Money and Good, Liquidity and Acceptability

Atsuyuki Fukaura
Faculty of Economics, Nagasaki University

ABSTRACTThis paper analyzed the differences between the single money economy and themulti-money economy where all goods in the economy perform the function of
medium of exchange. We concluded that single money economy is the ultimate
outcome of the process wherein the less liquid goods are excluded in turn. The mostimpressive finding is that moneyness, the total exchange feasibility of the economy,
does not change during this exclusion process. However, the economy can attain amore efficient situation when one good supports the total moneyness and all othergoods can be consumed physically or put into the production process.

Money and Good, Liquidity and Acceptability

1. INTRODUCTION

This paper attempts to answer the question : What is the substantial difference between the single
money economy and the money-goods economy, where money and near-money coexist ? Needlessto say, the modern economy is the money-goods economy because we have a series of monetarydevices,cash, deposit, IOU, credit and so. To consider this problem we hereby introduce three
concepts deeply related with one another : liquidity, acceptability and moneyness.

２．MONETARY USE AND PHYSICAL USE

First of all, we consider the nature of a good in the exchange economy. All goods can be
consumed physically as well as monetarily (economically) in the sense that we use them as themedium of exchange. Take gold for instance. We can make some ornaments by using gold as the raw
material. On the other hand, it can be used as the medium of exchange . Needless to say, the extent towhich a good is regarded as the exchange medium deeply depends on the social custom or history,
however, theoretically we can say all goods has the potential of being used both for monetary use
and for physical use.
Here we can write the potential use of good i as(1, 1 ) i
where the first (second) 1 shows that the good has the potential monetary(physical) use. For the good
not used as the medium at all, j , we have(0, 1 ) j
and for the good not consumed physically(=used only as the exchange medium) , k, we have
(1, 0 ) k
By using these expressions we get(1,1 ) 1 + (1,1) 2 = (1,0 ) 1 + (0,1 ) 1 + (1,1) 2
The left-hand side shows the economy where all the goods are used for monetary and physicaluse .On the other hand, in the economy depicted by the right hand side, the 1st good is the only1

medium of exchange. Intuitively speaking, the left-hand side economy is the barter economy
because all good can be traded with one another by receiving their worth in the form of good. In theexchange economy where a good can be acquired by paying the medium of exchange , the 1st goodcorresponds to the right hand side. Historically our society has evolved from the barter economy tothe exchange economy, i.e.,from the left-hand side to right hand side.
If we assume the potential use of the goods is additive, the total benefit which can be derived
from the goods is same (=2) in both economies,(2 , 2) 1, 2 = all goods .
What has happened in this transition process ? This is the key question we will try to answer insucceeding sections .

3. BASIC CONCEPTS

Our discussion focuses on the monetary use of goods. Here we assume that monetary use has
two characteristics inherent in the good itself, liquidity and acceptability. The liquidity of i-th goods(denoted by li ) is defined as the ability to describe the credit/debt relationship. In the other words, atthe transaction, if one can use some good as the measuring device to determine how much one owes
the other, that good has got liquidity. For example, if you can say that you owe your friend “threedesks” when you buy something from him, these desks have some liquidity.
Suppose there are m goods in the economy and n goods of m have liquidity, we can line them up
in decreasing order.
l1 > l2 > l3 > … ln.

li >0, n<m(1)

However, your friend may not cancel your debt by taking three desks. Hence we need to define theacceptability of i-th good as the medium of exchange (denoted by ai). Acceptability is defined as
the possibility that one accepts some good in compensation for the goods he has given the others.
The liquidity and acceptability depend on not only the good’s character but also on the social custom
and tradition . Without loss of generality, we could assume
a1 + a2 + ... + an = 1(2)

In the exchange economy where some good is used as a medium of exchange, that good is said tohave the moneyness or the agents can enjoy the benefit from the monetary use. Here we define themoneyness of i-th good (mi) as follows:2mi = ai li
This means the moneyness is an increasing function of liquidity and acceptability. People can use
any good they prefer as the medium of exchange so the moneyness of the economy is
M = a1 l1 + a2 l2 + ... + an ln(3)

It should be noted that (2) does not mean that the public can use all good as the exchange medium
in transaction. Instead, it only shows the maximum moneyness the economy can provide.
The visual expression of moneyness is shown in FIG-1 and FIG-2.

