(no subject)

[timelets]

Allocation -> Labor x Capital -> Value -> Allocation.

According to the Brouwer theorems, if this map is continuous there exists a fixed point, i.e. a "fair allocation." Am I interpreting it right? It looks like that this might work only if the underlying end-to-end process is continuous, i.e. contains no uncertainties and has no side effects.

Comments

[gomberg]

Kakutani, not Brower (assuming the commodity space is finite-dimensional, anyway) so upper hemi-continuous and convex-valued, and it is the latter that is a problem sometimes, yes. Not a big deal. This is all stuff of standard textbooks. Why are you blogging about it?

Uncertainties and "side-effects" (externalities) are extremely important, but have no consequences for continuity of the demand or the supply (externalities sometimes do, but that is not the main problem). The benchmark model accommodates uncertainty beautifully, but with the notion of the market redefined. Mathematically everything works almost unchanged: that is the beauty of the benchmark (asymmetric information, of course, changes things a lot - but that is not mere uncertainty). Presence of externalities, naturally, is one of the many reasons that the conclusions of the welfare theorems may not hold.

Methinks, you would enjoy Debreu's Theory of Value, for starters. But I do not quite see the point of the whole post otherwise.

[timelets]

But I do not quite see the point of the whole post otherwise.

This is thinking aloud about how an intro textbook would look like if we used Category Theory, instead of "regular" math. I find that CT approach makes it easier to convey key concepts because it hides tons of mathematical detail underneath a map (arrow).

Thanks for asking, though. Helps formulate the purpose of the whole exercise.

[(Anonymous)]

But uncertainty still has nothing to do with continuity :)

[timelets]

Why? If we assume that a tech breakthrough is possible within the process, continuity cannot be built into the model.