Equilibrium with transaction cost and money in a single market exchange economy

JOURNAL

OF ECONOMIC

THEORY

7, 418-452 (1974)

Equilibrium with Transaction Cost and Money in a Single Market Exchange Economy MORDECAI

Fourth

KURZ *

Institute for Mathematical Studies in the Social Sciences, Floor, Encina Hall, Stanford University, Stanford, California

94305

Received September 14, 1972; revised May 29, 1973

The theory of money is perhaps the oldest of the branches of economic theory. Yet, in some fundamental sense, it has not been resolved as yet. Recent contributions to this theory have focused attention on two aspects of the theory: the structure of exchange with transaction cost and the function of money in a sequence of markets. Particularly close to our approach are the papers by Starr [12, 131. The fundamental differences between this paper and Starr’s papers are the description of the transaction technology and the functioning of the trading activity in a world with transaction cost. To motivate the structure of the present paper let us reconsider the function of money. The speculative demand for money is well understood and the study of money as an important element in portfolio theory is well known (see, for example, Arrow [l]). The question which, in our mind, has not been resolved is the transaction demand for money and the store of value function of money. Thus the basic issue which has been raised is the analysis of money as a facilitating agent of the structure of transactions allowing the economy to achieve an equilibrium with minimal use of resources. In a world with transaction and without uncertainty we wish to investigate the three basic functions of money: the provision of a unit of account, medium of exchange, and a store of value. In an important sense, each one of these functions represents a more sophisticated monetary system and we shall in fact think of it in such a way. * The author was supported by a Ford Foundation Faculty Fellowship (1971/1972). The work was done at the Institute for Mathematical Studies in the Social Sciences at Stanford University and the Institute of Mathematics, the Hebrew University. The author benefited greatly from extensive conversations with Professor Bezalel Peleg. The Seminar in Mathematical Economics at the Hebrew University lead by Professor R. Aumann provided very valuable critical suggestions. Thanks are due to Dr. Toshihiko Hayashi who helped in the preparation of the final draft.

418 Copyright All rights

0 1974 by Academic Press, Inc. of reproduction in any form reserved.

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Thus the idea of the present paper is to present a sequence of economies growing in the level of their monetary development. We start with a barter economy with a unit of account studying the equilibrium structure of transactions. Next we present a monetary economy where money is use strictly as a medium of exchange, and finally an economy where mane is used both as a medium of exchange and store of value. All the above ucted in economies with transaction cost and no ~~certai~~y.

ECQNOMY 4: A. BARTER EXCHANGE ECQNOMY WITH TRANSACTION COST This economy was introduced and analyzed in an earlier paper IS]. Since we build ou the concepts and structure of that paper, a short review of the development is in order. The economy consists of H consuming households each of which an endowment cP, h = 1, 2 ,..., H, with a preference ordering Xh over the hth consumption set. The act of transaction entails the use of real resources and the search for an equilibrium consists of the search for an optimal structure of transactions. It is important to note that in spite of the fact that individual preferences are defined over final consumption bundles, the primitive concept in the analysis is the bundles of exchange, i.e., the transactions. The final bundles are derived from the optim,al transactions. A. Notatisn

N = number of commodities ~9 = initial endowment of consumer h

xh = vector of purchases of consumer h X=gXh h-1

/vh = vector of sales of consumer h H

gn = vector of real resources used in exchange of ~9 for 9

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MORDECAI

ch = final consumption p = price vector

KURZ

bundle

The individual consumer, in his search for an optimal consumption bundle, wishes to exchange the vector yn for xh. This will entail transaction costs g” and such an exchange will be regarded as feasible if (xh, yh, g”> E Th,

(1)

where Th is a set in ETN which states all the technologically feasible transactions. Since we are discussing exchanges, it follows that Th already embodies all the rules of the dividion of transaction cost between buyers and sellers. Thus if individual h, exchanges 91 for yhl it is clear that someone buys yhl and sells xhl . ghl is regarded as the share of individual h1 in the cost of the exchange. The set Th embodies all the ways in which exchange takes place and thus embodies all the institutional and legal requirements of exchange. If, for example, the exchange requires the registration of a title to a commodity, then Th will embody not only the time and resources necessary to exchange possession, but also all the legal cost of exchanging titles. We shall call Th the “transactions technological set” and the following are the possible assumptions: (T.l)

Th is a closed convex set.

(T.2) If (xl&, yh, g”) E Th then xh’ < xh, yh’ < yiL and gh’ > g” imply (xh’, yh’, g”‘) E Th. (T-3)

(0, 0,O) E Th.

(T.4) There exist sh > 0, 5” and i/t such that f” - $” > 0 and (gh,jh,gh) E Th. Assumption (T.l) follows the pattern of recent literature. The assumption of convexity is questionable because of the possibilities of economies in transactions. Assumption (T.2) says that if gh are the resources needed to carry out a given set of transactions then any gh’ 3 gh will suffice to carry out transactions of smaller volume. Assumption (T.3) allows inaction and (T.4) allows some positive action. Assumption (T.4) is important since it states that there exists some way of exchange which will result in a situation in which transaction costs are not so large as to make all transactions undesirable. Thus this assumption says that there is at least one way in which all commodities can be exchanged leaving some surplus (after transaction cost) behind.

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C. Preferences and Endowmenfs

We shall assume (D.l) sh is a continuous preference ordering defined over E+N where the consumption set is taken to be E+N. (D.2) s,& is a convex preference ordering so that if A? and xh’ are in EeN and xh k2;n xh’ then hxh + (1 - A) xh’ & xh’. Assumptions (T.1)-(T.4) and (D.l)-(D.2) will be maintained t~ro~gho~~ this paper. However, we shall discuss below the existence question under other alternative conditions. For the sake of notation, we shall ~~trod~c~ them here. (El)

.!d > 0

for all h.

The first theorem below will be stated for the case 0.3 > 0 ah k (El). D. Budget and Demand Correspondences

The individual consumer buys the vector xh, sells the vector y”, and the exchange involves the cost vector gn. At the end he consumes ch = Wh + Xh - y”,

(2)

but his choices are restricted by his budget constraint (xh, I/‘“, g”) E T”.

PXh < PYh - pgh,

This leads to the definition Bh(p) =

i

(31

of the budget set correspondence: (a) (9, yil, g”) E Th

(x*, y”, g”) E E+N

i

(b) Pxh < pyh - pg” (c) d+xh-yh>O)

i ~

(44)

Given any bundle (xh, yh, g”) E Bh(p), we define Ch(Xh,yh, g”) = 03 + Xh - yh and this leads to the definition Ch(Xh,yh3g”)& r*(P)

=

CyXh’r yh’S gh’)

(x”, Yh, g”> E Bh(P)

1 Finally we define

.

for all (9, yh’, g”‘) E P(p)

)

(6)

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MORDECAI KURZ

DEFINITION.

(4 (b)

A pair (p, (9, yn, g”) h = 1, 2 ,..., H} is an equilibrium

if

(x7&,yhv gh) E Y”(P), x d y -g.

Remark. It may appear paradoxical to require the material balances to be x < y - g. The point to note is that an equilibrium in the regular sense would require c+g<‘u,

(7)

where H

C: =

H

c Ch, g=ph¶

H

,=;lah.

h=l

However, since c = w + x - y the above condition

becomes

or

(8)

XGY-g, which is precisely the condition E. Equilibrium

(b) above.