ai

ai

0.3
0.7

0.7

0.3

0.7

0.3

li

0.7

FIG-1

0.3

li

FIG-2

When a1 =0.7, a2 =0.3, l1 =0.7.l2 =0.3, the shadowed area in FIG-1 represents the moneyness. FIG-2
shows the case of a1 =0.3, a2 =0.7, l1 =0.7.l2 =0.3 . The upper-left area in FIG-1, for example, shows
the loss from the fact that 30% of all agents never use 1st good with li =0.7.
The 1st good with l1 =0.7 and a1 =0.7 dominates all other good in terms of liquidity andacceptability. However, what if the 1st good is characterized by li =0.7 and ai =0.3 ? The 1st goodcannot dominate the 2nd good because the two shadowed areas are equal. Which then is the medium
of exchange ? Both ?
Now we are in the stage to define money. To do so, we need the next theorem on the moneyness
and acceptability.

3

Theorem 1 : M has maximum value if and only if a1 >a2 >a3 …＞an .
Proof :
(sufficiency: a1 >a2 >a3 …＞an ⇒ max M)
STEP 1. the case of n = 2..
M is
M1 = a1 l1 + a2 l2 .

(4)Here M’, where the order of a does not correspond to l’s order, is
M1 ’ = a2 l1 + a1 l2 .

(5)

We have, by assumptions,M1 - M 1’ = (a1 l1 + a2 l2 ) – ( a2 l1 + a1 l2 ) = ( l1 - l2 )( a1 - a2 ) > 0(6)

Accordingly M1 is maximum in n=2 economy.
STEP 2. the case of n = 3.
The ordering of li is given. First, we consider the case where l1 corresponds to a1. Then (l1 l2 l3 )must correspond to (a1 a2 a3 ) or (a1 a3 a2 ). Then our problem is reduced to a comparison of
M2 = a2 l2 + a3 l3 and M2 ’ = a3 l2 + a2 l3.

(7)

M2 > M2 ’ is directly concluded in the above fashion. Accordingly, M3 = a1 l1 + M 2 > a1 l1 + M2 ’. Thisimplies M3 is the maximum in n=3 economy. For the case where l2 , ,l3 corresponds respectably to
a2 ,a3, the same discussions can be applied.
STEP 3 : In the case of n=4, assuming the same analogy as the above, we can easily derive that if l1> l2 > l3 > l4 and a1 > a2 > a3.> a4 , M4 is the maximum.
STEP 4 : In the same fashion, we find in the case of n = k. l1 > l2 > l3 … > lk

and a1 > a2 > a3 … >ak and in the case of n = k+1. l1 > l2 > l3 … > lk+1 and a1 > a2 > a3 >… ak+1 , Mk,Mk+1 are themaximums. This completes the proof.
(necessity: max M ⇒ a1 >a2 >a3 …＞a n )In n=2 economy, because M1 = a1 l1 + a2 l2 is maximum thenM1 = a1 l1 + a2 l2 > M 1’ = a2 l1 + a1 l2.
andM1 - M 1’ = (a1 l1 + a2 l2 ) – ( a2 l1 + a1 l2 ) = (l1 - l2 )( a1 - a2 ) > 0

4(8)

l1 > l2 leads immediately to a1 > a2 . In n>2 economy the same discussion yields the same conclusion,
thus yielding Theorem 1.
Because Theorem 1 guarantees that the situations like FIG-2 can not emerge, it is rational todefine money as follows.

Definition: Money is the good with the highest liquidity and the highest acceptability.

4 LIQUIDITY CHANGE

We now turn to the case where the i-th good’s liquidity changes. Starting from a simple case, wefirst study Theorem 2.

Theorem 2 : Even if the i-th good is monetised, the economy can not enjoy the increase in the totalmoneyness.
Proof: Suppose that the liquidity of the i-th good is increased in the fashion, for simplicity, by l1 = li
+ α . Suppose the i-th good is the desk then l1 = li + α means the desk are sold formoney(=monetised). The total moneyness could be written as
Mafter = l1 a1 + l2 a2 +… + (l1 -α) ai + … + ln an=(a1 +ai )l1 + l2 a2 +… + li-1 ai-1 + li+1 ai+1 + … + ln an -αai

(9)It must be noted that the i-th good is not used as the medium of exchange. Instead, the acceptabilityof money is increased by (a1 +ai ) . On the other hand, the economy incurs the cost (-αai ), the loss ofmoneyness originating from the i-th good . Then we have
Mafter – M = ( l1 - li -α)ai = 0.

