On Differentiated and Indivisible Commodities: An Expository Re-framing of Mas-Colell’s 1975 Model

Abstract

With a pure exchange economy and its Walrasian equilibrium formalized as a distribution on the space of consumer characteristics, Mas-Colell (J Math Econ 2:263–296, 1975) showed the existence of equilibrium in a pure exchange economy with differentiated and indivisible commodities. We present a variant of Mas-Colell’s theorem; but more than for its own sake, we use it to expose and illustrate recent techniques due to Keisler-Sun (Adv Math 221:1584–1607, 2009), as developed in Khan-Rath-Yu-Zhang (On the equivalence of large individualized and distributionalized games. Johns Hopkins University, mimeo, 2015), to translate a result on a large distributionalized economy (LDE) to a large individualized economy (LIE), when the former can be represented by a saturated or super-atomless measure space of consumers, as formalized in Keisler-Sun (Adv Math 221:1584–1607, 2009) and Podczeck (J Math Econ 44:836–852, 2008) respectively. This also leads us to identify, hitherto unnoticed, open problems concerning symmetrization of distributionalized equilibria of economies in their distributionalized formulations. In relating our result to the antecedent literature, we bring into salience the notions of (i)“overriding desirability of the indivisible commodity,” as in Hicks (A revision of demand theory. Clarendon Press, Oxford, 1956), Mas-Colell (J Econ Theory 16:443–456, 1977) and Yamazaki (Econometrica 46:541–555, 1978; Econometrica 49:639–654, 1981), and of (ii) “bounded marginal rates of substitution,” as in Jones (J Math Econ 12:119–138, 1983; Econometrica 52:507–530, 1984) and Ostroy-Zame (Econometrica 62:593–633, 1994). Our work also relies heavily on the technical notion of Gelfand integration.

JEL Classification: D51

This research is supported by a Grant-in-Aid for Scientific Research (No. 15K03362, Suzuki) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. This work is part of a joint project on “General Equilibrium Theory with a Saturated Space of Consumers” with Nobusumi Sagara of Hosei University; the authors thank him for all his input in this paper, and express their regret that he did not think it enough to accept co-authorship. A preliminary version was prepared when Khan was visiting the Shanghai University of Finance and Economics (SHUFE), and he thanks Haomiao Yu and Yongchao Zhang for stimulating conversation and encouragement. This final version has benefited substantially from an anonymous referee’s emphasis on the need for replacing the assumption of overriding desirability of “every homogeneous commodity” to a single money-like commodity; and the Editor’s insistence that the authors take a stand on reporting the work as a technical note for a narrow specialized readership or as an expository essay for a general audience.

Appendix

In this section, we collect some mathematical results for the readers convenience.²⁷ The following is easy to see from Diestel and Uhl [16], pp. 53–54.

Fact 1

If $g: A \rightarrow ca(K)$ is weak* measurable and $\boldsymbol{q}g(a)$ is integrable function for all $\boldsymbol{q} \in C(K)$ , then g is Gelfand integrable.

It is easy to prove

Fact 2

Let {g _n } be a sequence of Gelfand integrable functions from A to ca(K) which converges a.e. to g in the weak* topology. Then it follows that $\int _{A}g_{n}(a)d\lambda \rightarrow \int _{A}g(a)d\lambda$ in the weak* topology.

The next fact (Alaoglu’s Theorem, Royden [70, ch. 14]) is well known.

Fact 3

Let K be a compact metric space. The weak* topology of ca(K) is separable and norm bounded subsets of ca(K) are compact and metrizable, in particular, the set of probability measures on K is compact with respect to the weak* topology.

The next fact is also well known (Fatou’s lemma in ℓ dimensions, Hildenbrand [26, p. 69]).

Fact 4

Let $(g_{n})_{n\in \mathbb{N}}$ be a sequence of integrable functions of a measure space $(A,\mathcal{A},\lambda )$ to $\mathbb{R}_{+}^{\ell}$ . Suppose that $\lim _{n}\int _{A}g_{n}(a)d\lambda$ exists. Then there exists an integrable function $g: (A,\mathcal{A},\lambda ) \rightarrow \mathbb{R}_{+}^{\ell}$ such that

• (a) g(a) ∈ Ls(g _n (a)) a.e. in A

• (b) $\int _{A}g(a)d\lambda \leq \lim _{n\rightarrow \infty }\int _{A}g_{n}(a)d\lambda$.

Let (Y, d) be a complete separable metric space (Polish space), $(A,\mathcal{A},\lambda )$ a complete and atomless probability space and g a Borel measurable map from $(A,\mathcal{A},\lambda )$ to ca(Y ). The direct image measure $\lambda \circ g^{-1}$ is denoted by $g_{{\ast}}\lambda$. The operator $_{{\ast}}\lambda$ is a map from the set of all measurable maps of A to Y which is denoted by L ⁰(A, Y ) to $\mathcal{M}(Y )$ where L ⁰(A, Y ) is the set of all (Borel) measurable functions from A to Y with the topology of convergence in measure. Recall that if a sequence of measurable functions converges a.e., it converges in measure. We then have (Keisler and Sun [31, Lemma 2.1])

Fact 5

• (i) The map $_{{\ast}}\lambda: L^{0}(A,Y ) \rightarrow \mathcal{M}(Y )$ is surjective and continuous.

