A closed economic system with production and exchange modelled as a game of strategy

Journal of Mathematical Economics 4 (1977) 253-287. 0 North-Holland Publishing Company

A CLOSED ECONOMIC SYSTEM WITH PRODUCTION AND EXCHANGE MODELLED AS A GAME OF STRATEGY*

Pradeep DUBEY and Martin SHUBIK

Cowles Foundation for Research in Economics, Yale University, New Haven, CTO6520, USA

Received July 1976, final version received August 1977

1. Introduction

Models of a closed economic system with production and exchange have been studied both by methods of general equilibrium analysis and cooperative game theory. The conditions for the existence of a price system which clears all markets and is efficient have been investigated in recent years by Debreu and many others.

A completely different approach to the study of the existence of a price system is to use a model of the economy as a game in coalitional form and to apply the cooperative game solution of the core to this game. The formulation of a market with trade alone as a game in coalitional form is relatively straightforward with or without sidepayments. The work of Shapley and Shubik (1969), Shubik (1959), Debreu and Scarf (1963), Aumann (1964), and many others attests to this. The formulation of a game with trade and production is more difficult but has been done by Hildenbrand (1974), Dubey (1975), and others (1974). The difficulties lie more with the basic economic modelling rather than the mathematics.

Given decreasing returns to scale it becomes necessary to consider individually owned technologies. It is not the purpose of this paper to enter into a critique of the modelling of production in economies studied as games in cooperative or coalitional form. Our stress is upon the description of an economy with trade and production as a game in strategic form.

There have been no models constructed of an economy with trade and production as a game in strategic form. We construct one and analyze the resul- tant game for its non-cooperative equilibria.

*This work relates to Department of the Navy Contract NO001476-C-0085 issued by the Office of Naval Research under Contract Authority NR 947-006. However, the content does not necessarily reflect the position or the policy of the Department of the Navy or the Govern- ment, and no official endorsement should be inferred.

The United States Government has at least a royalty-free, non-exclusive and irrevocable license throughout the world for Government purposes to publish, translate, reproduce, deliver, perform, dispose of, and to authorize others so to do, all or any portion of this work.

2.54 P. Dubey and M. Shubik, A closed economic system

2. A strategic model of exchange and production

In order to model a closed economic system as a game of strategy it is necessary to be completely specific about the market mechanism leading to the formation of prices; the details of information and the rules of production and trade.

The sequence of trade and production must be spelled out. There are many choices in modelling in fine detail where different variations each have their counterparts in actual methods of trade and production. In our first model our stress is on simplicity rather than on realism.

For a positive integer I’ we shall denote by 1, the set (1, . . ., r}, by E’ the Euclidean space of dimension Y, and by 9’ the non-negative orthant of E’.

Let

1, = the set of traders,

I m+l = the set of commodities,

I,+,\Z,=thesetoffirms.

The initial allocation of trader i is a vector ai E CYtl, where a: is the amount of commodityj available to i (forj E I,), and afnt 1 represents the money held by i. In addition, trader i has shares in the profits of the s firms. This we will denote by vi E Sz” where of is i’s share in firm Z, n+ 1 Slsn+s. The traders’ utility functions are real-valued,

and are assumed to be concave, continuous and non-decreasing. We shall say that trader i ‘desires’ goodj if u’(x’) is an increasing function of the variable x$, for any fixed choice of the other variables xf. A trader i for whom a;+ 1 > 0 will be called ‘monied’; and when a; > 0 we will say that he is ‘j-furnished’.

When we drop an index and use a bar it will indicate summation over the indexing set. Thus for xi E O”, zi means Cj Elm xi, etc. We assume that C> 0 for allj E I,,,, and n,= 1 for all k E I,.

Each firm 1 is endowed with positive money M’ and with a production set y’ c E=‘+l. Any y E Y’ represents a possible production that k may engage in : the negative (positive) components of y stand for the inputs (outputs) of production. We make the standard assumptions that Y’ is convex, and that Y’ n SZm+l = {O}.l

The production economy is modelled as a non-cooperative game that consists of two stages. There are m trading-posts, one for every j E I,,,. In the first stage trader i offers a quantity 4j of j for sale in the jth trading-post, and bids b$ amount of money on it; firm I merely bids ei. We require, obviously, that

‘0 denotes the origin of 52” + I.

P. Dubey and M. Shubik, A closed economic system 25.5

The price pj of j E I,,, is obtained by dividing the amount bid on j by the total supply ofj,

Pj=(6j+~j)/qj if gj>O,

=o if ~j=O.

Next we calculate the allocation xi E Qmfl (y’ E D”“) that results for trader i

(firm 0,

X$ = a$ -4; + (b~/pj),’ for j E I,,,,

x:+l=ai+l-jg m

bj+jg Pjqi, m

yj=(eti/PjL

y6+1=M’- c e,!. isI,

In the second stage, firms produce with the resources they have purchased; thus firm 1 selects a vector 9’ E (y’+ Y’) n Rm+l. Firms then send all their products to the trading-posts for sale. Traders bid and offer as in the first stage. The market gets cleared as before. Let us denote by (6’, 4’) the bids and offers of trader i in the second-stage market, and by flj the price of j E I,,,. Then &Sa$-q;,and

fij is given by

The final allocation 11’ to trader i is

?We set bJ’/pj = 0 ifpJ = 0.

256 P. Dubey and M. Shubik, A closed economic system

where IT’ is the profit3 made by firm Z,

The payoff to trader i is the utility of his final bundle, z&Q’); that of firm 1 is il’. To consider this as a non-cooperative game we must define the strategy sets

of the players. We assume that at the end of stage 1, the players are informed about the aggregate bid and offer due to all others in each trading-post. Their second move is based on this information. We think of a strategy of a player as consisting of a move in the first stage, along with a set of moves in the second stage planned in advance for every possible market situation that may arise in the first stage. To state this precisely let r=max {a,, . . ., a,,,, ii,+,+A?f}+K where K is some positive constant; Y serves as an upper bound on the aggregate bids or offers that can be made in any trading post. The set of all possible aggregate bids and offers may be represented as a 2m-dimensional set,

4={(c11, a;, . . .) tl,, a~) E SZZm: lajl~r

and IcrjjSr for joI,,,),

where Ccj (a;) is thought of as the aggregate bid (offer) in the jth trading-post. Then we can describe a plan of trader i for second-stage moves by a function

where $i(a)= [6,(4 dl(a), . . ,, 6,(a), am(a)] corresponds to bids and offers of i in the second stage, made with the information (given by a) about the first stage. Thus a strategy of i is a pair {(b, q), $i} such that

bo!Y, qEf2”‘, 0” ?!,f22m, and

6+b(a)Sak+,,

qj+Qj(ol)sai.

