notes4 - Chapter 4 Topics in Consumer Theory 4.1 Homothetic...

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Chapter 4 Topics in Consumer Theory 4.1 Homothetic and Quasilinear Utility Functions One of the chief activities of economics is to try to recover a consumer’s preferences over all bundles from observations of preferences over a few bundles. If you could ask the consumer an in f nite number of times, “Do you prefer x to y ?”, using a large number of di f erent bundles, you could do ap re t tygoodjobo f f guring out the consumer’s indi f erence sets, which reveals her preferences. However, the problem with this is that it is impossible to ask the question an in f nite number of times. 1 In doing economics, this problem manifests itself in the fact that you often only have a limited number of data points describing consumer behavior. One way that we could help make the data we have go farther would be if observations we made about one particular indi f erence curve could help us understand all indi f erence curves. There are a couple of di f erent restrictions we can impose on preferences that allow us to do this. The f rst restriction is called homotheticity . A preference relation is said to be homothetic if the slope of indi f erence curves remains constant along any ray from the origin. Figure 4.1 depicts such indi f erence curves. If preferences take this form, then knowing the shape of one indi f erence curve tells you the shape of all indi f erence curves, since they are “radial blowups” of each other. Formally, we say a preference relation is homothetic if for any two bundles x and y such that x y ,then αx αy for any α> 0 . We can extend the de f nition of homothetic preferences to utility functions. A continuous 1 In fact, to completely determine the indi f erence sets you would have to ask an uncountably in f nite number of questions, which is even harder. 87
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Nolan Miller Notes on Microeconomic Theory: Chapter 4 ver: Aug. 2006 x 2 x 1 Figure 4.1: Homothetic Preferences preference relation º is homothetic if and only if it can be represented by a utility function that is homogeneous of degree one. In other words, homothetic preferences can be represented by a function u () that such that u ( αx )= αu ( x ) for all x and α> 0 . Note that the de f nition does not say that every utility function that represents the preferences must be homogeneous of degree one — only that there must be at least one utility function that represents those preferences and is homogeneous of degree one. EXAMPLE: Cobb-Douglas Utility : A famous example of a homothetic utility function is the Cobb-Douglas utility function (here in two dimensions): u ( x 1 ,x 2 x a 1 x 1 a 2 : a> 0 . The demand functions for this utility function are given by: x 1 ( p, w aw p 1 x 2 ( p, w (1 a ) w p 2 . Notice that the ratio of x 1 to x 2 does not depend on w . This implies that Engle curves (wealth expansion paths) are straight lines (see MWG pp. 24-25). The indirect utility function is given by: v ( p, w μ aw p 1 a μ (1 a ) w p 2 1 a = w μ a p 1 a μ 1 a p 2 1 a .
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notes4 - Chapter 4 Topics in Consumer Theory 4.1 Homothetic...

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