On dual regularity and value convergence theorems

Journal

of Mathematical

ON DUAL

Economics

8 (1981) 37-57.

North-Holland

REGULARITY AND VALUE THEOREMS* Hsueh-Cheng C.O.R.E., Universiti

Received

version

Company

CONVERGENCE

CHENG

Catholique

April 1978, fnal

Publishing

de Louvain, Belgium received

December

1979

We deal with the Value Convergence Problem in the piecewise smooth framework including rates of convergence and the number of (symmetric) value allocations. In this connection, also discuss the concept of dual regularity and give several equivalent ways of formulating regularity conditions.

the we the

1. Introduction The concept of Shapley value has been applied to the market exchange situation [Shapley (1969)] as a possible way of resolving conflict of interests. The equivalence of this solution concept with the competitive equilibrium in large markets has been demonstrated by a number of authors in the smooth framework. Several papers also deal with the non-smooth case. The table in Hart (1977b) gives a clear picture of the major contributions in this subject. One notices a special role played by smoothness assumptions. The work of Hart (1977b) suggests that without smoothness assumptions one can best hope for ‘generic’ equivalence results. The work of Mas-Cole11 (1977) further makes it clear the kind of generic conditions needed: the regularity conditions used by Debreu (1970) for other purposes. Since smoothness assumptions are rather restrictive in economic theory, one would like to see how far the value equivalence results are maintained by relaxing the smoothness assumptions. This work is a contribution in that direction. In this connection, we also develop a dual concept of regularity and several equivalent ways of formulating the regularity condition. More specifically, we shall extend the results of Mas-Cole11 (1977) to the piecewise smooth framework. In the process we are able to remove his restrictive boundary condition designed to make consumption bundles away *This paper is an outgrowth of some of the subjects I studied in my Ph.D. Dissertation. The author wishes to thank Professor Andreu Mas-Cole11 for his inspiration and guidance in all stages of the development of the paper. Thanks are also due to Gerard Debreu, Roy Radner, Leonard Hurwicz, Steve Smale, Richard Day, David Gale, and many others.

38

H.-C. Cheng, Valur conoergmce

theorems

from the boundary of the consumption sets. We also show how the rate of convergence is affected if utility functions are allowed to be unbounded. For simplicity of exposition, we shall fix the number of types of agents while the number of agents goes to infinity and consider only symmetric value allocations. Experiences tell us that the results hold for more general sequences. We also prove a conjecture by Mas-Cole11 that the number of symmetric value allocations is ‘eventually’ equal to the number of competitive allocations when the economy is large enough. The proof is done by strengthening a Co convergence property in Mas-Cole11 (1977) to C’ convergence (see fig. 1 in section 3). The dual concept of regularity is part of the dual model of competitive equilibrium put forward by Mas-Cole11 (1976). It was Negishi (1960) who first noted that the introduction of utility weights can help solve the existence problem of the competitive equilibrium. Mas-Cole11 (1976) pointed out that utility weights also have the same role to play in the finiteness and stability problem of the competitive equilibria studied by Debreu (1970). He also treated utility weights and commodity prices in a dual manner in the equilibrium model and gave them interpretations. In section 2 we shall carry out these ideas of Mas-Cole11 to a greater length and in greater generality (in the piecewise smooth framework) as the scope of the paper permits. Two points are worth mentioning here: First, in our equilibrium equations, utility weights and prices appear in a more symmetric manner and allow us to put them together in one vector and apply a general normalization which can then be specialized to normalization on prices or utility weights alone. This brings out more fully the dual roles of prices and utility weights. Secondly, section 2 can be regarded as an extension of the interior analysis in Smale (1976) to cover the boundary analysis and the presence of kinks. Section 2 ends with several equivalent ways of formulating the regularity condition. Section 3 states the value convergence theorems which are proved in section 4. To help connect the two subjects together, we shall make a conceptual analogy between the processes of determining a value allocation and a competitive allocation. To find a competitive allocation, one assigns a price to each commodity. Given the initial endowments, these prices generate individual demands, hence a total demand for each commodity. If the total demand for each commodity is equal to the total supply, then we have a competitive allocation. To find a value allocation, one assigns a weight to each individual. Given the initial endowments, these weights generate a game (with transferable utility), hence a value for each individual. If the total endowments can be re-allocated so that each individual gets his value, then we have a value allocation. Prices enable us to evaluate bundles of goods; utility weights enable us to evaluate coalitions of players. Equilibrium prices reflect the relative scarcity and desirability of commodities; equilibrium values reflect the relative strength and contributions of individuals.

H.-C. Cheng, Vulue convergence

theorems

39

One final remark before we present the details. Although it is possible to take a more ‘ordinal’ approach, we shall describe the consumers by utility functions and endowments, and fix them throughout the paper. The nature of the results is such that they do not depend on which utility functions you choose for the preferences in the beginning. To be safe, we can adopt the view that utility functions have cardinal significance in our problem, as they do in Mas-Cole11 (1977).

