Advances in Mathematical Economics Volume 20 pp 103-128

# 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.

### Keywords

Exchange economy Differentiated commodities Indivisible commodities Large distributionalized economy Large individualized economy Cardinality of the set of consumers### References

- 1.Arrow KJ (2005) Personal reflections on applied general equilibrium models. In: Kehoe T, Srinivasan TN, Whalley J (eds) Frontiers in applied general equilibrium modeling. Cambridge University Press, CambridgeGoogle Scholar
- 2.Arrow KJ, Hahn FH (1971) General competitive analysis. Holden-Day, San FranciscoMATHGoogle Scholar
- 3.Anderson RM (1991) Non-standard analysis with applications to economics. In: Hildenbrand W, Sonnenschein H (eds) Handbook of mathematical economics, vol 4. North-Holland, New York, pp 2145–2208Google Scholar
- 4.Anderson RM (1992) The core in perfectly competitive economies. In: Aumann RJ, Hart S (eds) Handbook of game theory, vol 1. North-Holland, New York, pp 413–457Google Scholar
- 5.Aumann RJ (1964) Markets with a continuum of traders. Econometrica 32:39–50MathSciNetCrossRefMATHGoogle Scholar
- 6.Aumann RJ (1966) Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34:1–17MathSciNetCrossRefMATHGoogle Scholar
- 7.Becker GS (1965) A theory of the allocation of time. Econ J 75:493–517CrossRefGoogle Scholar
- 8.Becker GS (1981) A treatise on the family, Enlarged edn. Harvard University Press, CambridgeGoogle Scholar
- 9.Bewley TF (1972) Existence of equilibria in economies with infinitely many commodities. J Econ Theory 4:514–540MathSciNetCrossRefGoogle Scholar
- 10.Bewley TF (1991) A very weak theorem on the existence of equilibria in atomless economies with infinitely many commodities. In: Khan MA, Yannelis N (eds) Equilibrium theory in infinite dimensional spaces. Springer, Berlin/New York, pp 74–101CrossRefGoogle Scholar
- 11.Brown DJ, Robinson A (1972) A limit theorem on the cores of large standard exchange economies. Proc Nat Acad Sci USA 69:1258–1260; A Correction in 69:3068Google Scholar
- 12.Carmona G, Podczeck K (2009) On the existence of pure-strategy equilibria in large games. J Econ Theory 144:1300–1319MathSciNetCrossRefMATHGoogle Scholar
- 13.Castaneda MA, Marton J (2008) A model of commodity differentiation with indivisibilities and production. Econ Theory 34:85–106MathSciNetCrossRefMATHGoogle Scholar
- 14.Chamberlin EH (1956) The theory of monopolistic competition, 8th edn. Harvard University Press, CambridgeGoogle Scholar
- 15.Debreu, G. (1959) Theory of value. Wiley, New YorkMATHGoogle Scholar
- 16.Diestel J, Uhl JJ Jr (1977) Vector measures. Mathematical surveys and monographs, vol 15. American Mathematical Society, ProvidenceGoogle Scholar
- 17.Dubey P, Mas-Colell A, Shubik M (1980) Efficiency properties of strategic market games: an axiomatic approach. J Econ Theory 22:339–362CrossRefMATHGoogle Scholar
- 18.Forges F, Heifetz A, Minelli E (2001) Incentive compatible core and competitive equilibria in differential information economies. Econ Theory 18:349–365MathSciNetCrossRefMATHGoogle Scholar
- 19.Fradera I (1986) Perfect competition with product differentiation. Int Econ Rev 27:529–538MathSciNetCrossRefMATHGoogle Scholar
- 20.Gorman WM (1956) The demand for related goods: a possible procedure for analysing quality differentials in the egg market. Journal Paper No. 2319, Iowa Agricultural Experiment Station, Nov 1956; Reprinted in Rev Econ Stud 47:843–856 (1980)Google Scholar
- 21.Hart OD (1979) Monopolistic competition in a large economy with differentiated commodities. Rev Econ Stud 46:1–30CrossRefMATHGoogle Scholar
- 22.Hart S, Hildenbrand W, Kohlberg E (1974) On equilibrium allocations as distributions on the commodity space. J Math Econ 1:159–166MathSciNetCrossRefMATHGoogle Scholar
- 23.Hart S, Kohlberg E (1974) On equally distributed correspondences. J Math Econ 1:167–174MathSciNetCrossRefMATHGoogle Scholar
- 24.Hervés-Beloso C, Moreno-García E, Páscoa MR (1999) Manipulation-proof equilibrium in atomless economies with commodity differentiation. Econ Theory 14:545–563MathSciNetCrossRefMATHGoogle Scholar
- 25.Hicks JR (1956) A revision of demand theory. Clarendon Press, OxfordGoogle Scholar
- 26.Hildenbrand W (1974) Core and equilibria of a large economy. Princeton University Press, PrincetonMATHGoogle Scholar
