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The rate of convergence of the core of an economy
Journal of Mathematical
Economics 2 (1975) 1-7. 0 North-Holland
Publishing Company
THE RATE OF CONVERGENCE OF THE CORE OF AN .ECONOMY Gerard DEBREV Universityof California, Berkeley, Calif: 94720, U.S.A. Received June 1974, revised version received November 1974
The core of a finite economy has been shown to converge, as the number of its agents tends to infinity, under conditions of increasing generality in a series of contributions, of which the first, by Edgeworth (1881), studied replicated exchange economies with two commodities and two types of agents, and the latest, by Hildenbrand (1974), considers sequences of finite exchange economies (with a given finite number of commodities) whose distributions on the space of agents’ characteristics converge weakly. However, information on the rate of convergence of the core seems to be contained in only two articles. In Shapley and Shubik (1969, section 5) an example is given of an Edgeworth replicated economy whose core converges like the inverse of the number of agents. Recently, Shapley (1975) provided examples of Edgeworth replicated economies whose cores converge arbitrarily slowly, but concluded with the conjecture that for any fixed concave utility functions only a set of initial allocations of measure zero will yield cores that converge more slowly than the inverse of the number of agents. The theorem stated below for replicated economies with arbitrary numbers of commodities, and of types, asserts that such is indeed the case provided that preference relations are of class C2, and satisfy the conditions listed in the definition of the economy. At the same time, the theorem implies that the set of exceptional allocations is closed as well as of measure zero. In order to state the definition of an economy, we introduce the notation P = Int R$;
s = {P EP I IIPII= 11
where jlpjldenotes the Euclidean norm ofp; and L = Int R, . Definition.
The economy d is a list of m preference preorders & on P, and m
*I wish to thank the Miller Institute of the University of California, Berkeley, for its support of my research, and Shii-Shen Chem and Stephen Smale for the very valuable conversations I had with them.
2
G. Debreu, Rate of convergence of the core
endowment-vectors ej in P such that for every i(= 1, . . ., m), si is monotone, convex, complete, and of class C2; every indifference hypersurface has everywhere a non-zero Gaussian curvature, andits closure is containedin P.l
Given a pair (p, w) of a price-vector p in S, and a wealth w in L, we denote by f;:(p, w) the consumption-vector demanded by the ith consumer. The preceding properties imply that the demand function fi is of class C’ on Sx L [Katzner (1968)]. With an initial allocation e = (ei, . . ., ed in Pm is associated the excess demand function F, defined on S by PHF,@)
= c
I
[J;hp~ei)-eil.
This function obeys Walras’ law, i.e., for everyp in S, one hasp-F,(p) = 0. An equilibrium price-vector is, by definition, a p in S for which F,(p) = 0. Given a vector z in R’, denote by 2 the vector in R Is1 whose coordinates are the first I- 1 coordinates of z. Because of Walras’ law it is equivalent to define an equilibrium price-vector as ap in S for which p,(p) = 0. Definition.
The economy 8 is regular if
0 is a regular value of pe.
This definition, due to E. and H. Dierker (1972), is equivalent to that of Debreu (1970), as a calculation on Jacobian determinants of Balasko (1975) shows [still other definitions of a regular economy were given by Smale (1974) and Balasko (1975)]. Thus, given the m preference preorders of the economy, the set of e for which d is not regular is a subset of Pm of Lebesgue measure zero, and closed relative to P”. Now consider gn,, the n-replica of 8. The following results are taken from Debreu and Scarf (1963). Any allocation in the core of 8” gives identical consumption-vectors to all the agents of the same type. It can therefore be represented by a point x in Pm. In this manner the core of 8” is represented by a subset C, of Pm. Let W be the set of Walras allocations of b, and therefore of 8”. The sets W and C,, are non-empty, and compact. They satisfy the relations W c C II+1 c C,. Moreover, as n tends to infinity, the Hausdorff distance 6(C’, W) of C, and Wtends to zero. In the regular case this last assertion can be strengthened as follows. Theorem
For
a regular
economy
8,
as n tends to injinity,
6(C,,,
W) =
0(1/n). ‘A study of preferencerelations eqjoyingthese propertieswillbe found in Debreu (1972).