(10)

Examining (10), we see that the total moneyness is not changed. Further, it is impossible to holdli +α> l1 because it violates the condition that 1st good is money. Therefore Theorem 2 is proved.
Theorem 2 states that the economy can enjoy the moneyness M without the i-th good if the 1st
good’s acceptability is increased. Of course the excluded i-th good is used as the physical good. Anyother implications are given in Section 7 together with the implications of the next section.

55 SINGLE MONEY ECONOMY

The above procedure can be applied to derive the moneyness of the single money economy. We
derive the single money economy by excluding the last good(n-th good) , n-1 th good , n-2 th goodand so on, from the series of medium of exchange.
The liquidity change is assumed to be subject to
l1 = l2 +α2 , l2 = l3 +α2 , l3 = l4 +α3 …

ln-1 = ln +αn-1(11)This means that the liquidity of i-th good is increased byαi-1 and equalized to the liquidity of the i-1
th good. We have
M1 = l1 a1 + ( l1 -α2 ) a2 + ( l2 -α3 ) a3 + … + ( ln-1 -αn ) an
= [ l1 (a1 +a2 ) + l2 a3 + l3 a4 + … + ln-1 an ] – [α2 a 2 –α3 a3 –… – αn an ](12)(12) states the n-th good is not used as the medium of exchange. Instead the acceptability of the 1st
good (=money) in increased with some costs(=2nd term in the right-hand side). Further Theorem 2
also holds because
M1 – M = [ l 1(a1 +a2 ) + l2 a3 + l3 a4 + … + ln-1 an ]–[l1 a1 + l2 a2 +… + ln an ] - [α2 a 2 –α3 a3 –… – αn an ]= (l1 a2 -α2 a2 ) +[l1 ( a3 - a2 ) -α3 a3 ] +… + [ln-1 ( an - an-1) --αn an] - ln an= a2 (l1 -α2) +[ a3(l2 -α3) – a2 a3 ] +… + [an (ln- 1 -αn)- an-1 ln- 1 ] - ln an= l2 a2 +[ l3 a3 - l2 a2] +[l4 a4 - l3 a3]+ … +[ ln an - ln-1 an-1] + ln an= ln an - ln an = 0(13)

Substituting (11) into (12) and arranging gives
M2 = [ l1 (a1 +a2 +a3 ) + l2 a3 + l3 a4 + … + ln-2 an ]–[α2 ( a2 +a3 ) –α3 ( a3 +a4 ) –… – αn-1 ( an-1+an ) -αn an ](14)

and M2 = M1 .
We found that the n-1 th good is excluded from the economy. Iterating the same calculation
eventually results toM n-1 = l1(a1 +a2 +a3 … an)- [α2 ( a2 +a3 … an)+α3 ( a3 +a4 … an ) +… + αn-1 ( an-1+an )+αn an ](15)(14) indicates the moneyness of the economy where only money is accepted as the medium of6exchange. Needless to say ,M n-1 = M n-2 = … = M ,(16)

the subscript represents the number of unused goods as the medium. Then we call the economy with
Mn-1 the single money economy and the economy with Mk(k<n-1) the money-goods economy .
To make clear the difference between Mn- 1 and Mk, we assumeαi =αj ≡α for all i,j bychanging the unit of li. Then we have
M n-1 ==

l1 (a1 +a2 +a3 … an ) -α[ ( a2 +a3 … an)+( a3 +a4 … an ) +… + ( an-1+an ) + an ]l1 (a1 +a2 +a3 … an ) -α[ a2 + 2a3 + 3a4 + … + (n-2) an-1 +(n-1) an ](17)

Consider the case when all good except money is directly and simultaneously monetised, i.e.,l1 = l2 + a = l3 + 2a = l4 +3a= … = ln +(n-1)α(18)

It is easily confirmed that (17) can be derived by substituting (18) into M. This implies that themarkets for all goods can be traded for money and contribute to improve the exchange feasibility.
Our method, where the last good is excluded one after another, describes the historical processthat the physical mediums have been eliminated from the transaction. For instance take the shell thatwas used as the medium in ancient China. When more convenient good was recognized, the shellterminated its role as the medium.
It must be useful to provide a simple numerical example to illustrate the nature of (17).

