friedman, Milton. "The Optimum Quantity of Money." In The Optimum Quantity
of Money and Other Essays. Chicago: Aldine Publishing Company, 1969.
Gale, Douglas. "The Core of a Monetary Economy without Trust." Journal of
Economic Theory 19 (December 1978): 456-491.
Hahn, Frank. "On Transaction Costs, Inessential Sequence Economies and
Money." Review of Economic Studies 40 (October 1973): 449-461.
Hildebrand, F.B. Introduction to Numerical Analysis 2d ed. New York: Dover
Publications, 1974.
Judd, Kenneth L. "Projection Methods for Solving Aggregate Growth Models."
Journal of Economic Theory 58 (December 1992): 410-452.
Levine, David K. "Asset Trading Mechanisms and Expansionary Policy."
Journal of Economic Theory 54 (June 19911: 148-164.
Manuelli, Rodolfo and Thomas J. Sargent. "Alternative Monetary Policies in
a Turnpike Economy." Manuscript. Stanford: Stanford University,
Graduate School of Business and Hoover Institution, April 1992.
Scheinkman, Jose A. and Laurence Weiss. "Borrowing Constraints and
Aggregate Economic Activity." Econometrica 54 (January 1986): 23-45.
Townsend, Robert M. "Models of Money with Spatially Separated Agents." In
John H. Kareken and Neil Wallace, eds. Models of Monetary Economies.
Minneapolis: Federal Reserve Bank of Minneapolis, 1980.
. "Financial Structure and Economic Activity." American Economic
Review 73 (December 19831: 895-911.
Zeldes, Stephen P. "Consumption and Liquidity Constraints: An Empirical
Investigation." Journal of Political Economy 97 (April 19891: 305346.
24
Table 1.--Moments for Endowments, Consumption, and Velocity
fi=o.95,(r=2
N
Variance of
Endowment
Process
Variance of
Consumption
Process
Nonmonetary
Equilibria
1
2
3
4
5
6
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.0657
0.0307
0.0182
0.0122
0.0089
Monetary
Equilibria
1
2
3
4
5
6
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.0142
0.0128
0.0101
0.0090
0.0077
0.0067
0.2500
0.0000
Pareto Optimum
Mean of
Price/Velocity
Process
0.2176
0.6472
1.1004
1.9817
3.4450
5.9124
Table 2.--Expected Utilities
p=o.95, s=2
N
Nonmonetary
Equilibria
Monetary
Equilibria
Equal Weight
Pareto Optimum
-32:;593
-24.3245
-21.9773
-20.9517
-20.4037
-22.4106
-21.3638
-20.6967
-20.4450
-20.2287
-20.0538
-19.0000
-19.0000
-19.0000
-19.0000
-19.0000
-19.0000
1
2
3
4
5
6
Fig.
7. N=l
0.8
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
m
Cl
c(m,o)
A
c(m.1)
0.8
0.6
0
0.1
0.2
0.3
0.5
0.4
m
0
mu(m.0)
A
mu(m,l)
Fig.
2. N=2
0.8
0.7
0.6
0.5
0.4
0.3
I
,
0
0.1
0.2
L
1
I
I
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.0
0.9
1
m
0
c(m.0)
A
c(m.1)
1.1
1
0.9
0.8
0.7
0.6
1
-
0.5
,
-
0.4
0.3
0.2
0.1
0
0.1
0.3
0.2
0.5
0.4
0.6
m
0
mu(m.O)
A
mu(m,l)
Fig.
3.
N=3
0.7
0.65 -
0.6 -
0.55 -
0.5 -
0.45 -
0.4 -
0.35 -
I
1
,
I
I
I
&
I
I
I
I
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.8
0.9
0.3
m
0
c(m.0)
A
c(m,l)
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
m
0
mu(m,O)
A
mu(m.1)
0.7
1
Fig.
0.66
4.
N=4
,
0.64
-
0.62
-
0.6
-
0.58
-
0.56
-
0.54
-
0.52
-
0.5
-
0.48
-
0.46
-
0.44
-
0.42
-
0.4
-
0.38
-
0.36
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
m
0
c(m.0)
A
c(m.1)
“’rl-
0.9
-
0.8
-
0.7
-
0.6
-
0.5
-
0.4
-
0.3
-
0.2
-
0.1
-
0
0
0.1
0.2
0.3
0.4
0.5
0.6
m
0
mu(mD)
A
mu(m.1)
0.7
0.8
0.9
1
Fig.
5.
N=5
0.64
0.62 0.6 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 0.38
1
0
1
0.1
,
0.2
I
0.3
I
0.4
I
0.5
I
I
,
I
I
0.6
0.7
0.8
0.9
1
m
0
c(m,O)
A
c(m,l)
0.8
0.6
0
0
0.1
0.2
0.3
0.4
0.5
0.6
m
0
mu(m.O)
A
mu(m.1)
0.7
0.8
0.9
7
Fig.
0.61
6. N=6
,
0.6
-
0.59
-
0.58
-
0.57
-
0.56
-
0.55
-
0.54
-
0.53
-
0.52
-
0.51
-
0.5
-
0.49
-
0.48
-
0.47
-
0.46
-
0.45
-
0.44
-
0.43
-
0.42
-
0.41
-
0.4
’
0
0.1
I
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
m
0
0.8
-
0.7
-
0.6
-
0.5
-
0.4
-
0.3
-
0.2
-
0.1
c(m.0)
A
c(m.1)
I-
O
m
0
mu(m,O)
a
mu(m.1)
I
0.8
I
I
0.9
1
@FedFRASER