Advances in Mathematical Economics Volume 20 pp 103-128 | Cite as

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

- 1 Citations
- 398 Downloads

## 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 FranciscozbMATHGoogle 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 32:39–50MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Aumann RJ (1966) Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34:1–17MathSciNetCrossRefzbMATHGoogle 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 144:1300–1319MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Castaneda MA, Marton J (2008) A model of commodity differentiation with indivisibilities and production. Econ Theory 34:85–106MathSciNetCrossRefzbMATHGoogle 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 YorkzbMATHGoogle 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 22:339–362CrossRefzbMATHGoogle Scholar
- 18.Forges F, Heifetz A, Minelli E (2001) Incentive compatible core and competitive equilibria in differential information economies. Econ Theory 18:349–365MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Fradera I (1986) Perfect competition with product differentiation. Int Econ Rev 27:529–538MathSciNetCrossRefzbMATHGoogle 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 46:1–30CrossRefzbMATHGoogle Scholar
- 22.Hart S, Hildenbrand W, Kohlberg E (1974) On equilibrium allocations as distributions on the commodity space. J Math Econ 1:159–166MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Hart S, Kohlberg E (1974) On equally distributed correspondences. J Math Econ 1:167–174MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Hervés-Beloso C, Moreno-García E, Páscoa MR (1999) Manipulation-proof equilibrium in atomless economies with commodity differentiation. Econ Theory 14:545–563MathSciNetCrossRefzbMATHGoogle 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, PrincetonzbMATHGoogle 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 12:119–138CrossRefzbMATHGoogle Scholar
- 30.Jones L (1984) A competitive model of commodity differentiation. Econometrica 52:507–530CrossRefzbMATHGoogle Scholar
- 31.Keisler HJ, Sun YN (2009) Why saturated probability spaces are necessary. Adv Math 221:1584–1607MathSciNetCrossRefzbMATHGoogle Scholar
- 32.Khan MA (1985) On the integration of set-valued mappings in a non-reflexive Banach space. Simon Stevin 59:257–267MathSciNetzbMATHGoogle 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 315:127–146zbMATHGoogle 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 118:502–505MathSciNetCrossRefzbMATHGoogle 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 24:239–248MathSciNetCrossRefzbMATHGoogle 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 24:218–225MathSciNetCrossRefzbMATHGoogle 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 211:113–122MathSciNetCrossRefzbMATHGoogle Scholar
- 48.Martin-da-Rocha VF (2004) Equilibrium in large economies with differentiated commodities and non-ordered preferences. Econ Theory 23:529–552CrossRefzbMATHGoogle Scholar
- 49.Mas-Colell A (1975) A model of equilibrium with differentiated commodities. J Math Econ 2:263–296MathSciNetCrossRefzbMATHGoogle Scholar
- 50.Mas-Colell A (1977) Indivisible commodities and general equilibrium theory. J Econ Theory 16:443–456MathSciNetCrossRefzbMATHGoogle 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 13:201–206MathSciNetCrossRefzbMATHGoogle Scholar
- 53.Mas-Colell A (1984) On the theory of perfect competition. Nancy Schwartz Memorial Lecture. Kellogg School, Northwestern UniversityzbMATHGoogle Scholar
- 54.Mas-Colell A, Vives X (1993) Implementation in economies with a continuum of agents. Rev Econ Stud 60:613–629MathSciNetCrossRefzbMATHGoogle 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, CambridgezbMATHGoogle Scholar
- 57.Milgrom P, Weber R (1985) Distributional strategies for games with incomplete information. Math Oper Res 10:619–632MathSciNetCrossRefzbMATHGoogle Scholar
- 58.Nikaido H (1968) Convex structures and economic theory. Academic, New YorkzbMATHGoogle Scholar
- 59.Ostroy JM, Zame W (1994) Nonatomic economies and the boundaries of perfect competition. Econometrica 62:593–633CrossRefzbMATHGoogle Scholar
- 60.Podczeck K (1992) General equilibrium with differentiated commodities: the linear activity model without joint production. Econ Theory 2:247–263MathSciNetCrossRefzbMATHGoogle 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 44:836–852MathSciNetCrossRefzbMATHGoogle Scholar
- 63.Podczeck K (2010) On existence of rich Fubini extensions. Econ Theory 45:1–22MathSciNetCrossRefzbMATHGoogle 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 52:21–45CrossRefzbMATHGoogle Scholar
- 66.Prescott E, Townsend R (1984) General competitive analysis in an economy with private information. Int Econ Rev 25:1–20MathSciNetCrossRefzbMATHGoogle Scholar
- 67.Radner R, Rosenthal R (1982) Private information and pure-strategy equilibria. Math Oper Res 7:401–409MathSciNetCrossRefzbMATHGoogle 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, LondonzbMATHGoogle 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 7:295–300MathSciNetCrossRefzbMATHGoogle Scholar
- 73.Sun YN, Zhang YC (2009) Individual risk and Lebesgue extension without aggregate uncertainty. J Econ Theory 144:432–443MathSciNetCrossRefzbMATHGoogle Scholar
- 74.Suzuki T (2000) Monopolistically competitive equilibria with differentiated commodities. Econ Theory 16:259–275MathSciNetCrossRefzbMATHGoogle Scholar
- 75.Suzuki T (2013) Core and competitive equilibria for a coalitional exchange economy with infinite time horizon. J Math Econ 49:234–244MathSciNetCrossRefzbMATHGoogle 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 YorkCrossRefzbMATHGoogle Scholar
- 81.Vind K (1964) Edgeworth allocations in an exchange economy with many traders. Int Econ Rev 5:165–177CrossRefzbMATHGoogle Scholar
- 82.Yamazaki A (1978) An equilibrium existence theorem without convexity assumptions. Econometrica 46:541–555MathSciNetCrossRefzbMATHGoogle Scholar
- 83.Yamazaki A (1981) Diversified consumption characteristics and conditionally dispersed endowment distribution: regularizing effect and existence of equilibria. Econometrica 49:639–654MathSciNetCrossRefzbMATHGoogle Scholar