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Unformatted text preview: Basics of Expected Utility Theory Finance decisions involve monetary outcomes, time , and uncertainty . We will now introduce the uncertainty aspect and abstract from the time element. There are three levels of generality for models of choice and resource allocation under uncertainty, depending on what is taken as the set of basic elements of choice or allocation: 1 Most general: State preference 2 Less general: Preferences over distributions of income 3 Even less general: Expected utility Y. F. Chow (CUHK) Financial Economics 2009&10 First Term 1 / 65 Models of Choice under Uncertainty State preference theory Elements of choice: Random incomes (prospects) contingent on ω 2 Ω . Concept of &set of states of the world¡: A random variable is a mapping from the set of states of the world to any other space, for our purpose, to the real line < . By de¢nition of states, there is no more general way of representing allocations/elements. Preferences over distributions of income Assumes that random incomes that are identically distributed are equally preferred, or even indistinguishable, by an individual. Expected utility Take the second approach and further assume that individual preferences satisfy particular restrictions. Y. F. Chow (CUHK) Financial Economics 2009£10 First Term 2 / 65 Still less general: Preferences over a limited number of moments of the distribution & meanvariance Preferences over all moments would be identical to preferences over distributions (for ¡nite moment distributions). In general, assume that only ¡rst two moments matter. Two justi¡cations: Quadratic utility function or normal distribution. Very useful in ¡nance. Remark: The above can be viewed as imposing assumptions of increasing strength on preferences . One can impose assumptions on the distributions of random income as well. For example, the Arbitrage Pricing Theory imposes assumptions on the number of ¢state variables£in continuoustime stochastic choice theory. Y. F. Chow (CUHK) Financial Economics 2009¤10 First Term 3 / 65 Preference Representation Historical motivation: Gambling theory St. Petersburg Paradox Consider the following model for uncertainty: Ω = f ( ω 1 , ω 2 ) ∞ g . Consider payo/s f X n ( ω ) : n 2 @g where X n ( ω ) = & 2 n if ω = ω 1 , X n + 1 ( ω ) otherwise. with Pr ( f ω 1 g ) = Pr ( f ω 2 g ) = 1 / 2. Then E ( X ∞ ) = lim N ! ∞ N ∑ n = 1 1 2 n 2 n = ∞ . What is the &value¡of this gamble? Y. F. Chow (CUHK) Financial Economics 2009¢10 First Term 4 £ 65 St. Petersburg Paradox (cont.) Daniel Bernoulli proposed to consider &expected utility¡given a logarithmic utility function for payo/s at each stage: U ( X ∞ ) = ∞ ∑ n = 1 1 2 n ln 2 n = 2 ln 2 Obviously we will not choose the gamble if 2 ln 2 is dominated by another lottery....
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 Fall '09
 Prof.Chow
 Finance, Utility, F. Chow, Y. F. Chow

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