We will denote by s”’ the set of all strategies of trader i. A plan for firm I is again aptly described by a function

where tjI(cl) is the output produced by 1 for sale in stage 2. But $l(a) must be feasible. To state this requirement precisely let R’= {e=(el, . . ., em) E Q”:

3We assume that the firms pay out not only their profits but also they distribute all of their assets. Thus we do not subtract the ML (even if we did the theorems would hold).

P. Dubey and M. Shubik, A closed economic system 251

2544’) be the set of all bids that 1 can make in stage 1. Let y(e, ol) E fimfl denote the input that accrues to I at the end of stage 1 if he bids e E R’, and the aggregate bids and offers are given by c1 E D. Then a strategy for firm k is a pair {e, &>,

with the constraint on $1,

t464 E Me, 4 + Y’) n f-2” + ‘,

for all a E b. The set of all strategies of firm I (for n+ 1 glgn+s) will be denoted by s”‘. Let s”= s”’ x . . . x !?” k. The payoff functions

were defined before for i E I,,+r. Thus a non-cooperative game p has been well defined.

A note on consumer strategies4

In order to give some insight into the nature of the strategy sets of an individual and their relationship to final outcomes we consider a market with one commodity being bought or sold with trade in a commodity money.

In fig. 1 the point with coordinates (A, M) indicates the initial endowment of an individual. Consider the rectangle given by Osqs A and 05 b s M. Any point in this rectangle is a bid and an offer.5 Consider the strategy6 (4, b); we wish to study how it transforms the initial point (A, M). Suppose that all other traders have offered Q units of the good and a total of B units of money. Then a bid of (q, b) takes the initial holding of (A, M) to a final holding of

(A-q+b/p, M-b+qp)

where

P=(B+W(Q+d, Q>@

4This note has been given in a previous paper [Dubey-Shubik (1976a)] but as it is directly relevant to proofs in section 3, for convenience it is given directly here.

5For simplicity in describing the strategies of a single individual in a single market the superscript i and subscript j has been dropped as the meaning should be clear.

6As our comments are limited to a single market we refer to the pair (9, b) as a strategy, although in the larger game it would only be a move.

258 P. Dubey and M. Shubik, A cIosed economic system

money

0 (QM/B,O) *

Fig. 1

We may observe that had the individual bid (q-b/p, 0) this too maps (A, M) into

(A-q+Q, M-b+&.

The bids which map (A, M) into itself are given by

A=A-q+b/p and M=M-b+qp,

or

b=q((B+b)/(Q+q)) or Qb=Bq.

All bids on the line connecting (QM/B, 0) to (A, M) map (A, M) onto itself. Any strategy to the right of this line maps AM onto a point on the curve PIP, to the right.

All points on the curve PIP2 can be obtained by bids on the ‘L-shaped’ boundary of the strategy set, i.e., bids of the form (q, 0) or (0, b) whereOSql_A andOSbSM.

Let (x, m) be a point on the curve PIP2 then it can be checked that the equation for this surface is given by

m=M+ (A-x)B

Q+A-x’

This is a concave curve.


3. The existence of non-cooperative equilibrium points

A ‘Nash equilibrium’ (N.E.) of F is a collection of strategies (?, . . ., ?‘+k) = 9 (where s”’ E s”3 such that

IZ’(s)=max ZZ’(slt’) for all i t&3

[(iIt’) stands for (al, . . ., ii-l, I’, Si+l, . . ., in+k)]. N.E.‘s for this game exist trivially. For example, take those strategies where

bids and offers of all traders are 0 in stages 1 and 2, and the bids and production of the firms are also 0. This choice of strategies clearly constitutes a N.E.

We will be interested in establishing the existence of ‘nice’ N.E.‘s which con- verge to the competitive equilibrium (C.E.) under replication.

First let us reformulate the N.E. problem in a way that makes it mathemati- cally more tractable. The transformed problem that we will look at was studied by Debreu (1952).

By ep (e>O) we will mean the modification of r that results if an external agency places a bid and an offer of E in each of the m trading-posts in both stages. This does not change the strategy sets of the players but it does of course change their payoffs.

We now define a ‘generalized game’ [see Debreu (1952)] “Z associated with ‘7. Let

0=(x E SZm+l:jxj[ sr, for 1 sjgrn+ l}.

Thus D contains any possible holding that a player may acquire at the end of the first stage in “f, provided E is less than K (which we henceforth assume it is). Define

Si={(qi,bi,~i,bi):qiEom,b’EQm,~iES2m,~iESZ”,

tii+Li5a6+l,q~+qj~a~ for jEZ,},

for 1 sisn. We will interpret si E S’ as a strategy of trader i in “Z (E 20) where qi (4’) and b’ (6’) denote his bids and offers in the first (second) stage. For n+lIZSn+s,define --

where

S’=R’x E’,

E’=(D+ Y’) n Q”+l.

An element s’=(e’, 9’) [where et E R’, 9’ E E’] of S’ similarly represents a strategy of 1 in “r (E&O): e’ is the vector of his bids in stage 1, and 9 his output produced for sale in stage 2.


LetS=S’x . . . xS”+S . For any E 2 0, we can define

"P':S+ R

exactly as before, given the interpretation of S’, and assuming an external bid and offer of .s in each trading post. For

let

s=(sl, . . ., s”+S) E s,

and

?=(?, . . .) d-1, dfl, . . .) sn+q,

S’=Px . . . x P-1 x s’fl x . . . x sn+s.

Consider any S’ E s’. Define’

[&~s,(ei)]j=e: [Q@)+El [Bj(Si)+e+e$]’

forj E I,,

for any e” E R’. Now we define a mapping “y’ from s’ into 2s’for all 1 sign +s8.

If isn,

“f($) = s’.

If i>n,

“$(f’) = {(e’, pi) : e’ E R”, 9’ E E’, $’ E [“tai(ei) + Y’) n SZ”‘+‘}.

Following Debreu (1952), we will define a Nash equilibrium of “r to be a collection s=(sl, . . ., .s”+‘) E Ssuch that

7Throughout we will use the notation si = (g’, b' , ij’; 6’) for 1 5 i 5 n; and si = (e’, 3’) forn+l I i 5 IZ+S.

sy’.(?)% the subset of S1 to which i is restricted when the other players choose strategies P.


From a N.E. for “r we can extract a N.E. for ‘F. Let s= (sr, . . ., Pfs), si E S’, be a N.E. for “r. Construct pi: i? + Q2m as follows. For 12 isn

$i(Cr)=(6’,Q’) if, foralljEJ,, clj= (

C b:+ C e: +e, id,\(i) idn+s\In >

=0 otherwise.

