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1、第 3 章 不確定下的選擇理論: 期望效用函數(shù),【Definition】 An individual s preferences have an expected utility representation if there exists a function such that random consumption is preferred to random consumption if and only if,where is the expectation under the individuals probability belief.,3.1 Expected Utility F
2、unction Suppose that : There are two dates, time 0 and time 1; There is a single consumption good available, Individual consume only at time 1. Uncertainty in the economy is modeled by uncertain states of nature to be realized at time 1.,: the collection of all the possible states of nature , : an e
3、lement of .,【Definition 3.1】An consumption plan is a specification of the number of units of the single consumption good in different states of nature.,: a consumption plan. : the number of units of the consumption good in state of specified by .,Table3.1 An consumption plan,Let : X be the collectio
4、n of consumption plans under consideration. represents an individuals preference relation defined on a collection of consumption plans X. A preference relation is a mechanism that allows an individual to compare different consumption plans. For example, given two consumption plans and , preference r
5、elation enables an individual to tell whether he prefers to , or to .,We use utility function H represented the preference relation: For , iff .,Expected Utility: Consumption plans is preferred to if and only if Denoting the expectation operator under P by , the above can be equivalently written as,
6、.,In the above expected utility representation, if and are both certain or sure thing, that is and , , for some constants and , then and In this sense, u compares consumption plans that are certain. We call the function u defined on sure thing a von Neumann-Morgenstern Utility Function ( 1953).,. .,
7、3.2 The Existence of Expected Utility Function,Let P be a probability defined on the space . A consumption plan is a random variable, whose probabilistic characteristics are specified by P . Define the distribution function for a consumption plan as follows: . So, the expected utility derived from i
8、s,.,Assume that an individual only expresses his preference on probability distribution defined on a finite set Z. That is,We can represent a consumption plan by defined on Z, where is the probability that x is equal to z . Thus, for all and . The distribution function for the consumption plan is th
9、en , and .,【Axiom 3.1】 is a preference relation on . 【Axiom 3.2】 Complete: For any two consumption plans , we have or . 【Axiom 3.3】 Transitive: For any , a binary relation is transitive if and imply .,We denote the space of probabilities on Z by P and its elements by p, q, and r. If , the probabilit
10、y of under is .,【Axiom 3.4】 Independence axiom: For all and , implies .,.,【Axiom 3.5】 Archimedean axiom: For all , if then there exists such that .,For , let be the probability distribution degenerate at in that That is, represents the sure consumption plan that has units of consumption in every sta
11、te.,.,Suppose that is a binary relation on P that satisfies the above axioms. Then and imply that and imply that there exists a unique such that . and and imply that ., and imply that and imply that , for all . There exist such that , for all .,.,【Proposition 3.1】 on P satisfying the above axioms ha
12、s an expected utility representation.,【Proof】We take cases. Case 1 . Then for all . Therefore any for a constant will be a utility function for sure things. Case 2 .,3.3 Risk Aversion Consider the gamble that has a positive return , , with probability and a negative return , , with probability .,【De
13、finition 3.3】The gamble is actuarially fair when its expected payoff is zero, or . 【Definition 3.4】An individual is said to be risk averse if he is unwilling to accept or is indifferent to any actuarially fair gamble. An individual is said to be strictly risk averse if he is unwilling to accept any
14、actuarially fair gamble.,Let be the utility function of an individual. we have: , where denotes the individuals initial wealth. Using the definition of a fair gamble, the above relation may be rewritten as,圖3-1 一個凹的效用函數(shù),3.4 The Measure of Absolute Risk Aversion Consider a portfolio choice problem of
15、 a risk averse individual who strictly prefers more to less ( has a strictly increasing utility function). If the individual invests dollars in the j-th risky asset and dollars in the risk free asset, his uncertain end of period wealth, , would be,Thus the individuals choice problem is We assume tha
16、t there exists a solution to this problem. Since is concave, the first order necessary condition are also sufficient. They are Since is strictly increasing , the probability that must lie in (0,1).,【Proposition 3.2】 An individual who is risk averse and who strictly prefers more to less will undertak
17、e risky investments if and only if the rate of return on at least one risky asset exceeds the risky-free interest rate. 【Proof】Let and denote the rate of return of the single risky asset and the amount invested in it, respectively. For the individual to invest nothing or even short sell the risky as
18、sets as an optimal choice, it is necessary that the first order condition evaluate at no risky investments be nonpositive: , . Equivalently, , .,By assumption the individual strictly prefers more to less, therefore, , and the above condition is equivalent to only if , . Thus, an individual with a st
19、rictly increasing and concave utility function will avoid any positive risky investment only if none of the risky assets has a strictly positive risk premium. When one or more risky assets have strictly positive risk premiums, the individual will take part in some risky investment. That is , such th
20、at if , such that . ,【Proposition 3.3】 Consider a strictly risk averse individual who prefers more to less in an economy where there is only one risky asset and one riskless asset and where the risk premium of the risky asset is strictly positive. For small risk, the minimum risk premium that is req
21、uired to induce the individual to invest all of his wealth in the risky asset must be not less than,【Proof】For an individual to invest all his wealth in the risky asset it must be that,Taking a first order Taylor series expansion of around , multiplying both sides by the risk premium, and taking exp
22、ectations gives,where denotes terms of smaller magnitude than,Risk is said to be small when is small and terms involving and higher orders can be ignored. Ignoring the remainder term, the minimum risk premium required to induce full investment in the risky asset may be determined by setting the righ
23、t-hand side of the above relation to 0 and getting . ,【Definition 3.5】 We call the measure of absolute risk aversion.,【Definition 1.6】If , an individuals utility function displays decreasing absolute risk aversion ; Similarly, when implies increasing absolute risk aversion; and , implies constant ab
24、solute risk aversion.,3.5 The Measure of Relative Risk Aversion,【Proposition 3.4】 Decreasing absolute risk aversion over the entire domain of implies that the risky asset is a normal good; i.e, the (dollar) demand for the risky asset increases as the individuals wealth increase. Increasing absolute
25、risk aversion implies that the risk asset is an inferior good, and constant absolute risk aversion implies that the individuals demand for the risky asset is invariant with respect to his initial wealth. That is ; ; .,【Proof】The proof for only the decreasing absolute risk aversion case is presented
26、as the proof for the other two cases follow the same structure. At the optimum, we have: . This process is referred to as implicit differentiation of a with respect to and gives where as usual denotes the individuals end of period (random) wealth. The denominator is positive because strict risk aver
27、sion implies Therefore,Under decreasing absolute risk aversion, , we take the cases: (A) If , we have , since the amount invested in the risky asset is strictly positive. Thus (B) If , we have , since the amount invested in the risky asset is strictly positive. Thus .,Multiplying both sides of the t
28、wo results by gives in the event that , and in the event that . The above two relations imply So we have . ,【Definition 3.7】 We call the measure of relative risk aversion. 【Proposition 3.5】 Denote the wealth elasticity of the demand for the risky asset, so If , then , that is, the proportion of the
29、individuals initial wealth invested in the risky asset will decline as his wealth increases., If , then , that is, the proportion of the individuals initial wealth invested in the risky asset will increase as his wealth increases. If , then , that is, the proportion of the individuals initial wealth invested in the risky asset will be constant as his wealth changes.,【Proof】 The wealth elasticity of the dem
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