Theoretical Economics Letters, 2013, 3, 226-228
http://dx.doi.org/10.4236/tel.2013.34038 Published Online August 2013 (http://www.scirp.org/journal/tel)
Inseparability of Transaction Medium and Store of Value
in the Role of Money
Institute of Social Science, University of Tokyo, Tokyo, Japan
Received April 27, 2013; revised May 27, 2013; accepted June 27, 2013
Copyright © 2013 Masayuki Otaki. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We analyze whether or not money can be circulated as a purely transaction medium in the search theory. We obtain the
following result: unless the role of a store of value added to the function of money, money does not circulate and the
economy degenerates into a barter economy.
Keywords: Search Theory of Money; Degeneration of Monetary Economy into Barter Economy; Inseparability of
Transaction Medium and Store of Value
The paper explores the function of money in the search
model. There exist two main functions of money: trans-
action demand for clearing and demand for store of value.
Kiyotaki and Wright , which is the seminal work on
the search money model, insist that money can be circu-
lated solely by the motive of transaction even though it
does not operate as a store of value.
However, if money is perishable while differentiated
goods are durable, some advantage exists on the side of
goods. Namely, the visiting opportunities of exchange in
a barter economy are more frequent those in a monetary
economy. This leads us to the following hypothesis:
unless money possesses both of the above properties, it is
unable to circulate. Based on Kiyotaki and Wright ,
we ascertain the validity of this hypothesis.
This paper is organized as follows. In Section 2, we
construct a simple model based on Kiyotaki and Wright
, and exhibit that money never circulates only as a
transaction medium. Section 3 proves that money does
not circulate until being attached by the function as a
store of value. Section 4 contains brief concluding re-
2. Model and Its Properties
2.1. Structure of the Model
Our model entirely depends on Kiyotaki and Wright .
The individuals are classified into three: producers, com-
modity traders, and money traders. A producer possesses
nothing and is searching for the opportunity of produc-
tion. A commodity trader has already finish ed pr oduction
and is searching for a counterpart of exchange. The
counterpart is admissible regardless of whether he/she is
a commodity trader or a money trader. To become a
money trader, it is necessary to first become a commod-
ity trader. A money trader seeks a commodity trader for
consumption. Money is assumed to be accepted with
probability one by any trader, whereas there never exists
such a guarantee in barter trade.
We now state the assumptions of the model.
i) A unit good produced by each producer is differen-
tiated in the interval
0,1z. The good is more
preferable when approaches 0. Namely, the utility
derived from the consumption of a unit good ,
is a decreasing function of . This implies that indi-
viduals are uniformly distributed around a circle of di-
ameter 2 and that prefer goods produced at nearer dis-
ii) Money is perishable. Namely, a money trader can
stay in its position only within and returns to a
producer thereafter. This means that money serves only
as a transaction medium and not as a store of value.
iii) The opportunity of production follows the Poisson
process with mean
. Further, the opportunity of ex-
change also follows the Poisson process with mean
opyright © 2013 SciRes. TEL
M. OTAKI 227
C is the cumulative distribution function of
the production cost . If is located above
producer waits for the next chance.
2.3. Analysis of the Model
Let us denote the expected lifetime utility of a producer,
commodity trader, and money trader as ,
, and ,
It is easy to show that
rVmu zzm yVVV V
m is the ratio of money trader to total traders which is
given exogenously. is the upper bound of the good
that the money trader admits to exchange .
The problem is the derivation in m
V. The transition
from a money trader to a producer is classified into four
i) To match with a commodity trader and exchange
money for a good;
ii) To match with a not-preferred commodity trader
and get nothing ;
iii) To match with a money trader and get nothing;
iv) To match with no one.
Summing these four cases, we obtain
The first and second terms of the right-hand side of (3)
correspond to Case 1, 2, and 3, and the second term cor-
responds to Case 4. Subtracting from both sides
of (3) and rearranging the terms, we have
, we obtain
In addition to (1), (2), and (4), the value-matching con-
VVuxV V .
The second equation is necessary for (4) to retain the
economic meaning. The reason why mp
is that the rate of return from being a money trader be-
comes negatively infinite if it differs, because the jump in
the value functions occurs with probability one within
any small interval. Economically, it implies that the loss
caused by the perished money is kept invariant while the
gain from the trade becomes infinitesimally small, when
the relevant interval approaches zero.
This model possesses five equations and five endoge-
VVVxy. As such, it is closed.
Nevertheless, it contains a contradiction for circulating
money. We thus have the following theorem.
Theorem 1. In the model and money never
circulat es u nde r Assumpti o n 2. ,
Proof. From 5, we have
implies that no commodity trader
wishes to become a money trader. Accordingly, money
In this case, the search money model degenerates into
the Diamond  model without externality.
3. Money as a Store of Value
In this section, we replace Assumption 2 and assume that
money is perpetually storable. Then, the transition from a
money trader to a producer is classified into four cases.
i) To match with a preferred commodity trader, ex-
change money for a good, and then become a producer,
ii) To match with a not-preferred commodity trader,
and continue being a money holder,
iii) To match with a money trader, get nothing, and
continue being a money trader,
iv) To match with no one, and continue being a money
Consequently, we have the following equatio n:
The first tem in the right-hand side corresponds to the
expected utility gained in Case 1, and the second term
corresponds to that gained in Case 2 and 3. The third tem
is the gain in Case 4. Rearranging terms and letting
, we obtain
(6) is the equation that Kiyotaki and Wright  actu-
ally use. Interchanging the value-matching condition
to and applying their
Theorem 2, we can ascertain that the equilibrium is uni-
quely determined. Thus, we ha ve the following theorem.
Copyright © 2013 SciRes. TEL
Copyright © 2013 SciRes. TEL
Theorem 2. When money serves as not only a transac-
tion medium but also as a store of value, it surely circu-
In addition, if both functions are required in the search
model, the overlapping generations (OLG) model seems
far tractable1, because money plays the same roles: the
young generation receives money as a store of value, old
generation uses it as a transaction medium.
4. Brief Summary and Discussion a bout the
REFERENCES Theorem 1 indicates that money never circulates only as
a medium of transaction. In conjunction with Theorem 1,
Theorem 2 shows that only if the function of a store of
value is added to money, it can circulate in an economy
that is described by search models. This is a new finding
against Kiyotaki and Wright  which emphasize that
money can circulate without having the property of store
 N. Kiyotaki and R. A. Wright, “A Contribution to the
Pure Theory of Money,” Journal of Economic Theory,
Vol. 23, No. 1, 1991, pp. 215-235.
 P. A. Diamond, “Aggregate-Demand Management in
Search Equilibrium,” Journal of Political Economy, Vol.
90, No. 5, 1882, pp. 881-894. doi:10.1086/261099
 R. E. Lucas Jr., “Expectations and the Neutrality of Mo-
ney,” Journal of Economic Theory, Vol. 4, No. 2, 1972,
pp. 103-124. doi:10.1016/0022-0531(72)90142-1
We have shown that money never circulates as a purely
transaction medium in the search model. For sustaining
the monetary economy, money should also be a store of
value. In this sense, both functions are inseparable and
innate natures attached to money.
 M. Otaki, “The Dynamically Extended Keynesian Cross
and the Welfare-Improving Fiscal Policy,” Economics
Letters, Vol. 96, No. 1, 2007, pp. 23-29.
1For example,although whether or not the neutrality of money holds
cannot be easily checked by the search model, the OLG model can do
so. See Lucas  and Otaki .