on equilibrium in hart's securities exchange model

JOURNAL OF ECONOMIC THEORY 41, 392404 (1987)

Notes, Comments, and Letters to the Editor

On Equilibrium in Hart’s Securities Exchange Model

FRANK H. PAGE, JR.*

Deparrmem of Finance, University of Houston, Houston, Te.uas 77004

Received November 2, 1984; revised April 8, 1986

In this paper, we use a no unbounded arbitrage condition to give a very direct proof of the existence of equilibrium in Hart’s unbounded securities exchange model (J. Econ. Theory, 9 (1974). 2933311). We also examine the relationship between the no unbounded arbitrage condition and the sufficiency conditions of Hart, ihid. and Hammond (J, Econ. Theory, 31 (1983), 17G175). We present an example to show that if traders are not sufficiently risk averse, then Hammond’s overlapping expectations condition is not, in general, equivalent to the no unbounded arbitrage condition or Hart’s sufficiency conditions, and therefore, is not sufficient to guarantee the existence of equilibrium. We also present an example to show that it is possible for the no unbounded arbitrage condition to hold without overlapping expectations, and therefore, it is possible for equilibrium to exist without overlapping expectations. Journal CJ/‘ Economic Literafure Classification Numbers: 021, 2 13, 313. 11 ’ 19X7 Academtc Press, Inc.

1. INTRODUCTION

The purpose of this paper is twofold. First, we present a greatly sim- plified existence proof for Hart’s [6] unbounded securities exchange model. Our proof rests upon some elementary facts from the theory of recession directions, and a mild assumption concerning the degree of similarity of trader’s preferences. The preference similarity assumption eliminates the problems introduced by unboundedness by eliminating the possibility that any one trader can find a mutually compatible trading partner (or group of trading partners) with whom to engage in unbounded and preference increasing trades. Thus, the preference similarity assumption implies that traders will be unable to engage in unbounded arbitrage. Besides allowing for a more direct proof, the methods used here allow us to prove existence under a weaker set of assumptions than those used by either Hart [6,

* 1 would like to thank an anonymous referee for helpful comments. and I would especially like to thank Peter Hammond for his many helpful comments.

392 0@22-0531/87 $3.00 Copynghl (<I 1987 by Academic Press, lot. All rights of reproductmn in any form reserved

EQUILIBRIUM IN HART'S MODEL 393

Theorem 3.3, p. 3021 or Hammond [S, Theorem 2, p. 1743. Moreover, the proof is carried out within the context of a securities exchange model more general than Hart’s original model. In particular, as in Hammond [IS], we allow each trader’s feasible set to depend on current securities prices.

The second purpose of this paper is to examine the relationships between the no unbounded arbitrage condition, Hart’s sufficiency conditions [6, p. 302, (a.l), (a.2), and (a.3)], and Hammond’s overlapping expectations condition [IS, p. 1733. We show that under a fat expectations assumptions (see CA-123 below), the no unbounded arbitrage condition is equivalent to Hart’s conditions. Moreover, we show that given fat expectations, if traders are sufficiently risk averse, then the no unbounded arbitrage condition is also equivalent to Hammond’s overlapping expectations condition, In order to underscore the importance of the risk aversion assumption, we give an example to show that if traders are not sufficiently risk averse, then even though the overlapping expectations condition holds, the no unbounded arbitrage condition may not. Thus, if traders are not sulliciently risk averse, Hammond’s overlapping expectations condition is not equivalent to the no unbounded arbitrage condition or Hart’s sufficiency conditions, and therefore, is not sufficient to guarantee the existence of equilibrium. We also give an example to show that the no unbounded arbitrage condition can hold without the fat expectations assumption (Hart’s sufficiency conditions can also hold without the fat expectations assumption). Thus, it is possible for equilibrium to exist without overlapping expectations.

We shall proceed as follows: In Section 2, we present Hart’s model, and introduce our assumptions and notation. In Section 3, we give some preliminary definitions and lemmas needed for the existence proof. In Set- tion 4, we present existence results for bounded and unbounded securities exchange economies. In Section 5, we examine the relationship between the no unbounded arbitrage condition and the sufficiency conditions of Hart and Hammond. Finally, in Section 6, we present two examples.