- 27.Hotelling H (1929) Stability in competition. Econ J 39:41–57CrossRefGoogle Scholar
- 28.Houthakker H (1952) Compensated changes in quantities and qualities consumed. Rev Econ Stud 19:155–164CrossRefGoogle Scholar
- 29.Jones L (1983) Existence of equilibria with infinitely many consumers and infinitely many commodities. J Math Econ 12:119–138CrossRefMATHGoogle Scholar
- 30.Jones L (1984) A competitive model of commodity differentiation. Econometrica 52:507–530CrossRefMATHGoogle Scholar
- 31.Keisler HJ, Sun YN (2009) Why saturated probability spaces are necessary. Adv Math 221:1584–1607MathSciNetCrossRefMATHGoogle Scholar
- 32.Khan MA (1985) On the integration of set-valued mappings in a non-reflexive Banach space. Simon Stevin 59:257–267MathSciNetMATHGoogle Scholar
- 33.Khan MA (1989) On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semicontinuous payoffs. Trans Am Math Soc 315:127–146MATHGoogle Scholar
- 34.Khan MA (2008) Perfect competition. In: Durlauf S, Blume L (eds) The new Palgrave. Macmillan, London/New YorkGoogle Scholar
- 35.Khan MA (2012) La Concorrenza Perfetta come Teoria D’ellequilibrio, Chapter 25. In: Bartocci C, Odifreddie P (eds) La Matematica, vol 4. Guilio Einaudi Editore, Roma, pp 875–947Google Scholar
- 36.Khan MA, Rath KP, Sun YN, Yu H (2015) Strategic uncertainty and the ex-post Nash property in large games. Theor Econ 10:103–129MathSciNetCrossRefGoogle Scholar
- 37.Khan MA, Rath KP, Yu H, Zhang Y (2013) Large distributional games with traits. Econ Lett 118:502–505MathSciNetCrossRefMATHGoogle Scholar
- 38.Khan MA, Rath KP, Yu H, Zhang Y (2015) On the equivalence of large individualized and distributionalized games. Johns Hopkins University, mimeoGoogle Scholar
- 39.Khan MA, Sagara N, Suzuki T (2014) On the core and equilibria of a saturated exchange economy with differentiated commodities. The Johns Hopkins University, mimeoGoogle Scholar
- 40.Khan MA, Schlee E (2015) On Lionel McKenzie’s 1957 intrusion into 20
^{th}-century demand theory. Canad J Econ (forthcoming)Google Scholar - 41.Khan MA, Sun YN (1994) On large games with finite actions: a synthetic treatment. Mita J Econ 87:73–84 (in Japanese) (English translation, In: Maruyama T, Takahashi W (eds) Nonlinear and convex analysis in economic theory. Lecture notes in economics and mathematical systems, vol 419. Springer, pp 149–161)Google Scholar
- 42.Khan MA, Sun YN (1995) Extremal structures and symmetric equilibria with countable actions. J Math Econ 24:239–248MathSciNetCrossRefMATHGoogle Scholar
- 43.Khan MA, Sun YN (2002) Non-cooperative games with many players. Handbook of game theory with economic applications, vol 3. Elsevier Science, Amsterdam, pp 1761–1808Google Scholar
- 44.Khan MA, Yamazaki A (1981) On the cores of economies with indivisible commodities and a continuum of traders. J Econ Theory 24:218–225MathSciNetCrossRefMATHGoogle Scholar
- 45.Kolm S (2010) History of public economics: the historical French school. Eur J Hist Econ Thought 17:687–718CrossRefGoogle Scholar
- 46.Lancaster K (1966) A new approach to consumer theory. J Polit Econ 74:132–157CrossRefGoogle Scholar
- 47.Loeb PA (1975) Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans Am Math Soc 211:113–122MathSciNetCrossRefMATHGoogle Scholar
- 48.Martin-da-Rocha VF (2004) Equilibrium in large economies with differentiated commodities and non-ordered preferences. Econ Theory 23:529–552CrossRefMATHGoogle Scholar
- 49.Mas-Colell A (1975) A model of equilibrium with differentiated commodities. J Math Econ 2:263–296MathSciNetCrossRefMATHGoogle Scholar
- 50.Mas-Colell A (1977) Indivisible commodities and general equilibrium theory. J Econ Theory 16:443–456MathSciNetCrossRefMATHGoogle Scholar
- 51.Mas-Colell A (1982) Cournotian foundations of Walrasian equilibrium theory: an exposition of recent theory, Chapter 6. In: Hildenbrand W (ed) Advances in economic theory. Cambridge University Press, CambridgeGoogle Scholar
- 52.Mas-Colell A (1984) On a theorem of Schmeidler. J Math Econ 13:201–206MathSciNetCrossRefMATHGoogle Scholar
- 53.Mas-Colell A (1984) On the theory of perfect competition. Nancy Schwartz Memorial Lecture. Kellogg School, Northwestern UniversityMATHGoogle Scholar
- 54.Mas-Colell A, Vives X (1993) Implementation in economies with a continuum of agents. Rev Econ Stud 60:613–629MathSciNetCrossRefMATHGoogle Scholar
