微观经济学课件：第4章 Utility

1、Chapter FourUtilityStructureuUtility function (效用函数）效用函数） Definition Monotonic transformation （单调转换）单调转换） Examples of utility functions and their indifference curvesuMarginal utility （边际效用）边际效用）uMarginal rate of substitution 边际替代率边际替代率 MRS after monotonic transformationUtility FunctionsuA utility fu

2、nction U(x) represents a preference relation if and only if: x x” U(x) U(x”) x x” U(x) U(4,1) = U(2,2) = 4.uCall these numbers utility levels.p pUtility Functions & Indiff. CurvesuAn indifference curve contains equally preferred bundles.uEqual preference same utility level.uTherefore, all bundles in

3、 an indifference curve have the same utility level.Utility Functions & Indiff. CurvesuSo the bundles (4,1) and (2,2) are in the indiff. curve with utility level U 4 4uBut the bundle (2,3) is in the indiff. curve with utility level U 6.uOn an indifference curve diagram, this preference information lo

4、oks as follows:Utility Functions & Indiff. CurvesU 6U 4(2,3) (2,2) (4,1)x1x2p pUtility Functions & Indiff. CurvesuComparing more bundles will create a larger collection of all indifference curves and a better description of the consumers preferences.Utility Functions & Indiff. CurvesU 6U 4U 2x1x2Uti

5、lity Functions & Indiff. CurvesuThe collection of all indifference curves for a given preference relation is an indifference map.uAn indifference map is equivalent to a utility function; each is the other.Utility FunctionsuThere is no unique utility function representation of a preference relation.u

6、Suppose U(x1,x2) = x1x2 represents a preference relation.uAgain consider the bundles (4,1),(2,3) and (2,2).Utility FunctionsuU(x1,x2) = x1x2, soU(2,3) = 6 U(4,1) = U(2,2) = 4;that is, (2,3) (4,1) (2,2).p pUtility FunctionsuU(x1,x2) = x1x2 (2,3) (4,1) (2,2).uDefine V = U2.p pUtility FunctionsuU(x1,x2

7、) = x1x2 (2,3) (4,1) (2,2).uDefine V = U2.uThen V(x1,x2) = x12x22 and V(2,3) = 36 V(4,1) = V(2,2) = 16so again(2,3) (4,1) (2,2).uV preserves the same order as U and so represents the same preferences.p pp pUtility FunctionsuU(x1,x2) = x1x2 (2,3) (4,1) (2,2).uDefine W = 2U + 10.p pUtility FunctionsuU

8、(x1,x2) = x1x2 (2,3) (4,1) (2,2).uDefine W = 2U + 10.uThen W(x1,x2) = 2x1x2+10 so W(2,3) = 22 W(4,1) = W(2,2) = 18. Again,(2,3) (4,1) (2,2).uW preserves the same order as U and V and so represents the same preferences.p pp pUtility Functions: Monotonic TransformationuIf U is a utility function that

9、represents a preference relation and f is a strictly increasing function,u then V = f(U) is also a utility functionrepresenting . f ff fGoods, Bads and NeutralsuA good is a commodity unit which increases utility (gives a more preferred bundle).uA bad is a commodity unit which decreases utility (give

10、s a less preferred bundle).uA neutral is a commodity unit which does not change utility (gives an equally preferred bundle).Goods, Bads and NeutralsUtilityWaterxUnits ofwater aregoodsUnits ofwater arebadsAround x units, a little extra water is a neutral.UtilityfunctionSome Other Utility Functions an

11、d Their Indifference CurvesuInstead of U(x1,x2) = x1x2 consider V(x1,x2) = x1 + x2.What do the indifference curves for this “perfect substitution” utility function look like?Perfect Substitution Indifference Curves55991313x1x2x1 + x2 = 5x1 + x2 = 9x1 + x2 = 13V(x1,x2) = x1 + x2.Perfect Substitution

12、Indifference Curves55991313x1x2x1 + x2 = 5x1 + x2 = 9x1 + x2 = 13All are linear and parallel.V(x1,x2) = x1 + x2.Some Other Utility Functions and Their Indifference CurvesuInstead of U(x1,x2) = x1x2 or V(x1,x2) = x1 + x2, consider W(x1,x2) = minx1,x2.What do the indifference curves for this “perfect

13、complementarity” utility function look like?Perfect Complementarity Indifference Curvesx2x145ominx1,x2 = 83 58358minx1,x2 = 5minx1,x2 = 3W(x1,x2) = minx1,x2Perfect Complementarity Indifference Curvesx2x145ominx1,x2 = 83 58358minx1,x2 = 5minx1,x2 = 3All are right-angled with vertices on a rayfrom the

14、 origin.W(x1,x2) = minx1,x2Some Other Utility Functions and Their Indifference CurvesuA utility function of the form U(x1,x2) = f(x1) + x2is linear in just x2 and is called quasi-linear （准线性）准线性）.uE.g. U(x1,x2) = 2x11/2 + x2.Quasi-linear Indifference Curvesx2x1Each curve is a vertically shifted copy

15、 of the others.Some Other Utility Functions and Their Indifference CurvesuAny utility function of the form U(x1,x2) = x1a x2bwith a 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 Curvesx2x1Margi

16、nal UtilitiesuMarginal 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 UtilitiesuE.g. if U(x1,x2) = x11/2 x22 thenMUUxxx1111 22212 /Marginal UtilitiesuE.g. if U(x1,x2) = x11/2 x22 th

17、enMUUxxx1111 22212 /Marginal UtilitiesuE.g. if U(x1,x2) = x11/2 x22 thenMUUxxx2211 222 /Marginal UtilitiesuE.g. if U(x1,x2) = x11/2 x22 thenMUUxxx2211 222 /Marginal UtilitiesuSo, if U(x1,x2) = x11/2 x22 thenMUUxxxMUUxxx1111 2222211 22122 /Marginal Utilities and Marginal Rates-of-SubstitutionuThe gen

18、eral equation for an indifference curve is U(x1,x2) k, a constant.Totally differentiating this identity gives UxdxUxdx11220 Marginal Utilities and Marginal Rates-of-Substitution UxdxUxdx11220 UxdxUxdx2211 rearranged isMarginal Utilities and Marginal Rates-of-Substitution UxdxUxdx2211 rearranged isAn

19、ddxdxUxUx2112 /.This is the MRS.Marg. Utilities & Marg. Rates-of-Substitution; An exampleuSuppose U(x1,x2) = x1x2. Then UxxxUxxx12221111 ( )()()( )MRSdxdxUxUxxx 211221 /.soMarg. Utilities & Marg. Rates-of-Substitution; An exampleMRSxx 21 MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1.x1x28616U = 8U = 3

20、6U(x1,x2) = x1x2;Marg. Rates-of-Substitution for Quasi-linear Utility 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

21、slope of indifference curves for a quasi-linear utility function is constant 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 isconstant. MRS =- f(x1)MRS = -f(x1”)x1x1”Monotonic Transformations & Marginal Rates-of-Substitution