A strategic market game with transactions costs

M~lthematlcal Social Sciences 11 (1986) 139-160 Not lh-Holland

139

A S T R A T E G I C M A R K E T G A M E WITH T R A N S A C T I O N S COSTS J. R O G A W S K I and M. S H U B I K Department of Economws, Yale Umversity, New Haven, CT06520, U.S.A. Communicated by K.H. Kim Received 18 February 1985 Revised 29 April 1985

Transactions costs in terms o f the physical consumption o f goods are introduced into a strategic market game model of exchange with complete markets. The existence o f noncooperative equdlbria is established. The possibility that the activity of markets wdl depend upon the physical properties of a c o m m o d i t y which might serve as a money are considered. Trade m m commodities with a money calls for m - I markets as contrasted with re(m- 1)/2.

Key words: Market game; noncooperatwe equilibria; transactions costs.

I|ul th~s barter introduced the use of money, as might be expected; for a convement place from ~*hence to ~mport what you wanted, or to export what you had a surplus of, being often at a great distance, money necessardy made its way into commerce; for ~t ,s not everything which ~s naturally most useful that is easiest of carriage; for which reason they invented something to exchange w~th each other which they should mutually gwe and take, that being really valuable itself, should have file additional advantage of being o f easy conveyance, for the purposes of life, as iron and silver, ,~t anything else of the same nature: and this &t first passed in value s~mply according to its weight ,~r s~ze; but m process of time it had a certain stamp, to save the trouble of weighing, which stamp expressed its value - Aristotle, Pohtws.

I. Introduction I. 1. Money and markets Money is a complex institutional p h e n o m e n o n with m a n y features depending delicately upon the structure o f markets, customs of society and other rules o f the game which fully specify how trade may take place. There are three broadly recognized features of m o n e y (by no means the only ones; see Shubik, 1984, ch. 1) which are considered here: m o n e y as a numeraire, as a store of wealth and as a means of p a y m e n t . Using strategic market games, we attempt to m a k e these properties of a m o n e y precise and easy to identify in the m a t h e m a t i c a l structures being studied. Implicit in the idea o f a m o d e r n m a r k e t is that, at least to a good first approxima0165-4896/86/$3.50 © 1986, Elsevier Science Pubhshers B.V. (North-Holland)

140

J. Rogawski, M. Shubik / Strategic market game

tion, the structure of a transactions technology and the price system should be anonymous. Market trade in essence, given enough traders to avoid important oligopolistic effects, ignores both names and numbers of individuals. For example, the proof of the existence theorem in Debreu (1959) for market prices is independent of the numbers of consumers or firms. In Section 2.3 we also introduce a notion of competitive equilibrium in the context of a strategic market game. The Walrasian budget set is replaced by the set of strategies which, though possibly infeasible, lead to a feasible outcome. The ques= tion of whether or not a competitive equilibrium in this sense is feasible then pin= points the relationship of efficiency to the institutional market mechanism. It is necessary to consider replicated sequences of strategic market games or to consider games with a continuum of traders in order to establish a relationship between the prices formed in the strategic market game and efficient prices. A strategic market game is characterized by a price formation mechanism as part of the trading technology. This technology imposes constraints on the strategy sets of all individuals. The special strategic properties of a good used as money can be examined in terms of conditions in a strategic market game. A numeraire. Consider an economy with trade in (m + 1) commodities. We may set one price in advance, say p,,, +1 = 1, and have a mechanism determine all others. In strategic games this assumption highlights the nonsymmetry of treatment between a money and other goods. Efficient trade in m + 1 goods requires only m markets. Strategic price formation will determine only m exchange rates. A store of wealth. We may distinguish a commodity money from a fiat money in the sense that the former has an intrinsic value independent of its use in exchange. The latter does not. We confine our remarks to a commodity money which enters into the utility function of each trader. We assume that the marginal utility of a money to any individual is always greater than some positive number E. The introduction of a fiat money calls for an elaboration of the laws or rules of society which enable it to be used strategically. This includes describing how it is issued and withdrawn. This is discussed elsewhere (Shubik and Wilson, 1977; Dubey and Shubik, 1979). A money usually requires properties such that it is easily transportable, identifiable and divisible into small enough units and that it can be held in sufficient quantity by all traders. We return to this last point in Section 1.3. A means of payment. Consider an economy with m + 1 goods. We define a simple market as a market in which good i may be exchanged directly for good j. We define a good to be a money if it is connected directly to all other goods via simple markets, i.e. if it can be exchanged directly for ail other goods. Fig. l(a) shows an economy with four goods where the first is a money, and Fig. l(b) shows an economy with four goods where all four are monies. Under this definition the associated strategic market game which yields the Arrow-Debreu results is one with complete simple markets, i.e. with all goods as money. When all goods are money there is always enough money, when fewer than

j. Rogawski,

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/ Strategic

141

market game

1

2

lx 3

4

Fig. 1.

all goods are the money ;I +ortage

of money may occur. This is made more precise in

Section (I 1)

1.2 A strtegqic market

game with complete markets

Although the bid- offer model

of Dubey and Shubik (1978) had m markets and only one money. II I\ straightforward to extend this model to one with m(m + 1)/2 markets in which every commodity is a money. In such an economy the price of every commodity may be quoted in terms of its exchange rate with every commodidity, but it is reasonable to fix p,, = 1 for i= 1, . . . , m + 1. Furthermore, the relationship P,, I ,p,, lllust hold for all i, j. Thus, we need consider only m(m + 1)/2 markets and prices Suppose that the markets are designated by (2, I), (3, I), (4, l), . . . , (m + 1, m). A strategy by the trader i is a vector or dimension m(m + 1) of the form (b;,, q;,, bj,, q! I, 1 ,,,* 4, t I,,,,)~ where I?, + I