2. HART'S SECURITIES EXCHANGE MODEL

In Hart’s model there are n traders, indexed j= 1, 2,..., n, each seeking to choose a portfolio, x = (x i ,..., x,), of m securities, indexed i= 1, 2 ,,.., m, so as to maximize expected utility,

subject to the budget constraint (p, x) G (p, w’), and the feasible set constraint Y E Xl{ p), where p -+ X-l(p) c R” is a set-valued function of current relative security prices, p E A. Here Q denotes the nonnegative orthant in

394 FRANK H. PAGE. JR.

R”, ui the jth trader’s utility function, and CLJ(. 1 p) the jth trader’s probability beliefs concerning the securities return vector r = (r, ,..., r,). The probability measure P~(. 1 p), defined on the Bore1 subsets, B(Q), of Sz, is conditioned by the current relative security prices. As usual, d denotes the unit price simplex, and oi denotes the jth trader’s initial endowment portfolio. We shall also use the following notation throughout:

P,(P, xl = 1-x’ E R” I vj(p, x’) > Vj(p, xl>,

Cl pj(J’T -x) = {X’ E R” I vj(P, X’) 2 Vj(P, X)},

~‘(P)={~~R~I(P,x)~(P,~~)),

H’(p) = B’(p) n X’(p).

Also, we shall let

H’(A, p) = H’(p) n E(I) and P,(i, p, x) = Pj(P, x) n E(I),

where E(I)={xER”I--(l+i)~<x}n{x~R”~x<(l+~)~), and [is a positive vector in R” with - [ < oi< [ for j = 1 to n, and i E [0, cc 1. Thus E(i) is a closed box with oj~int E(L) for i E [0, ~YI] and j= 1 to n (int denotes interior). Finally, note that E( co) = R” so that

ff’(G P) = H’(P) and pj(m, P, x)=pj(P3 x).

In our existence proof we shall use Hart’s first four assumptions:

LA-1 1 each trader’s utility function, uj: R + R, is concave,

[A-2] each trader’s utility function is increasing,

[A-3] there is a bounded subset C of 52 such that pj(CI p) = 1 for each p E A and j = 1, 2 ,..., n,

[A-4] for each j = 1, 2,..., n, the function p + p,(. / p) is continuous in the topology of weak convergence of probability measures.

We shall also use the following assumptions:

[A-13] For each trader j, the set-valued mapping, p --+ X’(p), is convex-valued, and lower hemicontinuous with a closed graph such that d~X’(p) for all PEA. Moreover, for each 130 and PEA, Xj(p)n int B’(p) n int E(L) # fa.

[A-14] For each trader j and PEA, if x~Hj(l, p) and (p, x) < (p, o j), then there is a vector y E R” with nonnegative components (i.e., y>O), so that x+y~Hj(;i,p) and (p,x+y)=(p,oj).

[A-15] For each trader j and p E A, int 52 n K,(p) # fzr. Here, Kj(p) denotes the convex cone generated by the support of the probability


measure pj(. ( p), and int Q denotes the interior of the nonnegative orthant in R”.

In Hart’s existence proof [6, Theorem 3.33, nine assumptions are used (labeled [A-l ]-[A-9]). Hammond adds three assumptions to Hart’s list (labeled [A-lOI-[A- 12]), and states a subset of the augmented list in his existence theorem [S, Theorem 21. Here, we are adding three more to the list--hence the lables [A-13]-[A-15].

