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C hoice under U ncertain t yJonathan LevinOctober 20061 IntroductionVirtually every decision is made in the face of uncertainty. W hile w e often rely onmodels of certain inform atio n as you’ve seen in the class so far, m any economicproblem s requ ire that we tackle uncerta inty hea d on. For instance , ho w shou ld in-dividuals sa v e for retirement when they face uncertainty about their future income,thereturnondiﬀerent investm ents, their health and their future preferences? Howshould firms c hoose what products to introduce or prices to set when demand isuncertain? What policies should go vernmen ts c hoose when there is uncertain tyabout future productivity, gro wth, inflation and unemploymen t?Our objective in the next few classes is to dev elop a model of choice beha v iorunder uncertain ty. We start with the von Neumann-Morgenstern expected utilitymo del, w hich is the workhorse of modern econom ics. We’ll consider the foundation sof this model, and then use it to dev elo p basic properties of preference and cho icein the presence of uncertainty: measures of risk a version, rankings of uncertainprospects, and compara tive statics of ch oice under uncertaint y.As with all theoretical models, the expected utility model is not without itslimitatio n s. One lim itatio n is that it trea ts u n certaint y as objective risk – that is,as a series of coin flips where the probabilities are objectiv e ly kno w n . Of course, it’shard to place an objective probability on whether Arnold Schw arzenegger w ouldbe a good California gov ern or despite the uncerta inty. In response to this, w e’lldiscuss Savage’s (1954) approac h to choice under uncertain t y, which rather thanassuming the existen ce of objective probabilities attach ed to uncerta in prospects1makes assum p tions about choice beha vior and a rgues that if th ese assumptions aresatisfie d, a decision-m aker m ust act as if she is m ax im izing expected utility withrespect to some subjectively held probab ilities. Finally w e’ll conclude by lookingat some beha vioral criticisms of the expected utilit y model, and where they lead.A few comments about these notes. First, I’ll stic k p retty close to them inlectures and problem sets, so they should be the firstthingyoutacklewhenyou’restudying. That being said, you’ll probably want to consult MWG and ma ybeKrep s at various poin ts; I certaintly did in writing the notes. Second , I’v e follow edDavid Kreps’ style of throwing random poin ts into the footnotes. I don’t expecty o u to ev en read these necessarily, but they migh t be starting points if you decideto dig a bit deeper in to some of the topics. If you find yourself in that cam p, K reps(1988) and Gollier (2001) are useful places to go.2 E x pected U tilityWe start by conside ring the expected utility model, wh ich dates back to DanielBernoulli in the 18th century and w as formally dev eloped b y John von Neumannand Oscar Morgenstern (1944) in their book The ory of Games and Economic Be-havior. Rem arka bly, they viewed the dev elop me nt of the expected utilit y modelas something of a side note in the development of the theory of games.2.1 Prizes and LotteriesThe starting point for the model is a set X of possible prizes or consequences.Inmany economic problems (and for m uch of this class), X will be a set of monetarypa y oﬀs. But it need not be. If we are considering who will win Big Gam e thisy ear, the set of consequences migh t be:X = {Stanford wins, Cal wins, Tie}.We represen t an uncer tain prospect as a lottery or probability distribution overthe p rize space. For instan ce, the prospect that Stanford and Ca l are equally likelyto win Big Gam e can be written as p =(1/2, 1/2, 0) . P e rhaps these probabilities2depend on who Stanford starts at quarterback. In that case, there migh t be tw oprospects: p or p0=(5/9, 3/9, 1/9), depending on who starts.We w ill often find it useful to d epict lotteries in the p ro ba bility simple x, asshown in Figure 1. To understand the figure, consider the depiction of three-dimensional space on the left. The point p is meant to represent (1/2, 1/2, 0)and the point p0to represent (5/9, 3/9, 1/9). In general, any probabilit y vector(p1,p2,p3) can be represen ted in this picture. Of course, to be a valid probabilit yv e ctor , the probab ilities have to sum to 1. The triangle represents th e space ofpoints (p1,p2,p3) with the property that p1+ p2+ p3=1. The righ t hand sidepicture simply dispenses with the axes. In the right figure, p and p0are just asbefore. M oving up means increasing the likelihood of the second outcom e , mo v ingdo wn and left the likelihood of the third outcome and so on.p1p2p31110p pp1=1p2=1p3=1p’ p’Figure 1: The Probability SimplexMore generally, giv en a space of consequences X , denote the relevant set oflotteries o ver X as P = ∆(X ). Assum ing X = {x1, ..., xn} is a finite set, a lotteryover X is a v ector p =(p1,...,pn),wherepiis the probability that outcome xioccurs. Then:∆(X )={(p1, ..., pn):pi≥ 0 and p1+ ... + pn=1}3For the rest of this section, w e’ll main t ain the assum ptio n that X is a finite set.It’s easy to imagine infinite sets of outcome s – for instance, an y real n umberbet ween 0 and 1 – and later w e’ll w av e our hands and beha ve as if we haddeveloped the theory for larger spaces of consequences. You’ll have to take iton faith that the theory goes through with only minor amendmen ts. For those ofy o u who are interested , and relativ ely math-oriented, I recommend taking a lookat Kreps (1988).Observe that given t wo lotteries p and p0, any convex combination of them:αp+(1− α)p0with α ∈ [0, 1] is also a lottery. This can be viewed simply as statingthe math em atical fact that P is con vex. We can also view αp +(1− α)p0moreexplicitly as a compound lottery, summarizing the overall prob abilitie s from tw osuccessive events: first, a coin flip with weight α, 1 − α that determines whetherthe lottery p or p0should be used to determine the ultima te consequences; second,either the lottery p or p0. Following up on our Big Game example, the compoundlottery is: first the quarterbac k decision is made, then the game is pla yed.pp2=1p3=1p’p1=1αp + (1-α)p’α1-αpp’Figure 2: A Compound LotteryFigure 2 illustrates the idea of a compound lottery as a two-stage process, and asa mathematical fact about con vexity.42.2 Preference AxiomsNatura lly a rational decision-maker

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