11: ,+

C

d=3.d=2

_(a;,

(1)

and all I):Lz 0, qik>O. (See Section 2.3 for a more detailed description of the bid offer model .) It is a direct extension of the proofs given in Dubey and Shubik (1978) to establish both the existence of active noncooperative equilibria (N.E.) and the convergence ofthe N.E. to competitive equilibria (C.E.).’ As our prime purpose is to consider the effect of transactions costs on the strategic In;lrket games we do not dwell further on this game with complete markets but no transactions costs. however, proved the existence of an active N.E. with all markets active. When an individual strategy IS multidimensional in each commodity one can lose concavity of the payoff function and the proof that holds for markets with only one money no longer holds. We have not,

J. Rogawskt, M. Shubik / Strategic market game

142

1.3. P a r e t o o p t i m a l i t y ,

limited markets and enough money

The concept of enough money is really that of enough liquidity. When, as in the model in Section 1.2, every good can be directly exchanged for every good, then an individual's complete wealth and liquidity are of the same size. If there is only one money, this is not so. We may contrast the boundary conditions imposed by having m + 1, or 1 moneys, on an individual with endowment (al, a~,..., am + 1) facing prices Pl, P2, . . . ,Pm, 1 where the m + 1st commodity has been selected as numeraire; hence, Pm + 1 ~- l . With all goods as money, the individual attempts to maximize

U'(Xl, subject to m

E

J=l

-

4)--

a m' + 1 - x ; .' + I ,

(2)

where u' is i's utility function and xj the final holding of good j. When only the m + 1st good is a money, then the constraint becomes pj max[(x~ - a~), O] <_ a'm + ~ .

(3)

J=l

This states that purchases cannot exceed cash in hand. We observe that the imposition of condition (3) instead of (2) can limit the ability of the individual to maximize. In particular, if an individual has no money, he cannot buy anything! (This condition is extremely strong and possibly unreasonable. In most societies those who have assets but are illiquid are able to obtain credit but the granting of credit brings in a host of new problems such as bankruptcy and insolvency, which are not dealt with here.) An economy which uses a money may be said to have enough money when the strategic market game has an interior solution, i.e. when condition (3) is no more binding than (2). In essence, this means that the individuals all have enough money to finance a float that may exist between when they sell resources and when they are paid. Let us denote a strategic market game where the traders have initial resources al, . . . . a m for the first m goods and ka,,,+~ for the m + l s t good, by / ' ( a l , . - . , am, k a m + I ) . We may consider a sequence of games in which all individuals have more of all commodities in their initial endowments by multiplying through by k. A question commonly asked of exchange structures is: Why do certain markets not exist? The usual, common-sense, and to a great extent accurate answers are transactions costs, trust and a variety of other factors concerning uncertainty. But is is useful to separate difficulties and observe that we may answer questions concerning the implications of restricted numbers of moneys and markets separately from ques-

J. Rogawskr,

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/ Strategic market game

143

tions about why certain markets do not exist. We may assume axiomatically that certain markets do not exist, without offering anexplanation of why. Utilizing the bid-offer mechanism for forming prices in any market we can define a strategic market game for every network of simple markets. Optimality cannot be achieved with fewer than m simple markets for m+ 1 goods, bu’t with only one money either there has to be enough or credit must be introduced. As the number of markets is enlarged, the strategy sets of the individuals are enlarged until a strategy attains the dimension of m(m + 1) (m(m + 1)/2 bids and m(m + 1)/2 offers). A.4 the number of moneys is increased, the feasible set of exchanges is increased, bu’t it is only guaranteed that the feasible set of the strategic market game with m + 1 monies will touch the Pareto-optimal surface of the unconstrained exchange at a competitive equilibrium.

2. On transactions 2.1’. On

costs and efficiency

modeling transactions costs

It is suggested that there are at least six or seven qualitatively different factors involving transactions costs. They can be divided into two classes: (1) the production technology of transactions, and (2) the information and organization structure. There are at least four ways to characterize the physical effects of transactions costs. They are: (1) Simple constraints on the existence of markets, i.e. assume that certain markets do not exist. This has been discussed in Section 1.3. (2) Associate with each transaction a technology which consumes resources, assuming that the transactions technology can be described by a convex production set. (3) Assume that the transactions technology is described by an ‘approximately convex’ set, i.e. a set which has only small nonconvexities such as those caused by setput costs large to the individual, but small to the economy as a whole. Upper and lower bounding convex sets can be used to replace and approximate the actual set. (4)The transactions technology may display large increasing returns to scale which have to be considered directly. The information aspects of transactions pose problems in: (5) The costs of search, data-gathering and processing. (6) The appropriability, purchase and sale of information. (7) The public good aspects of communication and organizational networks. In the remainder of this paper, we concentrate on the second item noted in this listing. We consider that transactions utilize resources, but we do not tackle increasing returns.

144

J. Rogawskr,

M. Shubrk / Strategic market game

2.2. Pareto optimality and institutional efficiency The introduction of transactions costs even in their simplest form causes an ima portant modification to the attainable set of final distributions. Pareto optimality, or efficiency cannot be described without first specifying the set of feasible out= comes. Without transactions costs, Pareto optimality is defined independently from the distribution of resources. With transactions costs the feasible set of outcomes and, hence, the Pareto-optimal set depends upon the initial distribution (see Arrow, 1979; Dubey and Rogawski, 1982). Although the definition of Pareto optimality is applicable to any set of outcome which may or may not be dependent upon initial conditions, it appears to b( desirable both for clarity and emphasis to refer to the Pareto set of a distributior, dependent feasible set as the institutionally efficient set. Once distribution costs are regarded as a fact of life, then the relevant criterion of efficiency, i.e. taking institutions as given, must take these costs into account. If we are permitted to vary institutions (game theoretically change the rules of the game), then we may compare two or more sets of institutions to investigate their relative efficiency. 2.3. A strategic market game with transactions costs We take the strategic market game defined and analyzed by Dubey and Shubik and introduce transactions costs in a one-period model. We establish the general existence of a noncooperative equilibrium with active trade if transactions costs are not too prohibitive. We then observe that if the commodity used as a money in the m market economy has low transactions costs and is in sufficient supply and adequately distributed, whereas the others all have positive transactions costs, the corn-’ modity will emerge as a money in the game with m(m + 1)/2 markets if two fairly simple triangular inequalities hold between it and all other goods. Specifically, if Tk(i,j) stands for the transactions cost of trading i forj by individual k, then if for every triad i, j, m + 1,

Tk(i,j)l

Tk(i, m+ l)+ Tk(m+ 1,j)

(4)

and

P(j, i)2 Tk(j, m+ l)+ Tk(m+ l,j), then m + 1 will emerge as a money. This is made more precise in Section 2.4, and the relationship between transactions costs, markets, increasing returns to scale and institutional change is discussed in Section 3. Assume that there are n traders and (m + 1) commodities, where the (m + 1)st commodity is singled out as a money. Let R’, = ((Xi , . . . ,x,) E R’ : x, 2 0 for all j}

J.