EXAMPLE 1: n=4, l1 =10, l2 =8, l3 =6, l4 =4,α=2, a1 =0.4, a2 =0.3, a3 =0.2, a4 =0.1
M = (10×0.4)+(8×0.3)+(6×0.2)+(2×0.1) = 4 + 2.4 + 1.2 + 0.2 = 8.0
M3 = 10 – 2 [(0.3+0.2+0.1)+(0.2+0.1)+(0.1)] = 10 – 2(1) = 8.0

FIG-4 represents the moneyness of the money-goods economy. By sliding three upper rectangles
in FIG-4 to the left we get FIG-3 for the single money economy. In FIG-3, the size of ① is 2×0.1
because l1 -l2 =2.Similarly, ②=2×(0.2+0.1),③=2×(0.3+0.2+0.1). Hence, M3 =M.

7

① ② ③
1

↓ ↓ ↓

0.9

0.7

0.4

10

18

24

28

FIG-3(single money)

10

18

24

28

FIG-4(money-goods)

In Fig-4, all four goods are utilized for the monetary use as well as physical use. On the other
hand, the economy depicted by FIG-3 assigns only the 1st good as the exchange medium and all theother goods are physically consumed.
What happens if n is increased, i.e., the n+1 good is added as the medium? If an +1 is added,atleast one ak ( k< n) must be decreased so as to keep (2). It is natural to assume k = n because n+1-th
good is a close substitute for the n-th good as medium. Therefore,

a n+1 ’ + a n ’ = an

(19)where prime(’) implies “after”the n+1-th good is added. Using (19), we obtainM(n+1) -1 = l1 a1 + l2 a2 +… + l n an ’ + ln+1 an+1 ’ , l n > ln+1

(20)

M n-1 - M(n+1) -1 = ln an - l n an ’ - ln+1 an+1’ = ln( an - an ’ ) - ln+1 an+1 ’ = an+1 ’(ln- ln+1 ) > 0

(21)

and

(21) indicates that the moneyness is decreased as n is increased.
The added good takes the acceptability from the n-th good, however, its liquidity is smaller than
the n-th good. So the moneyness lost is bigger than the gain.
On the other hand, if the acceptability of the new good is located between li and li+1 (i<n+1), themoneyness does not decrease. Suppose the new good has lnew( li > lnew > li+1) . The moneyness after
introducing the new good isM’(n+1) -1 = l1 a1 ’ + l2 a2 ’+… + li ai ’+ lnew ai+1 ’+ li+1 ai+2 ’+… + ln an+1 ’

8where ai ’> ai ’ > ai+1 ’,

a1 ’+ a2 ’+ ... an+1 ’= 1

(22)

If the acceptability is adjusted for all ai like (21), we can not decide the sign of Mn-1 -M’(n+1) -1 . For
simplicity, we assume the newly introduced good takes the acceptability from li+1 , i.e.,ai ’=ai for all
i<n. In other words, condition (22) should be kept only by the last two terms is kept. As a result,
we have
M n-1 – M’(n+1) -1 = -lnew ai+1 + (ai+1 - ai+2 ) li+1 + (ai+2 - ai+3) li+2 +… +(an-1 - an ’) ln-1 +(an - an+1 ’) ln= -l newai+1 + (ai+1 - ai+2 ) l i+1 + (ai+2 - ai+3) li+2 +… +(an-1 - an ’) ln-1 +(an ’) ln= ai+1 (li+1 - lnew ) + ai+2 (li+2 - li+1 ) +… + an-1 (ln - ln-1 )+ an ’(ln - ln-1 )(23)

All those in parentheses are negative because of the nature of liquidity-order. It shows thatmoneyness is increased, although it depends on the acceptability profile.
Fig-5 and FIG-6 show the case whereby the newly good(l2 > lnew >l3 .) is added into the 3 moneygoods economy. The moneyness is
M2 = l1 a1 + l2 a2 +l3 a3 and M2+1 = l1 a1 + l2 a2 +lnew a3 ’+ l3 a4 ’

where a3 = a3 ’ + a4 ’