When C.IJ~ > Ofor All h

The first fundamental THEOREM 1.

result can now be stated:

The exchange economy satisfying the following

set of

assuwrptions: (a) (T.l), (T.2), (T.3), and (T.4), (b) (D.11, @4, (c) (E.l), i.e., uh > 0 for all h, has a competitive equilibrium. Proof. See Kurz [5, Theorem 13. The condition uh > 0 is a strong condition it with the rather reasonable condition (E.1’)

w = god

If (E.l’) is supplemented (D.3)

and one wishes to replace

> 0. with the “strong desirability”

if xh > xh’ then xh >& xh’ all h,

condition,

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then equilibrium can be proved to exist in the “classical” theory. Wnen transaction costs are present, the standard reasoning fails and we shah now analyze the two points at which the conventional theory fails in order to demonstrate the issues involved. Probbm a. When pw” > 0 it does not follow that the budget set has an interior point. This is so because even whenpwh > 0 (i.e., the individual has positive wealth) he should be able to sell a portion of his wealth in order to buy positive quantities of all commodities. This means that if he chooses, say yh = +uh, then with pyh > 0 we would like him to find an 2 > 0 and gh such that (x”, yh, g”) E Tk and pyh - p,gh > 0. This may or may not be true with our assumptions on Th so far. On a more intuitive level, even if the individual has positive wealth, it may be that the transaction cost of exchanging any portion of his wealth for positive quantities of all goods may be so high that it may exceed the value of that portion of wealth which is offered for sale.

Problem b. Now suppose that some price pj = 0. The conventionaij method of showing that this is not possible is to have the individual buy xjh + h of the jth commodity with no extra cost but the new point is more desirable than the previous one. This procedure is not applicable here since the act of buying that extra X amount of comm.odity j may require some positive transaction cost, thus the act of buying that extra X may not be feasible. F. Existence of Equilibrium

When Qnly f wh > 0 i&=1 In order to complete the investigation of the existence of equilibrium under the weaker condition M h=l

new restrictions must therefore be introduced. We fust introduce following equivalent (For proof see Kurz [S, Lemma 51.) conditions:

the

(T.4’) For any yh there exist (xx, y”, g”) E Tn such that xh > 0 and pyh - pgh > 0 for all p. pyh > pgh whenever py” > 0. (T.4”) For any yh there exist (xh, yh, g”) E: Th such that xh > 0 and y” > gh with yih > gih whenever yih > 0. There is one special case of (T.4”) which may be of interest and it is based on the special structure of the endowment vectors ah. it let us delineate commodity 1 as a special commodity-call it “leisure services” (This really calls for a redefinition of the consumption set, but

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MORDECAI KURZ

we shall ignore it here.) and assume that all individuals are endowed with that commodity, i.e., wlh > 0 all h. Now we may formalize the condition that some exchanges may be accomplished with labor only. Thus if yn = ( ylh, yc,..., yNh) is any vector with ylh > 0, denote by yhh the vector Ynh = (Ylh, hYzh, ~Y3h,-, AYN”),

O
(9 (TX) For any yh with ylh > 0 there exist 1 3 h > 0, xh > 0 and gh such that (a> (xh, Y,?, gh> E Th, (b) gh = (g,“, 0, O,..., 01, (cl Ylh > ah. The assumption (T.4”‘) says that if a vector yh with ylh > 0 is available for sale, then by maintaining ylh at its level but by reducing the amount of all other commodities one can find a way of exchanging this vector utilizing a small amount of labor such that some labor services were not used up in the transaction. We note in passing that any of the assumptions (T.4’), (T.4”), or (TM”) will ensure the existence of an interior point whenpWh > 0 if wlh > 0. Turning now to the question of purchases of free goods we can now introduce the last two assumptions: (D.3’)

Xh > Xh’ with Xlh > X,“’ * X” >h xh’ all h.

(T.5) For any (xh, yh, gh) E Th and for any h > 0 there exists a p > 0 such that if fh = xh + (h, 0,O ,..., 0) jh = gn + (EL,0, 0 ,..., 0) then (Sh, yh, 2”) E Th. Assumption (D.3’) is a weak form of the “strong desirability” condition (D.3). It says that if 9 > xh’ but xlb > x,“’ then xh is strictly preferred to xh’. (T.5) states that one can always purchase labor services using only labor services for transaction cost. We can finally state: THEOREM 2. The exchange economy with transaction cost satisfying the following Jet of assumptions:

(1) (2) (3) (4)

(T.l), (T.4’) (D.l), mh >

(T.2), (T.3), and (T.5), or (T.4”) or (T-4”‘), @.2), and (D.3’), 0 all h, ulh > 0 all h, and w = ~fffl uh > 0,

has a competitive equilibrium.

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Proof.

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See Kurz ES, Theorem 21.

G. Some Con&ding

Remarks on Interpretation

Our Model I is a natural extension of the Arrow-Debreu model to include transaction cost. In the Arrow-Debreu model we write the consumer budget constraint pxh f pmh, and the consumer is allowed to choose any x which will balance the above account. This means that “money” exists in the model as a unit of account, Our restriction pxh < pyh - pgh, (xk, yh, gh) e Th assumes, in the same way, the existence of a unit of account and in this sense our model provides a characterization of a barter economy with transaction cost. We interpret our model as that of a barter economy only to the extent that individuals are not engaged in acts of “buying” or “selling” but rather in “exchanging.” Moreover, they carry out their own exchanges using the transaction technologies available to them. This is an intuitive description of a trading process which allows all possibk exchanges between an individual and the market. Thus the existence of a single generalized market and a general unit of account to measure values are the essential characteristics of our Model I.

ECONOMY

II:

AN

EXCHANGE ECONOMY WITH TRANSACTTON a MEDIUM OF EXCSANGE

COST AND

Introduction We proceed now to the analysis of a more developed economy where a medium of exchange is introduced. It is, however, our desire to distinguish between a monetary economy in which money serves as a unit of account and an economy where it is used as a medium of exchange and store of value. Keeping in mind the fact that we are dealing with an economy without uncertainty, any manetary structure arises only because of the flow of transactions in the economy and this means that we a.re dealing here with the transaction demand for money. A. Monetary and Barter Economies We explained in Section (I) above our conception. of a barter economy with a market: it is an economy where exchange is the order of the day and *where the “‘market” provides the pool of resources needed to carry

426

MORDECAI

KURZ

out transactions. Needless to say, these resources are not free and are charged against the budgets of the transactors. A higher level of commercial development will give rise to a retailing or trading function. There are three essential elements that characterize the emergence of the new economic activity called “trading” or “retailing.” (a) The “transaction technology” of the trader is the most efficient relative to the “transaction technology” that each household has. Moreover, due to relative advantage, specialization in trading occurs and this leads to the emergence of a new economic activity which pulls the various trades of the different members of the community; this is a natural result of competition. (b) The essential characterization of the notion of “trading” rather than “exchanging” entails a separation between the act of selling and the act of buying. But-and here is the crucial fact-such a separation between buying and selling cannot be accomplished without a medium of exchange. Thus, in this level of commercial development the “retailer” or the “trader” can also act as a “banker” who creates “inside money” which can then be used as a medium of exchange and may or may not disappear at the end of trading. Thus, the separation of the acts of buying and selling leading to the creation of a “trading” activity and a medium of exchange is only the first step towards the creation of money and credit. (c) In a competitive equilibrium with the acts of selling and buying separated, the household should be allowed to sell as much as it wants and buy as much as it wants subject to its own budget constraints. However, with transaction cost involved, the onIy way the retailing activity can be provided is by having different buying and selling prices when the difference between them reflect the transaction cost involved. Thus the last characterization of a monetary economy with money as a medium of exchange (and separation between buying and selling) is the difference between buying prices and selling prices. In some sense, perhaps historical more than logical, the points made above are artificial in nature. Historically speaking, the function of money as a store of value, medium of exchange, and unit of account developed with the emergence of a specific commodity like gold which was generally accepted for exchange purposes. But this historical fact became possible as a result of the fact that those commodities had an intrinsic utility for consumption purposes and that was the reason why they had any value to begin with. The act of establishing money as money without an intrinsic utility value is fundamentally an act of social choice. No simple decentralized mechanism is known to explain how such a social contract

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develops. owever, once a medium of exchange has been estab~~s~~d the trader acts as a regular profit maximizing agent. The fact that money is not used here as a store of value is reflected in the fact that at e~u~~~b~~urn households do not desire and are not allowed to have any net monetary assets.

