Lecture I: Linkage Between Utility in Finance and Utility in Economics and Axioms of Expected Utility I. Utility in Finance and Economics A. From your first course in economics, you were exposed to the idea of a utility function. Some may have heard the terms cardinal versus ordinal utility. Historically, the cardinal or absolute utility measure is older, but requires more stringent assumptions. Ordinal, on the other hand is sufficient for most supply and demand derivations. 1. Axioms of Ordinal Utility Axiom 1 (Completeness): For every pair of vectors X 0 ú + and Z 0 ú + either X š Z or Z š X . Axiom 2 (Reflexivity): For every X 0 ú + X š X . Axiom 3 (Transitivity): If X š Y and Y š Z , then X š Z . Axiom 4 (Continuity): For every X 0 ú + , the two subsets of all strictly preferred and all strictly worse bundles are both open. 2. Given these four axioms, there exists an ordinal utility function which describes the individual’s preferences. U( X ) > U( Z ) ⇔ X > Z U( X ) = U( Z ) ⇔ X = Z 3. Coupled with the assumption that U( X ) is twice differentiable and strictly concave, these axioms are typically sufficient for the derivation of the consumer’s problem. In addition, the emphasis on ordinal functions leaves the optimal solution invariant to positive monotonic transformation. B. The Basic consumer problem is to maximize utility subject to a budget constraint. The problem also has a mathematical dual which is to minimize expenditures to obtain a given level of utility. This duality is of frequent use in financial economics. Hence, we will spend some time developing the concept of duality in consumption. In addition, the indirect utility function can be viewed as a forerunner of the cardinal utility of wealth which will be developed later.
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