G. Debreu, Rate of convergence of the core
3
That is to say, n 6(C,, W) is bounded. In other words, the distance between the core of 8, and its set of Walras allocations converges to zero at least as fast as the inverse of the number of agents. It would be very desirable to extend this result to the general framework of Hildenbrand (1974, chs. 2 and 3). There is little doubt that such an extension is possible. Before proving the theorem we note the following lemma: Lemma
The function Fe is proper.
Proof. We must check [Dieudonne (1972, p. 243)] that if K is a compact subset of R’, then H = F;‘(K) is compact. Let (pq) be a sequence in ZZ.Since S is compact, there is a subsequence (p”) converging top’ in 3. Assume thatp’ $ S. In this case our assumptions on preferences and on initial endowments imply that llF,@q)II tends to + co, a contradiction of FJjq) E K. Thus p” E S, which Q.E.D. implies F,(p’) E K. Thereforep’ E H. Proof of the theorem.
Let
Xf=(zEP\zkiee,
and
zsCej>. i
For every n, and for every allocation x in C,, Xi belongs to Xi. The subset pi of P is compact. Therefore so is u i Xi. Select a compact subset X of P containing U i Xi in its interior. Thus there is an open ball V in R’ with center 0 and radius u > 0 such that for every n, for every i, and for every allocation x in C,,, the open ball xi+ Vis contained in X. Given a point z in P, denote by Ii(z) the indifference hypersurface of the ith type through z, and by gi(z) the unit normal to Ii(Z) at z oriented in the direction of preference. Our assumptions imply strong monotony of preferences. Consequently, for every i, and for every pair (z, z’) of points in P, the angle of gi(z) and g*(z’) is strictly smaller than (n/2). Since this angle depends continuously on (z, z’), its maximum 8 over all i, and over all pairs (z, z’) in Xx Xis also strictly smaller than (n/2). We now investigate curvature properties of the indifference hypersurfaces, the importance of which for this analysis was anticipated by Nishino (1971) and Shapley (1975). For a type i, and a point z in P, consider the Z-l principal curvatures at z of Ii(z). By convexity, all these principal curvatures have the same sign; none of them vanish since the Gaussian curvature is everywhere different from zero. Denote by y&) the largest of their absolute values. The function yi is continuous ; let 7 be its maximum over all i, and over all z in X. Consider a unit vector t tangent to Ii(Z) at z in uiXi, and the curve Z intersection of Z,(z) and of the plane Z7 generated by gi(z) and t. At a point z’ of
4
G. Debreu, Rate of convergence of the core
Z n X, let t’ be a unit vector tangent to Z, and let u be the unit normal to Z contained in ZZand oriented in the direction of preference. Also, let 0 be the angle of u and gi(z’), the unit normal to Ii(z) at z’. Clearly 8 is at most equal to the angle of gi(z) and gi(Z’). Hence 8 5 8. Now by Meusnier’s theorem [Laugwitz (1965, p. 53)], at z’ the curvature of Z equals the normal curvature of Z,(z) determined by t’ divided by cos 8. Moreover, by Euler’s theorem [Laugwitz (1965, p. 55) states the theorem in R3; Eisenhart2 (1926, p. 154) states it in R’], at z’ the normal curvature Of Ii(z) determined by t’ is at most equal to yi(z’). Summing up, the curvature of Z at z’ is at most equal to (ri(z’))/cos 8. Consequently the curvature of Z at z’ is at most equal to j+os 8, which we denote