Assumption [A-13] is a strengthened version of Hart’s assumption [A-5] concerning the closedness and convexity of each trader’s feasible set. Since we are assuming that each feasible set is a function of current relative security prices, [A-13] is needed in order to ensure that each set-valued function is continuous. Assumption [A-14] implies that each feasible set correspondence, p -+ X-l(p), is such that if for a given price vector, p, the trader is off the boundary of his budget constraint, B’(p), then the trader can take a position on the boundary by moving in some nonnegative direction. Thus, p + X’(p) is such that no trader is constrained to be off the boundary of his budget set. Hammond’s assumption [A-12] also implies this, but [A-14] is weaker. Assumption [A-15] implies that for any portfolio x E R’“, and any nonzero vector y E R” with y 3 0, V,(p, .X + y) > V,(p, X) for p E d and j= 1 to n. This follows since given [A-15], ~i({rl(?:r)~O}l~)=l, and pj((rl(y,r)>O]lp)>O for any nonzero ~3 0. The nonsatiation implication of [A-15] represents a weaker version of Hart’s Assumption [A-8]. Also, it should be noted that Hammond’s “fat expectations” assumption implies [A-15] (i.e., Hammond’s assumption [A- 121 that int K,(p) is nonempty for j = 1 to n and p E d, implies [A- 151, see C5, p. 1721).

Let lE(i)= {E(A), H’(;1, .), P,(n, .)lj, j,~ [0, 031, denote a securities exchange economy residing in the set E(1”), populated by n investors with constraint correspondences H’( IV, . ), and preference correspondences P,(1.;;), j=l ton. For 2~~3, E(A), denotes a bounded securities exchange economy, while lE(co) denotes an unbounded economy.

2.1. DEFINITION. An equilibrium for E(i), ;IE [0, co], is an array (.u*, p*) where (x*, p*) = (xl*,..., xn*, p*), with xj* E H’(A, p*) for j= 1 to II, and p* EA. More importantly, (x*. p*) is such that

(1) c,x’*<c,w’,

(2) (p*, xi*) = (p*, 0’) for allj, and

(3) P,(& p*, XI*) n H.‘(& p*) = @ for allj.

2.2. OBSERVATIONS. (1) If (x*, p*) is an equilibrium for the exchange economy lE(A’) with xj* an element of the interior of E(A’) for all j, then

396 FRANK H. PAGE-JR.

(x*, p*) is an equilibrium for any exchange economy [E(A) with I > 2’. In particular, (x*, p*) is an equilibrium for the unbounded securities exchange economy [E( co ) (see Hart’s [6, Proposition 1, p. 3031).

(2) It follows from [A-l], [A-2], [A-14], and [A-15] that if (x*, p*) is an equilibrium for lE(A), ,J 3 0, then p* has strictly positive components. Thus since(p*, -xi*) = (p*, wj) for all j, p* > 0 implies that c, x’* = xi co’.

3. Two LEMMAS

In our equilibrium proof we shall need two lemmas. To begin, we need the definition of a recession direction of a convex set.

3.1. DEFINITION. Let S be any convex subset of R”. We say that y E R”’ is a recession direction of S if x + ry E S for real, positive t and x E S. The set of all recession directions of S is called the recession cone of S and is denoted by 0 + S.

It should be noted that if S is a closed convex set, then O+S is a closed convex cone containing the origin, and O+ cl S = 0 + ri S, where cl denotes the closure and ri relative interior (see Rockafellar [S, Sect. 8, pp. 6&71 I). Now we have the following lemma.

3.2. LEMMA. Let r he a set-valued mapping defined on D c Rk with non- empt-bl, convex values in R”. Suppose r is lower hemicontinuous (lhc) with a closed graph. Let (p,, x,), he any sequence in D x R” with p, -+ p* E D, Il.x,J -+ co, and x, E T(p,). Further, let (t,), be any sequence of positive numbers with t,, -+ 0. Then an-v cluster point qf the sequence (tnx,)n is an element sf OfUP*).

ProoJ Without loss of generality, suppose t,,x, + y*. Let .Y* be any element of T(p*) and t any positive number. Since r is lhc, there is a sequence (XT),, with x,* +x* and x,* E T(p,). Since r is convex valued, (1 - tt,) +xf + tt,x, E T(p,) for all n sufficiently large. Moreover, (1 - tt,) x,* + tt,,x,, -+ X* + tv*. Since r has a closed graph, x* + ty* E T(p*). Q.E.D.

The proof of the second lemma follows immediately from elementary facts on combinations of sequences (see Bartle [2, p. 100-J ).

