Intermediate Microeconomics Ch4 notes

Intermediate Microeconomics Ch4 notes - Ch 4. Utility We...

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1 Ch 4. Utility We studied in Ch 3 consumer preferences and their associated ICs, and this chapter will discuss how utility functions can be used to describe preferences. Note that a utility function is an algebraic expression of preferences while ICs are a graphical way to illustrate preferences. 4.1 Ordinal Utility A utility function (UF), u , is a way of assigning a # (called utility) to every possible bundle such that more-preferred bundles get larger #s: for X = (x 1 , x 2 ) and Y = (y 1 , y 2 ) , we have () () Y u X u Y X > f . A utility assignment is used to order the bundles of goods, and the size of the utility difference between any two bundles does not matter. That is, the magnitude of utility is not important, and only the ranking matters; this kind of utility is called ordinal utility . See a table on page 55 of the text Thus, if finding one way to assign utility #s to bundles, we can have an infinite # of ways to do it. E.g., if u(X) is a way, then () X u α (for any 0 > ) is just as good a way to assign utilities. A monotonic transformation (MT) is to transform a set of #s into another set of #s in a way that preserves the order of the #s. Let f(.) denote such a transformation that for any u 1 and u 2 , we have () () 2 1 2 1 u f u f u u > > . E.g., f(u) = 3u , f(u) = u – 17, f(u) = u 3 . In this case, 0 > u f and the graph of f always has a positive slope. See figure 4.1.
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2 Result : a MT of a UF represents the same preferences since () () () () ( ) () ( ) ( ) Y v X v e i Y u f X u f Y u X u Y X > > > _ ., . _ f A UF is a way to label ICs by assigning larger #s to higher ICs, and a MT is just a re- labeling of ICs and hence represents the same preferences. Intransitive preferences cannot be represented by any UF. Except this perverse case, we can find an ordinal UF to describe any reasonable preference ordering. See figure 4.2 for monotonic preferences. 4.2 Cardinal Utility If attaching an operational significance to the magnitude of utility, the size of the utility difference between two bundles will make sense. This kind of theory deals with cardinal utility . First, assign a higher utility to a more preferred bundle
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This note was uploaded on 02/19/2009 for the course ECIF ECIF201 taught by Professor Gu during the Spring '09 term at University of Manchester.

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Intermediate Microeconomics Ch4 notes - Ch 4. Utility We...

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