by l/p. Next we study the intersection f of r and of the ball z+ V. Taking t and gi(z) as unit coordinate-vectors in the plane ZZ, we look at P as the graph of a C2 convex function. Since f has everywhere a curvature at most equal to l/p, it is not difficult to show that f does not intersect the open ball in ZI with center z + pgi(z) and radius p. Since this assertion holds for any unit vector t tangent to Z,(z) at z, the set Z,(z) n (z+ V) does not intersect the open ball Bi(Z) in R’ with center z+pg,(z) and radius p. Thus z’ E (z+ V) n B,(z) implies z’>i z for every i,fOreVeryZinUiXi.3 We now examine an allocation x in C,. Let A, be the set of agents of 8,. Denote by a a generic element of A,, and by T(u) its type. Write, also, by an abuse of notation, ,-$,, e,, . . . for ST(o), eT(.), . . . . Consider a non-empty coalition E, writing #E for the number of its elements. Since E does not block x, one has
or
In particular, select an agent Fj of the ith type and let E = A,\2 [a type of coalition considered by Edgeworth (188 l), Hansen (1969), and Nishino (1971)]. Since -x0) = 0, 2 (ea one has 1,-x0) = x,-e,. Thus from (l),
ZA reference I owe to S.S. Chern. 3This assertion is easily seen to imply the assumption of ‘uniform smoothness’ of Nishino (1971, p. 41).
G. Debreu, Rate of convergence of the core
Lettingyi
= Xi- ei, andq, = #A,, one obtains
Since \]yi]l I_ ]\~~zI ej(\, the norm of [l/(q,-l)]y, is bounded by a,, = rom now on, we choose II large enough to satisfy the [l/(~n-l>lII~j ejll. F inequality a, < min {v, 2~). Therefore [l&q,- l)]yi E V. Since x is a Pareto optimum, all the gi(Xi) have a common value p. Denote by B the open ball in Rz with center pp and radius p. As we have shown earlier, for every a E A,, B n V is contained in {z E P 1z >-. a> -x,,. From (2), we conclude that [l/(q,,- l)]yi $ B n V.Therefore [l/(q,- l)]yi # B.4 The maximum ofpez for llz]l 5 a, and z 4 B is easily found to be a$2p. Thus for every i, p.Yi
=G*
Hence, 1 P’Yi <=2p(qn-1)
IIC j ejli2-
However, cy=I p*yI = 0. Consequently, for every i,
In-l IICj ejl12. p’yi >=-2p(qn-1) Summing up,
or for every i, p*ei--p-xi
= 0(1/n).
(3)
As xi belongs to X, and p = gi(X3, the vector p belongs to the set gi(x>, a compact subset of S. Similarly, p*x, and p-e, belong to X*gi(X), a compact subset of L. Thus by using the mean-value theorem of the calculus, we obtain from (3)
?Some of the ideas of this paragraph, especially to Lemma 4, pp. 44-45).
and of the next one are related to Nishiio
(1971,
G. Debreu, Rate of convergence of the core
6
which yields successively:
Hence, F,(P) = 0(1/n).
(4)
Since 0 is a regular value of Ne,,the set r’,-‘(O) is discrete. But Wis compact and non-empty. Therefore p;‘(O) is finite and non-empty. Let pl, . . ., p’, . . ., p” be its elements. For each r, there is in S an open neighborhood U’ ofp’, and there is in R'-la non-empty open ball U with center 0 such that the U’ are pairwise disjoint, and for every r, U’ and U are C’-diffeomorphic under the restriction of pe to U’. Consider now the set S\u,V, which is closed relative to S. Since F, is proper by the lemma, the set Q = F,(S\u,U’) is closed [Dieudonne (1972, p. 243)]. The origin 0 of R’ does not belong to Q. Consequently from (4), for n large enough, F,(p) $ (2; hence p E UT U’. Thus p belongs to some Uk, and from (4), by another application of the mean-value theorem, one obtains p-pk
= 0(1/n).