3.3. LEMMA. Let (x,), and (y,), be any two sequences of vectors in R” such that llxnll + Ily,/j -+ ~JJ, and x, t- yH < z E R” for all n. Let x# be any cluster point of the sequence {x,/( lixnll + II y,II)},, and y# any cZuster point


qf the sequence ((Y,/ll-%ll + II?‘A ))n. Then I# + y# ~0, and IIx#I/ + lly” // = 1. Moreover, if x,, + y,, = .z E R’” for all n, then x# + y# = 0, and /Ix#Il + ll.v#II = 1.

4. EXISTENCE RESULTS

4.1. OBSERVATIONS. We begin with the following observations:

( 1) Given [A-l]-[A-4] and [A-l 51, each preference correspondence, P,(,I, ., ), exhibits the following properties:

(a) For 1~ [0, co]. and for each (p, x)E d x int E(A), P,(& ., .) has nonempty, convex values in E(j”), and x E cl P,(I, p, x)\ P,(L P, xl.

(b) For E.E [0, co], the graph of I’,(& ., .) is open in [A x E(i)] x E(I).

(c) For 1. E [0, co], the set-valued mapping p + cl P,(& p, wj) is continuous.

(2) Given [A-l 33 and I E [0, co), each constraint correspondence, H/(2, . ), exhibits the following properties:

(a ) For each p E A, Hj(ji, ) has nonempty, convex, compact values in E(I).

(b) H’(%, .) has a closed graph in [Id x E(A)] x E(A).

(c) H’(n, . ) is lower hemicontinuous (see Hildenbrand [7, p. 351).

(3) Given [A-l], [A-2], [A-14], and [A-15], it follows that if x’E I?‘(%‘, p’), and I’,(,%‘, p’, s’) n H’(J’, p’) = a, I”‘>, 0, then (p’, x’) = (p’. oi). Also, since O-/E Xi(p) for all PEA, it follows that X’EC~ P,(A, p’, oi) for any 2 3 A’.

The following proposition establishes the existence of equilibria for the bounded securities exchange economies.

4.2. PROPOSITION. Suppose [A-l]-[A-4], and [A-13]-[A-15] hold. Then E(d) has an equilibrium for each ;1 E [IO, co ).

Proof: Given [A-13], and Observation4.1(2), it follows that the constraint correspondences, p + Hj(,I, p), are continuous (see Hildenbrand [7, Lemma 1, p. 331). Thus the correspondences, p -+ Hj(,I, p), are continuous with nonempty, convex values in E(i). As observed in 4.1(l) above, it follows from [A-I]-[A-4], and CA-151 that the preference correspondences, (p, x) -+ P,(& p, x), have nonempty, convex values in E(A) with open

398 FRANK H. PAGE, JR.

graphs in [d x E(i)] x E(A). The existence result follows from a theorem due to Shafer and Sonnenschein (see [9, p. 347]), and observation 4.1(3).

Q.E.D.

In order to prove existence for the unbounded case, we need only add the following assumption to the list:

[PI For each PEA, if C y,=O with y,EO+PJa, p, &) then Y,=o.

The recession cone 0 ‘P,( co, p. w’) contains all portfolios preferred to the initial endowment portfolio, oi, given current relative security prices. In [PI, then, we are assuming that trader’s preferences are similar enough to eliminate the possible formation of trading groups or coalitions in which members engage in unbounded and preference increasing trades (i.e., [P] is a no unbounded arbitrage condition). For example, if yj # 0 is a direction of unbounded arbitrage for the jth trader (i.e., y,~O+P,(co, p, wj)), then under [P], - y, cannot be a direction of unbounded arbitrage for any other trader (i.e., - .v, is not contained in O+P,(co, p, &), for k #j, k = 1 to n). Now we have our main existence result.