2. Dual regularity The consumption set is Q = {x E R’ :x 2 0). In the present cardinal utility context, a consumer is a pair (u, w) where ~$0 is the initial endowment vector and u :Q+ R the utility function. Given x E Q, let J, be the set of coordinates j such that xj=O. We assume: (*) At each x E Q, there is a neighborhood B, of x in which u is of the form each h is C2 with afi $0, a2f u=min{f,,..., f,>> u(x)=f1(x)=...=fk(x), negative definite, and {Zf2 - afl,. . ., afk - af,} u (ej: je J,} are independent. (When k= 1, the independence condition is understood to be vacuous.) We always let u (0) = 0. The independence condition in (*) is equivalent to a general position condition and is condition on the graphs of f;,. .., fk. It is a symmetric equivalent to, say, (af, - af,, . . ., afk_ 1 - af,} u {ej :j E J,} being independent. It is easy to see that u is strictly concave and monotonic. We shall use con(x) to denote the cone generated by {afl (x), . . ., a_&(x)} u {ej:jEJx}. We call x a regular point for u if {af, (x), . . ., afk(x)J u (ej: j E J,} are independent. The C’ manifold M near x defined by the equations fi (y) =. . . =fk(y), $ =0 for all jEJ, is called the demand manifold at x [so named because if x is the demand at some price-income pair, and the price belongs to the relative interior of con(x), then for all nearby price-income pairs, demands stay in

Ml. Remurk 1. It is obvious that the complement of the set of closed and has Lebesgue measure 0, but more importantly, that the set of price-income pairs (p,o), whose demand x belongs to the relative interior of con(x), is open and its Lebesgue measure 0.

regular points is it can be proved is regular and p complement has

A finite economy consists of a set of consumers {(ui, wi)}2= 1. An attainable allocation (x,, . . ., x,), x1 +. . . +x,=wl +. . . + w,, is Pareto optimal if and only if there exists p&O such that p E coni for all i. We shall restrict ourselves to the case when:

40

H.-C. Cheng, Value convergence theorems

(a) for all i, xi is a regular (b) for all i, p belongs

point;

to the relative

interior

of con,(+);

(c) x2= 1 Txi (Mi) = R’y where Mi is the demand the set of tangent vectors of Mi at xi.

manifold

at xi and TJM,)

is

Notice that the optimal prices associated with an optimal allocation are not necessarily unique, and that an optimal allocation near (x,,. . .,x,) with an associated optimal price near p must be in M, x . . . x M, and satisfy the same conditions. For each i, there exists a unique number /zi >O such that xi maximizes liui(y)-p ‘y on Q. Let 1=(1,, . . .,A,). We shall fix some (no, PO,x:, . . ., x,“) and carry out the analysis near it. Let the unique maximizer of iiui(y) -p . y be denoted by qi(iLi, p). An easy application of the implicit function theorem together with the above conditions (a), (b), (c) shows that: Lemma

1.

=ni($p), is negative D,ni,

The map ni is C’ and yi(Ai, p)~ Mi near ($,p’). Moreover, let xi the derivative with respect to p, D,ni, restricted to T+(Mi)+TJMi) definite,

and therefore

of degree

the

orthogonal

x1= 1 D,ni

complement

is negative

of

definite.

T,JMi)

is the

kernel

of

The map yi is homogeneous

1, hence Dni (& p) (Ai, p) = 0.

Let n be a neighborhood of (A”,po). We are mainly interested in two kinds of normalizations of (2,p)~A:Jl A II= 1, or 1)p II= 1 (11 II is the Euclidean distance). At no additional cost, we shall use r&p)= 1 to denote a general normalization, only requiring that Dr (A, p) (A, p) # 0. The space of parameters of the economy is W={(W~,...,W,,):W~+...+W,=S}, s is a fixed total resource vector. Let X=M, x . . . x M,. Define C={(3b,p,x,w)~A~X +x,=s, p.xi=p’wi, xi=ni(Ai,p), i=l,..., n}. x W:r(;i,p)=l, x1+... Proposition

1.

C is a C’

space of C at (A,p,x,w) x TX.(M,) x R”’ satisfying

manifold is the

Dr(Ap)@,P)=O,

with set

f;

of

dim(C) =dim(W), all

i

x,=0,

p.(xi-wi)+p.(Xi-Wi)=O, (the nth equation

is redundant),

xi-Dri(ni,P)(~i,P)=O,

Proof

When

wi=o,

i=l

i=l

i=l ,..., n, and

i=l , . . ., n.

n= 1, it is quite easy to check.

and

the tangent

(LP,~,~)ER”“xT,~(M~)x...

Hence let n> 1.