Forn+l jisn+s,

= 0 otherwise.

Then it is clear that if we put

Si=[(qi, b’), $i] for 1 SiSn,

= [e’, *i) for n+l sisn+s,

then (S’, . . ., T’+s) is a N.E. for ‘ji. [Note also that if ET were to be defined as a game with perfect information

(i.e., each player’s second move is made after knowing the detailed individual bids and offers due to all others in the first stage), even then a N.E. for “r can be converted in a similar manner to a N.E. of ‘T”.]

Lemma I. A N.E. exists for every “r, E > 0.

Proof. Observe that the sets S’ are convex, and that eyi is clearly continuous for every i and maps s’ into compact subsets of S’. In Lemma 2 we verify that the sets

E

“j+(S’)= (v* E y:(2): P~(slv*)= max Pi(s]v)} Uey,‘S’

262 P. Dubey and iU. Shubik, A closed economic system

are contractible.g Then, by the theorem in Debreu (1952), a N.E. for “r exists. Q.E.D.

Lemma 2. Thesets ej?(li)are contractibleforany 1 jijn+s, andany? E 3”.

(We will say that a set S in E’ is contractible to 5” c S if there exists a continuous function

F: [0, l]xS+ S,

such that

F(0, s)=s, for all s E S,

and

{F(l, s): s E S}=S’.

Recall that S is said to be contractible if it is contractible to some single point in S [Dubey-Shubik (1975)]).

Before the proof of Lemma 2 we will need some other results. Let for isn,

If the offers and bids due to all the others except i (including the external agency) are kept fixed at (Qj, Bj) in the first stage, and at (Qj, Bj) in the second stage, then gj and hj” depend only on si E S’. Denoting by g [h] the vector (gl, . . ., g,)

[(hr, . . ., 41, we will think of (g, h) as a (vector-valued) function f(s’) of si E s’.

Claim 1. The set T= {f (s’): si E S’} is convex.

Proof. Let (g’, h’) E T and (g”, h”) E T; and let (g*, h*)=A(g’, h’)+p(g”, h”) wherelLO,~LOO,A+~=l.Wewanttoshowthat(g*, h*)ET.

Supposef ((b’, q’, 6’, 4’)) = (g’, h’) andf((b”, q”, h”, 4”)) =(g”, h”). Consider (q, 6, g,G) defined as follows :

ij = min {q’, q”},ll

4 = min {d’, f$“},

E=Ab’+pb”,

G=nbl+@.

gFor the definition of ‘contractible’, see Lemma 2. “‘We drop the superscript isince no confusion will result. “I.~.,c& = min {q,‘, q,“} for each j E I,,,.


It is easily checked that (y, 6,3, $) E S’. Let (9, h) =f((Zj, 6,4, g)). We assert that fzf*, and hz:h*. To see this first observe that, if we keep bj [6j] fixed, then gj [hi] is a decreasing function of qj [ej]. Hence

and

(Y, h’)=f((% b’, 3, b’))Zf((q’, b’, 4’, @))=(g’, h’),

(g”, A”) =f((q, b”, 4, 6”)) zf((q”, b”, cj”, 6”)) =(a”, A”).

But if we keep qj [4j] fixed then gj [hi] is a concave function Of bj [~j]. Hence

(Q, Q=_M, 6,4, F))

Z.nf((C b’, $3 g))+,U-((q, b”, s, ~‘0)

2 nf((a’, b’, d’, 6’)) + pLf((q”, b”, 4”, 6”))

= I(g’, h’) + p(g”, A”)

=(g*, h”).

Put

4”=Iq’+pq”

g=nQ’+pQ”.

Note that (4, 0,4, 0) E S’. Let (g, h”)=f((& 0, 4, O)).Then

gj=aj-(fj

= a j - (Jqj + pqy )

=t?(aj-@+/L(aj-45’)

s &$ + pg;

* =g.

J’

Similarly

NOW gj [hj] is a continuous function of (qj, bj) [(4j, bj)] for which gj(qj, ~j)~ g;, gj(gj, O)sgT. By the intermediate value theorem in calculus there is some 2jj jq? 5 cjj and 0 5 b; 5 sj for which


Similarly, there is a @is@ sgj and O<&Tg$j for which h,(eT, br)=hj*. Hence

f(q*, b*, 4*, 6*)=(g*, h*). It is obvious that (q*, b*, 4*, 6*) E S’. Therefore (g*, h*) E T. Q.E.D.

Claim 2. The set f -l(t) is convex for any t E T.

ProoJ Consider gj as a function of the variables qj and bj

where 05qj5aj, and OSbjsa,,,+,. It was shown in Section 3.1 of Dubey and Shubik (1976b) and noted earlier in this paper that for any t in the range of gi, gr’(t) is a convex set of dimension 1. [So is h;‘(t) where we define hi with Qj and Bj in place of Qj and Bj.] Then

f-‘(t)=S’n [g;‘(t,)x . . . xg,-l(t,)xh;‘(t,+l)

X . . . X h,‘(bJl,

where t=(t,, . . ., tz,). This is clearly a convex set. Q.E.D.

Remarks

(1) f-‘(t) is obviously not empty for t E T. (2) For any t E T, we will denote by $(t) the unique point in f -l(t) for which

q,b,=O and gjgj=O for every j E I,. (See fig. 1.) It is easy to verify that $ is a homeomorphism between T and $(T).

(3) Let z?,‘(s) be the final holding ofj E Z,,,+l by i E I,* when he uses his strategy

s (other strategies being fixed as before).

Claim 3. Denote xksl, &fi by Cj. Then

%n+l(S)=%+l+ c Bj(aj-gj(s>) +C

Bj(aj + cj -h,(s))

jer,,, Qj+aj-gj(s) jflm Qlj+aj-hi(s) ’

Proof. Straightforward [see Claim 3 in Dubey-Shubik (1976b)].

Remark. (4) )2,+ 1 is a concave function of (g, h) E T. To show this, note that the second derivative (w.r.t. hi) of

F(hj) = Bj(Uj + Cj - hj) . 2Bj(Cj- Qj)

Qlj+aj-hj ls (Qj+aj-hj)3 *


But hj ~ Qj+aj since Qj + aj is the total amount of j available; also ~j=

c. l&l ai + cksr, fi 2 xkol, &i$ = Cj, since q: s 1. Therefore F”(hj) 5 0, and E is a concave function Of hj, hence of (g, h) as well. Similarly [Bj(aj-gj)]/[Qj+a,;-gj] is a concave function of (g, h). Thus z$,,+~ being a sum of concave functions of (g, h) E T is itself concave.

