Indifference Curves 2 The Marginal Rate of Substitution

無異曲線(Indifference Curves ) 2. 邊際替代率(The Marginal Rate of Substitution) 3. 效用函數(The Utility Function) l 邊際效用(Marginal Utility) 4. 特殊函數型態(Some Special Functional Forms) 1. 2

y • A x 5

y A • • B IC 1 x 9

y A • • B IC 1 x 13

y A • (. 5 A, . 5 B) • C IC 2 • B IC 1 x 15

Definition: The utility function measures the level of satisfaction that a consumer receives from any basket of goods. 21

l The utility function assigns a number to each basket l More preferred baskets get a higher number than less preferred baskets. l Utility is an ordinal concept l The precise magnitude of the number that the function assigns has no significance. 22

Ordinal ranking gives information about the order in which a consumer ranks baskets l E. g. a consumer may prefer A to B, but we cannot know how much more she likes A to B l Cardinal ranking gives information about the intensity of a consumer’s preferences. l We can measure the strength of a consumer’s preference for A over B. l 23

Example: Consider the result of an exam • • An ordinal ranking lists the students in order of their performance E. g. , Harry did best, Sean did second best, Betty did third best, and so on. A cardinal ranking gives the marks of the exam, based on an absolute marking standard E. g. Harry got 90, Sean got 85, Betty got 80, and so on. 24

Implications of an ordinal utility function: Difference in magnitudes of utility have no interpretation per se l Utility is not comparable across individuals l Any transformation of a utility function that preserves the original ranking of bundles is an equally good representation of preferences. eg. U = xy + 2 U = 2 xy all represent the same preferences. l 25

y Example: Utility and a single indifference curve 5 2 0 10 = xy 2 5 x 26

y Example: Utility and a single indifference curve Preference direction 5 20 = xy 2 0 10 = xy 2 5 x 27

Definition: The marginal utility of good x is the additional utility that the consumer gets from consuming a little more of x MUx = d. U dx l It is is the slope of the utility function with respect to x. l It assumes that the consumption of all other goods in consumer’s basket remain constant. 28

Definition: The principle of diminishing marginal utility states that the marginal utility of a good falls as consumption of that good increases. Note: A positive marginal utility implies monotonicity. 29

Example: Relative Income and Life Satisfaction (within nations) Relative Income Lowest quartile Second quartile Third quartile Highest quartile Percent > “Satisfied” 70 78 82 85 Source: Hirshleifer, Jack and D. Hirshleifer, Price Theory and Applications. Sixth Edition. Prentice Hall: Upper Saddle River, New Jersey. 1998. 30

We can express the MRS for any basket as a ratio of the marginal utilities of the goods in that basket Suppose the consumer changes the level of consumption of x and y. Using differentials: d. U = MUx. dx + MUy. dy l Along a particular indifference curve, d. U = 0, so: 0 = MUx. dx + MUy. dy l 31

Solving for dy/dx: dy = _ MUx dx MUy l By definition, MRSx, y is the negative of the slope of the indifference curve: MRSx, y = MUx MUy l 32

l Diminishing marginal utility implies the indifference curves are convex to the origin (implies averages preferred to extremes) 33

Example: U= (xy)0. 5 MUx=y 0. 5/2 x 0. 5 MUy=x 0. 5/2 y 0. 5 • • Marginal utility is positive for both goods: => Monotonicity satisfied Diminishing marginal utility for both goods => Averages preferred to extremes Marginal rate of substitution: MRSx, y = MUx = y MUy x Indifference curves do not intersect the axes 34