4.3. PROPOSITION. Suppose [A-l]-[A-4], [A-13]-CA.151, and [P] hold. Then E( cc) has an equilibrium.

Proof Let (x” ,..., .Y”‘, pk)li be a sequence of equilibria corresponding to the bounded economies lE(1+,) with lk + a. Given Observation 2.2 (1) above, it is sufficient to show that there is an integer k such that ,ykie int ,!?(A,) for each j. Assume the contrary. Then for each k there is some trader j with xk’ on the boundary of E(A,). Thus, x, Ilxkil( -+ GO as k + co. For each j, let x#j denote any cluster point of the sequence (x“j/& [Ix~~I~)~. Since xi llxkill -+ cc and c, x4= cj oi, it follows from Lemma 3.3 that c, x #j=O and C- I~x#‘II = 1. By Observation4.1(3), we have xkj~ cl P,( co, pk, oi) for kach j and k. Moreover, the set-valued mapping p + cl Pj( co, pk, 0’) is continuous and hence lhc with a closed graph. By apply- ing Lemma 3.2 to (pk xki) cl Pj(oo, ., oj), and (tk)k, where tk = l~,ll.~“ll, and assuming’ ,.l.Arg. that pk + p# E A, we can conclude that .K#jEo+ cl P,(q p#, wj) for each j. Moreover, since Of cl P,( 03, p #, wj) =O+ Pj( co, p#, 0’) (see Rockafellar [8, Corollary 8.3.1, p. 63]), .~#je O+ P,( cc, p#, oi) for each ,j. But given [P], it follows that x#j= 0 for each j, and this contradicts the fact that c, 11x #jll = 1. Thus, we must conclude that for some integer k, .xki~ int E(%,) for each j. Q.E.D.

Note that Hart’s assumptions 5 through 9 are not needed for the proof above, nor are Hammond’s assumptions 10 through 12. Moreover, by strengthening Hart’s sufficiency conditions (see [4, Theorem 3.3(a.l), (a.2),


(a.3)]) along the lines of our condition [PI, we are able to give a very direct existence proof for a securities exchange model more general than Hart’s original model. In the next section, we shall show that given fat expectations, Hart’s sufficiency conditions and [P] are equivalent. We shall also show that under the fat expectations assumption and some additional assumptions concerning risk aversion, Hammond’s overlapping expectations condition and [P] are equivalent.

5. EQUIVALENCE RESULTS

It follows from Rockafellar’s Theorem 8.7 (see [8, p. 701) and Hart’s Lemma 1 ([6; also see 11) that yeO+ cl P,(co, p, oi) if and only if

where D(+, y)= (rEf2 (r, y)30}, and I%, y)= {rEQ1 (r, y)<O). and where

si( + ) = lim du,(c)/dc, c++x

s’( - ) = lim du,(c)/dc. “4 I

Thus si( + ) and s-j( - ) denote the asymptotic derivatives of the jth trader’s utility function u,( . ).

Since O+ cl P,( “o, p, 0’) = O’P,( cc, p, 01) (5.1) also describes the elements of O+ Pj( co, p, oj). We can now restate Hart’s conditions as follows:

[Ht] For each p’~d, if c,y,=O with Y~EO+P,(GO, p’,~‘), then p,( (r-1 ( y,, r) = 0) I p) = 1 for j = 1 to n, and p in some neighborhood of p’ (See [6, p. 3021).

The condition, pj({rl(y~,,r)=Ojlp)=l for j=l to it, and p in some neighborhood of p’, can be restated as, y, is orthogonal to K,(p) for j= 1 to n, and p in some neighborhood of p’. This in turn is equivalent to the condition, y, is orthogonal to A,(p) for j= 1 to n, and p in some neighborhood of p’, where ,4,(p) denotes the smallest affine hull containing K,(p) (i.e., the smallest afline hull containing the support of the jth trader’s probability measure, pj(. I p)). Since A,(p) contains the origin in R”, Ai is also the smallest subspace containing K,(p). If each trader’s expectations are fat, that is, if int Ki( p) # 0 for j = 1 to n and p E A, then A j( p) = R” for

400 FRANK H. PAGE.JR.

j= 1 to n and p E A, and yj is orthogonal to A,(p) if and only if yj = 0. Thus, given fat expectations, the condition,

pj((ll(y,,r)=O}lp)=l forj=ltonandpinsomeneighborhoodofp’,

holds if and only if y,j = 0 for j= 1 to n. This brings us to our first equivalence result. The proof of this result follows immediately from the discussion above.