(5)
This relation and (3) yield P’Xi-pk*ei
= p’(Xi-eJ+(p-pk)*ei
=
0(1/n).
Hence by a last application of the mean-value theorem, for every i,
Thus,
denoting by $ the equilibrium associated with pk, for every i, xi-x;
= 0(1/n).
Q.E.D.
consumption-vector
of the ith type
G. Debreu, Rare of convergence of the core
7
Balasko, Y., 1975, Some results on uniqueness and stability of equilibrium in general equilibrium theory, Journal of Mathematical Economics 2, forthcoming. Debreu, G., 1970, Economies with a finite set of equilibria, Econometrica 38,387-392. Debreu, G., 1972, Smooth preferences, Econometrica 40,603-615. Debreu, G. and H. Scarf, 1963, A limit theorem on the core of an economy, International Economic Review 4,235-246. Dierker, E. and H. Dierker, 1972, The local uniqueness of equilibria, Econometrica 40, 867881. Dieudonne, J., 1972, Treatise on analysis III (Academic Press, New York). Edgeworth, F.Y., 1881, Mathematical psychics (Paul Kegan, London). Eisenhart, L.P., 1926, Riemannian geometry (Princeton University Press, Princeton, N.J.). Hansen, T., 1969, A note on the limit of the core of an exchange economy, International Economic Review 10,479-483. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, N.J.). Katzner, D.W., 1968, A note on the differentiability of consumer demand functions, Econometrica 36,41s18. Laugwitz, D., 1965, Differential and Riemannian geometry (Academic Press, New York). Nishino, H., 1971, On the occurrence and the existence of competitive equilibria, Keio Economic Studies 8,33-67. Shapley, L.S., 1975, An example of a slow-converging core, International Economic Review 16. Shapley, L.S. and M. Shubik, 1969, Pure competition, coalitional power, and fair division, International Economic Review 10,337-362. Smale, S., 1974, Global analysis and economics IIA, Journal of Mathematical Economics 1, 1-14.
Economics 2 (1975) 1-7. 0 North-Holland
Publishing Company
THE RATE OF CONVERGENCE OF THE CORE OF AN .ECONOMY Gerard DEBREV Universityof California, Berkeley, Calif: 94720, U.S.A. Received June 1974, revised version received November 1974
The core of a finite economy has been shown to converge, as the number of its agents tends to infinity, under conditions of increasing generality in a series of contributions, of which the first, by Edgeworth (1881), studied replicated exchange economies with two commodities and two types of agents, and the latest, by Hildenbrand (1974), considers sequences of finite exchange economies (with a given finite number of commodities) whose distributions on the space of agents’ characteristics converge weakly. However, information on the rate of convergence of the core seems to be contained in only two articles. In Shapley and Shubik (1969, section 5) an example is given of an Edgeworth replicated economy whose core converges like the inverse of the number of agents. Recently, Shapley (1975) provided examples of Edgeworth replicated economies whose cores converge arbitrarily slowly, but concluded with the conjecture that for any fixed concave utility functions only a set of initial allocations of measure zero will yield cores that converge more slowly than the inverse of the number of agents. The theorem stated below for replicated economies with arbitrary numbers of commodities, and of types, asserts that such is indeed the case provided that preference relations are of class C2, and satisfy the conditions listed in the definition of the economy. At the same time, the theorem implies that the set of exceptional allocations is closed as well as of measure zero. In order to state the definition of an economy, we introduce the notation P = Int R$;
s = {P EP I IIPII= 11
where jlpjldenotes the Euclidean norm ofp; and L = Int R, . Definition.
The economy d is a list of m preference preorders & on P, and m
*I wish to thank the Miller Institute of the University of California, Berkeley, for its support of my research, and Shii-Shen Chem and Stephen Smale for the very valuable conversations I had with them.