Note that for any (g, h) E T, the final allocation is given by

lj(g, h)=-Uj+gj+hj for je I,.

52 m+l is given by the function in Claim 3. So the payoff to player i at (g, h) E T is u’(@.

Claim 4. Let p be the subset of T on which ui is maximized. Then ? is convex.

Proof. Suppose (g’, h’) E p, (g”, h”) E p. Let (g, h)=A(g’, h’)+p(g”, h”) for A 2 0, p 2 0, A+ ,U = 1. By Claim 1, (g, h) E T. Moreover, by Claim 3, A,,,, I(g, h) 2 AR,,,, r(g’, h’) + pLR,+ I(g”, h”). Finally, for j E I,,,,

2j(g, h)= -aj+g+h

= -aj + (Ag’ + pg”) + (Ah’ + ph”)

=A(-aj+g’+h’)+p(-aj+g”+h”)

= lAj(g’p h’) + pAj(g”, h”).

Therefore

t(g, h) 2 Wg’, h’) + pt(g”, h”).

Since ui is concave, this implies

u’[R(g, h)] Llu’[R(g’, h’)] +pu’[R(g”, h”)].

The result easily follows. Q.E.D.

Proof of Lemma I. Since f -l(t) is convex for every t E i?, and contains +(t), there is a continuous mapping

H’: [0, I] xf -l(t) + f -l(t)

such that

H’(O, v) =Y,

H’V, Y> = W,


for every Y of -r(t). Define

H: [O, l] xf -i(F) + f _‘(P)

by

H(x, Y) = H’(x, Y) if Y E f -l(t).

[Since f -l(t) n f -‘(t’)=O if t # t’ this is well-defined.] H contracts f -l(F) to I+@). But I,@) is homeomorphic to p, and p is convex and thus contractible. It can now be easily shown that I,@) is contractible; hence so is f -‘(rf). This proves the lemma for isn.

To prove it for i>n, let us first consider the set Y of initial resources that i can purchase. Denote by Qj (oj) and Bj (Bj) the aggregate supply and bid due to the others. From the equations

Yj= Bsj for jE I,,

YIII+~=M- 1 ej,

jsIm

we deduce, as in the proof of Claim 3, that

Ym+l=Mi- C Bjyi jeI, ej_yj =h(ylp . . .) YA w

which is clearly a concave function of (yl, . . ., y,). Moreover, it is easily verified that the set

s= (Yl, * * -3 y,)ElP: yj= g$ c =} ej<M i J joL

is convex. Thus the set H of initial holdings that the firm can purchase before production is a surface defined by the concave function h on the convex domain s.

DehneF: 0”” --+ Rby

We check that F is concave. Let


and note that it is concave. Suppose z=Ix+py, where 120, ~20, J+P= 1, XESlm+l, yoSlm+? Suppose F(x) =f(x’) where x’ E (x+ Y’) n U”fl ; and F(y)=f(y’) where y’ E (y+ Y’) n !G?“‘+r. Put z’ = Ix’+ cry’. Since Y’ is convex, z’ E (z+ Yi) n sZmcl. But

Fo~f(z’)~~f(x’)+~f(Y’)

= izF(x) + pF( y).

Let H = (x* E H: F(x*) = max,,, F(X)}. We claim that R is convex and in fact consists of a single point. For let y’ E B, y” E R, y” #y’. Consider y = Ay’ + w”, for I > 0, p > 0, A + p = 1. Then since S is convex, andi y’ 1 nm and y” I Rm are in S, yin- ES. Hence y=(ylnm, h(y)) E H. Moreover 9,+1=h(yI.,)>I(y’Inm)+

CL(Y”ls&=Y,+l, since h is strictly concave. But then Fis strictly increasing in the (m + 1)st variable, therefore

a contradiction. Let A be the set

{y* E (R+ Y') n cJm+l :f(y*)= yE(ji:;:am+Lf(Y))* A is convex because (a+ Y’) n GY”+’ is convex, and because f is concave. It now follows that

&?)=H x A,

which is also convex. Q.E.D.

To obtain a ‘nice’ N.E. of r”, we will establish the existence of a ‘nice’ N.E. of r. We consider a sequence of N.E.‘s of eT, E -+ 0, and show that any cluster point of this sequence is a N.E. of r. A N.E. of P is then derived from this N.E. as described earlier. To distinguish such a N.E. from other kinds of N.E.‘s we will call it an E.P. (Equilibrium Point) of r.

Our first step in this direction is:

Lemma 3. For each j E I,,, suppose that there are at least three monied traders who desire j, and at least three j-furnished traders who desire money. Moreover suppose that qf <+(l E I”+S\IS) if i is a trader of the latter type. Let “pj, “fij be the prices at a N.E. of “r. There exist positive constants K, Cj and Dj (for j E I,,,) such that


for any E 6 K.

Proof.’ 3 W.1.o.g. letj=m, and assume that 1,2 and 3 are monied and desire m; and that 4, 5, and 6 are m-furnished and desire money. Put

Hk=max (1x1: x E (D+ Y”) n SZm+l},

H= c H”+K for some K>O. ksl,

(Thus H is an upper bound on the amount of any commodity or money that a player may hold at the end of the game “r, if E < K. We will assume in what follows that E < K.)

Also put

h=min [h(u’, m, H): i= 1, 2, 3],14

A= ---& min [a:,,: i= 1, 2, 31,

h=min [h(u’, m+l, H): i=4, 5, 61,

AI=+ min [ai: i=4, 5, 61,

/?=max [qf: i=4, 5, 6, ZEI,].

Note that these are all positive under our assumptions (except for /I which may be 0), and independent of E. Also note that /3< l/2, i.e., l/2-/3>0. First we establish the existence of C,,,. Put

and

6 = &pm = (6, f zrn + E)/(& + E),

8= “~,=(~~+&)/(~,+~,+E).

Suppose first that the condition

13This proof takes its cue from Shapley’s ingenuous proof of Lemma 4 in Shapley and Shubik (1976) which we use in this proof.