H.-C. Cheng, Value convergence

Let

CZER. PER’,

arbitrarily (&;,,X,W):

given.

yi~R,

If we can

i=l,..., solve

41

theorems

n-l, and si~TJMi), the following system

i=l,..., n, be of equations in

Dr(kp)(Xp)=u,

(1)

i xi=/?,

(1’)

i=l

i,

wi=o,

(1”)

i=l

p.(xi-wi)+p.(xi-li+)=yi,

i=l ,...,Iz-1,

(2)

xi -

i = 1,. . ., n,

(3)

Dyi (Ai, p)

(Ii, p) = di,

then a counting of dimensions gives us the proposition. By Lemma 1, there exists p’ such that I;= 1 Dyi(Ai, p)(O, p’) =p - x1= 1 6,. Choose PER, so that Dr(i,p)(t(A,p))=a-Dr(A,p)(O,p’), and let (&p)=(O,p’) + t(A, p). Let Xi = Dqi(&, p)(;Z, p) + hi, i = 1,. . ., n. Then (l), (1’) and (3) are satisfied. We can easily choose Wi, i= 1,. . ., n- 1, so that (2) is satisfied, and finally W, satisfying (1”). Q.E.D. The above proof also shows the following Proposition +w,, tangent

2.

Define

xi=qi(&,p), space

~~=lXi=O,

i=l,..., of 9:

at

resuh

~‘,={(~,~,x)E/IxX:II~II=~, n}. gw is a manifold

x,+...+x,=w,+... of dimension n- 1 and

(A,p,x)

(X&I?)

and Xi=Dqi(Ai,p)(&,p),

is the set i=l,...,

of all

satisfying:

the

I .x=0,

n.

Consider the projection from il xX x W to W and denote its restriction to C by lI:C+IY An economy w is called regular if each competitive equilibrium (p,x) satisfies the above conditions (a), (b), (c) and w is a regular value of 17. We shall see from the following theorem that this concept of regularity is independent of different ways of normalization and is equivalent to three kinds of full rank conditions. If we use the normalization 1I il I ( = 1, and solve the equations in (p, x) for a given 1,:~~ =qi(li, p), i= 1,. . ., n, c;= 1 xi =s, by Lemma 1 the solution (p(A),x(A)) is a C’ function of A. If we use the I I p 1I = 1, and solve the equations normalization in (A, x) for a given p:xi n, the following lemma implies that =qi(ki,p), i=l,..., n, p’xi=p.wi, i=l,..., the solution (i(p),x(p)) is also a C’ function of p: Lemma

2.

If&-Dqi(Ai,p)(&,O)=O

and p.Xi=O

then (&,&)=O.

42

H.-C. Cheng, Value concergence

theorems

Proof

If Ji =O, then xi =0 and nothing is to be done. Suppose Ii # 0, then p . Dyi (& p) (&, 0) = 0 and hence p . Dyi (;ii, p) (Ai, 0) = 0. Since Dyi (Ai, p) (Ai, p) = 0, we have p . Dvi(&, p)(O, p) =O. By Lemma 1, p I T,;(M,). We will show that this is contradictory to the assumption that xi is regular. At xi, let ui = min(f,, . ., f,}, by the definition of Mi and vi, p and &afi,. . ., AicY& have the same orthogonal projection on Txi(Mi). Hence Txi(Mi) is perpendicular to and, by the regularity of xi, we have a af1,...,8fk and (ej:jgJx,}, contradiction on the dimension of Mi. Q.E.D. Proposition

3. The set Q,={(&p,x)~/1xX:IIpl[=l, = 1,. . ., n, pxi =pw,, i = 1,. . ., n} is a mantfold of dimension

space

of

Q,

at

(A,p,x)

is

i=l,...,

=Dni(&p)(Ii,p),

the

set

of

all

n,p.(xi-wi)+p.Zi=O,

xi=ni(&p),

i

l- 1 and the tangent (&p,X) satisfying: p. p=O, Xi i=l,..., n.

Proof.

It is sufficient to show that for any given X~ET,i(Mi), pi E R, i=l,..., n, we can choose (I,;,%) such that p .p=O, Xi-Dni(&p) (&~P)=cc~~ and p.(~~-w~)+p.?~=/$, i=l,...,n. By Lemma 2, we can find (&Xi) such that xi -D~I~(&p)(&,o)=a, and p .Xi=fli. Then let p=O, and the proof is complete. Q.E.D. Theorem

equivalent

(1)

The

1.

An economy w is regular conditions holds:

derivative

of

the

Let

be

11A[l=l

F, :A-(p(I).

the

W)E C, where

normalization.

The

IIpII=

(x1 (A)- w,),. . .,p(L).

1 be

the

+~~= 1 xi(p)-s

has

(~(p),P,X(P),w)~~P’ normalization being Proof:

equations (a) (b)

normalization.

full where used.

The

rank l- 1 the subscript

By Proposition 1, w is regular if and - in (2, p, x, W) E rl;*,_, Wj(A x X x W), Dr(A,p)(Lp)=O, i i=l

x,=0,

at

m,=dim(M,).

(x,(L)1, such that (2, p(2), x(n), w) E C,, where particular normalization being used.