There are H households enumerated h = I, 2,..., 4-p each with an e~d~wrne~t ~0~E E+N and a preference ordering sh defined over the commodity space E+“. Each household can either use the barter process described in Section I above or else use the monetized retailing process by which it can buy and sell separately. Thus we allow the barter technology of transactions and the monetary technology of transactions to coexist and under certain circumstances they may both be viable ways of exchanging and trading. The budget constraints of the household wit1 then be composed of two sets: the one related to barter exchanges and the other related to monetary tradmg. Let us first introduce some notation: h = vector of household h purchases through barter h sales through barter Yb h = vector of household gbh = vector of household 12transaction cost in barter x’n,7L= vector of household h purchases in the monetary sector Yn h = vector of household h sales in the monetary sector ch = vector of household h consumption ms7z= number of units of account received from sales by i’z 112~~~ = number of units of account paid for purchases by h wh = vector of initial endowment of Iz y = vector of barter exchange rates (prices) pJ = vector of selling prices pb = vector of buying prices h = price of the medium of exchange. Xb

Then household 12can carry out any barter exchanges which satisfy Phh G PYbh - P&lh,

CXbh,Yah, g,hj E Tn.

As ,for the monetary side, the household is confronted ps and pb ; when it sells ysh it receives ly~,~which must satisfy P,msh < ps ymh,

00

with prices (121

428

MORDECAI

and when it buys commodities

KURZ

xmh it must pay mbh which satisfy

Finally when all trades are completed is defined by

the consumption

of household h

Ch = clJh+ X&h+ Xmh - y*h - y,n”.

(14)

The critical question of the formulation is related to the relationship between msh and mph. Let us first suppose that money as a store of value is allowed in our model. In this case let mh be the initial money holding and mob be the terminal money holding of household h. In such a case, we would have to add to Eqs. (11)-(14) the equation mbh < msh + mh - mob.

(15)

The point is that we do not allow money as a store of value thus we assume mh = mOh = 0 and hence (15) is reduced to mbh < msh.

w

Suppose an equilibrium is established with pm = 0. It is then clear that no purchases of value can be made thus pbxmh = 0 and no sales of value are desirable for the optimizing consumer, in which case ps ymh = 0. It is then clear that pm = 0 is the mathematical way of describing the act of closing down the retailing activities or what is equivalent-the abolition of the monetary sector. Since it is our desire to investigate the functioning of the mixed monetary-barter economy there are two ways to approach the problem. (a) To proceed with the constraints as stated and search for equilibrium prices (p, pm , pb ) p,) and then prove that pm > 0. (b) Noting that msh and mbh are to be chosen by the household, the relevant variables are pmmsh and pmmDh. It is then clear that we can set pm = 1 and under this restriction proceed to combine (12) and (13) into one constraint (16)

and thus neither pm nor (mbh, m, “) has to be determined. After equilibrium is established one can calculate pmmbh = pbxbh and pmrnsR.= pa ym”, then these two equations simply determine m,h and msh for any arbitrary choice of pm > 0 and equilibrium choice of (pb , ps) E P,, . L

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Note that if money as. a store of value were allowed, the proposal (b) above couId not work. The reason can be seen by combining (12), (B3), and (15) and obtaining Pbx,’

< ps ylnh + pm(mh - mCiL).

(17)

In such a case the household budget constraint is invariant under multiplication of p by a positive constant and (pm , pb , pp) by a positive constant. In this casep must be chosen from an N simplex and (pm , pb j pJ must be chosen from a (2N + 1) simplex. Setting:p, = 1 leaves no natural compact set from which to choose (pa, pJ. In such a case (I/& has the correct interpretation of “price level” and in equilibrium it must be established. If, however, equilibrium is estab~shed with pm = 0 and > 0, p8 ymn > 0 [which is possible with (17)] then we have achieved P&nh a contradiction whithin the model: pnL= 0 means that the monetary economy does not function while paxmh > 0 and P)~JL’~ > 0 means that the monetary economy does function! ! The above is a ghmpse of the paradoxes various writers have run into (see Starr 1131, Hahn 131, and Sontheimer [9]) while trying to investigate the functioning of money as a store of value. We return to this topic in Section 131 of this essay.

Generality of analysis would have called for the introduction of a large number of traders and then assume competition among them. For simplicity of exposition, however, we consider only one central trader and the reader may think of this entity as the sum of small and identical traders.(Section III below discusses the case of R traders; this extension is very simple indeed.) Let us first introduce some additional notation: u = vector of purchases of the trader from households 2; = vector of sales of the trader to households zb = vector of transaction cost of buying 2, = vector of transaction cost of selling. The trader has a “transactions technology” sets” Tb and P for buying and selling. Thus

or “transactions

Tb is the set of all feasible pairs (24,z,), T” is the set of all feasible pairs (0, z,).

possibility

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MORDECAI

KURZ

The “transactions possibility sets” have interpretations similar to those given in Section I above and reflect our earlier discussion on the level of monetization of the economy. Since pb and pS are buying and selling prices for consumers, it is then clear that pb and pS are selling and buying prices (respectively) for the trader. Thus for any set of technologically feasible trades (u, z,) E Tb (v, ZJ E TS we let m,,’ = number of units of the medium of exchange paid out by the trader for buying goods. mST = number of units of the medium of exchange paid to the trader for selling his goods. Then we have the natural restrictions psu

<

pmmbr9

Pbv

\(

pmmsr.

The process behind the above restrictions is that the trader acts like a bank in creating inside money. The trader issues units of money which we shall call “shkalim” and pays consumers with these shkalim whenever he buys any commodities from them. It is then clear that money as a medium of exchange in this model is nothing but the obligation of the trader to provide goods from his inventory in exchange for those shkalim held by consumers. Similarly the trader expects to be paid with shkalim whenever he sells any goods to consumers. Thus p&&,’ is the total value of the shkalim which the trader either issues or pays out for purchases from consumers while plnmsr is the total value of the shkalim which he expects to receive from consumers in exchange for delivery of his goods. The critical fact that we wish to emphasize is that in the above formulation it is impossible to operate the monetary economy when pm = 0. Put differently, the reason why the monetary economy may function and may be more efficient than the barter economy is because it has a medium of exchange which makes the trading process so much simpler. If this medium of exchange is not available, the technology of transactions with money cannot be operated. Now it is clear that the trader must take into account the fundamental material balances which is involved in any act of trading and that is u - v - zb - z, > 0.

(18)

Consider any technologically feasible (u, v, z b , 23 that satisfies the material balance condition as well. We can define the profit of the trader by 17 = p,u - p$.

