Advanced Microeconomics-Uncertainty Essay

4139 WordsDec 20, 201117 Pages
Expected Utility Theory Decisions under risk Outcome of at least one of the options is uncertain (& the probabilities of the possible outcomes are known.) Keep £ 1 N 1 14M Buy a lottery ticket r HH H £ 7 million HH H H Y H r H 1 1HHH 14M H H £0 () September 26, 2008 1 / 66 £0 N Rob a Bank r H HH £ 40 million H H HH Y H r 0.2 H HH H 0.8 H H Life in prison Any decision under risk can be represented by choice among lotteries. () September 26, 2008 2 / 66 The General Form of a lottery: [(x1 , p1 ), (x2 , p2 ), (x3 , p3 ), . . . , (xn , pn )] x1 , x2 , . . . xn can be any objects, n any number p1 + p2 + + pn = 1. e.g., £ 1 for sure: [(£ 1, 1)] Rob a bank: [(£ 40m, 0.2), (prison, 0.8)] How do individuals choose among lotteries? () September 26, 2008 3 / 66 Expected Utility Theory (Hypothesis): Each individual has a utility function u ( ), Given P = [(x1 , p1 ), . . . , (xn , pn )], Q = [(y1 , q1 ), . . . , (ym , qm )] P Q () p1 u (x1 ) + + pn u (xn ) q1 u (y1 ) + + qm u (ym ) (EU of P (calculated using u ( )) EU of Q) u ( ): von Neumann-Morgenstern (VNM) utility function. “EU maximizers”: individuals who behave according to the EU hypothesis. () September 26, 2008 4 / 66 Attitudes towards risks [(w1 , p1 ), . . . , (wn , pn )], (w1 , . . . , wn : sums of money). Expected value (EV) = p1 w1 + p2 w2 + + pn wn . An individual is risk-averse if he prefers getting the expected value of any lottery with certainty to getting the lottery itself; risk-loving if the opposite is true; risk-neutral if he is indi¤erent between any lottery and its expected value. Equivalent De…nition : An individual is risk-averse if he rejects all fair gambles. Fair gamble : a lottery whose EV is zero or a lottery whose price is equal to its EV. 1 e.g. [( 50, 1 )(50, 2 )] 2 [(0, 1 )(100, 1 )] and the

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