Competitive equilibria with indivisible goods. (English) Zbl 0552.90010
We analyse conditions which could guarantee a competitive equilibrium in an exchange economy where indivisible goods play an essential role. The indivisible goods are such that every individual has to consume exactly one of them, e.g. houses or jobs. It is proved that a competitive equilibrium exists if preferences are continuous and monotonic orderings and if, in addition, there is one divisible good. It is also proved that one cannot in general find an equilibrium if preferences are intransitive or if there are more than one divisible good.
This type of economy has been studied earlier by D. Gale [”The theory of linear economic models” (1960; Zbl 0114.122)] and in a previous paper of the author [Econometrica 51, 939-954 (1983; Zbl 0526.90017)]. The problem in the present paper is the same as that in Gale. The difference is that Gale makes an independence assumption; independently of what an individual holds of the divisible good, he always ranks the indivisible goods in the same order. In the above- mentional paper of the author the model is the same as in the present study while the problem is the existence of fair division of various kinds.
The problem in the present paper has recently also been studied by D. Dale [Int. J. Game Theory 13, 61-64 (1984; Zbl 0531.90011)] and M. Quinzii [ibid. 13, 41-60 (1984; Zbl 0531.90012)]. In Gale a direct proof of existence of equilibrium is given under weak assumptions about preferences; e.g. utility may depend on prices and must not be monotonic in the divisible good. The basic mathematical tool is a lemma of Knaster, Kurtowski and Mazurkiewicz. In Quinzii preferences are monotonic and continuous as in the present study. The core is first proved to be non- empty and then it is proved that the set of core allocations and the set of competitive allocations coincide. The present study takes a third approach. It is first proved that a convexified version of the model possesses a competitive equilibrium. This result follows from the existence of equilibria in an economy without ”free disposability” [see T. C. Bergstrom, J. Math. Econ. 3, 131-134 (1976; Zbl 0348.90029)]. It is then shown that within the set of equilibrium in the convexified economy there is also an equilibrium for the economy with indivisible goods.
This type of economy has been studied earlier by D. Gale [”The theory of linear economic models” (1960; Zbl 0114.122)] and in a previous paper of the author [Econometrica 51, 939-954 (1983; Zbl 0526.90017)]. The problem in the present paper is the same as that in Gale. The difference is that Gale makes an independence assumption; independently of what an individual holds of the divisible good, he always ranks the indivisible goods in the same order. In the above- mentional paper of the author the model is the same as in the present study while the problem is the existence of fair division of various kinds.
The problem in the present paper has recently also been studied by D. Dale [Int. J. Game Theory 13, 61-64 (1984; Zbl 0531.90011)] and M. Quinzii [ibid. 13, 41-60 (1984; Zbl 0531.90012)]. In Gale a direct proof of existence of equilibrium is given under weak assumptions about preferences; e.g. utility may depend on prices and must not be monotonic in the divisible good. The basic mathematical tool is a lemma of Knaster, Kurtowski and Mazurkiewicz. In Quinzii preferences are monotonic and continuous as in the present study. The core is first proved to be non- empty and then it is proved that the set of core allocations and the set of competitive allocations coincide. The present study takes a third approach. It is first proved that a convexified version of the model possesses a competitive equilibrium. This result follows from the existence of equilibria in an economy without ”free disposability” [see T. C. Bergstrom, J. Math. Econ. 3, 131-134 (1976; Zbl 0348.90029)]. It is then shown that within the set of equilibrium in the convexified economy there is also an equilibrium for the economy with indivisible goods.
Keywords:
order conditions for preferences; competitive equilibrium; exchange economy; indivisible goods; existence of equilibrium; free disposability; convexified economyReferences:
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