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Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 THE EXPECTED UTILITY CRITERION 1. Introduction We finally turn to the main subject matter of this course: choice under risk and uncertainty, and in particular expected-utility theory . I will first review the notions of preference relation and representation are in the context of choice among arbitrary objects. I say review because this is essentially the notion of preferences relevant to the theory of the consumer. [If you havent seen consumer theory, then by all means look closely at Section 2!] Next, I will subject you to a short (well, not so short really) philosophical discussion of the difference between risk and uncertainty . Feel free to skip it if you are so inclined... but at least skimming it would be good for you! We will then (finally!) turn to expected-utility theory. I will introduce it as a criterion that helps us construct reasonable preferences over random variables . Alternatively, we can view it as a possible description of preferences over random variables . The first view is normative: we use the theory to inform choices and behavior. The second is descriptive: we observe preferences and try to make sense of them in some coherent way. But, in both cases, it is important to emphasize that the objects of choice are random variables. 2. Choice Theory: a review Recall your intermediate micro consumer theory: the consumer is faced with a choice among a set of alternatives, or consumption bundles. At the most basic (and abstract) level, a choice problem is simply a set O of objects, and a binary relation < on O that captures the individuals preferences: x < y means that the individual prefers x to y . In our setting, O will be a collection of random variables we may be interested in: e.g. as- sets delivering different returns with different probabilities, consumption-savings decisions when production is subject to shocks, etc. Howeve, before we tackle choice under risk and uncertainty, I wish to remind you of two important aspects of abstract choice theory (equivalently, of choice under certainty). First, I mentioned that the most basic object of interest is the DMs preference relation over alternatives O . Formally: Definition 1. Let O be a set (of alternatives). A preference on O is a binary relation R OO . A preference < on O is complete if, for all x,y O , either x < y or y < x (or both) must hold; it is transitive if, for all x,y,z O , x < y and y < z imply x < z . That is: formally, a preference is a set of ordered pairs of elements from the set O . So, we could write ( x,y ) < instead of x < y , but typically we opt for the latter, clearer notation. Moreover, these pairs are ordered: that is, x < y is not the same as y < x . For instance, let O represent the members of your family, and let < be defined so that x < y means x is the father of y : then x < y actually implies that it is not the case that y < x !...
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