Theory of Value, Chapter 5...

Chapter 5 of Debreu's Theory of Value is on economic equilibrium.  First some definitions...let xi be the vector of consumptions of for consumer i with i indexed from 1,...,m; yj be the vector of production for producer j with j indexed from 1,...,n; let wi denote the resource endowment of consumer i (the sum of these endowments over 1,...,m equals w or the total resources of the economy); finally, sij is consumer i's share in the profits of firm j. Net demand is x-y, excess demand is defined to be x-y-w.

The first few sections in the chapter define what an economy is, what market equilibrium is, and what attainable states of an economy are.  Basically attainable states of the economy are states where consumers are choosing a consumption bundle that is possible for them (i.e., satisfies their wealth constraint), producers are choosing a production that is possible for them, and that the market is in equilibrium (i.e., the net demand must equal the total available resources).  Economy equilibrium is then defined to be some subset of these obtainable states where consumers are maximizing utility and firms are maximizing profit.

Some questions about private ownership...
Private ownership of the means of production, I think, is simply Debreu's way of allocating the pure profit that producers make in equilibrium when production processes exhibit decreasing returns to scale.  How are the shares determined?   This does not seem to be addressed.  If all firms are identical, then it doesn't matter.  However if firms are not identical, then it seems to me that consumer i's wealth (and by extension his choice of consumption) would change depending on his idiosyncratic portfolio of shares in the producers.  There seems to be a missing market for stocks in this world... 

On an unrelated note: I have always felt that the shape of the production possibilities set was technologically determined, and that it was shares in this technology that where owned by the consumers (who I suppose may or may not also be the workers).  These shares then give a consumer claims to the proceeds from the sale of the output produced.  The point is, I have always thought of the consumers owning the production technology itself and not simply owning the output of production.  Anyone have thoughts on this? Do consumers own the technology of production? Or do they simply own the output?  If it is the latter, then who owns the technology of production? 

Proof of Existence: I will not comment on the proof, except to say that Debreu proves only existence and does not show uniqueness of stability.  

Also...in the end of chapter notes Debreu cites a paper by L.W. McKenzie called "Competitive Equilibrium with Dependent Consumer Preferences" which looks really interesting...unfortunately I can not seem to find in cursory Google search...