*Theory of Value: An Axiomatic Analysis of Economic Equilibrium*. It has rekindle my interest in abstract mathematics, and as an economist it has so far proved helpful in understanding the particularities of General Equilibrium theory in more detail.

Right now I am reading Chapter 2: Commodities and Prices. From my MSc I was aware that Arrow-Debreu general equilibrium assumed the existence of markets for all commodities, where commodities are completely specified by their intrinsic characteristics, the time that they are acquired, and their location (i.e., Red Winter Wheat, today, in Chicago is a different good from Red Winter Wheat, a year from now, in San Francisco, etc.).

I was not aware however, that the commodity space that defines all possible combinations of these commodities has finite dimension and that time is also taken to be finite. Even the claim that the commodity space has finite dimension for a

*fixed*moment in time seems to be implausible. Intuitively, economic growth would seem to require (or be driven by) continual innovation of new commodities, but I am also not sure how one is to think of commodities that have not been created yet with this framework. Are they to be accommodated by allowing the dimensionality of the commodity space to increase with time? Or perhaps this is being abstracted from in the general equilibrium framework.

Debreu addresses some of these critiques in his end of chapter notes. Note 2 says that it is the assumption of finite time that allows the commodity space to be of finite dimension. He goes on to say that many of the results to follow can be extended to an infinite dimension commodity space. I am still not sure whether this addresses my concern about the ability of the theory to deal with commodities that have yet to be invented...

I should mention that I do find the theory quite elegant. It kind of cool the way you derive the exchange rates, interest rates, and discount rates from the price system as long as you have a unit of exchange. Here it is assumed that there exists some unit of exchange (I suppose that this is why so many economists have devoted their careers to developing theories of where money comes from...which is something else that I have never understood!)

There is a well-established literature extending GE existence results to infinite dimensional commodity spaces, which I believe starts with Aumann, "Existence of a Competitive Equilibrium in Markets with a Continuum of Traders", 1966, Econometrica. In principle there is no problem with allowing the number of commodities to increase with time, since commodities are already indexed by date. I would imagine one problem with extending GE to incorporate commodities that have not yet been invented is that one has to make some assumption about agents' beliefs about what commodities/ technologies will be invented in the future.

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