‘%ee Appendix A for the definition of h(u’, m, H), and the statement of Lemma 4 in Shapley and Shubik (1976).


and

a:+,-j&mb;-j& QLA m

(1)

holds for at least one of i= 1, 2, 3; say for i= 1. Then an increase A in l’s bid for m in stage 1 would be feasible if 0~ A gmin (E, A), and would have the following incremental effect on his final holding:

Ilj(d)-RJ=O for j~1,,,_~;

A’(A)_A’ =(%+s)(b:+A) Gm+s)b: m

m 6,+z,+s+d - &,,+?,,,+a

=(ijm+E)A b,+q,,+c-b;

(&,,+~,+E)(~,+Z,+E+A) 1 2 G, + &)A &,,/2 + ?,,,I2 i- E/2 + A/2

(&,+~,+~)(b,+i$,,+~+d) 1 Gm+4A A

=2(6,+Z,+E) = s *

(The ‘ 2 ’ above follows from

2 6,/2 + C&/2 + 42 + A/2);

Define

z= -228emt1,

and note that we have the vector inequality

x’(A)tx’+$ (z+e”‘). (2)

We are now in a position to apply Lemma 4 in Shapley and Shubik (1976) takingf=u’,j=m,andy=x’+z. We havexl oDmfl and IIx’IjsH.So,by the lemma, if both x1 + zz 0 and zsh, then

270 P. Bubey and M. Shubik, A dosed economic system

u’(xl+z+em)>ul(xl).

Since u1 is concave, this implies that

22(x’ + $ (z + em)) > &x1)

holds for sufficiently small A, and hence, by (32) and the monotonicity of ui, that

u’(x’(A)) > u’(xl)

for such A. But this means that trader 1 could have improved, a contradiction. Hence either x1 +z< 0, or I IzI I> h. If the former, we have

9;+,-2C5<0.

But R,,, 2 A/2; hence

] 26>A/2 1 (3)

If the latter, we have

l-l 26>h . (4)

Similarly, an increase in A in l’s bid for m in stage 2 would produce the following incremental effect:

R:(A)-Rj=O for ~EZ,_~;

%(A) -_$:zd . 28 ’

Defining z= -28emt1, (2) holds as before; and, arguing as before, we get that either

(5)


or

I I 28>h . (6)

Now consider the case where (1) fails for i= 1, 2, 3. W.1.o.g. we may assume that

From the failure of (1) for i= 1 we therefore have

jc bj+jz 6fZa,!,+l-AZmA. m m

Hence either

or

bf > A/2 for some j E I,,,,

&,! > A/2 for some j E I,,,.

First suppose that (7) occurs. Ifj= m, then b: > A and so a fortiori

lQA/2HI.

(7)

(8)

(9)

Ifj# m, trader 1 could then decrease bj by a small A >O and increase bi by the same amount with the incremental effect:

$(A)-Rb=O for p E Z,\{m,j};

tz’(A)_jzt =(bf-A)(iIj+E) bj(qj + E) j

j (6j+Zj+E-A) - (6j+~j++E)

>(b_i-A)(qj++) _ bj(qj+E) = (6j+Zj+E) (tij+sj+E)

A(Gj +E) =- (6j+Zj+E) ’


=-

(The ‘ 2 ’ above follows because p,(A) >p,,,.)’ 5

R:(A)-R&,

by the same calculation as earlier. If we define

then (1) is satisfied. Hence, arguing as before, either $‘+z<O or l[zlI >h. If 2’+z=O, then either IZjl>)2f or Iz,+,I>R~+~. But we have

A! 2 (4j+&lbj' (4j + EM ’ -(6j+Zj+E) ’ 2(5j+Zj+&) ’

and

which gives

1 26>A/2 / . wo

If llzll>h, then either lzjl>h, or Iz,,,+~ I P-/Z. In the first case the inequalities Zj+E,+E>bf>A/2,andqj+&<Hyield

26 >hA/2H ; I

in the second, rif 5 ij j yields

(11)

I I 26>h . (12)

ISHere pm(d) = (6, + &, + &+ d)/ (q, + &),pJ (d) = (6J + CJ + &- d)/(gJ + e), ek.


The bid A could, however, also have been transferred from bf (even whenj=m) to 6: with the incremental effect:16

$(A)-$=0 for p E I,\{j, m},

But &,(A) -&LO since an increase of the bid on m only increases the price of m. Hence

Then again we derive conditions bounding s^ from below:

(13)

I 28>hA/2H , (14)

or

)2b>h . (15)

Next suppose that (8) occurs. Ifj=m, clearly we have

( 8zA/ZH( . (16)

161f j = m, then these terms are the changes in the holdings of j in the first and the second stages. But the argument goes through in this case too.


Otherwise, a transfer of a small A from 6: to 6; leads to

,

=- ad, say.

(Note that as 1, since q: S 1 fork E I,.) Put

Then

R’(A)#+;3(z+em), (17)

and arguing as before we get that either 2’ +z<O or 1 IzI I> h. If A’ $ z< 0, theneither lzjI>R:orlz,+,I>R~+,.But

and

> Aa = -. 2


Therefore

(18)

If IlzlI>h, then either jzjl >h, or Iz~,+~ >h. In the first case, the inequalities

6j+a>6: >A/2andjj+Fj++<Hyield

I 2ik-hAj2H ; I

(19)

in the second, a 5 1 yields

I I 28>h . (20)

Finally, a transfer of d from 6; to bk (the argument holds also whenj=m) leads to the incremental effect:

=- LXA.

Put

Then (2) holds, and as in the previous case, we derive lower bounds for 6,

--- I 26>hA/2H ,

I (21

I I 26>h , (22


12&A/21. (23)

Pick C,,, to be the minimum of A/4, h/2, A/2H, hA/4H.

The existence of D, is established in a similar manner, We refer the curious reader to Dubey and Shubik (1976b) where we have ground through all the details. Here we will just quote that it suffices to pick D, to be the maximum of

Q.E.D.

Remark. (5) Note that if, for each j E I,,,, there is at least one monied trader in r1 who desires j, and one j-furnished trader who desires money, then the hypo- theses of Lemma 2 are satisfied for ir, k 2 3, where r, and ir are defined below.

Let 8,’ be a N.E. of “r. Consider the sequence {.s~~}~~~ where Ei + 0. Lemma 3 enables US to pick a subsequence {s,,} such that pi(s,) + pj where Cj~pj~ D,. Let s be a cluster point of this subsequence. Then s is a N.E. of r since it is a point of continuity of the payoff functions n’. (If the total bid and supply at s* are both 0 in market j, i.e., if the market is inactive, there is no problem because a trader will lose j if he only supplies j, will lose money if he only bids on j, and will merely retain what he has if he both bids and supplies.)

The N.E. s of r corresponds to an N.E. of r”. We have proved Theorem 1: An E.P. of F exists.

With this E.P. we will associate the prices (pi, . . ., pm, 1, PI, . . ., pm, 1)” obtained for s as a limit of “‘pj (“i@j) at the N.E. s,, of ‘*r.