Rogawskl,

M.

Shublk

)(x,, . . . . x,)~f?‘:x~>O

/ Strategrc

market

game

14.5

for all j>_

wherej

I\ given a vector aJ = (a{, . . . , a,{,+,) E R’Ti+’ of commodities as an *I ment For each trader j, let Y, = R):I’ *; we view Y, as the jth trader’s space ,tnd the utility function uJ of j is defined on Yj. We assume II the paper that uj is a restriction to I?‘:” of a c’-function on R”“’ strictly concave and strictly increasing in all variables, i.e.

Y=

(_q

,...)

_y,,)E Y, x...

x Y,7:

jiY,Yj= f aJ]. J=I

Y is [he space of all reallocations of the initial endowments among the n Given strategy sets S, forj=I,...,n, let S=S,x... Xs,,. A map ga:S+ Y 11form~(S)=(~t(S),... ,v,,(s)) with ~,((S)E y/ for YES. Such a map is called I I ~mechanism and defines a strategic market game: SD/(S)is thejth trader’s III< detemmred by the strategy choice s and u,(~~((s)) is the utility of the outA strategy choice S= (s,, . . . ,s,,) is called a Nash equilibrium (abbreviated 1 II for all j:

(3 1s]‘) denotes the strategy choice obtained by replacing thejth coordinate of with s]‘, In this section we consider a bid-offer mode1 with transactions costs. The jth I submits a strategy s, = (!I[, q{, . . . , bi,, qn/l)E Ry, where B,! is a bid of money buy commodity k and qkJ is an offer of an amount of commodity k for sale. A , \ _@I,..., s,) defines a price vector p= (p,, . . . ,pk) with the price pk of the commodity given by:

P/C= 0 I

if f qxJ=O. /=I

A transactions cost function is a vector-valued function L : II?:” E ,‘:+I and of transacting the bids bJ I(/‘,, q1,.*., b,, 4,)) represents the cost in commodities andoffersq, at the markets. We shall assume for ease of exposition that the funcnon L is linear, i.e. is given by a (2m) x (m + 1) matrix with non-negative entries. With somewhat more labor we could let the transactions costs sets be convex. For j= 1, . . . . n and S=(s,, . . . . s,) a vector of bids and offers for each of the

J, Rogawski, hf. Shublk / Strategic market game

146

traders,

let &(
xi=ai-qk/+

-

in lRm’i whose components

(xi, . . . ,xA+ i) are:

for 1
Pk

xi?+*

=47+,

-

i

bii+

k=l

Thus QJ is the jth costs. Set

payoff

f

&k.

k=I

function

for the bid-offer

model without

transaction!

Pj (S) = @,(3) - L (SJ) . The functions p, define the bid-offer game the jth trader’s strategy set is

model

sJ=

with transactions

f

k=l

and qi+Lk(s,)sai

costs,

and in this

bk+&,,+,(J”)-(a;+,

for k= 1, . . . . m ,

where &(s,) is the kth component of the vector L(s,). Note that the linearity of L implies that s, is convex. We call the game defined above r. An N.E. s of r is active if trade takes place, i.e. if S is not the zero vector. It is clear that S= 0 is an N.E. However, as in Dubey and Shubik (1978), we wish to investigate the existence of nontrivial N.E. and the convergence of N.E. to competitive equilibria (abbreviated C.E.) supported by a price system under replication of the economy. We first establish some preliminary lemmas. Forj= l,..., n and k= l,..., m, let Bi= C,,, bkand Qk= C,,, qi In dealing with the strategies of a single trader, we will drop the superscript j when the meaning is clear.

Fix j and assume strategies sk E Sk are fixed for k +j. Let ) E YJ and let v, (x) = {sJE S, : p, (s,, . . . , s,) = x} . Then the set w,(x) is x=(x*,...,x,+1 a convex subset of S,. Lemma

Proof.

1.

Define two functions

of variables

fdb,d=c-q+b

(E >

fi@,q)=c-b+q

($$ >,

and

b and q by:

J

where c, B and Q

M Shubrk i’ Strategrc market game

Rogawskl,

are constants.

It is easy to check

that

147

if O
if

f,(b, 4) f,(b’, 4’) for a= 1 or 2, then &Ab+(l

-A)q’)=f,(b,q)=f,(b’,

-A)b’,Aq+(l

4’).

Some the transactions cost function L(s,) is linear, the lemma follows immediately from the form of the payoff function ~0,. We recall.some observations made in Dubey and Shubik (1978) regarding the structure of the strategy sets in the bid-offer model without transactions costs. Consider a Iliarket with one commodity being bought or sold in a commodity money. In Fig. 2 the point with coordinates (A, M) indicates the initial endowment of an individual. The strategy set is the rectangle (q, b) with 0s q
m=M+

(Q+a-x)

’

and all final holdings lie on this curve, where (x, m) is the point on the curve. let B(resp. Q) be the total bid (resp. offer) of the other traders. Then a bid of (q, 6) takes the initial holding of (A, M) to a final holding of (A -q+ b/p, M-b+ qp), where p= (b + B)/(q + Q). The set of strategies in Fig. 2 which leads to the outcome (x, m> is the intersection of the rectangle with the line joining (x, m) to I Am/(B+Mm), 0). In particular, the curve PIP2 is concave and if sl and s2 two strategies in the rectangle, then the strategies s(A) = As, + (1 -A&, with A I 1, map onto the portion of the curve joining f(s,) to f(sz), where f(s) notes the outcome associated to a strategy s in the rectangle. The concavity of the Money (b)

A, Ml

i

(A-q,M-b)

/ \Good

G

A-

Qm B+M-m

)(A_@!

