Lec11-Uncertainty_Expected_Utility_Theory

Lec11-Uncertainty_Expected_Utility_Theory - Lecture Note...

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Lecture Note 11: Uncertainty, Expected Utility Theory and the Market for Risk David Autor 14.03, Microeconomic Theory and Public Policy, Fall 2005 1 Risk Aversion and Insurance: Introduction A huge hole in our theory so far is that we have only modeled choices that are devoid of uncertainty. That’s convenient, but not particularly plausible. Prices change Income f uctuates Bad stu f happens Most important decisions are forward-looking, and depend on our beliefs about what is the op- timal plan for present and future. Inevitably, these choices are made in a context of uncertainty. There is a risk (in fact, a likelihood) that the assumptions we make in our plan will not be borne out. In making plans, we should take these contingencies and probabilities into account. If we want a realistic model of choice, we need to model how uncertainty a f ects choice and well-being. This model should help to explain: How people choose among ‘bundles’ that have uncertain payo f s, e.g., whether to f yonan airplane, whom to marry. Insurance: Why do people want to buy it. How(andwhy)themarketforr iskoperates . 1.1 A few motivating examples 1. People don’t seem to want to play actuarially fair games. Fair game: E ( X )= Cost of Entry = P win · [ Payo f | Win ]+ P lose · [ f | Lose ] . 1
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Most people would not enter into a $1 , 000 dollar heads/tails fair coin f ip. 2. People won’t necessarily play actuarially favorable games: You are o f ered a gamble. We’ll f ip a coin. If it’s heads, I’ll give you $10 million dollars. If it’s tails, you owe me $9 million. Its expected monetary value is : 1 2 · 10 1 2 · 9=$0 . 5 million Want to play? 3. People won’t 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? E ( X )= 1 2 2+ 1 4 4+ 1 8 8+ ... 1 2 n 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 . Thefactthatagamblewithpositiveexpectedmonetaryvaluehasnegative‘utilityvalue’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 payo f sarethesame. We need to be able to say how people make choices when: Agents value outcomes (as we have modeled all along) Agents also have feelings/preferences about the riskiness of those outcomes 2 Three Simple Statistical Notions [background/review] 1. Probability distribution: 2
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De f ne states of the world 1 , 2 ...n with probability of occurrence π 1 2 ...π n .
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This note was uploaded on 04/11/2008 for the course ECON 14.03 taught by Professor Autor during the Spring '08 term at MIT.

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Lec11-Uncertainty_Expected_Utility_Theory - Lecture Note...

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