4. Limit equilibrium points and competitive equilibria

4. I. The convergence proof

Let us consider a ‘replication sequence’ of economies rl, . . ., rk, . . . there are a fixed number t, of types of traders, characterized by their utilities ui, endowments a’, and shares vi. Also there are a fixed number t2 of firms characterized by their production sets Y’ and money M’. The economy rk has kt, traders and kt, firms, k of each type. (The shares of traders of any fixed type are divided equally. Thus in r, the share of a trader of type i in a firm of type I is ,~f = vi/k.) An E.P. in which traders and firms of the same type choose the same strategies is called a type-symmetric E.P. (denoted T.S.E.P.)” and such a T.S.E.P. v can be represented as a vector in )(f~+t S’ where S’ is the strategy set of trader i

“We take the price of money to be 1 in both stages. ‘*That such a T.S.E.P. always exists is shown in appendix B.


(firm i) in rI for 15 is tl (for t, + 1 gis tz). Thus, for each k, a T.S.E.P. kv in

rk giVt?S us a price ,$ and an allocation ,$ in rI via kV in S. We will say that kv is an interior T.S.E.P. if, for kV=(~l, . . ., stl+‘*), and si=(qi, b’, 4’, b’),

Claim 6. Let Cj and Dj be as in Lemma 2. Then, for all k,

CjjkPj~ Dj and Cj~‘k~j~ Dj.

Proof. First note that ifs < K, then in $r the maximum amount any commodity or money that a player can hold is bounded above by H. Suppose not. Let ;x$ > H. Now there are k players of the same type as i, hence at the T.S.E.P. \v they have identical final holdings. Then the total amount ofj in ir is greater than kH. This is clearly impossible.

Note that all the other constants in Lemma 2 do not change with replication. Therefore the upper and lower bounds hold for the prices ip, ;#, and con- sequently Cj~~pj~ Dj and Cj~ekhj~ Dj for all k, if &SK. But kPj=lim,,, ipj, and kpj=e+e ifij. The lemma follows by taking limits. Q.E.D.

Remark. If kv is a sequence of T.S.E.P. such that lim,,, kpj exists and is pi, say, then the claim shows that Cjipj~ Dj. (So also for lim,,, k~j.)

Theorem 2. Suppose we have a symmetric, interior sequence kv of T.S.E.P., with associated prices (kpl, . . ., &,,, 1, kfif, . . ., &,, l), such that dp,, . . ., !$,,,t

19 ICAP . . ., &,,, 1) converges to (pl, . . ., p,, 1, fiI, . . ., j?,,,, l)> 0 ask -+ CO. Then

(PI, ..*,Pm, 1,31,. . ., p,, 1) are competitive prices (see appendix A for definition ofaC.E.)forrk,k=l, . . ., n, . . . .

(That such a sequence kv can be chosen is justified by Claim 6.) Before the proof of the theorem we will need some other results. Let ir stand

for the s-modification (E 5 K) of rk; and let ;v (;p, $), ix denote respectively the N.E. of ir, and the prices and allocation at this N.E., so chosen that ip + kp, ifi + k, as E + 0. Further, denote (for 15 is tl) by ;Bj (i&j) and :Qj (i&) the aggregate bid and supply at iv due to all the players except i in the jth trading- post in stage 1 (stage 2). I9 If we drop the superscript i we will mean the aggregate due to all the players. Finally put ;Cj =cr knf $f, where 1 varies over the kt, firms in rk, and ij$ is of course the produced output of I at iv.

‘%cluded in r’J3,’ (k’B,) is the external bid of 8, etc.

F

278 P. Dubey and M, Shubik, A closed economic system

Claim 7. At iv trader i maximizes his utility subject to the constraint that he buy andsell in the 2m markets at fixed prices f p;, “,a; (j E I,,,) given by

Proof. Consider the ‘holdings-surface’ H of trader i when all others keep their strategies fixed at ;v. The equation of this surface $,,+r =F(g, h) is of the form given in Claim 3, from which we observe that it is strictly concave and differenti- able. Let H’ be the unique hyperplane through ix’ E H, such that H lies below it. H’is determined by the partial derivatives of Fat ixi, and it can be easily checked that

and

To establish the claim, we must show that &xi maximizes ui on H’. Suppose not. Let y’ E H’ (see fig. 2) be such that y’ can be attained by i (if he sells and buys at the fixed prices ;p$, :@f .) and u’(y’) > #‘(ix’). By the continuity of ni we can pick a y” below H’ such that u’(y”) > uiGxi). Consider the straight line joining y” and ixi. Since His concave and H’ is the unique hyperplane that supports it at lxi, the straight line must intersect H at some point z E H. But then u’(z) > ui(ixi), because ui is concave. This contradicts the fact that z&x’) =maxyEH u’(y). Q.E.D.

Claim 8. For any 6 >O, there is a k, such that lim,,, ip$ and lim,,, i#$ exist for all traders i and all j E I,,,, and kpj - 6 5 lh,, 0 lpi skpj -I- 6, if k 2 ka. (so also

for 8.)

Fig. 2


Proof. First observe that

15;Qj/gQ$=(Q+(k+l)&$)/(Q+k;q$) for some Q>O,

51+1/k.

Similarly,

1 s;B,/;B; 5 1 + l/k,

1 j ;Bj/$; 5 I + 1 /k,

1$&j/;&51+1/k.

Finally,

O~~Cj/~Qj~l/k,

since

NOW, recall that ipj=;Bj/;Qj, ~fij=~i?j/~~j, and that kEPj~e-10 kpjr ;fij+s+O k~j.

The claim easily follows. Q.E.D.

Proof of Theorem 2. By the continuity of the payoff functions, it follows that kXi=lim,,, i i x is optimal for trader i if he trades at the fixed prices kpj, kflj* But since he does not bid all his money, ,$ is optimal for i even if we were to

relax his money constraint. For (#, k 6’) is the solution to the following prob-

lem:

Maximize u’(~(w, ti)),

subject to ai+ I - jg (wj+Gj)20, WLO, $20, m

where 0(w, @) is the final bundle that accrues to i if he purchases commodities at fixed prices Lpi, &) with bids (w, fi), keeping his sales fixed at (kqi, kdi).

Since ui is clearly a concave function of (w, G), by the Kuhn-Tucker theorem there exists a non-negative number A such that the above problem is equivalent to:


Maximize u’(f?(w, fi)) +3, &+r - 1 9 subjectto ~20, ~$20.

But (#, $) is the solution, therefore

which implies 1= 0 since kv is an interior T.S.E.P. Thus in fact ,$ is optimal for i on B’&p’, &). Then from the fact that ,$ +k+m xi, &pi, ,$‘) +k_+a, (p, b), it is easy to deduce that xi is optimal for i on his budget set.20 In a similar manner we can show that firms maximize profits in the limit. To ensure that the limiting prices and allocations is a C.E. for FL, we need to check that the balance conditions on trade and production hold. This holds at each N.E. kV, and hence in the limit at G.

