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Rubinstein2005-page130

Rubinstein2005-page130 - Problem 3 Standard Consider a...

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October 21, 2005 12:18 master Sheet number 128 Page number 112 Problem Set 9 Problem 1. ( Standard. Based on Rothschild and Stiglitz 1970. ) We say that p second-order stochastically dominates q and denote it by pD 2 q if p q for all preferences satisfying the vNM assumptions, monotonicity and risk aversion. a. Explain why pD 1 q implies pD 2 q . b. Let p and ε be lotteries. Define p + ε to be the lottery that yields the prize t with the probability α + β = t p (α)ε(β) . Interpret p + ε . Show that if ε is a lottery with expectation 0, then for all p , pD 2 ( p + ε) . c. (More difficult) Show that pD 2 q if and only if for all t < K , t k = 0 [ G ( p , x k + 1 ) G ( q , x k + 1 ) ][ x k + 1 x k ] ≥ 0 where x 0 < . . . < x K are all the prizes in the support of either p or q and G ( p , x ) = z x p ( z ) . Problem 2. ( Standard. Based on Slovic and Lichtenstein 1968. ) Consider a phenomenon called preference reversal . Let L 1 = 8 / 9 [ $4 ] ⊕ 1 / 9 [ $0 ] and L 2 = 1 / 9 [ $40 ] ⊕ 8 / 9 [ $0 ] . a. What is the maximal amount you are willing to pay for L 1 ? For L 2 ? b. What lottery do you prefer? c. Discuss the “typical” answer that ranks L 1 as superior to L 2 but attaches a lower value to L 1 (see Slovic, Tversky and Kahneman 1990).
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Unformatted text preview: Problem 3. ( Standard ) Consider a consumer’s preference over K-tuples of K uncertain assets. De-note the random return on the k th asset by Z k . Assume that the random variables ( Z 1 , . . . , Z K ) are independent and take positive values with proba-bility 1. If the consumer buys the combination of assets ( x 1 , . . . , x K ) and if the vector of realized returns is ( z 1 , . . . , z K ) , then the consumer’s total wealth is ∑ K k = 1 z k x k . Assume that the consumer satisFes vNM assumptions, that is, there is a function v (over the sum of his returns) so that he maximizes the expected value of v . Assume that v is increasing and concave. The con-sumer preferences over the space of the lotteries induce preferences on the space of investments. Show that the induced preferences are monotonic and convex....
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