5.1. PROPOSITION. If each trader’s expectations are fat, then [P] and [ H t ] are equivalent.

Let Z(p) = nj int K,(p). Hammond’s overlapping expectations condition can be stated as follows:

CHdl I(P) Z 521 for PEA.

In order to show that [Hd] and [P] are equivalent, we need the following lemma:

5.2. LEMMA. If for j = 1 to n, the following conditions hold.

[A-16] s’(+)=O and s’( - ) > 0; or sq-)= +a, then O+Pj(cx3, p, oj)=ZCi+(p)= {yERmI (y, r) 20 for all rEKj(p)), for j= 1 ton (i.e., the recession cone is equal to the polar cone corresponding to Kj(P)).

Proof: Given [A-16], the result follows directly from the fact that p E O+ cl Pi( UCI, p, wj) if and only if

s~(+)~~(+ ?‘, (ryy)~j(drIp)+si(-)~ OO( -, v)

<r, .v> pjLi(drI P)B@

where D(+, y)= {reQ/ (r, y)bO} and D’(--, Y)= {reQI (r, y)<O). Q.E.D.

Assumption [A-16] is an assumption concerning trader’s risk aversion. In particular, it has been pointed out by Bertsekas ([3, p. 238]), that si* = s’( - )/s’( + ) can be viewed as a measure of risk aversion. In fact, if we assume that uj(. ) is a twice continuously differentiable utility function, with first and second derivatives denoted by u,“(. ) and ~j’(. ), and index of absolute risk aversion denoted by rj(c) = u~(c)/uj(c), it can be shown that

In sj* = lim s rj( c) dc.

a- +cc [-“,a]

More importantly, it should be noted that if [A-16] does not hold, then K:(p) is a proper subset of OfPj(m, p, oi) (i.e., K,+(p) is strictly smaller


than 0 + P,( co, p, w’)). It is this fact that allows us to construct an example in which overlapping expectations holds (i.e., [Hd] holds), but the no unbounded arbitrage condition does not (i.e., [P] does not). Thus even under the fat expectations assumption, if traders are not sufftciently risk averse (i.e., if [A-16] does not hold), then [Ht] and [Hd] are not equivalent, and [Hd] is not sufftcient to guarantee the existence of equilibrium (see Example 1 below). Now we have our final proposition.

5.3. PROPOSITION. If each trader’s expectations are-fat and CA-161 holds,

then [P] and [Hd] are equivalent.

Proof Given the lemma above, under [A-16] we have, O+ Pj( co, p, 0’) = K,+(p). Moreover, for the case O+Pj(~, p, 09) = K,+(p), Hammond has shown that under fat expectations [Ht] and [Hd] are equivalent (see [S, Theorem 1, p. 1731). The result thus follows from Proposition 5.1. Q.E.D.

6. EXAMPLES

EXAMPLE 1. In this example, traders possess fat expectations and the overlapping expectations condition is satisfied. However, [A-16] fails to hold, and, as a result, the no unbounded arbitrage condition also fails to hold. To begin, consider a securities exchange economy in which two traders (denoted by j and j’) trade in two securities (denoted by 1 and 2). Suppose that both traders have identical utility functions with s( -) = 2, and s( + ) = 1 (e.g., a continuous, piecewise linear utility function with a kink at the origin). Suppose that trader j has probability beliefs concerning the securities return vector given by

with probability 0.7,

with probability 0.3,

and trader ,j’ has probability beliefs given by

with probability 0.5, with probability 0.5.

To keep matters simple, we will assume that probability beliefs are independent of current security prices. Thus it is easy to see that expectations are fat and overlapping (see Fig. 6.1).

M2,4,12-13


FIGURE 6.1

Consider the portfolio J’ = ( JJ , , y2) = (- 1.06. 1.05). This portfolio gives a direction of unbounded arbitrage for thejth trader since,

.~;(+)joi+.~~(r.?l),~,(drl~)+.~‘(-)j 0% -..s)

(r, .Y> Pjildrl PI

= (1 )[( - 1.06)( 1.05) + (1.05)( 1.10)](0.7)

+ (2)[( ~ 1.06)( 1.05) + (1.05)( 1.03)](0.3)

= 0.0105 3 0.