(3) Let

three

map F,:(~,p,x)+(x,-n,(A,,p),...,x,--nn(2,,p), p.(x,-w,)) hasfull rank n+l-l+~~=,mi

Cy=l xi- s, P.(X1-W1),..., each (&p,x) such that (;l,p,x,

(2)

if and only if one of the following

derivative

of

the

map

w,)) has full rank n- 1 at each the subscript i, indicates the

derivative

of

at each p indicates

only

the

map

F,:p

p such that the particular

if the system

of linear

43

Cc)

p’(xi-wi)+p.(xi-wi)=o,

i=l ,...> n 9

(4

Xi - hi

i=l

,...? n >

(e)

wi=o,

i=l

,...,n-1.

(4, P) (42 0) = O,

[one of the equations in (b) or (c) is redundant], has only a trivial solution 0. By counting the numbers of equations and variables, this is in turn equivalent to condition (1). The map (&p,x,w)+(VjIA 11,p/II AlI, x, w) from C to C, is a C’ map and an application of the implicit function theorem shows that the map has a C1 inverse and hence is a diffeomorphism of C and C,. Through this diffeomorphism, the regularity condition in terms of the special normalization 11J.l(= 1 is e q uivalent to the regularity condition in terms of the general normalization r (I,, p) = 1, in particular, the normalization I I p II= 1. Now we use the normalization I I 3,I I = 1, and replace (a) by (a’), I . ;T=O. By Proposition 2 the system of equations (a’)-(e) has only a trivial solution if and only if the map (A,p,x)+(p.(x,-w,),...,p.(x,-w,)) from Y’, to R” has full rank n-l. Since the map A+(i,p(A), x(A)) is C’ and has a C’ inverse (& p,x)-+i, it gives us a diffeomorphism between normalized 2 and Y’,. Through this diffeomorphism, condition (2) is equivalent to the regularity of w. Now we use the normalization I I p 1)= 1, and replace (a) by (a”), p . I-,= 0. By Proposition 3, the system of equations (a”-(e) has only a trivial solution if and only if the map (i,p,x)-+~~= 1 xi--s from Q, to R’ has full rank l- 1. Similarly we have a diffeomorphism p+(,i(p),p, x(p)) between normalized p and Q,“. Through this diffeomorphism, condition (3) is equivalent to the regularity of w. Q.E.D.

3. Value convergence theorems From now on, the paper is a continuation of Mas-Colell’s (1977) work, and I shall refer to the article by ‘M.C.‘. I shall not repeat his definitions and his notations will be retained as far as possible. The reader has only to keep in mind the following: (1) The space of consumer’s characteristics d is finite, LZZ= {(u,, w,)}F= i. Consumers of the same type are assigned the same utility weight, and we restrict ourselves to symmetric value allocations. (2) Assumptions on utility functions are different. We use piecewise smooth utility functions (as stated in section 2) without the boundary condition of MC. Utility functions are not required to be bounded but the following weaker condition is used:

H.-C. Cheng, Value convergence

44

theorems

Marginal utility condition. a,:u(x)-+O as xj+co for j= l,..., 1. Here J,: denotes the left-handed jth partial derivative. The condition expresses the idea that when a commodity is available in large quantities, marginal utility contributions of that commodity are small. It is satisfied if utility functions are bounded. This condition is affected by a monotone transformation of the utility function. We also describe modifications of the results without this condition. (3) The limit economy v is specified by the population weights 8,, . . ., ?I,, 0, >O for all t. These should not be confused with utility weights A,, . . ., A,. A finite economy d:Z-+d is completely specified by its cardinality #(I) and CT=, 8,= 1 is used. A population weights o1 (a), . . ., d,(a). The convention sequence of finite economies &,:I,+& converges to v (written c?~-v) if and only if # (In)-+co and &(&,)+B, for all t. The competitive allocations w(&‘,) and symmetric value allocations V+*(F,) are subsets of RT’. (4) As part of the regularity condition, we assume that for all competitive equilibrium (p, xi,. . ., xT) of the limit economy v, x, is regular and p belongs to the relative interior of con,(x,) for all t. We further assume CT= l TJM,) =R’ (see section 2). These are generic conditions, i.e., they are satisfied by most economies. The genericity is rather intuitively clear but due to the limitation of space we cannot offer a formal proof here. Skeptical readers can take these as assumptions and find some comfort in Remark 1 (which demands less of his faith). He might be converted into a believer if he reads the passages from Smale (1974). Under these assumptions, Theorem 1 in section 2 then tells us several equivalent ways of defining regularity. Due to the population weights &‘s, some obvious modifications in the analysis of section 2 are needed. For example, the total resource equation becomes is again a generic CL 1 0,x, =s. By Sard’s th eorem, the regularity condition condition. We will prove under

the piecewise

smooth

framework

the following

results:

Theorem 2. If v is regular and b, :I, -+sQ, #I, =n, is a sequence of finite economies (with a fixed number of types of agents) converging to v, then d(V*(6,),

%‘-(a,)) =0(1/n+)

and # V*(d,)= Theorem

2’.