70

B

0

’

(4)

PI >

Fig. 2.

J. Rogawski,

148

M. Shubik / Strategic market game

curve PIP2 implies that f@(A)) = A/@,) + (1 - A)f(.sJ + <(A), where r(A) is a vector ([i, &) with <,>O for j= 1, 2. We now return to our bid-offer model with transaction costs. For s E S, let r,(S)= (~,(.+;):s;ES,}. Thus, y/ (3) is the set of outcomes that trader j can obtain by changing his own strategy if the other traders remain fixed at s, (k#j). Let Y,“(S) be the set of out comes x E y/ (s) such that qN=Xm~;ju’(x’). ,

For k=O, . . . . m-t 1 define k=l,...,m+l let

Y,‘(S) as

follows.

Set

Y,‘(3)= Y,“(J) and for

Let q(S)=I;“+’ - Then for all xi, x2 E q’(S), there exist s1 E y/(x,) and s2 E I&) (s). such that L(q) =L(s2). In fact, we have the following lemma: Lemma 2. Y;(S) consists of a single point. Proof. Suppose xi, x2 E 5: (3) are distinct and let sJ E v/(x,) for j = 1, 2 be such that L (st ) = L(.s2). For 0 I AI 1 let s(A) = Ast + (1 - A).s2.Then s(A) E SJ since S, is convex

and L@(A)) =L(s,) =L(s2) since L is linear. Let l=L(.s,). Define the vector [(A) by ~JJ(s(~))=~~J(s~)+(~-~)~~(sz)+~(~), so that pJ (s(A)) = AqJ(s,) + (1 - A)&(s2) + ((A) - 1. It follows from our previous remarks that ((A) 10 and that at least one component of <(A) is strictly positive, Since uJ is strictly concave and strictly increasing in all variables, ‘JbJ

@tA))bu,

<@J

6,)

+

t1

>AuJ(~J(s,))+(l

and this is a contradiction,

-‘)@J&)

-

I)

=

uJ

(@J

tsd

+

t1

-

‘)yl,

(‘2))

-n)uJ(~J,,s2))=UJ(CO/(S,)),

This proves the lemma.

Following Dubey and Shubik (1978), we define an e-modified game r, in which an (n + 1)st player places a fixed bid of E> 0 and a fixed offer of E> 0 in each of the m markets. The strategy sets of the original n traders remain the same but the outcome functions for r,, which we denote by ull,“,are different. Proposition Proof.

3. For aN E>O, a N.E. of I-, exists.

Consider

the map 0

: S --) S

defined

by

J. Rogawskr, M. Shubik / Strategrc market game

where

xj is the unique point

in Y,‘(S) for j = 1, . . . , n. Then

149

G(S) is a nonempty

where x-valued correspondence by Lemma 1 and is easily seen to be upper semicontinous, Hence a fixed point exists by Kakutani’s theorem and a fixed point is a Convergence. We wish to consider the convergence of Nash equilibria of the game as tends to zero. Let P(E) = (pi (E), . . . , P,(E)) be the price vector at a N.E. of Although it may happen that p,(e) tends to zero as E tends to zero, we consider two approaches to transactions costs which lead to the conclusion that there exist positive constraints C and D such that C
for all bids and offers (b, q) in the kth trader’s strategy set. j and let k be such Let A =(m+ I)-’ min k (ak,,,+ i). Fix a commodity ?5! c:,, bf. Obviously one of the following two inequalities holds:

0)

i b,krA477+,1=l

(ii)

a:+,-

that

t,,,+,

f b;k
where &,,+ 1 is the maximum of L,, 1(b, q) over all (b, q) in k’s strategy set. By our assumption A - &, + , > 7> 0 for some positive 7 which is independent of k. We also have xi> 7’ for some 7’ which is independent of k when transactions costs are not loo high. Suppose that inequality (i) holds. Let xk be k’s final holding at the N.E. of r, with price vector p(c) = (pl (e), . . . , p,(e)) and let xk(d) be k’s final holding under an increased bid of d on thejth commodity. For d sufficiently small, this is feasible if (i) holds. Let lad and l,,,+ 1A be the transactions costs in the zeroth and (m + 1)st commodities of the increased bid of d. For i = 1, . . . , m, let 5, and Q, denote the sum of the bids and offers on commodity i, respectively. We have:

J. Rogawskl, M. Shubtk / Strategic market game

t50

X0k(A)- Xg = -t0A, x,k(A)-x3=O

for i = 0 , j , m + l , (O, + e)(/~ + e - b~)

A

2pj(e) ' k 1(A)--Xkm + 1 = ( (qjq+f C) Xm+

1-lm+~)A>-A(l +lm+~).

