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Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 RISK AND RISK AVERSION 1. Introduction The main reason why expected-utility theory is interesting is the phenomenon of risk aversion . The key idea is that people are typically unwilling to take up fair bets, i.e. monetary lotteries that yield zero in expectation. This lecture develops these ideas further, and makes them precise. Doing so uncovers important details, such as the fact that your willingness to accept a bet may depend on your wealth. A typical pattern we shall follow throughout this course emerges here, as well: we begin by formalizing an intuitive idea, risk aversion, in terms of simple but precise behavioral properties. Then, we seek a tractable characterization of these properties. Please keep this pattern in mind. Terminology and notation . Henceforth, we use the abbreviation EU for expected utility and DM for decision maker. The set X of prizes (the values that random variables of interest can take) will be some subset of R . Also, and crucially, when we say that a preference relation < on a set F of random variable is an EU preference with Bernoulli utility u : R R we mean that, for every X,Y F , X < Y if and only if E[ u ( X )] E[ u ( Y )]. 2. Risk Aversion Defined Recall that, in the St. Petersburg Paradox, a person who ranks random variables according to their expected value would be willing to pay an infinite amount of money in order to receive the benefits of the bet we described. Of course, most people would be willing to pay a relatively small sum for the privilege of participating in that lottery; in particular, an expected-utility DM with logarithmic utility is exactly indifferent between the certain sum $2 (equivalently: the degenerate random variable that equals $2 with probability 1) and the St. Petersburg bet. This leads to the following general idea: an individual is risk-averse if, for any r.v. X , she (weakly) prefers the certain quantity E[ X ] to X itself. We can restate this a bit more formally as follows. Our main object of interest is a preference relation < on a set F of random variables. It is tempting to write something like E[ X ] < X to mean that the certain quantity E[ X ] is at least as good as X , but this would be imprecise: E[ X ] is a number, not a random variable! So, we need the following, standard notation: the symbol x indicates the degenerate r.v. that takes up the value x R with probability one. Thus, to formalize the assumption that the DM prefers E[ X ] to X , we can write E[ X ] < X . This works! It is useful to notice that, since x is a r.v., we can compute its expected utility E[ u ( x )]; however, since the probability distribution of x is simply ( x, 1), we have E[ u ( x )] = u ( x ). Be sure you understand why this is the case....
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