- 55.McKenzie LW (1956–1957) Demand theory without utility index. Rev Econ Stud 24:185–189Google Scholar
- 56.McKenzie LW (2002) Classical general equilibrium theory. MIT, CambridgeMATHGoogle Scholar
- 57.Milgrom P, Weber R (1985) Distributional strategies for games with incomplete information. Math Oper Res 10:619–632MathSciNetCrossRefMATHGoogle Scholar
- 58.Nikaido H (1968) Convex structures and economic theory. Academic, New YorkMATHGoogle Scholar
- 59.Ostroy JM, Zame W (1994) Nonatomic economies and the boundaries of perfect competition. Econometrica 62:593–633CrossRefMATHGoogle Scholar
- 60.Podczeck K (1992) General equilibrium with differentiated commodities: the linear activity model without joint production. Econ Theory 2:247–263MathSciNetCrossRefMATHGoogle Scholar
- 61.Podczeck K (1998) Quasi-equilibrium and equilibrium in a large production economy with differentiated commodities. In: Abramovich Y, Yannelis NC, Avgerinos E (eds) Functional analysis and economic theory. Springer, Berlin/Heidelberg/New YorkGoogle Scholar
- 62.Podczeck K (2008) On the convexity and compactness of the integral of a Banach space valued correspondence. J Math Econ 44:836–852MathSciNetCrossRefMATHGoogle Scholar
- 63.Podczeck K (2010) On existence of rich Fubini extensions. Econ Theory 45:1–22MathSciNetCrossRefMATHGoogle Scholar
- 64.Pollak RA (2012) Allocating time: individuals’ technologies, household technology, perfect substitutes, and specialization. Ann Econ Stat 105/106:75–97CrossRefGoogle Scholar
- 65.Prescott E, Townsend R (1984) Pareto optima and competitive equilibria with adverse selection and moral hazard. Econometrica 52:21–45CrossRefMATHGoogle Scholar
- 66.Prescott E, Townsend R (1984) General competitive analysis in an economy with private information. Int Econ Rev 25:1–20MathSciNetCrossRefMATHGoogle Scholar
- 67.Radner R, Rosenthal R (1982) Private information and pure-strategy equilibria. Math Oper Res 7:401–409MathSciNetCrossRefMATHGoogle Scholar
- 68.Robinson J (1933) The economics of imperfect competition. Macmillan, LondonGoogle Scholar
- 69.Rosen S (1974) Hedonic prices and implicit markets: product differentiation in pure competition. J Polit Econ 82:34–55CrossRefGoogle Scholar
- 70.Royden HW (1988) Real analysis, 3rd edn. Macmillan, LondonMATHGoogle Scholar
- 71.Samuelson P (1969) The monopolistic competition revolution. In: Kuenne R (ed) Monopolistic competition theory: studies in impact. Wiley, New YorkGoogle Scholar
- 72.Schmeidler D (1973) Equilibrium points of non-atomic games. J Stat Phys 7:295–300MathSciNetCrossRefMATHGoogle Scholar
- 73.Sun YN, Zhang YC (2009) Individual risk and Lebesgue extension without aggregate uncertainty. J Econ Theory 144:432–443MathSciNetCrossRefMATHGoogle Scholar
- 74.Suzuki T (2000) Monopolistically competitive equilibria with differentiated commodities. Econ Theory 16:259–275MathSciNetCrossRefMATHGoogle Scholar
- 75.Suzuki T (2013) Core and competitive equilibria for a coalitional exchange economy with infinite time horizon. J Math Econ 49:234–244MathSciNetCrossRefMATHGoogle Scholar
- 76.Suzuki T (2013) Competitive equilibria of a large exchange economy on the commodity space \(\ell^{\infty }\). Adv Math Econ 17:1–19MathSciNetCrossRefGoogle Scholar
- 77.Suzuki T (2014) A coalitional production economy with infinitely many indivisible commodities. Econ Theory Bull (in press). doi:10.1007/s40505-015-0067-7Google Scholar
- 78.Theil H (1952) Qualities, prices and budget enquiries. Rev Econ Stud 19:129–147CrossRefGoogle Scholar
- 79.Varadarajan VS (1965) Measure on topological spaces. Mathematicheskii Sbornik 55:35–100 (in Russian) (English translation, Am Math Soc Trans Ser 2 48:161–228). Reprinted in The selected works of Varadarajan VS (1988) American Mathematical Society, ProvidenceGoogle Scholar
- 80.Vick JW (1994) Homology theory. Springer, Berlin/New YorkCrossRefMATHGoogle Scholar
- 81.Vind K (1964) Edgeworth allocations in an exchange economy with many traders. Int Econ Rev 5:165–177CrossRefMATHGoogle Scholar
- 82.Yamazaki A (1978) An equilibrium existence theorem without convexity assumptions. Econometrica 46:541–555MathSciNetCrossRefMATHGoogle Scholar
- 83.Yamazaki A (1981) Diversified consumption characteristics and conditionally dispersed endowment distribution: regularizing effect and existence of equilibria. Econometrica 49:639–654MathSciNetCrossRefMATHGoogle Scholar