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In view of (I ES),this is reduced to

where equality holds only if ps(u - L’ - zb - zs) = 0. And this is reasonable since (pb - pJ is the gross profit margin on sales while pS(zb + zS$ is the transaction cost. There is an alternative way of formulating the profits of the trader in terms of shkalim. If we write p,u = pmmbr, pbv = pmmsr, ther?

and [m,’ - mbl‘] is the total number of shkalim which the trader will pay out as dividends. We must turn to this issue now.

We shall work with a private ownership economy so that there exist numbers f h,, 0 < fh < 1, and ~~zI f n = 1, which represent the h household’s share of profits. This will require a modification of the household budget constraints. Thus let s denote the vector (u, 2;, zi, , z,) and efine the feasible set Y to be Y = (s = (u, u, zb , z&u,

zb) E Tb(u, z,) E T”, u - v - zL - z, 3 0)

and let the profit function n(pb , p,) be defined by n(pb

where n(p,

i ps)

=

y$$

[PbL’

-

,?sd,

4pJ is allowed to be + co. Now we define ?1(Pb

, Ps)

=

1s E

y

i P&

-

Ps”

=

nTPb

2 @&

(22)

as the supply correspondence. Let us now return to the household. Our basic economy calls for the budget constraint (including the share of profits)

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MORDECAI KURZ

This leads to the definition Bh(p,

pm

? Pb

, Ps>

=

((Xbhy

Ybhp

gbh,

Xmh,

Ymh,

m,h,

mbh>

1 (23a)+23e)

are satisfied).

(24)

Similarly fh(p,

Pm

5 Pb

(25)

2 Ps)

ch 2, ch’ for any $’ = wh + x;’ + x$ - y;’ - y; =

I cXbhT

Ybh,

E Bh(p,

gbh, pm

Cmh,

.hh7

3 Pb

, ps>

msh3

i /’

mbh>

such that

(x!‘, y;‘, g;‘, xg, yg, m:‘, mi’) E Bh

1

1

E. Assumptions We shall present here a set of assumptions which we maintian out. They are:

through-

(a) (TJ), (T.2), (T.3), (T.4), (D. l), (D.2), and (D.3) in Section I above. (b) We shall also assume (E.l) (w” > 0 all h) for mathematical convenience. It can be relaxed as in Section I but this will be discussed later. (c)

(T.1’) Both Tb and TSare closed convex sets. (T.2’) If (U, zb) E Tb, (U, zs) E Ts, and 0’ < U, U’ < U, z,,’ > zb and z,’ > z, then (u’, zb’) E Th and (v’, zi) E T”. (T.3’) (0,O) E Tb and (0,O) E T”. (T.4’) There exist (zi, sb) E P and (6, a,) E TS such that zi - 6 - .&, - ,& > 0 and 6 > 0. (T.5’) The set Y is unbounded in at least its first two components, i.e., if (u, D, zb , z,) E Y there exists (u’, z)‘, zb’, zs’) E Y such that u’ > u and v’ > v. Assumption (T. I’)-(T.4’)

(T.5’) is new in this section. It would have been implied if Th and TSwere assumed to be cones.

by

F. Equilibrium-Preliminary

With the aid of Section I we can now establish one immediate

result.

THEOREM 3. There exists a competitive equilibrium such that pm = 0 and ymh = xmn = 0, all h.

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433

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The equilibrium which is asserted here is one in which the Renzark . monetary economy is not functioning. PP.oo~. Consider the barter economy of Section B above. Since C.O~ > 0 Theorem I of Section I applies and thus there exists (p, (xbh, ygh, gbh) is = 1, 2,..., W> which constitutes an equilibrium in the barter economy. Wow choose pm = 0, pa = ps > 0, xmh = ymh = 0 2nd u = v = zb = z, = 0, and this set of values constitutes a competitive ~qu~~ibr~~rn* The theorem above is a natural consequence of the fact that the barter economy is in fact a subeconomy in our expanded economy. But the theorem brings out one fundamental point which we have started elaborating on earlier: the introduction of money either as a store of value or as a medium of exchange requires a social contract or a social decision Bn this sense, money-as a commodity without a utiSty value--is a public good whose existence as a viable economic entity requires a fundamental act sf social choice. Theorem 3 clarifies the fact that in our economy there may exist two basic equilibria: One is the barter equilibrium in which the monetary does not function, and the other is an equilibrium in which the monetary economy functions while the barter economy may or may not function In order to investigate the existence of a monetary equihbrium we follow the second suggestion we made earlier. According to this we as pm > 0 and proceed to eliminate pm ) mbfi, and mSh from the bo~seho~d budget constraint and pm , mbr, and PYZ,Tfrom the trader’s constraints. Naturally, that procedure is applicable only when ,pnz> 0. If we establish the existence of equilibrium in the reduced economy we shall in fact be provmg the existence of a monetary equilibrium. e shall discuss this interpretation later on. For the moment let us redefin he budget, demand, ly correspondences. First let

1 PXbh tXbh,

hh,

gbh,

-%nh,

Ymh)

<

1 pbXmh ‘dh

f

pybh

-

d

P&n”

xbh

+

pgbh, +

Xmh

&bh>

hPL,

.f”n(Pb -

ybh

h=l

E T’k

9 PSI -

Next we introduce a large rectangle KOwhich is defmed by

where w = c ‘@la.Wow let

g,“)

ymh

>,

Lb

434

MORDECAI KURZ

Next we define

Ch(Xbh,Ybh, gbh,

xmh, Ym”) = rrh + Xbh + xmh - Ybh - ymh

and Yoh(P,

Pb

7 Ps)

(29)

Finally let (30) (31) and %(Pb

5 Ps>

=

ts

E y,

1 pba

We can now introduce the definition

-

=

17,(pb

, pa)>.

(32)

of a monetary equilibrium.

3. A monetary equilibrium

DEFINITION

psu

is a price vector (p*, pb*, ps*)

and allocations {(x$h, y,*“, gch), (xzh, ygh), h = 1,129 .... If, (u*, o*, zb*’ zs*>>

such that (a>

p”

E PN

@I

(xch, Y,*“, g,*“, x:‘, Y:“) E Y~(P*, pb*, P,*),

(c)

xm*

=

Ym*= xb*

<

, (Pb”?

f h=1

Ps*)

xi”

<

-

gb**

E pin

)

u*,

h=l yb”

Remark. It appears that Condition (c) of the above does not really cover the true material balance condition. The “real” condition should be

c* + g,” + zb*+ z,* < 0.

(331

A SINGLE

But we can immediately that we can write.

MARKET

EXCHANGE

ECBNBMY

$35

show that (c) above implies (33). To see this note

&I * + xb* - y,* < v* - zd*$ y," -- ga*? thUS c&n* -I- xb*) - (yn2* + y,") < Urn- 21* - gb*.

w

Now since c” = w + x,”

+ xb* - y,”

- yb*,

(35)

we can combine (34) and (35) to have c* - w < v* - IA* - g,* Or

c* - (v* - u*> + g,* < w. Now since (a*, D*, z8*, z,*) E -g(pb*, ps*) we have

Introducing

this inequality

into (36) we obtain

which is (33). Condition (c) does have a deeper economic inter~ret~tio~~ it the economy into three markets. In the barter market xb* is the demand and yb* - g,* is the supply and, therefore, x0* < yb* - gb* is nothing but the demand < supply condition in the barter market. Next, in the monetary economy there are two markets: one in which households buy and the other where they sell. The condition x~* < D* is requir total household demand (purchases) not to exceed total retailing sup (sales) while the condition ym * > u* reverses the two functions. G. Existence of Monetary Equ~~~br~~~When C& > 0 All h

In the search for equilibrium

we can now restrict ourselves:

436

MORDECAI KURZ

where

LEMMA at

all

(p,

1. (Pb

The budget set correspondenceBh(p, pb ,p,) is continuous E PN x P2N , such that ps& + fhIL7(pb ,p,) > 0.