2
G. Debreu, Rate of convergence of the core
endowment-vectors ej in P such that for every i(= 1, . . ., m), si is monotone, convex, complete, and of class C2; every indifference hypersurface has everywhere a non-zero Gaussian curvature, andits closure is containedin P.l
Given a pair (p, w) of a price-vector p in S, and a wealth w in L, we denote by f;:(p, w) the consumption-vector demanded by the ith consumer. The preceding properties imply that the demand function fi is of class C’ on Sx L [Katzner (1968)]. With an initial allocation e = (ei, . . ., ed in Pm is associated the excess demand function F, defined on S by PHF,@)
= c
I
[J;hp~ei)-eil.
This function obeys Walras’ law, i.e., for everyp in S, one hasp-F,(p) = 0. An equilibrium price-vector is, by definition, a p in S for which F,(p) = 0. Given a vector z in R’, denote by 2 the vector in R Is1 whose coordinates are the first I- 1 coordinates of z. Because of Walras’ law it is equivalent to define an equilibrium price-vector as ap in S for which p,(p) = 0. Definition.
The economy 8 is regular if
0 is a regular value of pe.
This definition, due to E. and H. Dierker (1972), is equivalent to that of Debreu (1970), as a calculation on Jacobian determinants of Balasko (1975) shows [still other definitions of a regular economy were given by Smale (1974) and Balasko (1975)]. Thus, given the m preference preorders of the economy, the set of e for which d is not regular is a subset of Pm of Lebesgue measure zero, and closed relative to P”. Now consider gn,, the n-replica of 8. The following results are taken from Debreu and Scarf (1963). Any allocation in the core of 8” gives identical consumption-vectors to all the agents of the same type. It can therefore be represented by a point x in Pm. In this manner the core of 8” is represented by a subset C, of Pm. Let W be the set of Walras allocations of b, and therefore of 8”. The sets W and C,, are non-empty, and compact. They satisfy the relations W c C II+1 c C,. Moreover, as n tends to infinity, the Hausdorff distance 6(C’, W) of C, and Wtends to zero. In the regular case this last assertion can be strengthened as follows. Theorem
For
a regular
economy
8,
as n tends to injinity,
6(C,,,
W) =
0(1/n). ‘A study of preferencerelations eqjoyingthese propertieswillbe found in Debreu (1972).
G. Debreu, Rate of convergence of the core
3
That is to say, n 6(C,, W) is bounded. In other words, the distance between the core of 8, and its set of Walras allocations converges to zero at least as fast as the inverse of the number of agents. It would be very desirable to extend this result to the general framework of Hildenbrand (1974, chs. 2 and 3). There is little doubt that such an extension is possible. Before proving the theorem we note the following lemma: Lemma
The function Fe is proper.
Proof. We must check [Dieudonne (1972, p. 243)] that if K is a compact subset of R’, then H = F;‘(K) is compact. Let (pq) be a sequence in ZZ.Since S is compact, there is a subsequence (p”) converging top’ in 3. Assume thatp’ $ S. In this case our assumptions on preferences and on initial endowments imply that llF,@q)II tends to + co, a contradiction of FJjq) E K. Thus p” E S, which Q.E.D. implies F,(p’) E K. Thereforep’ E H. Proof of the theorem.