Without

rate of convergence

#-Itr(&,) the marginal

for all large n. utility condition,

nf replaced by n f.

the same is true, with the

H.-C. Chug,

Fig. 1.

F,G are Co close but have different

4. Proof of Theorems 4.1. An orientation

Value convergence

theorems

numbers

of solutions

2 and 2’

of the proof

Since the assumptions on the utility functions in Champsaur (1975) are weaker than ours, his results are applicable to our case. However, the reader should be aware that he uses replicated sequences of economies and our sequences are more general. This does not prevent us from quoting his results, because we only consider symmetric value allocations, and his results and proofs can be easily modified by putting population weights B,‘s at appropriate places. This is so mechanical that we feel free to do so without further explanations. In notations, we will follow M.C. as closely as possible. But assumptions in MC. are different from ours, hence many mathematical objects and lemmata have to be redefined and reproved. I will also follow the same notational conventions and omissions in MC. By Theorem 1 of Champsaur (1973, for all r>O, there is an integer E such that if nz fi’and x E V* (8”) then there is a competitive allocation y of the limit economy v with 11x - y 115E. By regularity of v, there are only finitely many competitive allocations. Hence we shall fix a competitive allocation j of v and focus our attention near J?. Let fi be an equilibrium price for j! (unique up to a constant multiple), we can find ;i, > 0 such that Jt maximizes

46

H.-C. Cheng, Valueconcergence theorems

&ut (x,) - ji. x, on Q. From now on we shall normalize j. instead of p. By Theorem 1 of section 2, p and X= (A,, . . ., &-), IIL[I = 1, are unique. It will be sufficient to restrict 1. to a neighborhood B(T) of J, in particular, 1” belongs to a compact set of strictly positive vectors. Given A, Cc& ZEQ, let

V(&A,C,z)=max

1 Aiui(yi):yieQ,

and x (gn, A, C, z) its unique

maximizer.

c yi5z

,

isC

itC

Let

r/;t&n’,, A c,z)’ C &%txi C&n3 4 c,z)), isC,

where C, is the set of type t agents in C. For simplicity of notation, some of the variables &,, i, C, z are omitted if there is no ambiguity from the context. However, when z is omitted, it is understood to be xiECwi. If C is omitted it is understood to be gn. Using Lagrange multiplier, we can find a vector p such that for all in C, Aiui(yi)-p . yi is maximized at xi(gn,A, z, C). By monotonicity of ui, p can be chosen to be $0. Such a vector is called a surgradient of (g,,,,A,C, z). When it is unique, it will be denoted by p(~$, A, C, z). Let o,(A, gn’,) be the Shapley value of the type t players of the game V (gn, A, . ) : C-t V (&n,A, C). The main task of the proof will be to show that in a neighborhood B(X) of 1, for large n, we can define

as C’ maps of A, F(i,&n)-G(A,8n)=O(l/nf) and F(A,&) and G&8”) are C’ close as n-co. That F (A,&“) is a C’ map of A has been shown in section 2. The C1 property of G(A, 8,) will be shown later (Lemma 9). The C1 closeness of F (3_,&) and G&J?,,) insures that # V*(&,,)= # %‘“(8”) for large n. Without it we can have the situation shown in fig. 1. The solutions of F (A, &) =O, G(i, 6,) =0 correspond to competitive allocations and value allocations respectively. By Theorem 1 of section 2, DF(A, c&) has full rank T- 1 at the solutions of F(l,&)=O. Once we have shown F-G=O(l/n7) and DF-DG--+O, G(i,d,) also has full rank T-l at the solutions of G(i,&?,,)=O. By a now standard argument [see the end of Grodal (1975)], F(l.,&‘,,)=O and G(A,&D)=O have the same number of solutions and the solutions are close to each other with the rate l/n?. Since

H.-C. Cheng. Value comxrgence

41

theorems

the allocations are smooth functions of A, and easy application of the mean value theorem completes the proof of the Value Convergence Theorem. From now on, we fix the sequence 8” -+v and whenever some property is asserted of the sequence, it should always be understood that the property holds eventually. We assume I,= { 1,. . ., n} ; 8 will denote a generic term of the sequence. 4.2. Lemmata Lemma 3.

Let x,:1,

-Q

be a sequence

such that

lim inf 11xni )1--t m

n-n,

Proof:

See MC.,

Lemma

4.

surgradient

For

i

Lemma ,fixed

2.

2, C,

V(z)

is continuous

and

concave.

Let

p be a

of (;_, C, z) then V(z’)s

V(z)+p.

Proqf:

See Champsaur

Lemma

5.

There

(z’-z)

(1975)

,for all

Section

z’EQ.