Finally, it is trivial to show that a C.E. for Fr (with allocations replicated) gives C.E.‘s for Fkr kz2. Q.E.D.

4.2. Two examples

In this section an example is provided to show that in this model with production even at the limit T.S.E.P. the price for a commodity in stage 1 will not necessarily be equal to the price in stage 2.

Furthermore, a second example is provided to illustrate the difference between totally competitive markets and a market with competitive traders but a monopolistic firm.

Example 1

In this two-stage model of production and trade, even in the large economy, prices of the same commodity before and after production need not necessarily be the same. It can be shown easily (see appendix A) that prices in the second stage cannot be higher than in the first, however it is possible that the reverse holds true. This state can be caused by a tightness in supply of an input commodity which, after production could be more abundant. An example with one type of consumer and two types of firms illustrates this.

Suppose that each consumer has an initial endowment of (25, 1, 2) of three commodities and each has an adequate supply of money so that no credit

?3ee appendix C for the definition of the budget set.

P. Dubey and M. Shubick, A closed economic system 281

constraints are active. Let each have a utility function of the form

where the x, y and z are the amounts of the three commodities and m is the amount of money.

Suppose that the firms of the first type control a production process

y = 2 min[x, 421,

and those of the second type control a process of the form

z=4 min[x, y].

Suppose that the following prices are announced (pi, pz, p3) = (I, 1, 4) and (PI, hz, f13)=(1, 1, l/2). We may check that the following is a competitive equilibrium: Firm 1 buys (8, 0, 2) and transforms it into (0, 16, 0). Firm 2 buys (1, 1, 0) and transforms it into (0, 0, 4).

Consumers sell (9, 1, 2) and buy (0, 16, 4). In making the comparison with the Walrasian competitive equilibrium model

we could consider a model with 2m rather than m commodities. The limit N.E. of this production and trading game with two-goods markets will approach the C.E. of the 2m commodity Walrasian model in general, but when input constraints are not binding the C.E. of the m commodity model will be app- roached.

Example 2

In our models all participants are modelled as strategic players. The effect of a mass market can be studied as a whole or in part by replicating all or only some of the players. In the examples given here we show a (1) competitive market, i.e., one in which all economic units are small, and (2) a market with competitive consumers but a monopolistic firm. We do not calculate the case where there are many firms and few consumers although the conditions are given in the model.

Consider two types of trader, each with the same utility function,

U=log x+log y+m.

The first have an initial endowment of (1, 3, 100) and the second of (1, 35, 100). There is a single type of firm with a production process that converts y -+ x,

i.e., one unit of the second commodity as inputs yields one unit of the first as output. The firms have as initial resources (0, 0, 100). The traders of the first type own all the shares of the firms equally.


Competition. Suppose that we assumed that there were so many traders and firms that all could treatp, andp, as fixed.

Monopoly. Suppose that as we replicate the original market there is only one firm whose resources grow in proportion to the replication. Thus its endowment at replication YE will be (0, 0, loon).

Table 1 presents a comparison of the full competition conditions with those of the monopolistic firm.

Table 1

Competition 0.1 0.1 1.8 18 0.9 0.9 0.7 25

Monopoly 0.316 0.0594 0.02568n 4.3246n 0.6838 0.6838 0.8218 18.165

Table 2

Competition

Monopoly

103.00 106.20 100

103.58 104.37 lOl.lln

5. Discussion and further problems

5.1. The trading of shares

In the models we have examined we have not included trade in shares. In- dividuals were given fixed endowments of shares which they were required to hold. A more general treatment would be to permit them to trade shares. Al- though we do not carry through a formal analysis of the market with the trading of shares in this paper, several observations and conjectures are made on the effect of having shares traded.

As information conditions are critical in models of this type we must be explicit about the sequencing of markets and the availability of information. The simplest model might be to have shares traded before or after the first round of trade prior to production, but with no information to the traders. This is equivalent to having the goods market and stock market decisions simultaneous.

We must specify the relationship between the date of ownership and the dividend receipt as well as the sequence of transfer and payment for shares.


Simple rules for this would be that there is one or two stock market trading sessions, say before the first - and second-goods markets. Dividends are paid to the final owners after the last stock trades.

We conjecture that for mass markets the price of the shares approaches the value of the dividends plus liquidation payment to be received from the shares at the limit N.E.

The major effect of the introduction of shares, if payment is received for selling them before all trade has taken place, is to increase the liquidity of some of the traders and hence to relax the conditions calling for ‘enough money’ or for credit.

At the limit N.E. which approaches a competitive equilibrium in the models with trade in shares it is conjectured that a new indeterminacy (or multiplicity) appears in the existence of equilibrium points. In particular if cash flow constraints are not binding then individuals will be indifferent between holding shares and obtaining dividends or selling them for a price equal to the dividends plus liquidation payment.

In the modelling of shares here we assume that they are non-voting and are of unlimited liability. Otherwise new analytical difficulties and different results are to be expected.

Even in the simplest of models with trade in non-voting shares only once, as is shown in fig. 3, when numbers are few we may expect new phenomena. In this model, if there are m goods and k firms a strategy by an individual trader consists of a vector of 2(m + k) numbers (a bid and offer for each good and each type of share) and a function for stage 2 which has as its argument which could be as high as any one of the vectors of 2(m + k)n + 2mk dimensions which could arise from the possible moves of all it traders and k firms in the first stage.

In this model we may expect new equilibria to appear for a finite number of traders. These equilibria will reflect the influence of ‘wash sales’ on both the goods markets and stock markets. An example of the influence of wash sales has been given in a previous paper by Dubey and Shubik (1976a).

Pl 2 , ,.*.

ll /

Trade in shares

and goods without information

Production and further trade in goods

Fig. 3


Another set of problems is encountered if we permit the firms to hold and trade in shares. In particular we must either rule out the control aspects of the holding of shares or open a new Pandora’s box. Even with this simplification a new dimension of oligopolistic manipulation is opened up. We must also decide whether treasury stock (i.e., stock of a company held by itself) obtains dividends. This raises a pro forma accounting problem in as much as the residual worth of the firm will be distributed on liquidation.

If a firm is permitted to own its own shares we must decide how to treat the possibility that it owns all of its shares. In this case we might require that the owner be a player with a utility function rather than a bloodless fiduciary maximizing profits for the share owners. Profit maximization in such a market cannot apriori be equated with utility maximization.