• (ii) Let $(g_{n})_{n\in \mathbb{N}}$ be a sequences in L ⁰ (A,Y ) and $(h_{n})_{n\in \mathbb{N}}$ a sequences in L ⁰ (A,Z), where Y and Z are Polish spaces. Suppose that $\lambda _{{\ast}}g_{n} \rightarrow \eta _{1} \in \mathcal{M}(Y )$ and $\lambda _{{\ast}}h_{n} \rightarrow \eta _{2} \in \mathcal{M}(Z)$ . Then some sub-sequence of $\lambda _{{\ast}}(g_{n},h_{n})$ converges (weakly) to $\theta \in \mathcal{M}(Y \times Z)$ such that $\theta _{Y } =\eta _{1}$ and $\theta _{Z} =\eta _{2}$ , respectively.

The next fact reveals a striking property of the saturated space (Keisler and Sun [31, Definition 2.2]).

Fact 6

Let Z and Y are complete separable metric spaces. Then a finite measure space $(A,\mathcal{A},\lambda )$ is saturated if and only if for every measure $\nu \in \mathcal{M}(Z \times Y )$ and measurable function $\mathcal{E}: A \rightarrow Y$ with $\mathcal{E}_{{\ast}}\lambda =\nu _{Y }$ , there exists a measurable function $\xi: A \rightarrow Z$ which satisfies $(\xi,\mathcal{E})_{{\ast}}\lambda =\nu$.

Let Z be a Hausdorff topological space. We denote the set of all closed subsets of a set Z by $\mathcal{F}(Z)$. The topology τ _c on $\mathcal{F}(Z)$ of closed convergence is a topology which is generated by the base

$$\displaystyle{[C;G_{1}\ldots G_{n}] =\{\, f \in \mathcal{F}(Z)\vert F \cap C =\emptyset,F \cap G_{i}\neq \emptyset,i = 1\ldots n\}}$$

as C ranges over the compact subsets of Z and G _i are arbitrarily finitely many open subsets of Z. It is well known that if Z is locally compact, the space $\mathcal{F}(Z)$ is a compact metric space; see Hildenbrand [26, pp. 15–19] for details. Recall that a set $K \subset Z$ is called $\sigma$-compact if it is a countable union of compact subsets of Z.

Fact 7

If F _n is a sequence of closed subsets of a locally compact and $\sigma$ -compact metric space K such that F _n → F in the topology of closed convergence and μ _n is a sequence of probability measures on K such that μ _n ( f _n ) = 1 for all n and μ _n →μ, then μ( f) = 1.

Proof

The proof is similar to that of Suzuki [75, Fact 5]. □ 

Let {K _n } be an increasing sequence of closed subsets of a compact metric space K converging to K in the topology of closed convergence. If $\boldsymbol{q}_{n}: K_{n} \rightarrow \mathbb{R}$ is continuous, we will write $(K_{n},\boldsymbol{q}_{n}) \rightarrow (K,\boldsymbol{q})$ if $\boldsymbol{q} \in C(K)$ and for every subsequence n _k and $t_{n_{k}} \in K_{n_{k}}$ with $t_{n_{k}} \rightarrow t$, $\boldsymbol{q}(t_{n_{k}}) \rightarrow \boldsymbol{ q}(t)$. We have from [49]

Fact 8

Let $\{\boldsymbol{m}_{n}\}$ be a bounded sequence in ca(K) with $support(\boldsymbol{m}_{n}) \subset K_{n}$ , and $(K_{n},\boldsymbol{q}_{n}) \rightarrow (K,\boldsymbol{q})$ . Then $\boldsymbol{q}_{n}\boldsymbol{m}_{n} \rightarrow \boldsymbol{ q}\boldsymbol{m}$.

Let $(K_{n},\boldsymbol{q}_{n})$ be a sequence as above. We will say that it is equi-continuous if for all ε > 0 there is a δ > 0 such that for all t, s ∈ K _n with d(t, s) ≤ δ, $\vert \boldsymbol{q}_{n}(t) -\boldsymbol{ q}_{n}(s)\vert \leq \ \epsilon$. Mas-Colell [49] proved

Fact 9

Let {K _n } be a sequence of closed sets of a compact metric space K with $K_{n} \subset K_{n+1} \subset \ldots \rightarrow K$ in the topology of closed convergence and $\{\boldsymbol{q}_{n}\}$ a sequence in C(K) with $\|\boldsymbol{q}_{n}\| \leq \hat{ q}$ for all n and for some fixed $\hat{q} > 0$ . If $(K_{n},\boldsymbol{q}_{n})$ is equi-continuous, then there is a subsequence n _k and $\boldsymbol{q} \in C(K)$ with $(K_{n_{k}},\boldsymbol{q}_{n_{k}}) \rightarrow (K,\boldsymbol{q})$.