Lecture 3 - Uncertainty - Econ 104B Choice Under...

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Econ 104B Choice Under Uncertainty
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Mathematical Statistics A random variable is a variable that records, in numerical form, the possible outcomes from some random event The probability density function ( PDF ) shows the probabilities associated with the possible outcomes from a random variable
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Mathematical Statistics The expected value of a random variable is the outcome that will occur “on average” ( 29 i n i i x f x X E = = 1 ) ( ( 29 dx x f x X E +∞ - = ) (
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Mathematical Statistics The variance and standard deviation measure the dispersion of a random variable about its expected value ( 29 [ ] ( 29 i n i i x x f x E x 2 1 2 = - = σ ( 29 [ ] ( 29 dx x f x E x x +∞ - - = σ 2 2 2 x x σ = σ
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Expected Value Games which have an expected value of zero (or cost their expected values) are called fair games a common observation is that people would prefer not to play fair games
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St. Petersburg Paradox A coin is flipped until a head appears If a head appears on the n th flip, the player is paid $2 n
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St. Petersburg Paradox The expected value of the St. Petersburg paradox game is infinite i i i i i i x X E = = = π = 1 1 2 1 2 ) ( = + + + + = 1 1 1 1 ... ) ( X E Because no player would pay a lot to play this game, it is not worth its infinite expected value
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Expected Utility Individuals do not care directly about the dollar values of the prizes they care about the utility that the dollars provide If we assume diminishing marginal utility of wealth, the St. Petersburg game may converge to a finite expected utility value this would measure how much the game is worth to the individual
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Expected Utility Expected utility can be calculated in the same manner as expected value ) ( ) ( 1 = π = n i i i x U X E Because utility may rise less rapidly than the dollar value of the prizes, it is possible that expected utility will be less than the monetary expected value
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The von Neumann-Morgenstern Theorem Suppose that there are n possible prizes that an individual might win ( x 1 ,… x n ) arranged in ascending order of desirability x 1 = least preferred prize U ( x 1 ) = 0 x n = most preferred prize U ( x n ) = 1
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The von Neumann-Morgenstern Theorem The point of the von Neumann- Morgenstern theorem is to show that there is a reasonable way to assign specific utility numbers to the other prizes available
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The von Neumann-Morgenstern Theorem The von Neumann-Morgenstern method is to define the utility of x i as the expected utility of the gamble that the individual considers equally desirable to x i
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The von Neumann-Morgenstern Theorem • Since U ( x n ) = 1 and U ( x 1 ) = 0
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Expected Utility Maximization A rational individual will choose among gambles based on their expected utilities (the expected values of the von Neumann-Morgenstern utility index)
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Expected Utility Maximization Consider two gambles: – first gamble offers x 2 with probability q and x 3 with probability (1- q )
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Expected Utility Maximization Substituting the utility index numbers gives expected utility (1) = q · π + (1- q ) · π
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