5.2. A comment on existence of oligopolistic equilibria

The model in this paper contrasts with the work of Arrow and Hahn (1971), Fitzroy (1974), Gabszewicz and Vial (1972), Laffont and Laroque (forthcoming), Marschak and Selten (1974), Negishi (1961) and Roberts and Sonnen- schein (1977) in several ways. In particular all traders and firms are treated as strategic players; a different market mechanism is used; a means of payment or a ‘money’ plays a crucial role and a two-stage trade is used in order to be able to deal with the procurement of inputs while preserving laws of conservation.

In multi-stage models of economic activity it may be desirable to distinguish between ‘perfect equilibria’ and more general non-cooperative equilibria. The former might not exist in systems in which the latter nevertheless exist.

5.3. A variation of the model

A variant of the model given here would allow the traders to use the money and commodities acquired at the end of the first stage as bids and offers in the second stage. We chose the simpler model for ease of presentation. The existence and convergence results nevertheless hold for the variant model as well. The proofs are in essence identical to the ones given here.

Appendix A

We quote, with proof, the following lemma that is invoked in the proof of Lemma 3 in section 3.

Lemma C (Uniform monotonicity). Let j E I,,,, let f (x) be a continuous, non- decreasing function from Q”’ to the reals that is actually increasing in the variable xj, and let H be a positive constant. Then a positive number h = h( f, j, H) exists such that for all x and y in Q”, if

IlxllSH and Ily-.4lSh,


then

f(v+4>fb-).

Proof. Let Q(a) denote the cube {x E 52”’ : 1 Ix 115 a}. The functionf (x + ej) -f(x) is positive and continuous, and so has a positive minimum p in Q(H+ 1). The function f itself is uniformly continuous in Q(H+ I), so there exists a OC 6 < 1 such that XE Q(H), YE P, and ll~-xll<6 imply that If(r)-f(x)I<p, and hence thatf(y+ej)-f(x)>O. So wemaytake h=& Q.E.D.

Appendix B2 ’

We wish to prove22 that there will always exist a T.S.N.E. of eT. We will assume that everybody desires money. (This assumption is required to guarantee the interiority of the N.E.‘s.) It can then easily be checked that p consists of a single point. Hence the sets ‘Yi (?) are in fact convex. Define S* to be the subset of S which consists of type-symmetric strategies, i.e.,

S* = {s E S: si=,sj if i and jare of the same type}.

S* is clearly convex. Consider the map

given by

0(s) is always non-empty because at any s E S* all players of any given type face the same optimization problem, therefore there will be identical strategies that lie in their ‘best-response sets’ ‘vi (5’). 0 is moreover convex-valued the U.S.C. We appeal to the Kakutani fixed point theorem, and see that the fixed point of 0 is a T.S.N.E. of “r.

Appendix C

The trading and production economy that corresponds to our model involves 2m + 1 commodities. The traders utility functions

Q2m+l 6’ --f fi’

“We adhere to the notation of section 3. “This proof is essentially the same as the one in Shapley and Shubik (1976).


are given by

~‘(Xl, . . .) x,,x:, . . .) X;,Xm+l)=t4yX1+XT, . . .) x,+2;,x,+1).

Their initial endowments are A i = (a;, . . ., at, 0, . . ., 0, a;+ &. Firm I has initial endowment (0, . . ., 0, Ml), and a convex production set P’ c RZm’l in which the first m commodities are inputs for production. We denote the vector of

prices by (pl, . . ., pm, PI, . . ., $,, 1). Let @(p, _r?) denote the maximum profit of firm 1 at the prices (p, 6). Then the budget set of trader i at these prices is

B’(p, b)={(x, x*, x,+& E SZZm+l: p*x+_B*x*+x,+l

QN.z’+a~+~ + c qf If’@, h)}. I=Iln + s\I.

Appendix D

Suppose “pj< “~j at a N.E. of “r. It follows immediately that q$=O and 6: = 0 for all traders i. But

and

which is a contradiction.

References

Arrow, K.J. and F.H. Hahn, 1971, General competitive analysis, ch. 6 (Holden-Day, San Francisco, CA) 151-165.

Aumann, R.J., 1964, Markets with a continuum of traders, Econometrica 32,39-50. Debreu, G., 1952, A social equilibrium existence theorem, Proc. Natl. Acad. Sci., U.S.A., 38,

886-893. Debreu, G., 1954, Theory of value (Wiley, New York). Debreu, G. and H. Scarf, 1963, Limit theorem on the core of an economy, International

Economic Review 4,235-246. Dubey, P., 1975, Some results on values of finite and intinite games, Technical Report (Cornell

University, Ithaca, NY). Dubey, P. and M. Shubik, 1975, A theory of money and financial institutions, Part 24: Trade

and prices in a closed economy with exogenous uncertainty, different levels of information, money and no futures markets, Econometrica 45, no. 7.

Dubey, P. and M. Shubik, 1976a, The noncooperative equilibria of a closed trading economy with market supply and bidding strategies, CFDP422, forthcoming in Journal of Economic Theory.

Dubey, P. and M. Shubik, 1976b, A closed economic system with production and exchange modelled as a game of strategy, CFDP 429.

P. Dubey and M. Shubik, A cIosed economic system 287

Fitzroy, F., 1974, Monopolistic equilibrium, non-convexity and inverse demand, Journal of Economic Theory 7,1-16.

Gabszewicz, J.J. and J.-Ph. Vial, 1972, Oligopoly ‘a la Cournot’ in general equilibrium analysis, Journal of EconomicTheory 4,381-400.

Hildenbrand, W., 1974, Core and equilibrium of a large economy (Princeton University Press, Princeton, NY).

Laffont, J.-J. and G. Laroque, forthcoming, Existence d’un equilibre general de concurrence imparfaite: Une introduction, forthcoming in Econometrica.

Marschak, T. and R. Selten, 1974, General equilibrium with price-setting firms, ch. 2 (Springer- Verlag, New York) 12-75.

Negishi, T., 1961, Monopolistic competition and general equilibrium, Review of Economic Studies 28,196-201.

Roberts, J. and H. Sonnenschein, 1977, On the foundations of the theory of monopolistic competition, Econometrica 45,101-114.

Shapley, L.S. and M. Shubik, 1969, On market games, Journal of Economic Theory 1, no. 1, 9-25.

Shapley, L.S. and M. Shubik, 1976, Traders using one commodity as a means of payment (Rand Corporation).

Shubik, M., 1959, Edgeworth market games, in: A.W. Tucker and R.D. Lute, eds., Contribu- tions to the theory of games, IV (Princeton University Press, Princeton, NY) 267-278.

Shubik, M., 1976, A theory of money and financial institutions, Part 30, The optimal bank- ruptcy rule in a trading economy using fiat money, CFDP 424.

A closed economic system with production and exchange modelled as a game of strategy

Documents

Transcript of A closed economic system with production and exchange modelled as a game of strategy