Since u k is strictly increasing in all variables and the range o f ~0j is b o u n d e d , it is not hard to see that there is a constant h > 0 which depends only on Uk such that for all outcomes x ~ R + +l and vectors y ~ R ~ +1, if []x-yl[ Ux(X) for all t = 0, ..., m + 1. (See L e m m a 5 for a precise statement.) Let z = - 2p(ej)(loeo + (1 + 1m +l)em +1). The above inequalities show that A xk(A)>_xk+ 2pj(c) (z+ej). If xk+z+ej>O and Ilzll-h, then uk(xk+z+ej)>uk(xk). cave, this w o u l d imply that uk

Since u k is strictly con-

@k+ 2pjA(e) (z +e~))>Uk(Xk),

contradicting the assumption that we are at a N . E . Hence, either [[z[I->h, in which case pj(e)>_½hIIloeo+(l+lm+l)em+ll[ -1, or some c o m p o n e n t of xk+z+ej is k negative, i.e. either Xm+ I - 2p(ej)(1 + lm+ l) < 0 or x k -- 2p(ej )/o< 0. In the first case, we obtain:

2p(cj) > (1 +/m+l)-l(A - £ m + 1)>(1

-t-/m+l)-lV,

since xkm+l>A-Lm+l when (i) holds; and in the second case, we have 2p(~j)> l o It'. Thus, if (i) holds, p(ej) is bounded below by a constant which depends only on the utility functions, the initial e n d o w m e n t s of the traders, and the transactions costs function. N o w suppose that (ii) holds. T h e n x,=mI b ~ > am+lk - A + L m+ I >_mA and, for some i, b~>_A. If i=j, then we obviously have A

&(~)>n

Otherwise trader k can decrease b k by A and increase bk by an amount A for A sufficiently small. Let IA =loeo+lm+mAem+l be the net transactions cost for this change of strategy. Let xk(A) denote k's new final holding. Then x0k(A ) - x0k = - / 0 A ,

Xk(A)--xk=O

for l=/=0, i , j , m + l ,

J

i:oguwskr,

\,b?-

I)

\,l,=

I

Shublk

I)-

151

game

@,+e)(b,k-E) (B+F)b,k-@,++e) !?,-A+&

XL1 +

1

\‘( 1)2x”+

market

3P(&j,)’

-

k ,(

/ Strategic

A

3 I II

’

M

9:

qJ =

-

-

qJ + ’

&(‘+

&+E

-

5,+&

’

A -4,. IA-

41-t E >

eJ)

J

*

_

’

we see that either llzII> h or some component of xk + z is negative. 0 then since b,k+c>b,k>A and q,ksQ,+cs CP=, a/+&, we easily obtain a ‘bound on pJ(&) which depends, as before, only on the initial endowments, fkctions , and transactions costs. If some component of xx-+ z is negative, have one of the following three inequalities: ‘

before,

\(: -2p(&J)I,r’I,-1,

(4i,+E)<0

_\;”- 2P(&,) 7 (b,+d -t-j -

,yk

>

2P(&,) (-&+i+,)
@,

I

x:1+,

’

+

&’ > (B,+ &)A

(b,+E) -

I

(&+e)

qfw> Aq,” (ql+&) - @,+d

’

,a11cases, we obtain, as before, a lower bound on pJ(&). We must now show that the prices pJ (E) are also bounded above. Fix a commodity and choose a trader k such that g,k’+g/. Then, if BJ denotes +min,(aj), then either(i) ajk - qJk> BJ or (ii) gJk- q,k < BJ. Suppose first that (i) holds and let (A) denote the final holding of k if k increases q; by A. This is feasible if A is insufficiently small and we have: x;(A)-x;(A)=

-loA,

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J. Rogawskt, M. Shubik / Strategic market game

x,k(A)-xi*=O 4(A)-4>_

ifi:~0, j, m+l, -A,

,(A)-x*m+ ,

l,.+ , A.

If pj (e) ___21m + 1, we are done. Otherwise Xm k + l (A) - Xm k + 1> 0 and if we set

z=

-2

Pj(e)-2lm+l

(loeo+ej),

we obtain:

xk(A)>_xk + A ( p J ( e ) --~2lm+ l ) ( Z + e m + l ) . As before, we conclude that either [Iz[I> h, in which case

pj(e)> ½hllloeo+ e~l1-1 +21re+l, or some c o m p o n e n t of x k + Z is negative. In this second case, we have either x°k -

2/0

<0

Pj (~) - 2lm + 1 or x' ~ _

2

< 0,

P j ( e ) - 21m+l and since x0k > r' and 4 - - - 4 -

qjk_ Bj, it is clear that we obtain an upper b o u n d on

pj(t). Finally, if (ii) holds instead of (i) qj*>af-By>_½a]>_Bj and thus 6j+g _< ET=l a m + l + e

We summarize the above discussion.

Proposition 4. I f transactions costs are not too high and involve only the zeroth and (m + 1)st commodity, there are positive constants M > N such that for all e > 0 and all N.E. o f F~ with price system (Pl (e),... ,Pro (e)), we have: N
for all j = l , . . . , m .

In our second a p p r o a c h to interior solutions of the strategic market game with transactions costs, we consider a sequence of games F t which differ f r o m the game F only in that for all k = 1 , . . . , n , the initial e n d o w m e n t of trader k i~ la k = (la~, •.., lam+~). k We will show the existence of interior N.E. of Ft as F t e n d s tc

J. Rogawski, M. Shubik / Strategic market game

under the

Assumption

1.

following

153

assumptions.

There is a positive constant y > 0 such that for all k = 1, . . . , n, for ail commodities i and j.

Assumption

2.

Transactions costs are not too high.

condition on the utility functions together with the requirement that the relative marginal utilities of any two commodities does not tend is zero (and hence also not to infinity). The second condition is precisely as follows. There is a vector T E I!??: ’ such that ak - (m + l)L(b, q)> T for all (b, q) in k’s strategy set for the game r. We are now assuming that L is given by an arbitrary (m + 1)x 2m matrix. Copsider an N.E. of r, with equilibrium price vector p(l) = (pl (I), . . . ,p,(f)) and (I) be the final holding of trader k at this N.E. We note first that for all i, tends to infinity with 1. To prove this, suppose that ~~~(1)
1 is a satiation

Pi(M

IA A Is,-M5T’

Where A = Cz=, a:,,

.

