# What is the definition of: 'commodity space'?

I have seen the concept of commodity space being used multiple times in economics, in particular within microeconomics, but I could not find a general definition of it. Based on the examples that I have seen, I would guess that it is a vector-space of n-dimensions, in which every dimension coincides with one specific commodity. Furthermore, n is equivalent to the cardinality of the set of commodities. Is this understanding correct? Is there even a universal understanding of a commodity space?

## 2 Answers

Yes, a commodity space is the set of all possible commodity bundles. The simplest Micro 101 example is typically the nonnegative quadrant of $$\mathbb{R}^2$$, but in general equilibrium theory it is often assumed to be infinite-dimensional since there are infinitely many commodities (each good is differentiated by time and space, etc.).

• You write "yes" (i.e. it is a vector-space), but then you immidiatly call it a set. I can see both concepts ultimately describing the same thing, but I would just really like to have a concrete defintion, instead of just a general understanding. So, is it a vector space, a set or something else? – Andre May 29 at 13:52
• Well, it's $\mathbb{R}^n_+$, which is a set and is endowed with the operations of addition between pairs of its elements and multiplication of its elements with real numbers (usually non-negative ones). So, it's a vector space over the reals (or to be precise, a subset of the vector space $\mathbb{R}^n$ with addition and scalar multiplication). Moreover, it is usually also endowed with the Euclidean norm and so, it is a normed space. The way authors view it depends on where they want to focus their attention. – pegasus May 29 at 23:14
• Thank you, I get it now. – Andre May 30 at 9:25

I’ll add just one caveat to the answers and comments already posted:

Vast majority of economists do no treat commodities as vectors as we define in physics. It is simply because there is no way to interpret the “direction” of a commodity bundle! So you should rather imagine elements of the commodity space to be simply columns of numbers denoting the quantity of each commodity.

So while all the addition and operations do work usual, we do not use the parallelogram law to determine the direction of the new vector.

P.S: You May come across a few papers that explicitly use directions, but they are very very few in number.

• Thanks, good to know! – Andre Jun 23 at 19:36