This paper examines an exchange economy with heterogeneous indivisible objects that can be substitutable or complementary. We show that a competitive equilibrium exists in such economies, provided that all the objects can be partitioned into two groups, and from the viewpoint of each agent, objects in the same group are substitutes and objects across the two groups are complements. This condition generalizes the well-known Kelso-Crawford gross substitutes condition and is called gross substitutes and complements. We also provide practical and typical examples from which substitutes and complements are both jointly observed. Copyright The Econometric Society 2006.

In their 1982 article, Kelso and Crawford proposed a gross substitutes condition for the existence of core (and equilibrium) in a two-sided matching model. Since then, this condition has often been used in the literature on matching models and equilibrium models in the presence of indivisibilities. In this paper we prove that a reservation value (or utility) function satisfies the gross substitutes condition if and only if it is an M♮-concave function defined on the unit-hypercube, which is a discrete concave function recently introduced by Murota and Shioura (1999).

We analyze a decentralized process in a basic labor market where finitely many heterogeneous firms and workers meet directly and randomly in pursuit of higher payoffs over time and agents may behave myopically. We find a general random decentralized market process that almost surely converges in finite time to a competitive equilibrium of the market. A key proposition en route to this result exhibits a finite sequence of successive bilateral trades from an arbitrary initial market state to a stable matching between firms and workers with a scheme of competitive salary offers.

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We formulate and prove several general theorems on existence of fixed (and zero) points for mappings defined on discrete sets in finite-dimensional Euclidean spaces. One of the major conditions imposed on the mappings under consideration is that of local gross direction preservation. In addition, economic applications are discussed. (2000). 47H10, 54H25, 55M20, 90C33, 91B50. Mathematics Subject Classification

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