,Pd

Proof. First we prove that Bh(p, pb , p,) has an interior point for all (p, (pb , pS)) E PN X PSNsuch thatp,wh + f h17(p, , pS) > 0. Since uh > 0

it is clear that there exist a small ymh > 0 and ejbh > 0 such that

By (T.4), (+,‘, ~j$,~, &,“) E Th and Epj&,’ > .&,‘. Also by (T.2) for all El < E, ($3Z’bh,@bh, @bh) E Th, thus for small enough El

Next consider the sale of yrnh > 0. Since pswh + f “n(pb , pS) > 0 by assumption and ymh > 0 it follows that there exists a vector x,~ > 0 such that Pbxm

’ < psymh+ f

hnfpb

, ps>*

Finally consider the next result: c = Wh + xmh f +?bh - ymh - ej$,“. Since by choice of ymh and ejJmh we have Wh - YnZh - Ej&h > &lJ” > 0, it follows that c > 0. Thus the budget set has an interior point if psmh+ fhn(pb , pS) > 0. Then using the same argument as in Debreu [2, pp. 64-651 we can conclude that the correspondence Bh(p,p, , pJ is lower semicontinuous at any (p, (pa)) E PN x PzN such that pswh + fhfl(pb ,p,) > 0, On the other hand, the correspondence is clearly upper semicontinuous at all this completes the proof. Q.E.D. (P, (Pb 9 Ps>> E pN x P2N - And We are thus motivated to define pn”(p,

Pb , ps>

=

YP’(P,

Pb

, Ps>

= B,h(~, Pb , I’s)

f

if

Psmh

f

if

psmh

+fh2?(pb

hn(Pb

, P,>

>

o

,p$>

=

0.

(37)

A SINGLE

MARKET

EXCIIANCE

437

ECONOMY

kFMMA2. (a) f,,“(p,pb , p$) is a noneinpty, compact, co%vex, and upper se~~contilluo~~correspondence. (b) q,,(pb , pJ is a nonempty, compact, convex, and upper semicontinuous correspondence.

ProaJ In both (a) and pactness are straightforward Upper semicontinuity of continuity of 9p follows from and Y, is compact.

(b) the nonemptiness? convexity, and comand follow the standard line of argument. 9” follows from Lemma I and upper semithe fact that the profit function is continuous ED.

e are now ready to state THEOREM 4. Under the maintained a~sz~~pt~o~~made if2 (11.5) there exists a monetary equilibrium.

Proof.

First define for the bounded economy

p maximizes p(& - & t &) and (pb ,p,) maximizes i. E PN x PpJ

(P,

=

(Pb

3 us>)

1P?J(“m- v> -i- Ps(u - Y,3

1

(40)

)

And, finally, consider the mapping $%h

9 Yb , gb =

&,

x

, &n

, Ym

‘%I

x

, u,

u,

dxb

zb

3 Yb

, d,

2 gb

6p,

(&J

> &L

2 Ym

> p&1

3 %

a,

zb

(41)

9 zsj.

It follows from Lemma 2 that $J, is a nonempty, convex, compact correspondence which satisfies all the hypotheses of the Kakutani fixed point theorem and therefore has a fixed point

This i;xed point satisfies the following conditions: @b*,

p*(&” Pb”hn” -

?+J*,

gb*,

&a*,

@*,

u*,

zb*,

-

Yb*

+

v*>

+

ps*(u*

gb”)

2 -

.?+n*> zs*)

&b* h*>

3

E ?2(p*, ’ Yb”

pbhn”

(42a)

Pb*,Ps*),

%(Pb”,

C433)

ps*)I,

+

all

&*) -

v*) df

+

p&* (pb

p E P, , -

, ps)

(‘=I

ym*) E P,,

.

(L?iZd)

438

MORDECAI

KURZ

Since xb

*

-

yb*

+

gb*

=

f

x$h

-

h=l

from (42a) it follows that P”h”

-

Yb*

+

gb*>

<

O-

Thus by (42c), p(&,* - y,* + gb*) < 0 for all p E PN and this proves x*-y*+g*

GO.

(43)

On the other hand, from (42a) it follows that &“X;h

< p,*yz” + f hnbb”, p,“>’

w

Summing this over all h and recalling (19) we obtain pb*(&*

- v*> +ps*&*

- ym*> < O-

(45)

Thus it follows from (42d) that p&m*

- u”) + ps(u* - j&*)

< 0

al1

(Pb

? Ps)

E pZN

and this means .-cm* e II*,

(464 (46b)

Conditions (42a), (42b), (43), (46a), and (46b) prove that there exists a quasiequilibrium in the bounded economy. Now in order to prove that this quasiequilibrium is in fact an equilibrium, it is sufficient to prove thatp,* > 0. Since UJ~> 0, then, Ps”Wh + fhmPb*,

ps*> > 0

for all h.

Suppose the contrary, i.e., ps * = 0. Then by (T-4’) and since 1 n n (% % zb , ‘%>E y we have 17(p,*,

O> =

z$xpb”v P

>pPa”v

>

0,

since pc > 0 for at least one j. Since II(p,*, p,*) > 0 it follows that p,*ah -I- fhn(pb*, ps*) > 0 for all h with fh > 0. Thus let H+ = {h If h > 0} and let #H+ be the number of elements in H+ .

A SINGLE

Finally let 0: = #N+/H.

MARKET

EXCHANGE

ECONOMY

439

It then follows that for any h E

qoh(P*, Pb*,ps*b

= ypYp”, pb*, ps*>.

Now suppose p& = 0 for some i". Then x$$ = pwiO and therefore X;,a 3 p(aH) w&f, and since 0 < l/H

< cx < 1 it follows that x”mio >’ P W.H. zo

Qn the other hand, y*h m zxz0 *h< \P Y 97%

for all h E H+ , w

for all 12$ H+ .

It thus follows that YZi, d PC1 - 4 f&l

3

and since cx > 0 it follows that

which contradicts the fundamental condition (46a)-(46b) which requires .x,~* < U* < U* ,< ym*. This proves that pb* > 0. Next it follows from Assumption (T.5’) that whenp,” > 0 ansIp,* = 0, for some i, ui* = pHwi . To see this assume ui* < pl-Hw, for all i. Then by (T.5’) there is a feasible (u’, zi’, z b’, zs’) such that U’ > U* and U’ > 2:*, which impliesp,*v’ > pb*v* becausep,” > 0. If u’ < pHw, the optimality of u* is contradicted. If uj’ > pHq , for some j, then we can find a convex combination (u”, 27, z: , 21) of (u*, zj*, zb*, z,*) and (LX’,vi, zb’, zS’) such that (u”, v”, &’ ,zi) E Y,, : u* < pHw and ui* = pHoi , for some i, of and that pb+vR > ph*v*- Again this contradicts the optimality (u”, v”, 2&*, z,*). Because of Assumption (D.3) we have yz” = 0 for any h E $J+ . Hence it follows that ym” < ~(1 - LX)Hw. By the market clearing condition we have established earlier, then we must have ui* = $3~~ < y& < ~(1 - a) Hwi . However, this implies 01< 0, which is impossible. Thus we have shown ps* > 0. Finally, in order to prove that we can let p tend to + co, we can use the QED. same argument as in Kurz [5].

