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

Producer Theory and Profit Maximization...where exactly are opportunity costs accounted for within the general equilibrium framework?  This question was posed to me by one of my first year undergraduates this year (in a slightly different form!) and I don't think I had a very good answer for him.

As I read through Debreu's axiomatic treatment of profit maximization I find myself asking the same question.  Where are opportunity costs taken into account?  Are opportunity costs essentially a special type of contingent commodity that exists in perhaps a different time and place with its own price?

This matters because the assumption of additivity and the possibility of inaction implies that the maximum profit of a producer either does not exist or is null.  Null profit in equilibrium makes sense to me IF one is talking about economic profits and not accounting profits.  Economic profits requires taking opportunity costs into account...

Anyone out there have any thoughts on this one...or is this discussion just too pedantic 

Theory of Value: Chapter 3...

Chapter 3 is on producer theory.  I was rolling right along without problems through the first few pages until I encountered the following:

"A production yi is classified as possible or impossible for the ith producer on the basis of his present knowledge about his present and future technology.  The certainty assumption implies  that he knows now what input-output combinations will be possible in the future (although he may not know the details of the technological process which will make them possible)."

How could you know the input-output combinations that are possible in the future without knowing the technology?  Seems a bit weird to assume certainty, but then to also assume that producers have perfect knowledge about everything except the technology used to produce things. 

A more interesting comment appears in Debreu's discussion of the various assumptions made on a producer's production possibilities set.  While discussing various interpretations of the additivity assumption, Debreu writes as follows:

"In so far as the [production possibilities set] for a producer represents technological knowledge, it is clear that two production plans separately possible are jointly possible.  Alternatively the jth producer can be interpreted as an industry rather than a firm; then the additivity assumption means that there is free entry for firms into that industry.  Under additivity if yj is possible than so is kyj, where k is any positive integer.  Therefore additivity implies a certain kind of non-decreasing (i.e., increasing or constant) returns to scale."

It is this last comment that additivity implies a certain kind of non-decreasing returns to scale that stopped me.  I see why k has to be an integer (additivity implies that yj + yj +...+yj  = kyj must also be possible). I suppose I had just forgotten that constant returns to scale act as lower bound when we assume additivity (i.e., that decreasing returns to scale are not possible).

The next assumption discussed is convexity.  Convexity implies non-increasing returns to scale (convexity plus the no-free-lunch assumption rules out increasing returns).  Thus if one wants to assume additivity and convexity of the production set for a particular producer, then the production technology must exhibit constant returns to scale.    

Theory of Value, Chapter 2...

So, as yet another side project, I am reading Gerard Debreu's 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!) 

On the Principal of Continuity of Approximation...

"If the conditions of the real world approximate sufficiently well the assumptions of the ideal type, the derivations from these assumptions will be approximately correct."

Many thanks go to Cosma Shalizi for posting this critique by  by Herb Simon and Paul Samuelson of Milton Friedman's infamous essay "Methodology of Positive Economics"

The quote above is from Simon's section.  Also be sure to read Samuelson's section where he posits

"that the non-positivistic Friedman has a strong effective demand which a valid F-Twist brand of positivism could supply"

Just brilliant...

After reading things like this it is very, very unclear to me why I should care at all about the economic and policy implications of a macro model where agents have rational expectations...

Brilliant Lectures on Finance...

In my spare time I am putting myself through Robert Shiller's course at Yale on Financial Markets...I just finished lecture 5 on insurance.  My main takeaway so far is that financial innovation is very closely related to institutional innovation.  Institutions significantly constrain and shape the types of financial innovations that are possible or achievable.  The link between institutions and finance was not one that I appreciated adequately prior to this course...

Just Finished Some Summer School Apps...

I just finalized my application for the Sante Fe Summer School on Complex Systems, and my application for a summer internship at the World Bank...this still leaves my applications for internships at the IMF and the Federal Reserve to finish.  If I fail to get accepted to the Sante Fe Summer School, I am going to apply to the Sante Fe Program on Computational Social Sciences as well...

Anyone else have ideas for summer internships?

Posting will be eratic this week...

I am working on my presentation for the upcoming workshop in Edinburgh on Network Fragility and as such posting will be highly limited this week.