One‐seller assignment markets with multi‐unit demands: Core and competitive equilibrium

• Published date: 01 June 2019
• Author: Francisco Robles,Marina Núñez
• Date: 01 June 2019
• DOI: http://doi.org/10.1111/ijet.12166

doi: 10.1111/ijet.12166

One-seller assignment markets with multi-unit demands:

Core and competitive equilibrium

Francisco Robles∗and Marina N´

u˜

nez

We consider an assignment market with one seller who owns several indivisible heterogeneous

goods and many buyers each willing to buy up to a given capacity.In this market, the core con-

tains the Vickrey payoff vector. However,core allocations may not be supported by competitive

equilibrium prices, even in a finite replication of the market. We first characterize the convex-

ity of the associated coalitional game and we show that it is a sufficient condition so that the

buyers-optimal core allocation is competitive.With respect to the seller-optimal core allocation,

we provide a characterization of competitiveness by means of buyers’ valuations. In addition,

we characterize in terms of the valuation matrix the coincidence between the core and the set of

competitive equilibrium payoff vectors.

Key wor ds many-to-many assignment market, core, competitiveequilibrium

JEL classification C71, C78

Accepted 24 November2016

1 Introduction

We study markets with several buyers and only one seller. The seller owns many indivisible and

potentially different objects on sale. Being heterogeneous, the objects are of the same type, for

instance different houses or different tasks. On the other side of the market, each buyer has a non-

negative valuation for each object and a desire to acquire a certain number of objects. This number

is known as the capacity of the buyer. Since we are thinking of objects such as houses, cars or jobs,

it makes sense to assume that a buyer, even if he values all of them positively, is not interested in

acquiring more units than those that can be of use to him. Weassume buyers have quasi-linear utility

functions and value packages of objects additively up to a given capacity.Side-payments are allowed.

Our aim is to determine the conditions under which all core allocations can be priced by means of

competitive prices.

This market is a particular case of that considered in Jaume et al. (2012) and Mass´

o and Neme

(2014), where there are several sellers, each with a set of heterogeneous objects on sale. It also a

particular case of the package auction of Ausubel and Milgrom (2002), where there is only one seller,

but buyers may not value packages additively. Two-sidedmarkets with one seller were also considered

in Cami˜

na (2006) and Stuart (2007). Two-sidedmarkets with tr ansferable utility are first considered

from the viewpoint of coalitional games in the assignment game (Shapley and Shubik 1972). In this

∗Department of Mathematical Economics, Finance and Actuarial Sciences,University of Barcelona. Email: frobles@ub.edu

The authors acknowledge the support of the research grants ECO2014-52340-P (Ministerio de Econom´

ıayCompeti-

tividad) and 2014SGR40 (Generalitat de Catalunya). The first author also acknowledges the support of the FPU grant

(Ministerio de Educaci´

on).

International Journal of Economic Theory xx (2018) 1–14 © IAET 1

International Journal of Economic Theory

International Journal of Economic Theory 15 (2019) 169–182 © IAET 169