现代观点讲义中级微观ch040utility theory v

1、Chapter FourUtilityPreferences - A Reminderux y: x is preferred strictly to y.ux y: x and y are equally preferred.ux y: x is preferred at least as much as is y.p pf fPreferences - A ReminderuCompleteness: For any two bundles x and y it is always possible to state either that x y or that y x.f ff fuR

2、eflexivity: Any bundle x is always at least as preferred as itself; i.e. x x.f ff ff ff fuTransitivity: x y and y z x z.Utility FunctionsuA preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.uContinuity means that small chan

3、ges to a consumption bundle cause only small changes to the preference level.Utility FunctionsuA utility function 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 indiff

4、erence curve contains equally preferred bundles.uEqual preference same utility level.uTherefore, all bundles in 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,

5、3) is in the indiff. curve with utility level U 6.uOn an indifference curve diagram, this preference information looks as follows:Utility Functions & Indiff. CurvesU 6U 4(2,3) (2,2) (4,1)x1x2p p0123Constructing a utility function from indifference curvesMeasures distance from originx1x2Utility F

6、unctions & Indiff. CurvesuAnother way to visualize this same information is to plot the utility level on a vertical axis.U(2,3) = 6U(2,2) = 4 U(4,1) = 4Utility Functions & Indiff. Curves3D plot of consumption & utility levels for 3 bundlesx1x2UtilityUtility Functions & Indiff. Curves

7、uThis 3D visualization of preferences can be made more informative by adding into it the two indifference curves.Utility Functions & Indiff. CurvesU 4 4U 6 6Higher indifferencecurves containmore preferredbundles.Utilityx2x1Utility Functions & Indiff. CurvesuComparing more bundles will create

8、 a larger collection of all indifference curves and a better description of the consumers preferences.Utility Functions & Indiff. CurvesU 6U 4U 2x1x2Utility Functions & Indiff. CurvesuAs before, this can be visualized in 3D by plotting each indifference curve at the height of its utility ind

9、ex.Utility Functions & Indiff. CurvesU 6U 5U 4U 3U 2U 1x1x2UtilityUtility Functions & Indiff. CurvesuComparing all possible consumption bundles gives the complete collection of the consumers indifference curves, each with its assigned utility level.uThis complete collection of indifference c

10、urves completely represents the consumers preferences.Utility Functions & Indiff. Curvesx1Utility 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

11、other.Utility FunctionsuThere is no unique utility function representation of a preference relation.uSuppose 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,

12、2).p pUtility FunctionsuU(x1,x2) = x1x2 (2,3) (4,1) (2,2).uDefine V = U2.p pUtility FunctionsuU(x1,x2) = 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

13、.p pp pUtility FunctionsuU(x1,x2) = x1x2 (2,3) (4,1) (2,2).uDefine W = 2U + 10.p pUtility FunctionsuU(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 th

14、e same preferences.p pp pUtility FunctionsuIf U is a utility function that 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

15、 preferred bundle).uA bad is a commodity unit which decreases utility (gives 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 lit

16、tle extra water is a neutral.UtilityfunctionSome Other Utility Functions and 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 Curves55991313x1x

17、2x1 + x2 = 5x1 + x2 = 9x1 + x2 = 13V(x1,x2) = x1 + x2.Perfect Substitution 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, cons

18、ider W(x1,x2) = minx1,x2.What do the indifference curves for this “perfect 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 58358

19、minx1,x2 = 5minx1,x2 = 3All are right-angled with vertices on a rayfrom the 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.uE.g. U(x1,x2) = 2x11/2 + x2.Quasi-line

20、ar Indifference Curvesx2x1Each curve is a vertically shifted copy 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)

21、 = x1 x23 (a = 1, b = 3)Cobb-Douglas Indifference Curvesx2x1All curves are hyperbolic,asymptoting to, but nevertouching any axis.Marginal 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

22、; 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 thenMUUxxx1111 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 U

23、tilities and Marginal Rates-of-SubstitutionuThe general 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

24、 Rates-of-Substitution UxdxUxdx2211 rearranged isAnddxdxUxUx2112 /.This is the MRS.Marg. Utilities & Marg. Rates-of-Substitution; An exampleMRSxx 21 MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1.x1x28616U = 8U = 36U(x1,x2) = x1x2;Marg. Rates-of-Substitution for Quasi-linear Utility FunctionsuA qua

25、si-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 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 constan