Let
Xf=(zEP\zkiee,
and
zsCej>. i
For every n, and for every allocation x in C,, Xi belongs to Xi. The subset pi of P is compact. Therefore so is u i Xi. Select a compact subset X of P containing U i Xi in its interior. Thus there is an open ball V in R’ with center 0 and radius u > 0 such that for every n, for every i, and for every allocation x in C,,, the open ball xi+ Vis contained in X. Given a point z in P, denote by Ii(z) the indifference hypersurface of the ith type through z, and by gi(z) the unit normal to Ii(Z) at z oriented in the direction of preference. Our assumptions imply strong monotony of preferences. Consequently, for every i, and for every pair (z, z’) of points in P, the angle of gi(z) and g*(z’) is strictly smaller than (n/2). Since this angle depends continuously on (z, z’), its maximum 8 over all i, and over all pairs (z, z’) in Xx Xis also strictly smaller than (n/2). We now investigate curvature properties of the indifference hypersurfaces, the importance of which for this analysis was anticipated by Nishino (1971) and Shapley (1975). For a type i, and a point z in P, consider the Z-l principal curvatures at z of Ii(z). By convexity, all these principal curvatures have the same sign; none of them vanish since the Gaussian curvature is everywhere different from zero. Denote by y&) the largest of their absolute values. The function yi is continuous ; let 7 be its maximum over all i, and over all z in X. Consider a unit vector t tangent to Ii(Z) at z in uiXi, and the curve Z intersection of Z,(z) and of the plane Z7 generated by gi(z) and t. At a point z’ of
4
G. Debreu, Rate of convergence of the core
Z n X, let t’ be a unit vector tangent to Z, and let u be the unit normal to Z contained in ZZand oriented in the direction of preference. Also, let 0 be the angle of u and gi(z’), the unit normal to Ii(z) at z’. Clearly 8 is at most equal to the angle of gi(z) and gi(Z’). Hence 8 5 8. Now by Meusnier’s theorem [Laugwitz (1965, p. 53)], at z’ the curvature of Z equals the normal curvature of Z,(z) determined by t’ divided by cos 8. Moreover, by Euler’s theorem [Laugwitz (1965, p. 55) states the theorem in R3; Eisenhart2 (1926, p. 154) states it in R’], at z’ the normal curvature Of Ii(z) determined by t’ is at most equal to yi(z’). Summing up, the curvature of Z at z’ is at most equal to (ri(z’))/cos 8. Consequently the curvature of Z at z’ is at most equal to j+os 8, which we denote by l/p. Next we study the intersection f of r and of the ball z+ V. Taking t and gi(z) as unit coordinate-vectors in the plane ZZ, we look at P as the graph of a C2 convex function. Since f has everywhere a curvature at most equal to l/p, it is not difficult to show that f does not intersect the open ball in ZI with center z + pgi(z) and radius p. Since this assertion holds for any unit vector t tangent to Z,(z) at z, the set Z,(z) n (z+ V) does not intersect the open ball Bi(Z) in R’ with center z+pg,(z) and radius p. Thus z’ E (z+ V) n B,(z) implies z’>i z for every i,fOreVeryZinUiXi.3 We now examine an allocation x in C,. Let A, be the set of agents of 8,. Denote by a a generic element of A,, and by T(u) its type. Write, also, by an abuse of notation, ,-$,, e,, . . . for ST(o), eT(.), . . . . Consider a non-empty coalition E, writing #E for the number of its elements. Since E does not block x, one has
or
In particular, select an agent Fj of the ith type and let E = A,\2 [a type of coalition considered by Edgeworth (188 l), Hansen (1969), and Nishino (1971)]. Since -x0) = 0, 2 (ea one has 1,-x0) = x,-e,. Thus from (l),
ZA reference I owe to S.S. Chern. 3This assertion is easily seen to imply the assumption of ‘uniform smoothness’ of Nishino (1971, p. 41).
G. Debreu, Rate of convergence of the core
Lettingyi
= Xi- ei, andq, = #A,, one obtains
Since \]yi]l I_ ]\~~zI ej(\, the norm of [l/(q,-l)]y, is bounded by a,, = rom now on, we choose II large enough to satisfy the [l/(~n-l>lII~j ejll. F inequality a, < min {v, 2~). Therefore [l&q,- l)]yi E V. Since x is a Pareto optimum, all the gi(Xi) have a common value p. Denote by B the open ball in Rz with center pp and radius p. As we have shown earlier, for every a E A,, B n V is contained in {z E P 1z >-. a> -x,,. From (2), we conclude that [l/(q,,- l)]yi $ B n V.Therefore [l/(q,- l)]yi # B.4 The maximum ofpez for llz]l 5 a, and z 4 B is easily found to be a$2p. Thus for every i, p.Yi
=G*
Hence, 1 P’Yi <=2p(qn-1)
IIC j ejli2-
However, cy=I p*yI = 0. Consequently, for every i,
In-l IICj ejl12. p’yi >=-2p(qn-1) Summing up,
or for every i, p*ei--p-xi
= 0(1/n).