440

MORDECAI KURZ

In the process of proving the existence of equilibrium, two important results: (4

pb* >

(b)

ps” > a

we also proved

0,

We can add: LEMMA

3. If (pb*, pS*) is an equilibrium vector then pbX > pS*.

Proof. If pg < p: then one can show that consumer optimization in the bounded economy would call for x;: > 0 and profit maximization would call for I.+* = 0. This is true for all p and thus cannot be an equilibrium. Q.E.D.

It is possible that in equilibrium no transactions will ever take place since optimal consumer allocation will call for ~2” = J$” = 0. This equilibrium is not excluded and implies pm” = 1 and rnc” = m:h = 0. If, however, xzh > 0, then pb*xzh > 0, and thus pS*yz’ > 0 as well. In any of these cases we can calculate from the equilibrium of the reduced economy the number of units of account used in the system. Since we can arbitrarily set pm = 1, we can calculate

(474 W’b) mb

*r = ps”u”,

m*r s = pb*v*,

C47c) (474

and this definition concludes the construction of the monetary equilibrium in the economy with a unit of account and a medium of exchange. Some remarks.

(1) It is important to note that inside money as a medium of exchange is nothing but a debt instrument of a financial institution: It is the obligation of the trader to accept the money back in exchange for goods. (2) The model of money as a medium of exchange expresses very well what is the “veil” concept of money; it is nothing but the ability to reduce the monetary economy to its “real variables” as expressed in the demand correspondence(19) and the supply correspondence(22).

(3) The reader may suggest that a similar situation exists in the barter economy I above in the sense that it is the reduced economy of some other

A SINGLE

MARKET

EXCHANGE

ECONOMY

441

monetary economy which can be reconstructed. We insist that this is noi the case since in Economy I the consumer must choose (xh, yh, g”) s Th for the act of an exchange. In order to reconstruct a monetary economy with a medium of exchange, one must find gob and g,” such that (Xh, 0, ghh) E Th, (Q, Yh, gs”> E Th, ghh ‘-t gsh = giL, and there is nothing in the model to make the above possible. This is the deeper meaning of our statement that in Economy I consumers must ““exchange” bundles while in Economy II they must ‘“bay” and “sel14” and these are separate activities tied together through a medium of exchange. . E~~i~~br~un~ When f

cob> 0 But Not ?Jecessarily~2 > 0 All h

h4

In the proof of Theorem 4 we used both the assumption CJ~> 8 for all h and strong desirability (D.3). This is in fact too strong a set of assumptions and Theorem 4 is, in fact, true under the weaker condition ZLi mh> 0 without adding any assumptionson Th and Ts. This may appear surprising in view of the fact that in Section I above we needed a rather elaborate set of assumptions to prove Theorem 2. In fact, the assumptions made in Theorem 2 are sufficient to establish the existence of a monetary equilibrium even when 69 > 0 for all h does not hold. We provide a brief description of the argument. If PsWh+ f -TPzI >PSI > 0,

then the consumer can buy strictly positive quantities of all goods in the monetary sector and then use them to exchange a strictly positive vector in the barter economy. This proves the existence of an interior point in the budget set. If psWh+ f hD(p, , ps> = 0, then we nse the argument used in Ref. [5], Lemmas 6, 7, and 8 to prove that the consumer can obtain a positive quantity of all goods in the barter sector. Next using the same argument as in Theorem 4 we prove that pb* > 0 and ps* > the positive quantity obtained in the barter sector the consumer can also sell positive value in the monetary sector. Putting the two cases together we have an interior point in the budget set and thus can establish the existence of monetary equilibrium. THEOREM 5. Let the maintained assumptionsin Section I1.E hoid with the following modifications:

442

MORDEC.41

KURZ

(a) h’l mh > 0 but not necessarily mh > 0 for ail h; (b)

the additional assumptions on Th made in Theorem 2.

Then a monetary equilibrium

exists.

I. The Dual Equilibria Combining Theorems 3, 4, and 5 we see that our economy has two sets of equilibria: a set of barter equilibria and a set of monetary equilibria. Under very mild assumptions of dominance it is clear that the monetary economy produces a strictly larger set of possibilities for the economy and thus the barter equilibria are not eficient. Using conventional methods, one can easily show that any monetary equilibrium is efficient. Thus we arrive at the peculiar conclusion which can be stated as follows: The economy under study has two sets of competitive equilibria: the set of barter equilibria which may not be efJicient and a set of monetary equilibria which are always efJicient. In order to clarify the meaning of the above conclusion let us note that the barter structure is a complete subeconomy of the monetary economy. There is no clear decentralized process which will lead to the establishment of the monetary equilibrium. More specifically, if a competitive dynamic mechanism is postulated then the following will probably be true: (a) If an ininitial disequilibrium pm = 0, the economy will never converge to a monetary equilibrium with pm > 0. (b) Even if in the initial disequilibrium we have pm > 0 the monetary equilibrium may “disintegrate” and the economy may converge to the barter equilibrium. The reason for our conjecture follows from our conviction that a monetary economy cannot arise out of a spontaneous decentralized competitive system. This is so because money as a commodity without intrinsic utility (i.e., fiat money) acts in a social system as a public good; it develops because there has been either explicit or an implicit social contract or an act of social decision to accept it as a medium of exchange and a unit of account. Thus the creation of fiat money is a collective act of the creation of “trust,” “confidence,” or any other such feelings on the part of the participants in the economy. Like any other public good, money exists because society has decided to create it collectively and no competitive decentralized system can generate this commodity. The creation of money-as our analysis has demonstrated earlier-can be extremely beneficial to society and thus the incentive for the creation of this public good is present.

A SINGLE MARKET EXCHANGE ECONOMY

ECONOMY

111: A SINGLE PERIOD ECONOMY WITH TRANSACTIOI\\: AND MONEY AS A STORE OF VALUE

443 COST,

We wish now to add to the economy of Section II above money as a store of value. It is well known that in a single market economy the “end” of trading presents a specially difficult problem due to the fact that individual consumers have no incentive to hold fiat money after the “‘close” of the market. This is also true in any economy with a finite sequence of markets in the sense that when the “last” market closes, no consumer wants to hold fiat money. Since the main issue of the existence of a monetary equilibrium in such an economy arises in the “last” market, it is clear that the study of the store of value function of money in a single market monetary equilibrium is equivalent to its study in any economy with a finite sequence of markets. Our main investigation is directed towards the issue of the ‘6cover’9 of fiat money. We note that all writers have recognized that in an exchange economy the only way to ensure the existence of a monetary equilibrium is to provide 100 % cover for fiat money. Thus writers like kerner [6], Starr [l&12], Hahn [4], Sontheimer [9], and others have either just “required” the consumer to demand fiat money at the end or motivated such holdings by the requirement that terminal money be used f~or tax purposes. The use of a finite economy is a reflection of the finite life of consumers who do not wish to hold money beyond their own life. It is equally undeerstood that for a monetary equilibrium to be established, some agent must be “future oriented” and thus the existence of such an equilibrium must be related the behavior of such agents. In this section of the paper we introduce into the model trading firms in order to explore the effect of ffirms’ behavioral patterns on the issue of monetary e~~i~~~~~~rn. We shall note later that the key issue is the motivational structure of firms. We reject the view that firms have utility functions which are defined over consumption bundles. Consumers have such utility functions and their refusal to hold money at the “end” reflects the fact that, with positive price, they can use money to acquire more consumption goods during the trading period; this strate,g is better than holding money, Firms do not have such utility functions; nor do we insist that a firm’s life terminate when trading stops. We start by adding the following notation: w’~ = initial vector of endowment of trader r MZ = initial money holdings of trader r

444

MORDECAI

KURZ

p = terminal money holdings of trader Y mh = initial money holdings of household h mob = terminal money holdings of household h m0

h=l mT

=

5

T=l m7

9-1

m.