We may conclude

that

x:+1 (,)rq~(~lp,(f)r(l7,-Mlp,(I). Let x;“(I, d) be k’s outcome WC have: x;“(/, d)-xJk((I)2_0

if q,k(l) -d

for j#i,

is offered

instead of q:(l). This is feasible

m + 1,

x~wl)-*~(o=Ll(lG),

Note that if x;” (1) q,k(l)~ It, -M). There are two possibilities: (a) p,(l) > o for some fixed o > 0 for all 1, or (b) pi(,) tends to zeros as I--+ 00. In case (a), XL + I (I) and hence also xi + , (I, A) tend to infinity with ! for fixed d, and Assumption 1 immediately implies that for 1 large, uk(xk(f, d))> u”(x”(/)), contradicting the assumption that we are at a N.E. (we omit the obvious idetails). In case (b), xi+,(/, d) approaches xi+,(r) while x;k(l, d) approaches

154

J. Rogawskr,

M. Shublk

/ Strategic

market game

xlk(l)+d, as I + 00 and

again it is clear that uk(xk(l, d)) > z.?(xk(/)) for 1 sufficien ly large. Therefore x:(r) must tend to infinity as I+ 00. Next we remark that there exist positive constants N and A4 such that N
on I?:+’ such that for all i and j, (af/ax,)/(af/ax,)>n for some positive constant n. Then therl isa constant hsuch thatforallj=l,...,m+l andallx, YEIT?:+‘, if I/x--_vIISh, then f (y + e,) >f (x).

Proof. Let y=x+6 and set &t)=f(x+t@+e,)). there is a to such that 01 t,,s 1 and

By the mean-value

theorem,

f (u+ eJ -f (x) = ~(1) - ~(0) = W0)

@+e,)

=Vf(x+t0(~+eJ)W m-t1 =

El

af -$

(x+t0@+e,)W,+~ I

and this is positive We summarize

af

(x+t0(~+e,h

J

if /a,/ < n/(m + 1). the above discussion

in the following

proposition,

Proposition 6. There are constants N, A4, P and a function f (1) which tends to infinity as I + m depending only on (ak >and (uk] such that for all 11:P and all N.E, of r, with price vector p(l) and final holding vectors x1 (1), . . . ,x”(l), the following inequalities hold:

Nf(l)

for j= l,...,m, for j+ l,...,

m and k+ 1, . . . . n.

We will denote by a subscript E (for E> 0) the game obtained from a game r by adding an (n + 1)st trader who places a fixed bid of e and supply of E in each market. By Proposition 3, N.E. of these E-modified games exist. An N.E. of the game rwill. be called a r will be called a G.N.E. (good Nash equilibrium) if the prices p and strategy choices s E S are obtained as a limit as e + 0 of prices p(e) and strategy choices s(e) of N.E. of the game r,. Since S is compact, by picking convergent subsequences of p(e) and S(E), the next proposition follows immediately from Propositions 4 and 6.

J. t:(q~lwsh~, M Shubrk / Strategrc market game Proposition

7. In the following

two cases, a G.N.E.

155

exists:

If transactions are costs are not too high and involve only the zero and (m + 1)st G. N. E for r exists. If transactions are costs are not too high and Assumption 2 on r(l) is satisfied, 1,. G. N. E exists for r, for ali sufficiently large 1. Proof >

,,I)

The type of. trader is characterized

by the utility function and the in‘ lowment I cl r(l) be the strategic market game with kn traders in which ‘. 1 I traders oftype (u’, aK) for k= 1, . . . . n. An N.E. of r(l) will be called abbreviated S.N.E.) if the strategies of traders of the same type are iden. : 11(II I Irat the proof of Proposition 3 yields a S.N.E. for I”(1) for all E>O - * players of the same type must solve the same optimization problem and ?. I I I( I I IIC map @ in the proof of Proposition 3 to the subset of symmetric L 1 III \ l+urthermore, the statement of Proposition 7 remains valid if G.N.E. 1 replaced by S.G.N.E. (symmetric good Nash equilibrium). A basic iproperty of the bid-offer market game without transactions costs is that . II, converge to competitive (Walrasian) equilibrium under replication. We 1, II this remains true after the introduction of transactions costs (which are not too high provided that the notion of budget sets is modified. The standard I) 11101 a budget set makes no sense in this context since the price of a final I : I( 111depends on the strategy used to obtain it. Thus, we define for each trader g and price vector p: P(p)

= (Sk E R2m :~.~~,(sk)+L(sE:)~~.ak;(P;(sk)+L(s~)EIR”:+’},

with the price of money normalized and (pL(sk) 1 I II\,~I holding of k if the prices are fixed at p. It is the set of strategies such is thejnulholding is feasible, even if the strategy leading to it is not. A comparative equilibrium (abbreviated C.E.) then consists of a price vector and for each strategy choice sk E Bk(p) such that uk(qk(sk)) is a maximum for uk on the set :i,‘i,(\~):a,EBk(p)}. A strategy choice is called interior if (Cy,, bf+L(sk)) , for all k, where sk = (6:, qf, . . . , bk, 4;). A C.E. is called symmetric (ababriviated S.C.E.) if the strategy choices of traders of the same type are identical. In the replicated game r(l), an S.C.E. will be denoted by (p, sI, . . . ,sn), where sk is the strategy used by all traders of type k. where

Proposition

(p, 1) is the price vector

8. Let s(l) E S be a sequence

of S.G.N.E.s with price vectors p(l) such that s(l) + s and p(l) -+p as I-+ 00. Then (p, sl, . . . , s,) is an S. C.E. for T(1) for all

We first note that the prices pJ (1) of the jth commodity satisfy N
J. Rogawski, M. Shubik /Strategtc market game

156

proof of Proposition 8 is based on the following two observations. Let Q,k(l, e) (resp. Be(l, e)) denote the total offer (resp. bid) on commodity i by the traders other than k at the N.E. of the game F(I)e with price vector p(l,e) (where p(l, e) ~p(l) for a suitable sequence of e ~ 0 defines the G.N.E. s(l)). Set

.ok (l, e)= p, (l, e)2(Qi k (l, e)/B~ (/, e)) and let

pk(l, e) =

e),..,

k g)). ,pro(l,

The first observation is that at the N.E. s(l, e) with price vector p(l, e) (converging to (p(l), s(l)), the associated final holding xk(l, e) of trader k satisfies the condition:

Uk(xk(l,e))>--Uk(k(Sk))

for

skeSkNBk(p~(l,e)).

