尼科尔森中级微观经济学课件 (19)

Expected Value
• Suppose that Smith and Jones decide to flip a coin
– heads (x1) Jones will pay Smith $1 – tails (x2) Smith will pay Jones $1
• From Smith’s point of view,
paradox game is infinite
E( X )

i xi
i 1

2i 1 i i 1 2
E( X ) 1 1 1 ... 1
• Because no player would pay a lot to play this game, it is not worth its infinite expected value
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St. Petersburg Paradox
• A coin is flipped until a head appears • If a head appears on the nth flip, the
player is paid $2n
x1 = $2, x2 = $4, x3 = $8,…,xn = $2n
– x1 = least preferred prize U(x1) = 0 – xn = most preferred prize U(xn) = 1
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The von NeumannMorgenstern Theorem
• The point of the von NeumannMorgenstern theorem is to show that there is a reasonable way to assign specific utility numbers to the other prizes available
E( X ) 1x1 2x2 ... n xn
n
E( X ) i xi i 1
• The expected value is a weighted sum of the outcomes
– the weights are the respective probabilities 3
• The probability of getting of getting a head on the ith trial is (½)i
1=½, 2= ¼,…, n= 1/2n
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St. Petersburg Paradox
• The expected value of the St. Petersburg
• There may be a few exceptions
– when very small amounts of money are at stake
– when there is utility derived from the actual play of the game
• we will assume that this is not the case
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Risk Aversion
• Two lotteries may have the same expected value but differ in their riskiness
– flip a coin for $1 versus $1,000
• Risk refers to the variability of the outcomes of some uncertain activity
• Note that this choice of utility numbers is arbitrary
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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
• 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
– a common observation is that people often refuse to participate in actuarially fair games
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Fair Games
• People are generally unwilling to play fair games
UNCERTAINTY AND RISK AVERSION
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Probability
• The probability of a repetitive event happening is the relative frequency with which it will occur
– probability of obtaining a head on the fair-flip of a coin is 0.5
– a flip of a coin for $1,000 promises a small gain in utility if you win, but a large loss in utility if you lose
– a flip of a coin for $1 is inconsequential as the gain in utility from a win is not much different as the drop in utility from a loss
– second gamble offers x5 with probability t and x6 with probability (1-t) expected utility (2) = t · U(x5) + (1-t) · U(x6)
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Expected Utility Maximization
• If a lottery offers n distinct prizes and the
probabilities of winning the prizes are i
(i=1,n) then
n
i 1
i 1
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Expected Value
• For a lottery (X) with prizes x1,x2,…,xn and the probabilities of winning 1,2,…n, the expected value of the lottery is
n
E( X ) iU(xi ) i 1
• 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
q · 2 + (1-q) · 3 > t · 5 + (1-t) · 6
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Expected Utility Maximization
• If individuals obey the von NeumannMorgenstern axioms of behavior in uncertain situations, they will act as if they choose the option that maximizes the expected value of their von Neumann-Morgenstern utility index
• When faced with two gambles with the same expected value, individuals will usually choose the one with lower risk
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Risk Aversion
• In general, we assume that the marginal utility of wealth falls as wealth gets larger
E( X ) 1x1 2x2
E( X ) 1 ($1) 1 ($1) 0
2
2
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Expected Value
• Games which have an expected value of zero (or cost their expected values) are called actuarially fair games
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Expected Utility Maximization
• Consider two gambles:
– first gamble offers x2 with probability q and x3 with probability (1-q) expected utility (1) = q · U(x2) + (1-q) · U(x3)