(24)

The net gain, accordingly, is
M2+1 - M2 = lnew a3 ’ + l3 a4 ’ - l3 a3 =lnew a3 ’ + l3 (a4 ’ - a3) = a3’ ( lnew - l3 ),(25)

The reason behind this is simple. The vertical area in FIG-5 ( l3 a3 ) represents the moneyness lost
and the horizontal areas in FIG-6 (lnew a3 ’ ) is the moneyness gained. The net gain is ① ‑ ②, whichequals to a3 ’ (lnew - l3 ), the last term of (23) or (25).

Next consider the case m→∞. FIG-7 and FIG-8 depict the case where li - lj =α<ε(forall i,j =i-1,ε is a sufficiently small positive number). The moneyness of a barter economy is aconcave-line because of Theorem 2 and that of a single money economy is the a-axis itself. Thisis the foundation of the widespreading knowledge that the evolution of the human economy ischaracterized by the process of improving the moneyness, i.e., the exchange feasibility, by
conversing into a single medium of exchange and by introducing the new monetary device.

9

FIG-5(before)

l1

l2

lnew

l3

a4 ’
a3 ’

↑

↑

①

②

FIG-6(after)

a

a

l∞

l∞

FIG-7(single money)

FIG-8(barter)

Now, suppose n = m . This leads us to the barter economy because all goods can be theexchange medium for all other goods. Therefore we know that the monetary economy and barter
economy are indifferent in the sense the moneyness of both economies is equal.

6

DISCUSSIONSIn modern society, all goods are private property and can be freely traded with one other. This
also means all goods can serve as the exchange medium if both trading parties agree. Simultaneously10

it should be noted that all goods have their own utilities if the owner consumes or utilizes them
directly. Then we can refer to the former as the monetary use and the latter as the physical use.

The benefit of monetary use comes from our two concepts, liquidity and acceptability. In
other words, moneyness, M, is the summation of the utility the economy can enjoy when all
goods are in monetary use. The liquidity change discussed in section 4 is the process todistinguish the monetary use and the physical use then integrate the former into the money(1stgood) and leave the latter to the good itself. As a result, the good used only in the physical senseis excluded from the list of monetary devices. However, it concentrates in the utilization of its
physical use, for example, as raw materials.
Here is an implication for the asset-liquidation or securitisation. If the i-th good is the claim
to the future income stream, the liquidity change in this case corresponds to the asset liquidation
by securitisation. The above explanation shows that in asset liquidation, the monetary use of thei-th good is transferred into money and the economy can use the i-th good in physical use. Thisis the reason why the asset liquidation is expected to be the remedy for the non-performing loan.
By securitising the non-performing loan(often backed by real estate), the economy can acquire
the monetary use of the backed real estate.

7 CONCLUSION

The explicit purpose of this study was to examine several important concepts of monetary
economics. If the moneyness could depend on the liquidity and acceptability of goods, the single
money economy and the money-goods economy are indifferent in the sense their moneyness areequal.
It is interesting to examine the social effects of the asset-liquidation or securitisation. Given thenumber of goods, the asset-liquidation can not increase the moneyness.
The only way to improve the moneyness is the injection of a new device whose liquidity
dominates one of the existing goods into the monetary system. In this sense, the liquidation of theexisting goods works not as the creation mechanism of moneyness, but as the liquidity-allocation
mechanism.

11論文要旨：
物々交換経済に見られるように、すべての財は交換手段として用いられる可能性を持つ
一方、財それ自体として消費されうる。それに対して、近代的な経済は特定の財のみが交換手段として機能する貨幣経済として特徴づけられる。物々交換経済はどのような過程を経て貨幣経済へて進化するのかを、すべての財に潜在的に含まれる交換手段としての２つの要素（流動性(liquidity)、受容性(acceptability)）およびそれによって決定される経済全体の交換可能性（貨幣性(moneyness)）を中心に考察した。そして、ある財の流動性が何らかの理由によって高められ、それに伴い受容性が高まるという過程を想定することにより、単一の財が貨幣として機能する経済を論理的に再現した。この分析は、近年注目を集めている債権流動化の機能について興味深い含意をもたらす。

12