An example of the Peleg and Yaari economy

Abstract

The purpose of this note is to present an example of a Peleg-Yaari [13] exchange economy. The example uses the order properties of the commodity space to concentrate the analysis on the principal ideal A,o generated by the endowment vector co. This will illustrate an idea of Aliprantis, Brown and Burkinshaw [1, p. 142]. They observe that if 8 is an exchange economy with Riesz dual system (E,E') , then the characteristics of ~~ restricted to A,o-define a new exchange economy with Riesz dual system (A,o, A~). Underlying every exchange economy with Riesz dual system (E, E'), there is an exchange economy with the same preferences and endowments and with (A~,, A~) as the commodity-price duality? The example will also illustrate the methods developed by Boyd [8, 9] to study recursive utility functions. The most important point about this example is that it shows how the order structure can be used to get the existence of Pareto optimal allocations and the second welfare theorem in a model that could not generate those results on the larger commodity space of all sequences of real numbers. Beals and Koopmans [6] gave an example of a utility function that is not continuous on the space of all real sequences. Their example is given below as (B1). However, the Beals and Koopmans example is continuous on a subset of the commodity space; this smaller set contains the principal ideal Ao, generated by the endowment vector. The restriction of the model to A,o is also in the spirit of Back's [5] exchange economy model where consumption sets may be strictly contained in the positive cone of the commodity space. My focus is on the second welfare theorem, so I will not assign endowments to consumers.2 The social endowment or a99reoate endowment vector is denoted co.