(3)
As xi belongs to X, and p = gi(X3, the vector p belongs to the set gi(x>, a compact subset of S. Similarly, p*x, and p-e, belong to X*gi(X), a compact subset of L. Thus by using the mean-value theorem of the calculus, we obtain from (3)
?Some of the ideas of this paragraph, especially to Lemma 4, pp. 44-45).
and of the next one are related to Nishiio
(1971,
G. Debreu, Rate of convergence of the core
6
which yields successively:
Hence, F,(P) = 0(1/n).
(4)
Since 0 is a regular value of Ne,,the set r’,-‘(O) is discrete. But Wis compact and non-empty. Therefore p;‘(O) is finite and non-empty. Let pl, . . ., p’, . . ., p” be its elements. For each r, there is in S an open neighborhood U’ ofp’, and there is in R'-la non-empty open ball U with center 0 such that the U’ are pairwise disjoint, and for every r, U’ and U are C’-diffeomorphic under the restriction of pe to U’. Consider now the set S\u,V, which is closed relative to S. Since F, is proper by the lemma, the set Q = F,(S\u,U’) is closed [Dieudonne (1972, p. 243)]. The origin 0 of R’ does not belong to Q. Consequently from (4), for n large enough, F,(p) $ (2; hence p E UT U’. Thus p belongs to some Uk, and from (4), by another application of the mean-value theorem, one obtains p-pk
= 0(1/n).
(5)
This relation and (3) yield P’Xi-pk*ei
= p’(Xi-eJ+(p-pk)*ei
=
0(1/n).
Hence by a last application of the mean-value theorem, for every i,
Thus,
denoting by $ the equilibrium associated with pk, for every i, xi-x;
= 0(1/n).
Q.E.D.
consumption-vector
of the ith type
G. Debreu, Rare of convergence of the core
7
Balasko, Y., 1975, Some results on uniqueness and stability of equilibrium in general equilibrium theory, Journal of Mathematical Economics 2, forthcoming. Debreu, G., 1970, Economies with a finite set of equilibria, Econometrica 38,387-392. Debreu, G., 1972, Smooth preferences, Econometrica 40,603-615. Debreu, G. and H. Scarf, 1963, A limit theorem on the core of an economy, International Economic Review 4,235-246. Dierker, E. and H. Dierker, 1972, The local uniqueness of equilibria, Econometrica 40, 867881. Dieudonne, J., 1972, Treatise on analysis III (Academic Press, New York). Edgeworth, F.Y., 1881, Mathematical psychics (Paul Kegan, London). Eisenhart, L.P., 1926, Riemannian geometry (Princeton University Press, Princeton, N.J.). Hansen, T., 1969, A note on the limit of the core of an exchange economy, International Economic Review 10,479-483. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, N.J.). Katzner, D.W., 1968, A note on the differentiability of consumer demand functions, Econometrica 36,41s18. Laugwitz, D., 1965, Differential and Riemannian geometry (Academic Press, New York). Nishino, H., 1971, On the occurrence and the existence of competitive equilibria, Keio Economic Studies 8,33-67. Shapley, L.S., 1975, An example of a slow-converging core, International Economic Review 16. Shapley, L.S. and M. Shubik, 1969, Pure competition, coalitional power, and fair division, International Economic Review 10,337-362. Smale, S., 1974, Global analysis and economics IIA, Journal of Mathematical Economics 1, 1-14.