1、Chapter Four,Utility,Preferences - A Reminder,x y: x is preferred strictly to y. x y: x and y are equally preferred. x y: x is preferred at least as much as is y.,p,Preferences - A Reminder,Completeness: For any two bundles x and y it is always possible to state either that x y or that y x.,Preferen
2、ces - A Reminder,Reflexivity: Any bundle x is always at least as preferred as itself; i.e. x x.,Preferences - A Reminder,Transitivity: If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i.e. x y and y z x z.,Utility Functions,A preference r
3、elation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function. Continuity means that small changes to a consumption bundle cause only small changes to the preference level.,Utility Functions,A utility function U(x) represents a preference relation
4、 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. 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
5、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 pr
6、eferred bundles. Equal 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 4 But the bundle (2,3) is in the indiff. curve with utili
7、ty level U 6. On an indifference curve 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,Another way to visualize this same information is to plot the utility level on a vertical axis.,U(2,3)
8、 = 6,U(2,2) = 4 U(4,1) = 4,Utility Functions & Indiff. Curves,3D plot of consumption & utility levels for 3 bundles,x1,x2,Utility,Utility Functions & Indiff. Curves,This 3D visualization of preferences can be made more informative by adding into it the two indifference curves.,Utility Functions & In
9、diff. Curves,U 4,U 6,Higher indifference curves contain more preferred bundles.,Utility,x2,x1,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 Functions & Indiff. Curves
10、,U 6,U 4,U 2,x1,x2,Utility Functions & Indiff. Curves,As before, this can be visualized in 3D by plotting each indifference curve at the height of its utility index.,Utility Functions & Indiff. Curves,U 6,U 5,U 4,U 3,U 2,U 1,x1,x2,Utility,Utility Functions & Indiff. Curves,Comparing all possible con
11、sumption bundles gives the complete collection of the consumers indifference curves, each with its assigned utility level. This complete collection of indifference curves completely represents the consumers preferences.,Utility Functions & Indiff. Curves,x1,x2,Utility Functions & Indiff. Curves,x1,x
12、2,Utility Functions & Indiff. Curves,x1,x2,Utility Functions & Indiff. Curves,x1,x2,Utility Functions & Indiff. Curves,x1,x2,Utility Functions & Indiff. Curves,x1,x2,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions &
13、Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,x1,Utility Functions & Indiff. Curves,The collection of a
14、ll 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 representation of a preference relation. Suppose U(x1,x2) = x1x2 represents a preference re
15、lation. Again consider the bundles (4,1), (2,3) and (2,2).,Utility Functions,U(x1,x2) = x1x2, so U(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 Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2. The
16、n V(x1,x2) = x12x22 and V(2,3) = 36 V(4,1) = V(2,2) = 16 so 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) (4,1) (2,2). Define W =
17、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,If U is a utility function that represents a preference relation and f is a strictly increasing functio
18、n,then V = f(U) is also a utility function representing .,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 which does not chang
19、e utility (gives an equally preferred bundle).,Goods, Bads and Neutrals,Utility,Water,x,Units of water are goods,Units of water are bads,Around x units, a little extra water is a neutral.,Utility function,Some Other Utility Functions and Their Indifference Curves,Instead of U(x1,x2) = x1x2 consider
20、V(x1,x2) = x1 + x2. What do the indifference curves for this “perfect substitution” utility function look like?,Perfect Substitution Indifference Curves,5,5,9,9,13,13,x1,x2,x1 + x2 = 5,x1 + x2 = 9,x1 + x2 = 13,V(x1,x2) = x1 + x2.,Perfect Substitution Indifference Curves,5,5,9,9,13,13,x1,x2,x1 + x2 =
21、 5,x1 + x2 = 9,x1 + x2 = 13,All are linear and parallel.,V(x1,x2) = x1 + x2.,Some Other Utility Functions and Their Indifference Curves,Instead 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 complementarity” utility function
22、look like?,Perfect Complementarity Indifference Curves,x2,x1,45o,minx1,x2 = 8,3,5,8,3,5,8,minx1,x2 = 5,minx1,x2 = 3,W(x1,x2) = minx1,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 ray from the origin
23、.,W(x1,x2) = minx1,x2,Some Other Utility Functions and Their Indifference Curves,A utility function of the form U(x1,x2) = f(x1) + x2 is linear in just x2 and is called quasi-linear. E.g. U(x1,x2) = 2x11/2 + x2.,Quasi-linear Indifference Curves,x2,x1,Each curve is a vertically shifted copy of the ot
24、hers.,Some Other Utility Functions and Their Indifference Curves,Any utility function of the form U(x1,x2) = x1a x2b with a 0 and b 0 is called a Cobb-Douglas utility function. E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2) V(x1,x2) = x1 x23 (a = 1, b = 3),Cobb-Douglas Indifference Curves,x2,x1,All curve
25、s are hyperbolic, asymptoting to, but never touching any axis.,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,E.g. if U(x1,x2) = x11/2 x22 then,Margin
26、al Utilities,E.g. if U(x1,x2) = x11/2 x22 then,Marginal Utilities,E.g. if U(x1,x2) = x11/2 x22 then,Marginal Utilities,E.g. if U(x1,x2) = x11/2 x22 then,Marginal Utilities,So, if U(x1,x2) = x11/2 x22 then,Marginal Utilities and Marginal Rates-of-Substitution,The general equation for an indifference
27、curve is U(x1,x2) k, a constant. Totally differentiating this identity gives,Marginal Utilities and Marginal Rates-of-Substitution,rearranged is,Marginal Utilities and Marginal Rates-of-Substitution,rearranged is,And,This is the MRS.,Marg. Utilities An example,Suppose U(x1,x2) = x1x2. Then,so,Marg.
28、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 F
29、unctions,MRS = - 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-l
30、inear Utility Functions,x2,x1,Each curve is a vertically shifted copy of the others.,MRS is a constant along any line for which x1 is constant.,MRS = - f(x1),MRS = -f(x1”),x1,x1”,Monotonic Transformations & Marginal Rates-of-Substitution,Applying a monotonic transformation to a utility function repr
31、esenting a preference relation simply creates another 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 positive monotonic transformation.,