6.

is K >O such that Jbr all i in a compact

set and C cb

we

have ( 1xi@, C) I( SK for all i E C. Proof Let A be the compact set of A’s. If the lemma is false, we have a sequence 1,~ A, C,, and iE C, such that xi(Ln, Cn)-+=. Without loss of generality, let xi (A,, C,)+cc. By Lemma 3, we can find another i’E C, such that +(A,,C,) is bounded. Let H be a compact set containing such a sequence, G a number such that for all i E A, &/A,, 20 for all t, t’. Let 6 = minxsB 8: u,(x) where 8: denotes the right-handed derivative with respect to the first coordinate. By assumption, 8;ui(xi(;ln, C,))-+O. If n is large enough, Z;ui(xi) ~6. Hence iia;ui(xi)
6.

There

set and C ~8, Proof.

w,$O.

is a compact

set H c R’+ + such that for all A in a compact

and for any surgradient

This is an immediate

p of E., C, we have p E H.

consequence

of Lemma

5 and the assumption

48

Lemma

H.-C. Cheng, Value conaergence

7.

theorems

There is 5 > 0 such that for all 8, A, C and i y! C,

Proof: Let p be a surgradient of (A, C). Let z’=xjeC xj(C u {i]) and =cjSc wj, then z’-z=wi-xi(Cu {i}), and V(Cu (i))=V(C,z’) + /I,ui (xi (C u {i))). Hence by Lemma 4, OIV(Cu{i})-V(C)=V(C,z’)-V(C,z)+~iui(xi(Cu(i))) ~p’(Wi-Xi(CU

By Lemmata

{i})).

{i}))+AiUi(Xi(CU

5 and 6, the proof is complete.

Lemma 7’. Without the marginal utility such that for all 8, A, C, and i$ C,

Pro05

Q.E.D. condition,

Let m=#C,K=max{IIw,II:t=l,...,T}.

we have: There

We have

Let H be a number such that H 1 u;(x,) for all II x, I IzK U,(0) = 0, by concavity, U, ((l/m)x,) 1 (k/m)u, (x,). Hence

ui(xi(A,C))smui

-xi(A,C) (:

is t>O

and for all t. Since

5m.H. 1

We get V(&C)=

c &ui(xi(A,C))~m2

.H’

for all A,C.

Q.E.D.

isC

Lemma 8.

There is 5 > 0 such that for all 8, II, C and i $ C, I K(a,ACu

Proof:

ii\i)-~((a,n,C)li(#C)5.

This is an immediate

consequence

Lemma 8’. Without the marginal such that for all &, A, C, and i $ C,

utility

IV,(&,~,CU{~})-I/;(~,~~,C)I~(#C)~~. Proof

The same as in Lemma

7’.

of Lemma condition,

5. we have: There

is t>O

z

H.-C.Cheng,

V’lur

Lemma 9. V(J, C, z) is a C’ function all 1, C, z and t.

Proof: where

convergence theorems

49

of A and a,,V(& C, z)= K(& C, z)/I, for

By definition, m, is the

V (2, C, z) = max {CT= 1 &mtut (x,) :I;= I m,x, 5 z, x, E Q), C. Define A number of type t agents in m,x, 5 z}. Then A is a convex set, and ={(m,u,(~,),...,mTtlT(~=)):~,~e, ~f=l V(1, C, z) as a function of ,I is simply the support function of A. By Nikaido (1968, lemma lS.l(iii), p. 298), V(I) is C’ and c’i,V(i,C,z)=m,u,(x,(/l,C,z)) Q.E.D. = r: (4 C, z)l&. Let s=CT= 1 Btw, and B(S) a neighborhood of S. Let Cc&, m = #C and m, =the number of type t agents in C, define O(C)= (m,/m,. . ., mT/m). In the following, B (8), B(X), B(p) always denote properly chosen neighborhoods. A vector p is a surgradient of (&C, z) if and only if it satisfies 8,(C)q, (ir,p) . . . . + Or(C)qT(&.,p) = z/m. By Lemma 1 and the implicit function theorem, : is unique when (&O(C), z/m) E B(z) x B (8) x B(S), and ~(2, C,z) is a C’ function of i, z and O(C). So are x(;L, C, z) and V@,C,z). And O,V(;I, C, z) =p(I, C, z). .Hence I/ is also a C2 function of z. From Lemma 10 to Lemma 13, we restrict ourselves to B(l), B(g), B(-)s so that these smooth properties are satisfied. They also hold without the marginal utility condition. We might as well assume that if O(C)E B(0) then Citc”J#C~ B(S). It is also understood that #C> some properly chosen large ti. From

Lemma

the smooth

property

we immediately

10.

There exists K > 0 such that

Lemma 11.

There exists K > 0 such that

have:

(,I,O, z’) E B(X) x B(B) x B(s), let ~(2, 0, z’) be implicitly defined and let x,(i,e,z')=~,(Iz,,p(3,,8,~')), Then g(I,e,~')=CT=~e~~*u~(~,(~,e,~')). P(1,C,z)=P(~,e(C),z/#C), x,(n,C,z)=xt(~,e(C),z/#C) and v(n,c,z)=(#C)g(i,,B(C),z/#C). Apply Proof. by

Given

8191(~11P)+...+e,g,(ll,,P)=z’,

50

H.-C.

