北大微观经济学课件]ch4Utility.ppt

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1、Chapter Four,Utility效用,Structure,Utility function(效用函数）DefinitionMonotonic transformation（单调转换）Examples of utility functions and their indifference curvesMarginal utility（边际效用）Marginal rate of substitution 边际替代率MRS after monotonic transformation,Utility Functions,A utility function U(x)represents a

2、preference relation if and only if:x x”U(x)U(x”)x x”U(x)U(x”)x x”U(x)=U(x”).,p,p,Utility Functions,Utility is an ordinal(i.e.ordering)concept.序数效用E.g.if U(x)=6 and U(y)=2 then bundle x is strictly preferred to bundle y.But x is not preferred three times as much as is y.,Utility Functions&Indiff.Curv

3、es,Consider the bundles(4,1),(2,3)and(2,2).Suppose(2,3)(4,1)(2,2).Assign to these bundles any numbers that preserve the preference ordering;e.g.U(2,3)=6 U(4,1)=U(2,2)=4.Call these numbers utility levels.,p,Utility Functions&Indiff.Curves,An indifference curve contains equally preferred bundles.Equal

4、 preference same utility level.Therefore,all bundles in an indifference curve have the same utility level.,Utility Functions&Indiff.Curves,So the bundles(4,1)and(2,2)are in the indiff.curve with utility level U 4But the bundle(2,3)is in the indiff.curve with utility level U 6.On an indifference curv

5、e diagram,this preference information looks as follows:,Utility Functions&Indiff.Curves,U 6,U 4,(2,3)(2,2)(4,1),x1,x2,p,Utility Functions&Indiff.Curves,Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumers preferences.,Utility Fun

6、ctions&Indiff.Curves,U 6,U 4,U 2,x1,x2,Utility Functions&Indiff.Curves,The collection of all indifference curves for a given preference relation is an indifference map.An indifference map is equivalent to a utility function;each is the other.,Utility Functions,There is no unique utility function rep

7、resentation of a preference relation.Suppose U(x1,x2)=x1x2 represents a preference relation.Again consider the bundles(4,1),(2,3)and(2,2).,Utility Functions,U(x1,x2)=x1x2,soU(2,3)=6 U(4,1)=U(2,2)=4;that is,(2,3)(4,1)(2,2).,p,Utility Functions,U(x1,x2)=x1x2(2,3)(4,1)(2,2).Define V=U2.,p,Utility Funct

8、ions,U(x1,x2)=x1x2(2,3)(4,1)(2,2).Define V=U2.Then V(x1,x2)=x12x22 and V(2,3)=36 V(4,1)=V(2,2)=16so again(2,3)(4,1)(2,2).V preserves the same order as U and so represents the same preferences.,p,p,Utility Functions,U(x1,x2)=x1x2(2,3)(4,1)(2,2).Define W=2U+10.,p,Utility Functions,U(x1,x2)=x1x2(2,3)(4

9、,1)(2,2).Define W=2U+10.Then W(x1,x2)=2x1x2+10 so W(2,3)=22 W(4,1)=W(2,2)=18.Again,(2,3)(4,1)(2,2).W preserves the same order as U and V and so represents the same preferences.,p,p,Utility Functions:Monotonic Transformation,If U is a utility function that represents a preference relation and f is a

10、strictly increasing function,then V=f(U)is also a utility functionrepresenting.,Goods,Bads and Neutrals,A good is a commodity unit which increases utility(gives a more preferred bundle).A bad is a commodity unit which decreases utility(gives a less preferred bundle).A neutral is a commodity unit whi

11、ch does not change utility(gives an equally preferred bundle).,Goods,Bads and Neutrals,Utility,Water,x,Units ofwater aregoods,Units ofwater arebads,Around x units,a little extra water is a neutral.,Utilityfunction,Some Other Utility Functions and Their Indifference Curves,Perfect substitute V(x1,x2)

12、=x1+x2.Perfect complementW(x1,x2)=minx1,x2Quasi-linearU(x1,x2)=f(x1)+x2Cobb-Douglas Utility FunctionU(x1,x2)=x1a x2bWhat do the indifference curves for these utility functions look like?,Perfect Substitution Indifference Curves,5,5,9,9,13,13,x1,x2,x1+x2=5,x1+x2=9,x1+x2=13,All are linear and parallel

13、.,V(x1,x2)=x1+x2.,Perfect Complementarity Indifference Curves,x2,x1,45o,minx1,x2=8,3,5,8,3,5,8,minx1,x2=5,minx1,x2=3,All are right-angled with vertices on a rayfrom the origin.,W(x1,x2)=minx1,x2,Quasi-Linear Utility Functions,A utility function of the form U(x1,x2)=f(x1)+x2is linear in just x2 and i

14、s called quasi-linear（准线性）.E.g.U(x1,x2)=2x11/2+x2.,Quasi-linear Indifference Curves,x2,x1,Each curve is a vertically shifted copy of the others.,Cobb-Douglas Utility Function,Any utility function of the form U(x1,x2)=x1a x2bwith a 0 and b 0 is called a Cobb-Douglas utility function.E.g.U(x1,x2)=x11/

15、2 x21/2(a=b=1/2)V(x1,x2)=x1 x23(a=1,b=3),Cobb-Douglas Indifference Curves,x2,x1,Marginal Utilities,Marginal means“incremental”.The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes;i.e.,Marginal Utilities,If U(x1,x2)=x11/2 x22 then

16、,Marginal Utilities and Marginal Rates-of-Substitution,The general equation for an indifference curve is U(x1,x2)k,a constant.Totally differentiating this identity gives,Marginal Utilities and Marginal Rates-of-Substitution,rearranged is,This is the MRS.,Marg.Utilities An example,Suppose U(x1,x2)=x1

17、x2.Then,so,Marg.Utilities An example,MRS(1,8)=-8/1=-8 MRS(6,6)=-6/6=-1.,x1,x2,8,6,1,6,U=8,U=36,U(x1,x2)=x1x2;,Marg.Rates-of-Substitution for Quasi-linear Utility Functions,A quasi-linear utility function is of the form U(x1,x2)=f(x1)+x2.,so,Marg.Rates-of-Substitution for Quasi-linear Utility Functio

18、ns,x2,x1,MRS=-f(x1)does not depend upon x2.,MRS is a constantalong any line for which x1 isconstant.,MRS=-f(x1),MRS=-f(x1”),x1,x1”,Monotonic Transformations&Marginal Rates-of-Substitution,Applying a monotonic transformation to a utility function representing a preference relation simply creates anot

19、her utility function representing the same preference relation.What happens to marginal rates-of-substitution when a monotonic transformation is applied?,Monotonic Transformations&Marginal Rates-of-Substitution,For U(x1,x2)=x1x2 the MRS=-x2/x1.Create V=U2;i.e.V(x1,x2)=x12x22.What is the MRS for V?which is the same as the MRS for U.,Monotonic Transformations&Marginal Rates-of-Substitution,More generally,if V=f(U)where f is a strictly increasing function,then,So MRS is unchanged by a positivemonotonic transformation.,