Inseparability of Transaction Medium and Store of Value in the Role of Money

doi:10.4236/tel.2013.34038

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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

Masayuki Otaki

Institute of Social Science, University of Tokyo, Tokyo, Japan

Email: [email protected]

Received April 27, 2013; revised May 27, 2013; accepted June 27, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

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

1. Introduction

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 [1], 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 [1],

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

[1], 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-

marks.

2. Model and Its Properties

2.1. Structure of the Model

Our model entirely depends on Kiyotaki and Wright [1].

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.

2.2. Assumptions

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 ,





uz,

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-

tance.

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



M. OTAKI 227

iv)



C is the cumulative distribution function of

the production cost . If is located above

C C

, the

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 ,

respectively. m

It is easy to show that



pcp

CrVVVF C











 (1)

and









mcc

rVmu zzm yVVV V



 

 

 

.







(2)

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

cases.

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











e1d 1

VmuzzV



 





 



 (3)

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

erm









e1 ()d

ypm

muzz

















 



















Letting 0



, we obtain





1dlim

rVmu zz









 

 (4)

In addition to (1), (2), and (4), the value-matching con-

ditions require



cp mp

VVuxV V .

(5)

The second equation is necessary for (4) to retain the

economic meaning. The reason why mp

is required

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.

VV

This model possesses five equations and five endoge-

nous variables





,,,,

pcm

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. ,

VV

Proof. From 5, we have





cp pm

VVuxVV

Inequality cm

implies that no commodity trader

wishes to become a money trader. Accordingly, money

never circulates.

VV

In this case, the search money model degenerates into

the Diamond [2] 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

trader.

Consequently, we have the following equatio n:



















e1 d

e1 .

Vmuzz

mmyV





 



 



 

yV

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









rVmu zVVz





 



.

(6)

(6) is the equation that Kiyotaki and Wright [1] actu-

ally use. Interchanging the value-matching condition

from mp



to and applying their

Theorem 2, we can ascertain that the equilibrium is uni-

quely determined. Thus, we ha ve the following theorem.



VVuy

M. OTAKI

228

Theorem 2. When money serves as not only a transac-

tion medium but also as a store of value, it surely circu-

lates.

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

Obtained Results

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 [1] which emphasize that

money can circulate without having the property of store

of value.

[1] 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.

doi:10.1016/0022-0531(91)90154-V

[2] 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

[3] 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

5. Conclusions

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.

[4] M. Otaki, “The Dynamically Extended Keynesian Cross

and the Welfare-Improving Fiscal Policy,” Economics

Letters, Vol. 96, No. 1, 2007, pp. 23-29.

doi:10.1016/j.econlet.2006.12.005

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 [3] and Otaki [4].