## Wednesday, December 8, 2010

### 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.