10Pratt's Theorem on Risk Aversion and Portfolio Demand

# 10Pratt's Theorem on Risk Aversion and Portfolio Demand -...

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Pratt’s Theorem on Risk Aversion and Portfolio Demand 1 For t = 0 , 1 let u t be a C 1 vN-M utility with positive ﬁrst derivative everywhere. Consider the following problem: α t arg max α 0 ±Z u t ( w + αx ) dF ( x ) ² . where R xdF ( x ) > 0, R x 2 dF ( x ) > 0 (so F is nondegenerate) and F has bounded support. Theorem Let u 0 be strictly more concave than u 1 : u 0 ( z ) = T ( u 1 ( z )) for all z R + , where T : Range ( u 1 ) R is strictly concave. Then α 1 α 0 . Proof : Let u 0 be strictly more concave than u 1 . Deﬁne V ( α,t ) = Z u t ( w + αx ) dF ( x ) , and note that wlog we may take the domain of V ( · ,t ) to be R ++ . (Why?) We have V α ( α, 0) = Z u 0 0 ( w + αx ) xdF ( x ) = Z T 0 ( u 1 ( w + αx )) u 0 1 ( w + αx ) xdF ( x ) ± T 0 ( u 1 ( w )) Z u 0 1 ( w + αx ) xdF ( x ) = Z h T 0 ( u 1 ( w + αx )) - T 0 ( u 1 ( w )) i u 0 1 ( w + αx ) xdF ( x ) + T 0 ( u 1 ( w )) Z u 0 1 ( w + αx ) xdF ( x ) < T 0 ( u 1 ( w )) Z u 0 1 ( w + αx ) xdF ( x ) = T 0 ( u 1 ( w )) V α ( α, 1)
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## This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.

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