Advanced microeconomics Topic 2: Individual Preferences Primary Readings: DL-Chapter 4; JR-Chapter 3; Varian-Chapter 7 In most economic models, we start with an agent's utility function. The utility function basically maps from bundles that the agent might choose, to the real line. The utility function is quite convenient: it can be maximized and manipulated using mathematical tools. But the question is are we imposing some hidden or desirable assumptions when we take this approach ?prefeent's Is it valid to reduce ple, real- value function, something as complicated preferences over a wide variety of bundles? What does it really mean about the agent's preferences? In this lecture, we will try to answer these questions by analyzing the relationships between axioms about an agent's preferences, and then establishing the existence of a utility function that represents the agent's preferences 2.1 The Consumer's Preferences We let the consumption set, X, represent the set of all alternatives, or complete consumption lans, that the consumer can conceive- whether some of them will be achievable in practice or not Every element of X is called a consumption bundle or a consumption plan X captures the universe of all possible choices a consumer may have. For this reason, the consumption set is also known as the choice set Normally, XcR +-the entire nonnegative orthant of the real space R We will always assume that X is a closed and convex set 2. 1. 2Basic Properties& Axioms of Preferences For x, y e X, when we write x& y, we mean that"the consumer thinks that the bundle x is at least as good as bundle y "We call a a preference relation on X We say, "x is(weakly) preferred to y It is clear that 2 is a binary relation defined on X As the final purpose of introducing a preference relation is to order the set of consumption bundles we need to assume a number of axioms. These axioms of consumer choice are intended to give formal mathematical expression to fundamental aspects of consumer behavior and altitudes toward the objects of choice AXIOM 1:( Completeness)Vx, yE X, (x y)v(y 2 x).(Note:v="or") To satisfy the completeness axiom, the preference 2 cannot be defined so that 12 yox2y, Vj.Reason: it is only a partial ordering. (Note: While this axiom appears innocuous, in combination with the usual confinement of the consumption set to the consumption of the individual only, it rules out externalities in consumption.)1 Advanced Microeconomics Topic 2: Individual Preferences Primary Readings: DL – Chapter 4; JR - Chapter 3; Varian – Chapter 7 In most economic models, we start with an agent's utility function. The utility function basically maps from bundles that the agent might choose, to the real line. The utility function is quite convenient: it can be maximized and manipulated using mathematical tools. But the question is: Is it valid to reduce a simple, real-value function, something as complicated as an agent's preferences over a wide variety of bundles? What does it really mean about the agent's preferences? Are we imposing some hidden or desirable assumptions when we take this approach? In this lecture, we will try to answer these questions by analyzing the relationships between axioms about an agent's preferences, and then establishing the existence of a utility function that represents the agent's preferences. 2.1 The Consumer's Preferences 2.1.1Consumption Set We let the consumption set, X, represent the set of all alternatives, or complete consumption plans, that the consumer can conceive - whether some of them will be achievable in practice or not. Every element of X is called a consumption bundle or a consumption plan. • X captures the universe of all possible choices a consumer may have. For this reason, the consumption set is also known as the choice set. • Normally, X  R m + - the entire nonnegative orthant of the real space Rm. • We will always assume that X is a closed and convex set. 2.1.2Basic Properties & Axioms of Preferences • For x, y  X, when we write x y, we mean that "the consumer thinks that the bundle x is at least as good as bundle y." We call a preference relation on X. • We say, "x is (weakly) preferred to y". • It is clear that is a binary relation defined on X. As the final purpose of introducing a preference relation is to order the set of consumption bundles, we need to assume a number of axioms. These axioms of consumer choice are intended to give formal mathematical expression to fundamental aspects of consumer behavior and altitudes toward the objects of choice. AXIOM 1: (Completeness)  x, y  X, (x y)  (y x). (Note:  = "or") • To satisfy the completeness axiom, the preference cannot be defined so that x y  xj  yj, j. (Reason: it is only a partial ordering.) (Note: While this axiom appears innocuous, in combination with the usual confinement of the consumption set to the consumption of the individual only, it rules out externalities in consumption.)