# Proofs

Spring 2018 Midterm Exam

Relevant Lemmas and Exercises:

Older documents, not entirely redundant, but in some parts strictly dominated by the ones above:

# Things I Should Have Said about Rederiving the Classic Arrow-Pratt Theory

1. Around w, I should have written that greater central risk aversion around w implies greater diffidence around w, and greater diffidence around w implies greater local Arrow-Pratt risk aversion at w. If true for all w, one can close the loop, since then (but not if only for one value of w) greater Arrow-Pratt risk aversion implies greater central risk aversion around all w.
2. The integral to get the implication that greater central risk aversion implies greater diffidence is very closely related to the picture I drew.
3. According to Polya’s principle in “How to Solve It” of building up one’s human capital by solving easier problems before banging one’s head too long against harder problems without such preparation, it would be a great idea to literally solve the differential equation (that is the counterpart with an = sign) before trying to “integrate” the differential inequality by rewriting terms as the derivative of the integral of something and doing monotonic transformations.
4. An easy way to get the implication that m>0 in the derivation of the condition for central risk aversion would be to define central risk aversion with two inequalities: Ms. 2 wants less than or equal to 1 unit of the risky asset if Ms 1 wants less than or equal to 1 unit. Nevertheless, the math problem of what happens if m is negative in the case we did is of some interest.
5.  The mathematical reference for this class is R. Tyrrell Rockafellar’s wonderful book Convex Analysis.  I strongly recommend it to you (and put the Amazon link in the last sentence). Tyrrell gives this statement of the separating hyperplane theorem on p. 97:

Theorem 11.3. Let C_1 and C_2 be non-empty convex sets in R^n. In order that there exists a hyperplane separating C_1 and C_2 properly [that is, without C_1 and C_2 both being contained in the hyperplane], it is necessary and sufficient that [the relative interior of] C_1 and [the relative interior of] C_2 have no point in common.

The key clarification is in the concept of “relative interior.” Here is what Tyrrell says about relative interiors in p. 44:

In the case of convex sets, the concept of interior can be absorbed into a more convenient concept of relative interior. This concept is motivated by the fact that a line segment or triangle embedded in R^3 does have a natural interior of sorts which is not truly an interior in the sense of the whole metric space R^3. The relative interior of a convex set C in R^n, which we denote by ri C, is defined as the interior which results when C is regarded as a subset of its affine hull aff C.

The concept of “affine hull” is probably somewhat more familiar, but here is what Tyrrell says about that on page 6:

Corrollary 1.4.1. Every affine subset of R^n is an intersection of a finite collection of hyperplanes. …

Obviously, the intersection of an arbitrary collection of affine sets is again affine. Therefore, given any S [that is a subset of] R^n there exists a unique smallest affine set containing S (namely the intersection of the collection of affine sets M such that M [is a superset of] S). This set is called the affine hull of S and is denoted aff S. It can be proved, as an exercise, that aff S consists of all the vectors [that can be formed by arbitrary linear combination of elements of S with weights that can each individually be any real number, but collectively must sum to 1].