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Unformatted text preview: Preferences and Utility Simon Board * This Version: November 18, 2008 First Version: November 18, 2008. These lectures examine the preferences of a single agent. In Section 1 we analyse how the agent chooses among a number of competing alternatives, investigating when preferences can be represented by a utility function. In Section 2 we discuss two attractive properties of preferences: monotonicity and convexity. In Section 3 we analyse the agent’s indifference curves and ask how she makes tradeoffs between different goods. Finally, in Section 4 we look at some examples of preferences, applying the insights of the earlier theory. 1 The Foundation of Utility Functions 1.1 A Basic Representation Theorem Suppose an agent chooses from a set of goods X = { a,b,c,... } . For example, one can think of these goods as different TV sets or cars. Given two goods, x and y , the agent weakly prefers x over y if x is at least as good as y . To avoid us having to write “weakly prefers” repeatedly, we simply write x < y . We now put some basic structure on the agent’s preferences by adopting two axioms. 1 Completeness Axiom: For every pair x,y ∈ X , either x < y , y < x , or both. * Department of Economics, UCLA. http://www.econ.ucla.edu/sboard/. Please email suggestions and typos to [email protected] 1 An axiom is a foundational assumption. 1 Eco11, Fall 2008 Simon Board Transitivity Axiom: For every triple x,y,z ∈ X , if x < y and y < z then x < y . An agent has complete preferences if she can compare any two objects. An agent has transitive preferences if her preferences are internally consistent. Let’s consider some examples. First, suppose that, given any two cars, the agent prefers the faster one. These preferences are complete: given any two cars x and y , then either x is faster, y is faster or they have the same speed. These preferences are also transitive: if x is faster than y and y is faster than z , then x is faster than z . Second, suppose that, given any two cars, the agent prefers x to y if it is both faster and bigger. These preferences are transitive: if x is faster and bigger than y and y is faster and bigger than z , then x is faster and bigger than z . However, these preferences are not complete: an SUV is bigger and slower than a BMW, so it is unclear which the agent prefers. The completeness axiom says these preferences are unreasonable: after examining the SUV and BMW, the agent will have a preference between the two. Third, suppose that the agent prefers a BMW over a Prius because it is faster, an SUV over a BMW because it is bigger, and a Prius over an SUV, because it is more environmentally friendly. In this case, the agent’s preferences cycle and are therefore intransitive. The transitivity axiom says these preferences are unreasonable: if environmental concerns are so important to the agent, then she should also take them into account when choosing between the Prius and BMW, and the BMW and the SUV....
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 Fall '08
 cunningham
 Utility, indifference curves, Convex function, Simon Board

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