If there are no transactions costs, this is proved in Lemma 4 of Dubey and Shubik (1978). That argument is valid mutatis mutandis, in the present context if it is remarked that for any two strategies sk, S'keBk(p,k(l, e)) and all 0_<2_< 1,

L(2Sk + (1 - 2)Sk) = 2L(sk) + (1 - 2)L(sk) by the linearity of transactions costs. The second observation is that as l ~ 0% Q~(l, e) and Bk(l, e) approach Qi(l, e) and B, (l, e), respectively, where Qi (l, e) and B, (l, e) are the total offers and bids on commodity i, and hence that pk(l, e) ~p(l, e) as l ~ oo. This is a trivial calculation which we omit. It is clear by continuity sk(l) = lim~-~0 sg(l, e) is optimal in Bk(p(l, e)) for all k. If sk(l) is interior, then sk(l) is still an optimal strategy of prices p(l) even if no strategic budget constraint is imposed (that is, if k maximizes over B k (p(l)) instead of Bk(p(l)) tq Sr).

2.4. A strategic market game with complete markets and transactions costs In Section 2.3 we derived prices for all goods and hence evaluated the resources utilized in transactions. Since we specified only m markets we noted only the physical costs (i.e. consumption of commodities) required for trade in these markets. If we consider the possibility that all m(m + 1)/2 markets exist, then if at prices (p~, ... ,p*, l) the equilibrium prices associated with the game with m markets, for all individuals and any two commodities i a n d j it is cheaper or as cheap to exchange i for j via money than directly, the N.E. of the game with m markets remain N.E. of the game with m(m ÷ 1)/2 markets. The above condition does not rule out the possibility of the existence of other N.E. in the larger game with more than m markets active. The interplay between :osts and the thickness of a market provide for the possibility of many equilibria. Dubey and Shubik (1978) have already noted that there was the possibility of a

J. Rogawski,

M. Shubik

/ Strategic market game

151

multiplicity of equilibria caused by the effect of wash sales (i.e. buying and selling in the same market thereby making the market ‘thicker’). This source of multiplicity is somewhat cut down by the presence of transactions costs.

3. Some economic

interpretations

3.1 General equilibrium or strategic market game analysis Foley (1970), Hahn (1971), Kurz (1974a, 1974b) and others have approached the problem of transactions costs by direct modifications of the general equilibrium model In contrast here we are explicit in the construction of a game in strategic form . In terms of the existence of different buying and selling prices to different agents as noted in the modeling of Foley and Hahn or in the introduction of a vector of real resources consumed in exchange as modeled by Kurz, our model is related to the earlier work. However, our emphasis is somewhat different. In particular, we are concerned with models with price formation mechanisms (even though they are relatively simplistic); furthermore, we are concerned with features such as the interactions among finite numbers of traders strategically trading off the possibilities of influencing the thickness of markets versus paying extra transactions costs. We are also concerned with the strategic meaning of enough money and enough of other resources; to avoid unreasonable restrictions on trade. Finally, we are concerned with efficiency properties of the equilibria of strategic market games with transactions

COStS.

It should be noted that although like both Foley and Hahn we have ‘own’ prices for buying and selling our model generates only F.O.B. prices at each market. As we describe physical processes for each market the only prices formed are market

prices.

J.2. Barter, complete or limited markets In the two models noted here we contrast an economy with m markets with one which may have as many as m(m + 1)/2 markets active. We do not attempt to characterize barter, although Kurz (1974b, p. 419) simultaneously refers to the model with m(m + 1)/2 markets as a barter exchange economy and as a natural extension of Arrow-Debreu; we wish to make a further distinction. An economy with complete markets is, if it is anything, a model of the ideal of modern economics and tinance where all commodities are perfectly liquid. The properties of mass, imonymous, aggregative markets forming prices remain. Barter, in contrast with the mass market, may in general be neither mass nor anonymous. Fewness of individuals, and who they, are all count. The number of rlifferent barter arrangements is far larger than the number of anonymous markets. It involves the combinations of individual groupings as well as the groupings of goods.

158

J. Rogawski, M. Shubik / Strategic market game

3.3. Enough goods, time and money Kurz (1974b, p. 423) noted that it is possible that even if an individual has positive, wealth, the transactions costs of exchanging goods may be so high that trade il prevented and a zero price exists for a good of positive worth. In the formulation and analysis of the strategic market game we encounter the difficulties noted by Kurz. In Section 2.3 we proposed two ways to avoid them. Both were mathematically adequate, but must be interpreted as crude first-order approximations for a complex phenomenon. In essence, in an ongoing society much of transactions costs are paid for in the use of the individual's own time, whether walk, ing or driving to the store or standing in line for a bargain. Among the other more common inputs utilized in transactions are transportation, packaging and paper. work. It is unlikely that the lack of paper bags or virtually any other input would stop trade; in actuality substitutes are virtually always available. Our first way to avoid the difficulties attempts to operationalize the idea that in essence 'one eats up time and money' in most transactions activities. This is similar to Kurz's 'leisure services' (Kurz, 1974b, p. 423). A different set of conditions sufficient (but not necessary) to sidestep the difficulties is obtained by inflating the size of the initial holdings of all traders, assuming all traders have strictly concave utility functions which approach saturation and that the ratio of marginal utilities between any two commodities is bounded. The precise conditions for there to be enough commodities to facilitate trade involve relationships among the distribution of commodities, their marginal utility and explicit roles in the technology of transactions. Societies work out many different institutional and technological arrangements to satisfy them. The detailed description of these conditions is not of prime concern here.

3.4. Efficiency, transactions costs and numbers In Section 2.3 we defined efficiency in the context of the game with transactions costs. We were able to suggest an analogue to the competitive equilibrium and to show that for a finite number of players the N.E. were not efficient but under replication institutional efficiency is achieved.

