Walrasian equilibrium in an exchange economy with indivisibilities

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may AbstractThis paper studies an exchange economy with indivisibilities. Our main goal is to see if a price system can function well in an economy (e.g., an economy with complementary preferences) that does not have a Walrasian equilibrium. We study the price adjustment processes governed by the Euler iterative scheme. We show that in an economy that has a Walrasian equilibrium, our price adjustment processes have a common uniform limit that is unique and converges to a Walrasian equilibrium price vector in finite time. Surprisingly, in an economy that does not have a Walrasian equilibrium, our price adjustment processes also have a common uniform limit that is unique and converges to a market equilibrium price vector in finite time. Moreover, market equilibrium prices coincide with Walrasian equilibrium ones in an economy that has a Walrasian equilibrium. Further, there are no prices other than the Walrasian or market equilibrium ones that have such a property of global stability.

Convergence of Markovian price processes in a financial market transaction model

“…Let D j denote the demand curve for buyer j 2 J and S i the supply curve for seller i 2 I. Then, for a large class of quasilinear economies, we have coD j ðyÞ ¼ Àog j ðyÞ and coS i ðyÞ ¼ of i ðyÞ for all y 2 Y, where coA denotes the closed convex hull of A (see Ma and Nie 2003). For an economy given in P, the following holds 0 2 X i2I coS i ðyÞ À X j2J coD j ðyÞ for all y 2 Y Ã , since 0 2 P m i¼1 of i ðyÞ þ P n j¼1 og j ðyÞ for all y 2 Y Ã .…”

“…R are real-valued (possibly non-differentiable) convex functions and Y is a nonempty convex subset of R d . For a large class of quasilinear economies for selling a single divisible good, f i ðyÞ is the producer's surplus for seller i at price y and g j ðyÞ is the consumer's surplus for buyer j at price y; the gradient 5f i ðyÞ is the quantity supplied for seller i at price y and the gradient À 5 g j ðyÞ is the quantity demanded for buyer j at price y; and a price y is Walrasian if and only if y is a solution to problem P (Ma and Nie 2003). These observations can be extended to an exchange economy for (multiple) indivisible goods in Bikhchandani and Mamer (1997), typified by the well-known job matching model of Kelso and Crawford (1982), when the duality gap is zero.…”

“…With the gross substitutes preferences in Kelso and Crawford (1982), a solution y to problem P is also a Walrasian equilibrium for such an exchange economy with (multiple) indivisible goods. For an exchange economy in Bikhchandani and Mamer (1997), the problem P is exactly the dual problem of a linear programming that finds an efficient allocation; and Bikhchandani and Mamer (1997) show that a Walrasian equilibrium exists if and only if the duality gap is zero (also see Ma and Nie 2003;Ma and Li 2011). We are interested in the convergence of the price processes generated by a double auction in Ma and Li (2011) in an economy that has a dual P specified above.…”