T-

I$

mop

A. The Households and the Traders The formulation of a household’s demand correspondence does not present any special problem. The natural constraints are PXbh < PYbh - Pd,

(~2, ybh, a”) E Th,

Pmmsh < PsYmh, pbxmh < pmhh

i- mh - mob> + 2

(48) (49)

frhD(pm, ps , ad,

0 < c = 03 + Xbh + xmh - ybh - ymh,

(50) (51)

where Dv(pm ,ps ,pJ is the pay-out function of firm r. This may not be equal to profits. To avoid confusion we note here that since fiat money does not enter the consumption set it is clear that there is no loss of generality in the understanding that mob = 0 all h. Thus there is a fixed stock of outside money: m shkalim. Before the market opens households own Cf=‘=, mh shkalim while the traders own Cf=, mnrshkalim. We do not require the households to hold the stock of shkalim at the end; moreover, we clearly allow the households to use the shkalim they have for buying commodities as long as the value of money is positive. A trading agent presents a more complicated issue. Such an agent has an initial capital in the form of an initial bundle wY of commodities. He buys the vector u’, sells the vector vT, and incurs transaction costs zsT and zbr, where (z/‘, z~‘) E (U, z,‘) E T,.” and (U, z,~) E T,*. The critical issue is related to the relations between the profit and pay-out functions of the firm. Before turning to the profit function we note the natural constraints of each trader: (UT,zb’) E Trb, (52) W, zsT>E KS, WY+ UT - VT- Zbr - z,T >, 0,

(53) (54)

A SINGLE

MARKET

EXCHANGE

5

ECONOMY

IT&)+”i YYiDT ,( rYzsT+ nlr.

152

In order to define the profits function of the trader we need to understa the valuation of his initial capital wr. To assume that this capital can evaluated in terms of simple market prices like pswT or pbw’ siqdy ignores the fact that wT must be sold and the act of selling will require transaction costs. If we think of the “net value” of the initial vector as a function MT(pb , pa) we can follow the argument of Section II and define the profits of the trader as

Thus the net value of the initial capital is charged as cost to the trader. This is perfectly reasonable but the issue remains open: how to define K(Pb

> t9sj

?

Without much further motivation we assert that Xr(pa, p,) should be defined as the increase in profits due to the additional commodity vector ~2”. Formally we let Y’ =

UT >, VT+ ZbV+ z,q I) (u’, 26) (v’, z,r) E T,” / 1 (u’, ZbT)E Tyb i

We note that profit maximization of the profits function:

(%a)

by the trader will call for the definition

which implies

and this means that the value of profits are the same as they would have been without the bundle wr. This is exactly what we wanted but the crucial

446

MORDECAP

KURZ

difference lies in the fact that if we employ (60) as the definition we obtain a different supply correspondence than if we use (61). The correct supply correspondence is obtained from the definition (60) which we shall proceed to employ. The fundamental issue can now be clearly stated: What is the relationship between the pay-out function DT(p, ,pJ and the profit function @(p, , ps) ? Since this is the essential behavioral issue on which the existence of monetary equilibrium depends let us explain this point in detail. B. Profit and Pay-Out Functions

We are investigating the effects of firm’s behavior on the existence of monetary equilibrium. With this in mind we propose to view the firm as an independent agent with unlimited life expectancy who expects trading to resume in the future, at which time its assets can be reactivated. For this reason the firm does not liquidate itself but rather pays out only its current profits; thus D’(p, ,pJ = ITT(p, ,pJ. This means that the firm retains its assets for future use. However, in evaluating the form in which to keep the assets, the firm is indifferent between commodities and money as long as money has equilibrium positive value. Given that we do not introduce risk of future prices, this neutrality of the mm is perfectly consistent with profit maximization. This neutrality allows the consumers to acquire all the commodities at the end of trading. A special case of the above is the single trading-banking firm which can be thought of as a central bank combined with a commodity exchange. Note also that the shares fh entitle the consumer to receive profits only; the equity which the consumer has is nil since the initial endowment was not paid out by him. One may take the orthodox viewpoint which will relate property ownership to consumers and require that the firm liquidate itself completely at the end of trading and turn its assets over the “stockholders.” Thus both consumers and firms have the same finite life, at the end of which the firm’s pay-out function is DQ,

, ps) = ni‘(p* , pJ + KQ,

, ps) = total revenues.

It will become clear that under this condition, consumers and firms are identified and thus in a finite economy there will be no monetary equilibrium. We hold a different viewpoint since we believe that the essence of the monetary economy with fiat money lies in its continuity resulting from some agent looking into the future. Ideally one would want to construct a complete model with an infinite sequence of markets, firms with infinite

A SINGLE

MARKET

EXCHANGE

447

ECONOMY

life, and consumers with a finite life. This analysis is not yet within our reach but the model which we propose here is the first approximation. We now proceed to examine the existence question when D’(p, 9pS) = WP* 3PJ. C. The Existence of Monetary Equilibrium Hn equilibrium, the conditions (55)-(57) would hold with equalities so we may as well write the profit functions

and the aggregate profit function fl = n&Q

- mT) - K(P, ,PJ,

where

K(P, 3~s) = 2 WP, r=1

,p,3.

We thus have LEMMA 4. Assume that there exists a monetary Cf=“=,mh > 0. Then

PTn* = Kb*, Broof.

p.*l/il

eq~~~ibr~~~~ with

mh.

($3

From the definition of the profits function it follows that 17”” = p,*(mz“

- mT) - KT(p,*,ps*).

Let II*

= p,“(m,*T

- mT> - K(P,*,P,*).

We also have from consumer optimization

that

N + pm* c mh + -wP,“,

J%*x* = Ps”Y”

A*),

h=l y*

=

u*,

x"

=

2's >

and thus H pa%”

-

ps*u* = pm” c ma 5 -D-*. k=l

448

MORDECAI

KURZ

It can be seen that we also have - p,u* = p,*(~$~

p,W

Thus combining

- mT> = IT* + &J,*,P,*~.

W)

(62) and (64)-(66) we have n*

+ K(P,“,P,“)

= Pm* 2 r@+IP h=l

or Pm” jjl mh = K(P,“, COROLLARY.

If in equilibrium

Ps”).

Q.E.D.

pb* > 0 and ps* > 0 then pnz* > 0.

ProoJ: It is sufficient to prove that Kr(p,*, ps*) > 0. Let (V’, ii’, Zbl; ZSr) be an optimal solution for the following problem:

By (T.5’) there exist (V’, 2,“) E Trs, (u”, ~9;‘) E T,” such that vr’ > B, U” > 3 and vT’ + 2;’ + z,” = ZF’. By convexity we have

[(a~ + (1 - a) vT’),(G; + (1 - 4 $‘>I ET,“, [(“ii’ + (1 - a) u”), (cxzbbr+ (1 - a) z;?l E Trb, and since mT > 0 we can choose a! close enough to 1 such that (1 - a) UT’ - (1 - a) z;’ < or;

(67)

with this choice we have [a~ + (1 - CX)z;‘, cGsr + (1 - IX) z;] E T,s, [ii’, z,‘] E TTb,

(016”+ (1 - 0) VT’, 3) E y+Jq.