Thus (- 1.06, 1.05) ~0 ‘p,( cc, p, w’) for any pod and wi. Observe, however, that ( - 1.06, 1.05) is not an element of K,’ (p). Also, since expectations are fat, it is possible to show that y = ( - 1.06, 1.05) is a direction of strictly increasing utility (i.e., V,(p, x + A>,) is increasing in i for any s E P). But now observe that - v = (I .06, - I .05) is an arbitrage direction for the j’th trader (i.e., - J;O’P,.( W, p, 0”) for any PEA and w”) since,

=(1)[(1.06)(1.05)+(-1.05)(1.01)](0.5)

+(2)[(1.06)( 1.05) + (- 1.05)( 1.07)](0.5)

= 0.0 1575 3 0.

Also, observe that - y = ( 1.06, -- 1.05) is not an element of K,?(p). Again, since expectations are fat, - J’ = ( 1.06, - 1.05) is a direction of increasing utility for the j’th trader. Graphically, the situation looks like Fig. 6.1.

If we assume that X’(p) = Xi’(p) = R” for all p E A, then this particular two securities, two trader exchange economy satisfies assumptions [A-l]-

403

FIC;UKE 6.2

EQUILIBRIUM IN HART'S MODEL

[A-4]. [A-13]-[A-15], and the overlapping expectations condition. However, [P] is not satisfied and, as a result, no matter what vector of relative security prices prevails, unbounded arbitrage is possible. Thus, there is no equilibrium in this unbounded exchange economy.

EXAMPLE 2. Again consider a securities exchange economy in which two traders (,j and ,j’) exchange two securities (1 and 2). Suppose this exchange economy satifies [A-1]-[A-4], [A-15], and [A-16]. Moreover, suppose X’(p) = X”( p) = R"', so that [A-13] and CA-141 are satisfied. Finally, suppose the supports of the trader’s probability beliefs (denoted by S,,(p) and S,(p)) are independent of security prices. and look like Fig. 6.2. In Fig. 6.2, the j’th trader has fat expectations, hence int K,.(p) # 0. However, since the support of thejth trader’s probability beliefs consist of two points in a straight line from the origin, K,(p) is equal to the line from the origin containing S,(p), and hence int K,(p) = 0. Thus, the overlapping expectations condition is not satisfied. Notice, however, that all the nonzero vectors in the J’th trader’s recession cone, O’P,.( co, p, ~9’) = K,?(p), are contained in the interior of the jth trader’s recession cone, 0’ P,(,x, p. oi)= F?(p). Thus [P] is satisfied (i.e., the are no unbounded arbitrage opportumtles), and thus, by Proposition 4.2, this particular two securities. two trader exchange economy has an equilibrium.

REFERENCES

1. K. J. ARROW AND G. DEBREU, Existence of an equilibrium for a competitive economy. E~~mmerr~ca 22 (1954). 265-290.

2. R. G. BARTL~, “The Elements of Real Analysis,” Wiley. New York, 1976. 3. D. P. BERTSEKAS, Necessary and sufficient conditions for existence of an optimal portfolio,

J. Gon. T/wor~ 8 ( 1974), 235-247. 4. J. R. GKE~N. Temporary general equilibrium in a sequential trading model with spot and

future transactions. Ec~~nomrrricc~ 41 (1973), 1103-I 124.


5. P. J. HAMMOND, Overlapping expectations and Hart’s condition for equilibrium in a securities model, J. Econ. Theory 31 (1983), 17G175.

6. 0. D. HAKT, On the existence of equilibrium in a securities model. J. Econ. Theory 9 (1974), 293-311.

7. W. HILDENBRAND. “Core and Equilibria of a large Economy.” Princeton Univ. Press, Princeton, N.J.. 1974.

8. R. T. ROCKAFELLAR, “Convex Analysis,” Princeton Univ. Press, Princeton. N.J.. 1970. 9. W. SHAFER AND H. SONNENSCHEYIN. Equilibrium in abstract economies without ordered

preferences, J. Math. Eron. 2 (1975), 345-348.

on equilibrium in hart's securities exchange model

Documents