9Arrow-Pratt Theorem

# 9Arrow-Pratt Theorem - The Arrow-Pratt Theorem For t = 0 1...

This preview shows pages 1–2. Sign up to view the full content.

The Arrow-Pratt Theorem For t = 0 , 1, let u t be a C 2 vN-M utility with u 0 t > 0 on R ++ . And let L be the set of cumulative distribution functions on R ++ with F (0) = 0 and F ( z ) = 1 for some number z < . 1 For t = 0 , 1, denote the Arrow-Pratt absolute risk aversion measure by r t ( z ) = - u 00 t ( z ) u 0 t ( z ) , the certainty equivalent by CE t ( F ) and the risk premium by RP t ( F ). Theorem. The following are equivalent. 1. r 1 ( z ) r 0 ( z ) for all z > 0 . 2. There is a concave, strictly increasing function T : Range ( u 0 ) R such that u 1 ( z ) = T ( u 0 ( z )) for all z > 0 . 3. CE 1 ( F ) CE 0 ( F ) for all F ∈ L . 4. RP 1 ( F ) RP 0 ( F ) for all F ∈ L . Corollary. Let u, F satisfy the hypotheses of the Theorem, and let CE ( F, a ) be the certainty equivalent and RP ( F, a ) the risk premium for vN-M utility u when the positive number a is added to the risk z (so final wealth is z + a ): u ( CE ( F, a )) = R u ( z + a ) dF ( z ) and RP ( F, a ) = R ( z + a ) dF ( z ) - CE ( F, a ) . The following are equivalent 1. r ( · ) is decreasing on R ++ . 2. CE ( F, a ) - a CE ( F, 0) for all F ∈ L and a > 0 . 3. RP ( F, a ) RP ( F, 0) for all F ∈ L and a > 0 . To prove the corollary, set

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern