DBJ Discussion Paper Series, No.1007

On the function of money in the search model

Masayuki Otaki(Institute of Social Science, University of Tokyo)

March

2011

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings, conclusions
or recommendations expressed in these papers are those of the authors and do not reflect
the views of the Institute.

On the function of money in the search model

Masayuki Otaki (University of Tokyo)
7-3-1 Hongo Bunkyo, Tokyo 113-0033, Japanohtaki@iss.u-tokyo.ac.jp
Keywords: Money in the search model, Transaction demand,
Store of value.
JEL Classiﬁcations: E41, E42

Abstract
We check whether or not money can circulate as a purely transaction medium
in the search model. We obtain the following result: unless the role of a storeof value is added to the function of money, money does not circulate and theeconomy degenerates into a barter economy.

1

Introduction

This paper explores the function of money in the search model. There exist
two main functions of money: transaction demand for clearing and demand
for store of value. Kiyotaki and Wright (1991), which is the seminal work on
the search money model, insist that money can circulate solely by the motiveof transaction even if 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 much frequent than those in amonetary 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 (1991), 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 (1991), and exhibit that money
never circulates only as a transaction medium. Section 3 proves that money
does not circulate without being a store of value. Section 4 contains brief
concluding remarks.

2

Model and its properties

2.1

Structure of the model

Our model depends entirely on Kiyotaki and Wright (1991). The individuals
are classiﬁed into three: producers, commodity traders, and money traders.
A producer possesses nothing and is searching for the opportunity of production. A commodity trader has already ﬁnished production and is searchingfor a counterpart for exchange. The counterpart is admissible regardless ofwhether he/she is a commodity trader or a money trader. To become amoney trader, it is necessary to ﬁrst become a commodity trader. A money
trader seeks a commodity trader for consumption. Money is assumed to be1

accepted with probability one by any commodity trader, whereas there neverexists such a guarantee in barter trade.

2.2

Assumptions

We now state the assumptions of the model.
1. A unit of good produced by each producer is diﬀerentiated in the interval z ∈ [0, 1]. The good z is more preferable when z approaches
0. Namely, the utility derived from the consumption of a unit goodz, u(z), is a decreasing function of z. This implies that individuals
are uniformly distributed around a circle of diameter 2 and that thepreferable goods are enumerated in the clockwise direction.
2. Money is perishable. Namely, a money trader can stay in its position
only within period ϵ ≈ 0 and becomes a producer thereafter. Thismeans that money serves only as a transaction medium and not as astore of value.
3. The opportunity of production follows the Poisson process with mean
α. Further, the opportunity of exchange follows the Poisson processwith mean β.
4. F (C) is the cumulative distribution function of the production cost C.
If C is located above x, 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 Vp , Vc , and Vm , respectively.
It is easy to show that∫ x

rVp = α0[Vc − Vp − C]dF (C)

2

(1)

and
rVc = β[1 − m]{∫ x0{u(z) − [Vc − Vp ]}dz} + βmy[Vm − Vc ].

(2)

m is the ratio of money traders to total traders, which is exogenously given.
y is the upper bound of the good that the money trader admits to exchange.
The problem is in the derivation in Vm . The transition from a money
trader to a producer is classiﬁed into four cases:
1. To match with a commodity trader and exchange money for a good,
2. To match with a not-preferred commodity trader and get nothing,
3. To match with a money trader and get nothing,
4. To match with no one.
Summing these four cases, we obtain
Vm = e−rϵ{βϵ{[1 − m]

∫ y
0

}

u(z)dz + Vp } + [1 − βϵ]Vp + o(ϵ).

(3)

The ﬁrst term in the RHS of (3) corresponds to Cases 1, 2, and 3, and thesecond term corresponds to Case 4. Subtracting e−rϵ Vm from both sides of(3), and rearranging the terms, we have∫ y{
Vp − Vm }[1 − e−rϵ ]Vm = e−rϵ β[1 − m][ u(z)dz] + [] + o(ϵ).
ϵϵ
0Letting ϵ → 0, we obtain
rVm = β[1 − m]∫ yVp − Vm].
ϵ→0ϵu(z)dz + lim[
0(4)

In addition to (1), (2), and (4), the value-matching conditions require
Vc = Vp + u(x),V m = Vp .

