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# Inv Dec Symmetr Inf BA 855

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This 9 page Class Notes was uploaded by Robyn Thiel on Thursday October 29, 2015. The Class Notes belongs to BA 855 at University of Michigan taught by Staff in Fall. Since its upload, it has received 48 views. For similar materials see /class/231618/ba-855-university-of-michigan in Business Administration at University of Michigan.

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Date Created: 10/29/15

5 Expected Utility Theory We have been talking about arbitrage models in discrete time Now we are going to begin talking about utilitybased models in discrete time In this section of the notes we review some results from the economics of uncertainty We are going to say that people maximize expected utility subject to budget constraints This material is covered in several places including Varian39s chapter 11 It should be review for most of you so we will cover it fairly quickly 51 Expected Utility A consumer39s expected utility function is not a primitive in economics We make assumptions about an agent39s preferences in order to derive an expected utility function for him or her We de39ne expected utility over the space of lotteries L Using Varian39s notation pix 1 i p iy means receiving x with probability p and y with probability 1 i p The operator gtgt implies indi erence while 0 implies weak preference We assume 1 Getting a prize with probability 1 is the same as getting the prize for certain 1 iX 1 i1iy gtgtx 2 The consumer doesn39t care about the order in which the lottery is described PiX 1iPigtgt1iPiy PiX 3 A consumer39s perception of a lottery depends only on the net probabilities in the lottery not on how the lottery is packaged q i p x 1 i p iy 1 i q iy gtgt qpx 1iqpy 4 Consumers39 preferences over lotteries are 2 complete either x O y or y 0 x or both 8x y 2 re exive x O x 8x 2 and transitive if x O y and y 0 2 then x O z 5 Preferences are continuous fp 2 01 p i x 1 i p i y 0 lg and fp 2 01 z O p i x 1 i p yg are closed sets for all x y and z in L 6 If people are indi erent about two goods they will be indi erent about lotteries overthosegoodsxgtgtypix 1ipizgtgtpiy 1ipiz Existence of Expected Utility Function lf L 0 satisfy the above axioms then there is a utility function u that ranks lotteries according to preferences PiX 1iPiyAqu 1 iQiZ 68 UPiX 1iPiYgtUqu 1MHZ and satis39es the expected utility property upix 1 i piy pux 1 i MW 69 You can read the proof of the theorem in Varian or in other references It can also be shown that expected utility functions are unique up to an aine or linear transformation These properties make expected utility maximization an extremely useful way to think about people39s behavior under uncertainty Why do we worry about the existence of an expected utility function Some of the attractiveness of expected utility maximization is driven by its mathematical tractabil ity We don39t want our models to be determined by tractability alone we want them to re ect reality as well Several economists have proposed alternatives to the expected utility paradigm Mark Machina has made a career out of his alternative to expected utility He drops some of the assumptions we made above and 39nds something like 33 local expected utility maximization Kahnemann and Tversky two psychologists have also been trying to replace expected utility with something that is more consistent with behavior Whenever economic models fail it is possible that people are simply not maximizing expected utility functions like we want them to For this reason many economists are more comfortable assuming that there is no arbitrage in the market than assuming that all agents are maximizing expected utility somehow 52 Risk aversion What is it about expected utility that makes it so useful for quotname Besides assuming that people are maximizing expected utility functions we usually assume that their utilities make them risk averse A risk averse person would rather take a certain amount of money than take a gamble with an expected payo that is slightly larger than the certain amount People can also be risk neutral or risk loving of course It turns out that people with concave utility functions are risk averse This result is expressed with an oftused inequality Jensen39s Inequality If fx is a strictly concave function like a riskaverse utility function then Efx lt fEx Again you can see the proof of this result in Varian The intuition behind this result is what comes out of the diagram that is usually explained in intermediate economics classes To say more about risk aversion we are going to have to de39ne a risk premium Suppose we were thinking about a random consumption bundle x 1 39i where 1 is a constant and 1 is a random variable with an expected value of zero For now a risk premium is de39ned as the value of 12 that makes true the statement Eux u i V2 70 