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Rubinstein2005-page121

Rubinstein2005-page121 - u represents For all α ∈ 0 1...

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October 21, 2005 12:18 master Sheet number 119 Page number 103 Risk Aversion 103 First recall some basic properties of concave functions (if you are not familiar with those properties, this will be an excellent oppor- tunity for you to prove them yourself): 1. An increasing and concave function must be continuous (but not necessarily differentiable). 2. The Jensen Inequality : If u is concave, then for any finite se- quence k ) k = 1, ... , K of positive numbers that sum up to 1, u ( K k = 1 α k x k ) K k = 1 α k u ( x k ) . 3. The Three Strings Lemma : For any a < b < c we have [ u ( c ) u ( b ) ] /( c b ) ≤ [ u ( c ) u ( a ) ] /( c a ) ≤ [ u ( b ) u ( a ) ] /( b a ). 4. If u is differentiable, then for any a < c , u ( a ) u ( c ) , and thus u ( x ) 0 for all x . Claim: Let be a preference on L ( Z ) represented by the vNM utility func- tion u . The preference relation is risk averse iff u is concave. Proof: Assume that u is concave. By the Jensen Inequality, for any lottery p , u ( E ( p )) Eu ( p ) and thus [ E ( p ) ] p . Assume that is risk averse and that
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Unformatted text preview: u represents % . For all α ∈ ( 0, 1 ) and for all x , y ∈ Z , we have by risk aversion [ α x + ( 1 − α) y ] % α x ⊕ ( 1 − α) y and thus u (α x + ( 1 − α) y ) ≥ α u ( x ) + ( 1 − α) u ( y ) , that is, u is concave. Certainty Equivalence and the Risk Premium Let E ( p ) be the expectation of the lottery p , that is, E ( p ) = ∑ z ∈ Z p ( z ) z . Given a preference relation % over the space L ( Z ) , the certainty equiv-alence of a lottery p , CE ( p ) , is a prize satisfying [ CE ( p ) ] ∼ p . (To justify the existence of CE ( p ) we need to assume that % is monotonic and continuous in the sense that if p  q , the inequality is maintained if we change both lotteries’ probabilities and prizes a “little bit”). The...
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