Taylor’s

theorem

Cheng,

Value convergence

theorems

on g (A, 8, .), we get

V&C,z)--V(A,C)--p&C)

(

1 wi

z-

ieC

)I

=(#c).lg(n,s(o,~)-g(~,e(c),~~) 1

-p(A,fI(C))‘P

z-C #C

(

Wi isC

>I 2

1 g#c)K.p (#W 2

K z-cwi

-/I #C

There are K, E, rii, B(n) such that whenever # C > ti, i q!C, I&(C) and /&(Cu {i})-8,(8”)IZE,for all t, AEB(X), we have

Lemma 12. -6,(8n)IS~,

Proof: (Xi (A,

Let C

u

(*)

Q.E.D.

.

I/

itC

Z=CjeCXj(Cu

{i])).

( v(C

By smoothness,

By Lemma U {i>)-

then

{il),

V(C u {i})= V(C, z)+Aiui

11, we have V(C)-AiUi(Xi(A,

C

U

{i}))-p(C).

we also have

(**)

Combining

(*) and (**), the lemma

is proved.

Q.E.D.

(Wi-Xi(C

U {i}))]

H.-C.

Lemma

Cheng,

Vurlur convergence

51

theorems

13.

Given 6 >O, there exist E, 61 such that for all #C> ti, i$ C, i oj’ type f 10,(C)-H,(&n))S~ and Ie,(Cu{i})-0,(&n)15~(Vt), we have

Proof: By Lemma 9, r/; (& z)/%, = L?i,I/ (A, z), (2, . d,,V(A, z))=~$,~B, V(i., z)=i.,c’,,p(i, z). Similar to the proof of Lemma 11, let g, (4 0, Z’) = b~4

hence

&1:(2, z)=(:=

(x,(a,8,Z’I),

then ~(~,C,z)=m,3.,u,(x,(~~,C,z)) =(,c,,,,,,,(x~(i.e(c~,~))=I#c)g,(l;B.~). and ~z,g,(~,e,~')=l,a,,p(3,,e,z'). Let z=CjtcXj(CV

as

{i)), w=CjEcwj,

II II z-w

40.

~ #C

Since z-w=wi-xi(Cu{i}),

Hence if t#f,

(*)

then

K(A,Cu

as #C+co,

IIz-wl//#C+O

(i})=1/(A,C,z),

we also have

and

II:(~~cu{~})-r:(~,c)-a,a,,p(~.,c)~(W,-X~(Cu{i}))~~O as

#C-+co.

52

H.-C. Cheng, Value conuergence theoreuls

If 11e(c

u

11+O,II e(C)-e(&,)ll+O,

{i})-0(&J

[uniformly

for A in some compact

B(x)],

(**I

1G%,PG

c u (i>,,

-b%*P@,

(2.

h-&(A

4,) .(w,-x,(4

Combining (*), (**), we have similarly proved. Q.E.D.

4.3. Probability

by continuity

we have

hence for t # f,

4,)) p.

the

lemma

for

t#f.

The

case

for

t=f

is

estimates

As in M.C., we need the following result on sampling [Hoeffding (1963, theorem 1 and sect. 6)]:

without

replacement

Let the population consist of n values cr, . . . , c, with ci =0 or l(Vi). denote a random sample without replacement, X,, X,, . . , X, R=(l/m)(X,+...+X,). Then Prob{lX-pII&} I*=(lln)C2=1 ci, and (P) Let

$2e-fmE2.

Proposition 4. Let n(C) be the probability assigned by i to a coalition C not containing i and ge= {C: i$ C, I e,(C)-8,(&,,) I5.s for all t}. There is H >O such that &+u,x(C)~H/ne2, for all 1>~>4/(n-1).

Proof: Let the population be &‘n\{i> with value and 0 assigned to others. Then p=

(0, (4) - lln)l(l - l/n),

1 assigned

or

w-2/(1-l/~)~ depending

on i of type 1 or not. Applying

(P), we have

Prob{lR-8,(~“))I&~~Prob{lg-~lI&E_I-e8,(~”)I} jProb{l~-~IZE/2}j2e-~“’ for all

.5>4/(n-

1).

to type 1 agents

H.-C.Cheng, Value convergence theorems

Let _@I be the set of coalitions

C with i $ C and / B,(C)-0,

for some H’ > 0, for all E> 4/(n - 1 ), e < 1. Repeat the above arguments for g2,. . ., ~8~. If C E %I u . . v ST. Therefore for all

~>4/(n-l),

5. Let x(C), %?, be the same as in Proposition such that for all 1 >e>4/(n1) we have

(#C)n(C)IH/ne4,

cz

CIIE

Proof:

Follow

(8”) 1>E, then

C $U,,

i # C,

then

Q.E.D.

&cl.

Proposition

1

53

4. There

is H>O

(#C)‘r(C)SH/nE6. E

the proof of Proposition

4, we get

For x E (- 1,1) we have the formulae

5 mx”=x/(l

-x)2,

Again, following Q.E.D.

