MIT14_03F10_lec14

MIT14_03F10_lec14 - Lecture Note 14: Uncertainty, Expected...

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Lecture Note 14: Uncertainty, Expected Utility Theory and the Market for Risk David Autor, Massachusetts Institute of Technology 14.03/14.003, Microeconomic Theory and Public Policy, Fall 2010 1 Risk Aversion and Insurance: Introduction A significant hole in our theory of consumer choice developed in 14.03/14.003 to date is that we have only modeled choices that are devoid of uncertainty. That’s convenient, but not particularly plausible. Prices change Income FIuctuates Bad stuff happens Most decisions are forward-looking, and these decisions depend on our beliefs about what is the optimal plan for present and future. Inevitably, such choices are made in a context of There is a risk (in fact, a likelihood) that not all scenarios hoped for will be borne out. In making plans, should take these contingencies and probabilities into account. If want a realistic model of choice, need to model how uncertainty affects choice and well-being. This model should help to explain: How people choose among ‘bundles’that uncertain payoffs, e.g., whether to FIy on an airplane, whom to marry. Insurance: Why do people to buy it. How (and why) the market for risk operates. 1
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1.1 A few motivating examples 1. People don’t seem to want to play actuarially fair games. Such a game is one in which the cost of entry is equal to the expected payoff: E ( X ) = P win · [ Payoff | Win ] + P lose · [ | Lose ] . Most people would not enter into a $1 , 000 dollar heads/tails fair coin fiip. 2. People won’t necessarily play actuarially favorable games: You are offered a gamble. We’ ll fiip a coin. If it’s heads, I’ ll give you $10 million dollars. If tails, you owe me $9 million. Its expected monetary value is : 1 1 2 · 10 2 · 9 = $0 . 5 million Want to play? 3. People pay large amounts of money to play gambles with huge upside potential. Example “St. Petersburg Paradox.” Flip a coin. I’ ll pay you in dollars 2 n , where n is the number of tosses until you get a head: X 1 = $2 ,X 2 = $4 3 = $8 ,...X n = 2 n . What is the expected value of this game? 1 1 1 1 E ( X ) = 2 + 4 + 8 + ... 2 n 2 4 8 2 n = . How much would you be willing to pay to play this game? [People generally do not appear willing to pay more than a few dollars to play this game.] What is the variance of this gamble? V ( X ) = . The fact that a gamble with inFInite expected monetary value has (apparently) limited ‘utility value’suggests something pervasive and important about human behavior: As a general rule, uncertain prospects are worth less in utility terms than certain ones, even when expected tangible payoffs the same. We need to be able to say how people make choices when: consumers value outcomes (as we have modeled all along) consumers also feelings/preferences about the riskiness of those outcomes 2
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± 2 Five Simple Statistical Notions Definition 1 Probability distribution Define states of the world 1 , 2 ...n with probability of occurrence π 1 2 ...π n .
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MIT14_03F10_lec14 - Lecture Note 14: Uncertainty, Expected...

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