Commodities and prices

We've seen examples of general equilibrium economic systems characterized by N commodities for N = 2 (Robinson Crusoe in Chapter 2; Edgeworth box in Chapter 3), N = 4 (2 × 2 × 2) in Chapter 4, and arbitrary positive integer N in Chapter 5. Chapters 6 through 9 summarized the mathematics suitable for analyzing these economies using RN as the commodity space. To represent a list of quantities of N goods, we'll use a point in RN. The expression x = (x1, x2, x3, …, xN) represents a commodity bundle. That is, x is a shopping list: x1 of good 1, x2 of good 2, and so forth through xN of good N. The coordinates xn (n = 1, 2, …, N) may be either positive or negative (subject to interpretation).

The price system consists of an N-tuple p = (p1, p2, …, pN). Let pn ≥ 0 for all n = 1, …, N. The value of a bundle x ∈ RN at prices p is p · x.

What are these N commodities? That turns out to be rather a deeper question than it appears, so a full discussion will be postponed until Chapter 20.

The formal structure of pure economic theory

The plan for the rest of this book is to develop a formal mathematical model of a market economy. Professor Debreu describes below some of the strengths of this approach.