(6 84 (68b) 6334

To prove (68~) we must show that (a$ + (1 - a) VT’) + (Kz; + (1 - LX)z;‘) + ZbT d U’ + UT. we first note that from (68a) we have a@- + zbr + 2,‘) + (1 - a)(v” + z;’ + z;‘) = aur + (1 - a) UT’

and thus K’(p,*,p,“)

> 0, which proves pm* > 0.

QED.

We are now ready to tnrn to the proof of the existence of a monetary equilibriums THEOREM 6. Under the condition of Theorem 4 and in addition for some r ho’> 0 and cfzl mh > 0, the economy (48)-(57) and (60) has a monetary equilibrium with pm* > 0.

Proof. We provide here only a sketch of the proof. In the search for equilibrium we first let ($9) artd then restrict the search for equilibrium to fp E BN and (pb , p,) E PzN n We then reduce our economy as in Theorem 4 rewriting the household constraints. (4X)-(50) as

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MORDECAI KURZ

and the trader’s constraints (55)-(57) as p&f

<

PbvT

+

[Kc&

, Ps)/h$l

n@]

(m*

-

f%‘h

r =

1, 2 ,..., R. (71)

We thus obtain a new economy which is similar to the one analyzed in Theorem 4 except for the modifications in (70) and (71). In order for the techniques of the proof to be applicable here we have to ensure that the modifications above do not change the properties of the correspondences. First we note that P(P, , p,) > 0. This follows from the optimization (61) and the condition (0, 0) E Yr, which leads to the conclusion max [p@ - p#] w, 213EYF

> 0.

Second, all the needed continuity properties have been preserved and therefore the methods of proof used in Theorem 4 or Theorem 5 are applicable. We thus show the existence of equilibrium in the reduced economy. Now under the hypothesis W? > 0 for some r we use the corollary to Lemma 4 to prove that KT(p,*,p,?) > 0 and, since c& mh > 0 by assumption, we prove p,” > 0. Q.E.D. The “peculiar” equation (63) brings out what happens in our monetary equilibrium: the households possess the sum of money equal to cf=r mh, which they exchange for the vector of goods in the hands of the traders, where transaction costs enter here through the fact that the vector is worth K(p,*, p,*) and not CFB1pswr. In equilibrium the value of money paid by the consumers isp,” Cf=, lnh and the value of commodities which they receive for it (net of transaction cost) is K(p,*,p,“). Thus we have (63). COROLLARY. Suppose wT = 0 all r then the only equilibrium in our economy is the one with pm* = 0.

possible

What we have shown above is that in an economy with a trading activity the inventories of the traders can provide as good a “cover” to fiat money as any other. This is so in a finite horizon economy. We can in fact go one step further and point out that in a finite economy money as a store of value will have a positive value in a competitive equilibrium only if it has some kind of “cover” in the real economy. This “cover” takes the form of goods which individuals may receive at the end of trading. Our Corollary is thus not surprising: Even if our traders are willing to hold money at the end, their willingness must be backed by valuable inventories in order to enable the money in circulation to have any value. In the case of payment of taxes to the government the “cover” is more

A SINGLE MARKET

EXCHANGE

ECONOMY

451

subtle since here individuals use money to pay obligations to the government and those debts are the “cover” for money. Note that if money itself was originally issued as an obligation of the government then the payment of taxes represents the disappearance of money from the economy. Regardless of what are the covers of the various currencies in the world, it is perfectly clear that there is no relationship between the value of money (i.e., the price level) and the explicit cover of money. This means that a useful theory of money should deal with fiat money and establish the price level on a deeper foundation than the ‘“cover.” It is, therefore, clear that the function of money as a store of value presupposes the reopening of the markets and the presence of future-looking agents in the economy. But we have argued in this paper that if the markets reopen only a finite number of times and money has no cover then it will have no value. This leads to the concludion that the natural way to devefog the theory further is to assume that the economy consists of an infinite sequence of markets.

IV.

GENERAL

Co~~~usro~s

We constructed in this paper a sequence of three economies all with transaction cost: a barter economy, a monetary economy with a medium of exchange but without a store of value, and fmally a monetary economy with a medium of exchange and a store of value. We can sum our conclusions as follows: (a) The transition from a barter to a monetary economy is a structural change that must be made with a fundamental act of socal choice since money is a public good. For this reason economies with money will always have a dual set of equilibria: the first is the one in which the monetary system does not function and we return to the barter competitive equilibrium and the second is the monetary equilibria with positive value of money. The first is inefficient and the second set is efficient. (b) The ““veil” theory of money is correct whenever we deal with economies with inside money which is used as a medium of exchange only. 4n such economies all money is ‘“cancelled” at the end of trading. In snch cases (Economy II above) the monetary economy can be ““reduced” to its real variables and the “veil” is thus removed. (c) When a fixed supply of money is introduced, a finite duration monetary economy will function only whenever money has a real cover. This is so since the use of this cover will in fact be experienced. By this we mean that a day always arrives in which the cover will be used up by

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KURZ

the holders of money. Thus the classical theory of the real “cover” for money is correct for economies of finite duration. (d) It is tentatively concluded that the natural way money exists as a store of value without full real cover is when the economy is one of infinite duration such that no time actually arrives at which equilibrium calls for the liquidation of the cover. REFERENCES 1. K. J. ARROW, “Aspects of the Theory of Risk Bearing,” Helsinki, 1965. 2. G. DEBREU, “Theory of Value,” Wiley, New York, 1959. 3. F. H. HAHN, On some problems of proving the existence of an equilibrium in a monetary economy, in “Conference on the Theory of Interest and Money,” (F. P. R. Brechling, Ed.), Royamount, France, 1962. 4. F. H. HAHN, Equilibrium with transaction cost, Econometrica 39 (1971), 417-439. 5. M. KURZ, “Arrow-Debreu Equilibrium of an Exchange Economy with Transaction Cost,” Working Paper No. 7, Institute for Mathematical Studies in the Social Sciences, Stanford University, Stanford, California, 1972. 6. A. P. LERNER, Money as a creature of the state, Proc. Am. Econ. Assoc. 37 (1947). 7. J. NIEHANS, Money in a static theory of optimal payment arrangements, J. Money Banking 1 (1969),706-726. 8. R. RADNER, “Existence of Equilibrium of Plans, Prices and Price Expectations in a Sequence of Markets I and II,” Technical Report No. 5, Center for Research in Management Science, University of California, Berkeley, California, 1970. 9. K. SONTHEIMER, “On the Determination of Money Prices,” State University of New York, Buffalo Discussion Paper No. 128, 1970. 10. R. M. STARR, “Equilibrium and Optimum in a Pure Exchange Monetary Economy,” Cowles Foundation Discussion Paper No. 317, Yale University, 1970. 11. R. M. STARR, “Structure of Exchange in Barter and Monetary Economies, Cowles Foundation Discussion Paper No. 295, Yale University, 1970. 12. R. M. STARR, “Equilibrium and Demand for Media of Exchange in a Pure Exchange Economy with Transaction Cost,” Cowles Foundation Discussion Paper No. 300, Yale University, 1970. 13. R. M. STARR, “The Prices of Money in a Pure Exchange Monetary Economy,” Cowles Foundation Discussion Paper, No. 310, Yale University, 1971.