# I p p 1 p n y income weak preference relation b

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Chapter 15 / Exercise C4
Introductory Econometrics: A Modern Approach
Wooldridge
Expert Verified
ip(p1; :::; pn)yincome<weak preference relationBrevealed preference relationUutility function¹utility levelTable 4.1: The Consumer: Basic NotationDe³nition 4.1A bundlexisrevealed preferredto a bundlex0(written insymbolsxBx0) ifxis actually selected whenx0was also available to the con-sumer.The idea is almost self-explanatory and is given operational content by thefollowing axiom.Axiom 4.1 (Axiom of rational choice)The consumer always makes a choice,and selects the most preferred bundle that is available.This means that we can draw inferences about a person´s preferences byobserving the person´s choices; it suggests that we might adopt the followingsimple µbut very powerful µassumption.Axiom 4.2 (Weak Axiom of Revealed Preference)IfxBx0thenx07x:In the case where purchases are made in a free market this has a very simpleinterpretation.Suppose that at pricespthe household could a/ord to buyeither of two commodity bundles,xorx0; assume thatxis actually bought.Now imagine that prices change fromptop0(while income remains unchanged);if the household now selectsx0then the weak axiom of revealed preference statesthatxcannot be a/ordable at the new pricesp0. Thus the axiom means thatifnXi=1pixi³nXi=1pix0i(4.3)thennXi=1p0ixi>nXi=1p0ix0i(4.4)If you do not choose today something that you chose yesterday (when today´sbundle was also available and a/ordable) it must be because now you cannota/ord yesterday´s bundle: see Figure 4.3.
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Chapter 15 / Exercise C4
Introductory Econometrics: A Modern Approach
Wooldridge
Expert Verified
74CHAPTER 4.THE CONSUMERFigure 4.3:xis chosen Monday;x0is chosen TuesdayFigure 4.4: Extension of the revealed preference concept
4.4.PREFERENCES: AXIOMATIC APPROACH75You can get a long way in consumption theory with just this. Indeed with alittle experimentation it seems as though we are almost sketching out the resultof the kind of cost-minimisation experiment that we performed for the ³rm, inwhich we traced out a portion of a contour of the production function. Perhapswe might even suspect that we are on the threshold of discovering a counterpartto isoquants by the back door (we come to a discussion of ±indi/erence curves²on page 77 below). For example, examine Figure 4.4: letxBx0, andx0Bx00,and letN(x)denote the set of points to whichxis not revealed-preferred. Nowconsider the set of consumptions represented by the unshaded area: this isN(x)\N(x0)\N(x00)and sincexis revealed preferred tox0(which in turn is revealedpreferred tox00we might think of this unshaded area as the set of points whichare µdirectly or indirectly µrevealed to be at least as good asx00: the set isconvex and the boundary does look a bit like the kind of contour we discussed inproduction theory. However, there are quite narrow limits to the extent that wecan push the analysis. For example, it would be possible to have the followingkind of behaviour:xBx0,x0Bx00,x00Bx000and yet alsox000Bx. To avoid thisproblem actually you need an additional axiom µthe Strong Axiom of Revealed