3.5. Increasing returns, setup costs, institutions and customs It is well known that transactions technologies may involve setup costs and increasing returns to scale. Although mathematical techniques exist which can 'handle nonconvexities which are not too severe', we wish to sketch at a less formal level a possible dynamic of increasing returns which accounts for a nonsymmetry in the growth and decline of market institutions. This nonsymmetry has its interpretation to some extent in terms of social custom, professional courtesy and threat noncooperative equilibrium in even large games.

J. Rogawskt, M Shubtk /Strategtc market game

159

S u p p o s e , for example, a p r o f e s s i o n such as medicine has a c u s t o m o f professional ~ourtesy where A will treat B w i t h o u t charge. In essence the m e m b e r s o f the class do not need to k n o w each o t h e r as individuals, but merely as m e m b e r s o f the class. I hey m a y need an identification only as violators o f the n o r m . But if all k n o w that ,l violator o f the code will be identified a n d ' p u n i s h e d ' by even an a n o n y m o u s upholder o f the code, then stable threat e q u i l i b r i u m can be established. If A is ~ eated by B this does not m e a n that B m u s t be treated by A. In essence a code o f behavior works on a clearing-house basis. In terms o f m a r k e t s and the need for m o n e y , the existence o f customs, codes or professional courtesy provide for p o c k e t s o f exchange outside o f explicit markets. I'his in t u r n makes the p r o b l e m o f m e a s u r i n g trade a n d the required a m o u n t o f money in an e c o n o m y s o m e w h a t m o r e difficult. The relationship between c u s t o m , threat equilibria and increasing returns comes ~n the d y n a m i c s o f the f o r m a t i o n o f social n e t w o r k s . A chance event such as a war can p r o v i d e an exogenous event bringing a great m a n y individuals together. This may provide the basis for a v e t e r a n ' s m a g a z i n e which in turn m a k e s it e c o n o m i c a l l y l easible to f o r m a discount or swap club. It is suggested that to a great extent the d y n a m i c s o f the f o r m u l a t i o n o f financial ~nstitutions m a y d e p e n d in an i m p o r t a n t way u p o n one or m o r e events which enable a few o r g a n i z a t i o n s to o v e r c o m e the barriers o f new institution building in a fashion characterized by increasing returns to c o m m u n i c a t i o n n e t w o r k size and v o l u m e o f trade. The type o f e x o g e n o u s event provides e n o u g h impetus m a y be a war, an innovation, a new tax or a c h a n g e in the law. T h e g r o w t h o f the m o n e y m a r k e t funds in the U n i t e d States in the 1970s provides an example. T h e constraints on b a n k i n g a n d high interest rates p r o v i d e d an incentive large e n o u g h for a new institution to go t h r o u g h the zone o f high costs a n d low v o l u m e . In 1982 b a n k i n g laws c h a n g e d and at first sight it might a p p e a r that the m o n e y m a r k e t funds could be wiped out as fast as they a p p e a r e d . But in the course o f a decade the institutions have built up an equity or a value in and o f themselves associated with their c u s t o m e r lists, interfacing with brokers and o t h e r aspects o f t r a n s a c t i o n s convenience.

3.6. Taxation and the quantity o f money We suggest that in a m o d e r n e c o n o m y with m actual e c o n o m i c goods, k financial i n s t r u m e n t s and a fiat m o n e y , a little m o r e t h a n m + k, but considerably less t h a n ( m + k ) ( m + k + 1)/2, m a r k e t s will exist. If a tax changes or law or t e c h n o l o g y influences the e c o n o m y a p p r o p r i a t e l y b o t h the n u m b e r o f financial i n s t r u m e n t s and m a r k e t s will change. W h e n a t t e m p t s are m a d e to c o n t r o l an e c o n o m y by taxation or legal restrictions there is a high probability that c o n d i t i o n s m a y be created for the f o r m a t i o n o f new i n s t r u m e n t s a n d markets. T h u s , even the d e f i n i t i o n of an o p t i m a l supply o f m o n e y c a n n o t be given in a satisfactory m a n n e r w i t h o u t specifying the transactions

160

J. Rogawski, M. Shubtk / Strategtc market game

technology and estimating how it may change as taxes or other controls or technological developments influence the number of markets and instruments.

Acknowledgment Research partially supported N00014-77-C-0518.

by

an

Office

of

Naval

Research

grant

References K.J. Arrow, Pareto efficiency with costly transfers, in: J. Los et al., eds., Studies in Economic Theory and Practice (North-Holland Publishing Company, Amsterdam). K.J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica 22 (1954) 265-290. G. Debreu, Theory of Value (Wiley, New York, 1959). P. Dubey and J. Rogawski, Inefficiency of Nash equilibria, CFDP #622 (1982). P. Dubey and M. Shubik, The noncooperative equilibria of a closed trading economy with market supply and bidding strategies, Journal of Economic Theory 17 (1) (1978) 253-287. P. Dubey and M. Shubik, Bankruptcy and optimality in a closed trading mass economy modelled as | noncooperative game, Journal of Mathematical Economics 6 (1979) 115-134. D.K. Foley, Economic equilibrium with costly marketing, Journal of Economic Theory 3 (1970) 276-291. F.M. Hahn, Equilibrium with transaction cost, Econometrica 39 (1971) 417-439. J.R. Hicks, Value and Capital (Oxford University Press, Oxford, 1938). J.M. Keynes, The General Theory of Employment, Interest and Money (Macmillan, London, 1936), M. Kurz, Arrow-Debreu equilibrium of an exchange economy with transaction cost, Internation|l Economic Review 15 (1974a) 699-717. M. Kurz, Equilibrium with transaction costs and money in a single market exchange economy, Journal of Economic Theory 7 (4) (1974b) 418-452. M. Shubik and C. Wilson, The optimal bankruptcy rule in a trading economy using fiat money, Zeitschrift fur Nationalokonomie 37 (1977) 337-354. M. Shubik, A Game Theoretic Approach to Political Economy (MIT Press, Cambridge, Mass., 1984),