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...

Quote of the Day...

"Death had to take Roosevelt sleeping, for if he had been awake, there would have been a fight."

-Thomas Marshall, U.S. Vice President,
Commenting on the death of Theodore Roosevelt 

Quote of the Day...

"I happen to have a talent for allocating capital. But my ability to use that talent is completely dependent on the society I was born into. If I'd been born into a tribe of hunters, this talent of mine would be pretty worthless...but I was lucky enough to be born in a time and place where society values my talent, and gave me a good education to develop that talent, and set up the laws and the financial system to let me do what I love doing - and make a lot of money doing it. The least I can do is help pay for all that."

-Warren Buffett

When I was younger, I think I way over-estimated the correlation between abilities and outcomes.  There really is a lot of randomness in the world, and this randomness holds some (perhaps considerable) sway over individual outcomes.  How does one deal with the possibility that your lot in life may depend on a certain degree of randomness and not necessarily on one's individual talent?  Is it the case that despite the influence of randomness in our lives, that the optimal behaviour is still to behave as if everything was completely deterministic and within our control?  Does working harder minimize the impact of randomness in some sense? Or does working harder simply provide us with the illusion of control over our lot in life?  Somehow I think that society in general benefits from having citizens who genuinely believe that they are the master of their fate...even if this isn't completely true.  The incentives are better...  

Slide from my trade networks presentation...

So, today I gave my first presentation of work that I have been doing on the evolution of hierarchy and community structure in international trade networks.  I have included my slides below...you can also skip down to just below my slides and find a short description of the various plots (if you are interested)...feedback and comments are appreciated as always...
Network Fragility Short)


Some additional thoughts on the CCC plots...
Below is the plot of the CCC for OECD countries from 1962-2009 using data from UN Comtrade.  Grey bars represent U.S. recessions as defined by NBER.   I will focus my discussion on this plot (the other is similar...one of the nice things about limiting analysis to OECD countries is that the results are not dependent on choice of data!).  Note that five countries have yet to report for 2009. 

The evolutionary theory that I am testing predicts that environmental change will cause an increase in the modularity of the trade network.  Here modularity (really hierarchy) is measured by CCC.  He (2010) supposes that U.S. recessions are an indicator of environmental change.  I have a couple of issues with this.  First, he is using U.S. recessions which may or may not be a good indicator of global recessions (to my knowledge there is not a universally accepted measure of global recessions).  More importantly, while the CCC does indeed increase during recessions, the largest increases in the CCC seem to occur outside of recessions...see for example the entire decade of the 1960's!  Generally speaking, the economic environment in which global trade is being conducted is constantly changing and forcing individuals and companies to adapt along with it.  I think a better measure of environmental change is needed.

Not that this is the answer, but I think I will plot some measure of real oil price shocks against CCC ans see what that looks like...           

Theory of Value: Chapter 4...

Having just finished putting the finishing touches on my lecture slides for my talk at tomorrow's workshop of network fragility here in Edinburgh, I am back to reading Gerard Debreu's Theory of Value. I am currently in the chapter on consumer theory.

I like the way Debreu emphasizes that the indifference relation is a complete binary relation that partitions the consumers choice set (i.e., the indifference relation is reflexive, symmetric, and transitive, and complete).  The interest in the utility function then follows from the fact that we would like to have some increasing function that associates each indifference class with a real number that can be used to distinguish it from other indifference classes.

The proof of existence of a utility function when one assumes a form of continuity of preferences is quite clever.  The proof shows that there exists a dense subset of a consumer's choice set.  Defines a clever increasing function on that subset, and extends the function from the dense subset to the entire choice set.  Then the function is shown to be continuous.

As in producer theory, convexity of the choice set is crucial.  Working through the three different types of convexity: weak-convexity, convexity, and strong convexity was worthwhile.  Weak-convexity allows "thick" indifference curves, convexity rules out such "thick" indifference curves, strong-convexity is the type of convexity that is taught to 1st year undergraduates as being one of the reasons that marginal rates of substitution decrease as one moves down an indifference curve. 

The wealth constraint. The proof of existence of equilibrium in a private ownership economy rests crucially on the continuity of the correspondence between the set of price-wealth pairs such that the set of possible consumption bundles is not empty and the choice set of our agent.  Why? I will let you know after I have read chapter 5 on equilibrium.  For now I cite Debreu...(I am going to guess that continuity of the correspondence is necessary to insure that our profit maximizing producers do not choose to produce an amount of output that falls into a "hole" so to speak in the set of consumer utility maximizing bundles...I will let you know if this intuition turns out to be correct or not!)   

Social Dynamics of Tribal Wars...

A very interesting post about an online game called tribal wars.  Here is the Wikipedia entry.  I wonder if such games will ever become a possible (and then eventually acceptable) source of data for academics?

Theory of Value: Chapter 3 (cont'd...Again!)...

Last post on producer theory.  In his end of chapter notes, Debreu makes underlines three things that are not covered by the producer theory that he has described.  I repeat them below as I think they merit attention:

1. External economies and diseconomies: the case where the production set of a producer depends on the production sets of the other producers (and or on the consumptions of consumers).  Both of these cases are, I think, incredibly likely to occur in the real-world.  In fact such interdependencies are well modeled by networks.
2. Increasing returns to scale: one of my new favorite pastimes...these can also be well modeled with networks (although they can be well-modeled via other methods as well).
3. The behavior of producers who do not take prices as given: monopolistic competition...I would go so far as to submit that some form of monopolistic competition is a superior null model of producer behavior than perfect competition.