## Thursday, December 9, 2010

### 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!)