4.4.

f

m2xm= (x(1 +2x))/(l

-x)3.

m=O

m=O

the reasoning

Co and C’ convergence

of Proposition

4, the results

are proved.

properties

The readers are reminded that we focus our attention in some compact neighborhood B(X) of X We have seen in section 2 that F,@,&‘,,) =p(i,~~).(w,-x,(I,~~)), t=l,...,T are C’ maps of I for large n. By Lemma 9. G (i, 8,,,,)= L’,(i, gH,)-J+, (x, (I., &Yfl)),t = 1,. . ., 7: are also C’ maps.

H.-C. Cheng, Value convergence theorems

54

Proposition

6.

max I1F(i,~“)-G(i,~“)II=O((l/n):). ItB(X)

Proof: Take a type f agent i, and let it be fixed. Let V be the set of coalitions not containing i. Let ij(C) = I/ (A, C u {i}) - V(A, C) - Aiui (xi (A, gn)) -p(;l,fTn). (wi-xi(Iz,B,)), we can write

We divide %?into three subsets: %?I={CE%?:

#C&j,

@?,={C~9?:#C>nf

and

Ie,(C)-e,(~“)(~(l/n)iVt},

%Ys=%?\(%Y,UVZ). By Lemma

7, I It/(C) 15 5 for a fixed 5. Hence for C E%?~,

For C E %a, by Proposition

4,

c

n(C)+(C) CEVJ For C E$?~, by Lemma

5;5

~~=~If~n-;

n.n-f

12,

Hence

Proposition

6’.

Without

the marginal

utility condition,

max IIF(i,g,)-G(i,8,)ll=O((l/n)t). AsB(;i)

55

H.-C. Cheng, Value convergence theorems Proo$

Follow the proof of Proposition 6, replace nf by nf in the definition of WI, q2, %TS. Apply Lemma 7’ instead of 7, Proposition 5 instead of Proposition 4. We get

)Cc,u71(C)~(C)(~r.n-S+i;H.n-f+K.n-f. Proposition

7.

max IIDF(~,~“)-DG(I,~“)))-*O

as

Q.E.D.

n--+co.

kB(;i)

Proof Let us calculate For f # f, we have

the derivatives

first. Let i be a fixed type f agent.

a,(G,(i,~“)-Ff(~,&n,)=~ c n(C)(Vf(l,Cu {i})-V,(A,C)) I CEI -a,Q(nY&n,)’

(wi-xi(A~n))

where Ii/(C) = &(A, C u (i> ) -

V,(A,

C)

-

&3,p

(A, ~9~) . (w,-

For t = f, we have

where q(C) = 5(& C u {i}) - &(A, C)- &t+(x,(A, 6Q)

-~,anp(n,~~).(W,-Xf(~,~")). Given

6 ~0, choose E; fi so that Lemma

13 is applicable.

Let

%Yg,={CE%:#CSrii}, ‘Z2={C~9?:#C>fi

and

Ie,(C)-8,(~“)I~;E;V1},

%?s=%?\(%?I u%J. By Lemma

13, l$(C)l~&

ltjS(C)jjS,

hence for CE%?~,

x,(A,

8”)).

56

H.-C. Cheng, Value convergence theorems

For C E %‘,, by Lemma

&

8,

n(C)*(C) 1

For CE%?~, by Proposition

S(,ti.fi/n56

when

n>@fi2/S.

5,

when

n > CHISEL.

Hence

so is

Proposition

7’.

Without

the marginal utility condition,

Proposition

7 holds.

Proof: In the above proof, replace Lemma 8 by Lemma 8’, and apply the second inequality of Proposition 5. The same argument gives us the proof.

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H.-C. Chug,

Value convergence

theorems

51

Mas-Colell, A., 1977, Competitive and value allocations of large exchange economies, Journal of Economic Theory 14, no. 2. Negishi, T., 1960, Welfare economics and existence of an equilibrium for a competitive economy, Metroeconomica 12, 92-97. Nikaido, H., 1968, Convex structure and economic theory (Academic Press, New York). Shapley, L., 1953, A value for N-person games, in: Contributions to the theory of games II (Princeton University Press, Princeton, NJ). Shapley, L., 1964, Values of large games VII: A general exchange economy with money (Rand Corporation, Santa Monica, CA). Shapley, L., 1969, Utility comparison and the theory of games, in: G. Guilband, ed., La decision (Editions du C.N.R.S., Paris). Shapley, L. and M. Shubik, 1969, Pure competition, coalitional power and fair division, International Economic Review 10, 3377362. Smale, S., 1974, Global analysis and economics IV: Finiteness and stability of equilibria with general consumption sets and production, Journal of Mathematical Economics 1, 119-129. Smale, S., 1976, Global analysis and economics VI: Geometric analysis of Pareto optima and price equihbria under classical hypothesis, Journal of Mathematical Economics 3, 1-14.