# 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?

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