In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) Group 2 Dr. S. Farshad Fatemi Chapter 6: Choice under Uncertainty
Expected Utility Theory In many cases, individuals face some degree of risk when making their decisions. The general theory of choice established earlier in the course can be applied to these situations. Definition (MWG 6.B.1): A simple lottery L is a list L = (p 1,, p N ) with p n 0 and n p n = 1; where p n is interpreted as the probability of outcome n occuring. Definition (MWG 6.B.2): Given K simple lotteries L k = p 1 k,, p N k, k = 1,, K and probabilities α k 0 with k α k = 1, the compound lottery (L 1,, L K ; α 1,, α K ) is the risky alternative that yields the simple lottery L k with probability α k for k = 1,, K. Microeconomics 2 Dr. F. Fatemi Page 2
For any compound lottery, a simple lottery can be calculated which is called the reduced lottery. Definition (MWG 6.B.3): The preference relation on the space of simple lotteries L is continuous if for any L, L, L L, the sets {α [0,1] αl + (1 α)l L } [0,1] and {α [0,1] L αl + (1 α)l } [0,1] are closed. The continuity axiom implies the existence of a utility function representing the preference relation. U L R L L U(L) U(L ) Microeconomics 2 Dr. F. Fatemi Page 3
Definition (MWG 6.B.4): The preference relation on L satisfies the independence axiom if for all L, L, L L, and α [0,1] we have: L L αl + (1 α)l αl + (1 α)l. Definition (MWG 6.B.5): The utility function U L R has an expected utility form if there is assignment of numbers (u 1,, u n ) to the N outcomes such that for every simple lottery L = (p 1,, p N ) L we have: U(L) = u 1 p 1 + + u n p n. A utility function U L R with the expected utility form is called v.n-m (von Neumann-Morgenstern) expected utility function. Microeconomics 2 Dr. F. Fatemi Page 4
Proposition (MWG 6.B.1): A utility function U L R has an expected utility form if and only if it is linear, that is, if and only if satisfies the property that K U α k L k = α k U(L k ) k=1 K k=1 for any K lotteries L k L, k = 1,, K, and the probabilities (α 1,, α K ) 0, k α k = 1. Note: Unlike what we have seen so far about the utility function properties, the expected utility property is a cardinal property of utility functions. Microeconomics 2 Dr. F. Fatemi Page 5
Proposition (MWG 6.B.2): Suppose that U L R is a v.n-m expected utility function for the preference relation on L. Then U L R is another v.n-m utility function for if and only if there are scalars β > 0 and γ such that U = β U + γ for every L L. Using this property the differences of utilities have meaning. If a preference relation is representable by a utility function which has the expected utility form, then since a linear function is continuous, is continuous and satisfies the independence axiom. Microeconomics 2 Dr. F. Fatemi Page 6
Proposition (MWG 6.B.3) Expected Utility Theorem: Suppose that on L satisfies the continuity and independence axiom. Then admits a utility representation of the expected utility form. That is we can assign a number u n to each outcome n = 1,, N in such a manner that for any two lotteries L = (p 1,, p N ) and L = (p 1,, p N ), we have N N L L u n p n u n p n n=1 n=1 Examples: Expected utility as a guide to introspection The Allais paradox Machina s paradox Induced preferences Microeconomics 2 Dr. F. Fatemi Page 7
Money Lotteries and Risk Aversion A monetary lottery can be described by a cumulative distribution F(x): F R [0,1] where x is the amounts of money. In terms of density function f(x), the distribution can be written as: x F(x) = f(t) dt From here we assume that the lottery space L to be the set of distribution functions over nonnegative amounts of money. Microeconomics 2 Dr. F. Fatemi Page 8
Then if represents a rational preference of a decision maker and satisfies the continuity and independence axiom, there is an assignment of utility values u(x) to nonnegative amounts of money with the property that any F(. ) can be evaluated by a utility function U(. ) of the form: U(F) = u(x) df(x) To distinguish between these two utility functions they are called: U(x) : The von Neumann-Morgenstern expected utility function and u(x) : The Bernoulli utility function. u(x) is increasing and continuous. Microeconomics 2 Dr. F. Fatemi Page 9
Definition (MWG 6.C.1): A decision maker i) is risk averse if for any lottery F(. ): u(x) df(x) u x df(x) ii) is risk neutral if for any lottery F(. ): u(x) df(x) = u x df(x) iii) is strictly risk averse if the indifference holds only when the two lotteries are the same. The inequality which is called Jensen s inequality is the definition of a concave function. Microeconomics 2 Dr. F. Fatemi Page 10
Definition (MWG 6.C.2): Given a Bernoulli utility function u(. ) we define the following concepts: i) The certainty equivalent of F(. ) is c(f, u) where: u c(f, u) = u(x) df(x) ii) For any x and positive number ε, the probability premium is π(x, ε, u) where: u(x) = 1 2 + π(x, ε, u) u(x + ε) + 1 2 π(x, ε, u) u(x ε) Note: For a risk averse decision maker (recall u(x) is nondecreasing): c(f, u) x df(x) u c(f, u) u x df(x) u(x) df(x) u x df(x) Microeconomics 2 Dr. F. Fatemi Page 11
Proposition (MWG 6.C.1): Suppose a decision maker is an expected utility maximize with a Bernoulli utility function u(. ) on amounts of money. Then the following properties are equivalent: i) The decision maker is risk averse. ii) u(. ) is concave. iii) c(f, u) x df(x) for all F(. ). iv) π(x, ε, u) 0 for all x, ε. Examples: Insurance Demand for a risky asset General asset problem Microeconomics 2 Dr. F. Fatemi Page 12
Definition (MWG 6.C.3): Given a twice differentiable Bernoulli utility function u(. ) on amounts of money, the Arrow-Pratt coefficient of absolute risk aversion at x is defined as r A (x) = u (x) u (x). Comparison of Payoff Distribution in Terms of Return and Risk Definition (MWG 6.D.1): The distribution F(. ) first-order stochastically dominates G(. ) if for every nondecreasing function u R R, we have u(x) df(x) u(x) dg(x). Microeconomics 2 Dr. F. Fatemi Page 13
Proposition (MWG 6.D.1): The distribution of monetary payoffs F(. ) first-order stochastically dominates the distribution G(. ) if and only if F(x) G(x) for x. Definition (MWG 6.D.1): For any two distributions F(. ) and G(. ) with the same mean, F(. ) second-order stochastically dominates(or is less risky than) G(. ) if for every nondecreasing concave function u R + R, we have u(x) df(x) u(x) dg(x). Microeconomics 2 Dr. F. Fatemi Page 14