Mathematical Modeling in the Social and Life Sciences

CHAPTER 9

Equilibrium in an Exchange Economy

The basis of political economy is non-interference. The only safe rule is found in the self-adjusting meter of demand and supply. Do not legislate. Meddle, and you snap the sinews with your sumptuary laws.

—Ralph Waldo Emerson

I. Introduction

We continue our exploration of axiomatic models in this chapter by investigating a model of an exchange economy in which consumers trade goods and services with each other. Each consumer enters the marketplace with an original endowment of commodities and personal desires for more or less of the available items that may be traded.

The principal questions we ask are

• Can we satisfy each consumer's demand with the available supplies?
• What assumptions do we need to make about consumers, goods, and services to guarantee there is some mechanism for matching supply and demand?
• Are simple natural assumptions about consumers inconsistent with each other so that no such mechanism is possible?

We begin with the simplest interesting case: an economy with two consumers and two commodities. We then turn to a more realistic situation with a large but unspecified number of consumers and commodities. We make some reasonable assumptions about the consumers and then investigate whether these assumptions are consistent and whether they necessarily lead to a set of prices for the commodities under which the available supply is adequate for the total demand of each good or service.

II. A Two-Person Economy with Two ...