哈佛大学：《中级微观经济学》（英文版）Lecture 6 Choice Under Uncertainty

The decision-maker is assumed to have preferences, denoted by 2, over lotteries For this to make sense- assume that each state n, which occurs with probability ph in lottery k has a fixed payoff xn The set of alternatives(i.e the equivalent of X in chapter 1)is denoted te is the set of all simple lotteries over outcomes or consequences c We will assume that the decision-maker has rational preferences(i.e transitive and complete)over￡

The decision-maker is assumed to have preferences, denoted by , over lotteries. For this to make sense– assume that each state n, which occurs with probability pn k in lottery k has a fixed payoff xn. The set of alternatives (i.e. the equivalent of X in chapter 1) is denoted  is the set of all simple lotteries over outcomes or consequences C. We will assume that the decision-maker has rational preferences (i.e. transitive and complete) over 

Assuming continuity(defined in MWG 6. B3)means that just as before there exists a utility function that will rank lotteries

Assuming continuity (defined in MWG 6.B.3) means that just as before there exists a utility function that will rank lotteries

Axioms of Expected Utility MWG Definition 6.B. 4 The preference relation on the space of simple lotteries t satisfies the independence axiom fora!L,L,L"∈anda∈(0,1) we have L≥ L if and only if aL+(1-a)"≥aL+(1-a)L The ranking is immune to adding on extra lotteries. Sometimes I think of this as saying that your utility from getting a higher probability of state j is independent of your probability of state k

Axioms of Expected Utility MWG Definition 6.B.4: The preference relation  on the space of simple lotteries  satisfies the independence axiom: if for all L,L ,L   and   0, 1 we have L  L if and only if L  1  L  L  1  L. The ranking is immune to adding on extra lotteries. Sometimes I think of this as saying that your utility from getting a higher probability of state j is independent of your probability of state k

Definition 6. B 5 The utility function U: >r has an expected utility form if there is an assignment of numbers(ul,.uN) to the N outcomes such that for every simple lottery L=(p1,pN)∈￡ We haveL(L)=∑ A utility function U: t-R with the expected utility form is called a von Neumann -Morganstern expected utility function

Definition 6.B.5: The utility function U :    has an expected utility form if there is an assignment of numbers u1,... uN  to the N outcomes such that for every simple lottery L  p1, .pN    we have UL  j1 N ujpj A utility function U :    with the expected utility form is called a von Neumann-Morganstern expected utility function

Proposition 6. B. 1 A utility function U:￡→ R has an expected utility form if and only if it is linear, i. e if and only if it satisfies the property that K K U∑ akLk ∑ akU(lk for any K lotteries, k=1,., K and weights Qk≥0wh∑ak=1 Proof: First, if the utility function has the expected probability form then

Proposition 6.B.1: A utility function U :    has an expected utility form if and only if it is linear, i.e. if and only if it satisfies the property that U  k1 K kLk   k1 K kULk for any K lotteries, k  1, . . . ,K and weights k  0 with k k  1 Proof: First, if the utility function has the expected probability form then