(5)

The second equation is necessary for (4) to retain the economic meaning.
The reason why Vm = Vp is required is that the rate of return from beinga money trader becomes negatively inﬁnite if it diﬀers, because the jump3

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 to zero.
This model possesses ﬁve equations and ﬁve endogenous variables, (Vp , Vc , Vm , x, y).
As such, it is closed. Nevertheless, it contains a contradiction for circulating
money. We thus have the following theorem.
Theorem 1 . In the model, Vc > Vm , and money never circulates under
Assumption 2.
Proof. From (5), we have
Vc = Vp + u(x) > Vp = Vm .
Inequality Vc > Vm implies that no commodity trader wishes to become amoney trader. Accordingly, money never circulates.
In this case, the search money model degenerates into a Diamond (1982)
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 isclassiﬁed into four cases:
1. To match with a preferred commodity trader, exchange money for agood, and then become a producer,
2. To match with a not-preferred commodity trader, and continue beinga money trader,
3. To match with a money trader, get nothing, and continue being amoney trader,4

4. To match with no one, and continue being a money trader.
Consequently, we have the following equation:
Vm = e

−rϵ

{

∫ y

βϵ{[1−m]0}u(z)dz+yVp }+{[1−βϵ]+βϵ{m+[1−m][1−y]}}Vm +o(ϵ).

The ﬁrst term in the RHS corresponds to Case 1, and the second term corresponds to Cases 2, 3, and 4. Rearranging the terms and letting ϵ → 0, we
obtainrVm = β[1 − m]

{∫ y
0

}

{u(z) − [Vm − Vp ]}dz .

(6)(6) is the equation that Kiyotaki and Wright (1991) actually use. Interchanging the value-matching condition from Vm = Vp to Vm = Vp + u(y) andapplying their Theorem 2, we can ascertain that the equilibrium is uniquelydetermined. Thus, we have the following theorem.
Theorem 2 . When money serves as not only a transaction medium but
also as a store of value, it surely circulates.

4

Concluding remarks

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.
However, if both roles are required in the search model, the overlapping
generations (OLG) model seems to be more tractable1 , because money plays
the same roles: the young generation receives money as a store of value andthe old generation uses it as a transaction medium.

1

For 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 (1972) and Otaki(2007).

5

References[1] Diamond, Peter. A. Aggregate-demand management in search equilibrium. Journal of Political Economy 1982; 90; 881-894.
[2] Kiyotaki, N. and Wright, R. A contribution to the pure theory of money.
Journal of Economic Theory 1991; 53; 215-235.
[3] Lucas, Robert E. Jr. Expectations and the neutrality of money. Journalof Economic Theory 1972; 4; 103-124.
[4] Otaki, Masayuki. The dynamically extended Keynesian cross and thewelfare-improving ﬁscal policy. Economics Letters 2007; 96; 23-29.

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根津智治「セキュリタライゼーションと公的部門の金融活動」1989年5月.
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8902
8901
8805
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8801

堀内昭義「証券化－金融システムの構造変化についての分析－」1989年5月.
桜井宏二郎「金融構造と実体経済」1989年5月.
堀内行蔵「レーガノミックスの乗数分析」1989年3月.
岡田羊祐「技術移転論の動向－プロダクトサイクル論を中心として－」1988年12月.
Suzuki, Kazuyuki, and Jyoji Tokui, "Misalignment of the Yen-Dollar Exchange Rate and the International
Competitiveness of the Manufacturing Industries in Japan and the United States," November 1988.
Murase, Hideaki, "Monetary Policy Games in a Stochastic Environment: Does Discretionary Monetary Policy
Cause Stagflation?," November 1988.
國則守生「経済的償却率の推計－我が国建設機械の例－」1988年10月.
石黒隆司・堀内行蔵「先進国の現状と問題点：1980年代－変化と不均衡の時代－」1988年9月.

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JDB Discussion Paper Series