1、Chapter FourUtility效用效用What Do We Do in This Chapter?u We create a mathematical measure of preference in order to advance our analysis.Utility FunctionsuA preference relation that is complete,reflexive,transitive can be represented by a utility function.Utility FunctionsuA utility function U(x)repre
2、sents a preference relation if and only if:x x”U(x)U(x”)x x”U(x)0 and b 0 is called a Cobb-Douglas utility function.uE.g.U(x1,x2)=x11/2 x21/2 (a=b=1/2)V(x1,x2)=x1 x23 (a=1,b=3)Cobb-Douglas Indifference Curvesx2x1All curves are hyperbolic,asymptoting to,but nevertouching any axis.Marginal UtilitiesuM
3、arginal means“incremental”.uThe marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes;i.e.MUUxii Marginal Utilities and Marginal Rates-of-SubstitutionuThe general equation for an indifference curve is U(x1,x2)k,a constant.Totally diffe
4、rentiating this identity gives UxdxUxdx11220 Marginal Utilities and Marginal Rates-of-Substitution UxdxUxdx11220 UxdxUxdx2211 rearranged isMarginal Utilities and Marginal Rates-of-Substitution UxdxUxdx2211 rearranged isAnddxdxUxUx2112 /.This is the MRS.Marg.Rates-of-Substitution for Quasi-linear Uti
5、lity FunctionsuA quasi-linear utility function is of the form U(x1,x2)=f(x1)+x2.so Uxfx11 ()Ux21 MRSdxdxUxUxfx 21121 /().Marg.Rates-of-Substitution for Quasi-linear Utility FunctionsuMRS=-f (x1)does not depend upon x2 so the slope of indifference curves for a quasi-linear utility function is constan
6、t along any line for which x1 is constant.What does that make the indifference map for a quasi-linear utility function look like?Marg.Rates-of-Substitution for Quasi-linear Utility Functionsx2x1Each curve is a vertically shifted copy of the others.MRS is a constantalong any line for which x1 isconst
7、ant.MRS=-f(x1)MRS=-f(x1”)x1x1”Monotonic Transformations&Marginal Rates-of-SubstitutionuMore generally,if V=f(U)where f is a strictly increasing function,thenMRSVxVxfUUxfUUx /()/()/1212 UxUx/.12So MRS is unchanged by a positivemonotonic transformation.The Key to this Chapteru The indifference curve of a consumer preference can be represented by a utility function based equation:U(x1,x2)=k,a constant.