Now for a particular realization of 39i we can use a taylor series expansion4 to argue that 2 u quot m u quotwoo 311 71 Therefore 3A w Eux M u u 1 72 Furthermore if 3A is smallquot then 12 is also small so using a taylor expansion again u i 12 M u i 12u gtt 73 which means that we can eXpress our risk premium as Zum 3 I uquotk RA39 74 VZMIi where RA is known as the absolute risk aversion coeicient The absolute risk aversion coeicient is a nice way to measure risk aversion People with higher coeicients are more risk averse than people with lower coeicients If you replace the additive error term that we assumed above with a multiplicative w fk n 4m fa f ax i a f Zax i a2 K fgtx i a akftwdtz error term the measure of risk aversion that results is known as relative risk aversion I uw 1 us RR RA 75 You can 39nd a derivation for relative risk aversion in Varian or elsewhere Next we are going to state but not prove an important theorem Pratt39s theorem Given 2 utility functions u1 and uz that are twice di erentiable strictly concave and increasing the following are equivalent 1 RAD Riot 2 V2101 quot 5 V2201 quot 3 u1 is more concave than uz Proofs of this theorem can be found in lots of places including Varian Pratt39s theorem tells us three di erent but equivalent ways to determine if one person is more risk averse than another 53 Utility Functions There are several utility functions that are used very frequently by economists We will discuss three of them here The 39rst type of utility function we will discuss is what is known as the constant relative risk aversion CRRA or power utility function It is parameterized as ux x 2 01 76 As the exponent of a particular version of the power utility function goes to zero it becomes the log utility function x i1 logX 77 where the logarithm in the function is a natural log This family of utility functions is called CRRA because its coeicient of absolute risk aversion is RA 1 i x 78 giving it a relative risk aversion that is constant A second family of utility functions that is commonly used in research is the constant absoulte risk aversion family CARA This family is parameterized ux 1 iei quot 5 gt 0 79 The absolute risk aversion coeicient for this utility function isjust 5 The last type of utility function we will discuss is the quadratic utility function This function is written ux a bx i cxz bc gt 0 80 You can calculate the risk aversion coeicients for this utility function as a homework assignment A quadratic utility function looks like an inverted parabola There is always a point at which marginal utility uquotx becomes negative You get to solve for this as a homework problem as well 54 Stochastic Dominance We have talked about ways to determine whether one person is more or less risk averse than another person Now we will shift our emphasis to asking whether a particular lottery is more or less risky than another lottery Probably the most general way to compare the risk of lotteries is in terms of what is called stochastic dominance First Order Stochastic Dominance The cumulative distribution of payo s F 39rst 37 order stochastic dominates FOSD G i Gx 5 F x 8x 2 I where l is the sample space of x First order stochastic dominance is an attractive property because it has been shown that for all increasing utility functions ux F FOSD G EFthH 5 EGhKXH 8D where E is the expectation taken under the assumption that F is the distribution of payo s To interpret FOSD remember that Gx Prx x and draw a picture A weaker concept than FOSD is second order stochastic dominance SOSD To de39ne SOSD we need to de39ne the function Z UMJHMiHMW Second Order Stochastic Dominance F SOSD G i Tx 5 0 8x 2 l and EGX EFX Second order stochastic dominance is a weaker concept than FOSD in the sense that FOSD implies SOSD but SOSD does not imply FOSD For all increasing and 38 strictly concave utility functions F SOSD G EFux EGux 83 Since SOSD is a weaker concept than FOSD we need the additional condition that ux is concave to get the result that people should prefer payo s that second order dominate Second order stochastic dominance is an attractive property to work with because it corresponds to a frequently used abstraction known as a mean preserving spread Economists often add a mean preserving spread to their models in order to introduce uncertainty They are sometimes described as a sprinklingquot of risk If we de39ne the random variable y as x plus a mean preserving spread then y x A 84 where EA 0 EiAJ39x 0 85 VarA gt 0 In this case the distribution of x will second order stochastic dominate the distribution of y A plot of a mean preserving spread can be instructive A useful theorem for interpretting SOSD is the RothschildStiglitz theorem Rothschild Stiglitz Theorem The following conditions are equivalent 1 F SOSD G 2 G F plus noise mean preserving spread 3 F and G have the same mean and all risk averters prefer F to G

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