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chap18and19 - Uncertainty and Consumer Choice: Chapter 18...

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1 Uncertainty and Consumer Choice: Uncertainty and Consumer Choice: Chapter 18 Chapter 18 Skip section on the state-preference approach at the end of the chapter Consider a lottery with outcomes X 1 , X 2 , …, X n , and probabilities of each outcome of π 1 , π 2 , …, π n The sum of all probabilities is one: all outcomes are specified. 2 Expected Value Expected Value The expected value of a lottery is the weighted average of all possible outcomes where the weight is the probability of that outcome: E(X) = π 1 X 1 + π 2 X 2 +…+ π n X n A lottery is defined as a fair game if E(X)=0 We expect that consumers will not play fair games unless the game has some consumption value
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3 Example 1: Fair Lottery Example 1: Fair Lottery Individual invests $10 75% of time wins and gets $11, so net gain is X 1 = 11 - 10 = 1 25% of time looses and gets $7, so net gain is X 2 = 7 - 10 = -3 Expected value of game is E(X)=.75(1)+.25(-3)=0 This is a fair game: the individual expects to just break even. 4 Example 2: Modified Lottery Example 2: Modified Lottery Individual invests $10 • 75% of time wins and gets $12, so net gain is X 1 = 12 - 10 = 2 • 25% of time looses and gets $7, so net gain is X 2 = 7 - 10 = -3 Expected value of game is E(X)=.75(2)+.25(-3)=.75 The individual will always “on average” always win.
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5 Von Neumann Von Neumann -Morgenstern Utility Morgenstern Utility Define an index function rank bundles by size of prize define U(X 1 ) = 0 and U(X n ) = 1 Now choose some X i . An individual will be indifferent between X i and some gamble with prizes X 1 and X n U(X i ) = π i U(X n ) + (1- π i ) U(X 1 ) = π i If X i is big, then the consumer will only accept the gamble if the probability of winning the best prize is high. 6 Risk Aversion Risk Aversion We expect that individuals will attach greater weights to losses than to equal size gains Holding constant the expected value of two games, consumers will generally prefer the game that is least risky, where least risky refers to the variance of possible outcomes
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7 Example of Risk Aversion Example of Risk Aversion Individual pays $10 to play game 50% of time wins and gets $15 50% of time loses and gets $5 Three key concepts expected utility of gamble: EU=.5U(15)+.5U(5) expected value of gamble: E(X)=.5(15)+.5(5)=10 utility of expected value of gamble=U[E(X)]=U(10) Risk aversion says that we expect U(10)>.5U(15)+.5U(5) 8 Graph of Risk Aversion Graph of Risk Aversion Utility Wealth 5 10 15 U(10) .5U(15)+.5U(5) w*
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9 More on Risk Aversion More on Risk Aversion Prefers the expected value of a gamble to the gamble itself Willing to pay “risk premium” up to 10-w* to avoid any gamble Utility Wealth 5 10 15 U(10) .5U(15)+.5U(5) w* 10 What Feature of Utility Preferences What Feature of Utility Preferences Leads to Risk Aversion? Leads to Risk Aversion?
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This note was uploaded on 05/26/2008 for the course ECON 101 taught by Professor Buddin during the Winter '08 term at UCLA.

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chap18and19 - Uncertainty and Consumer Choice: Chapter 18...

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