Utility Theory • For consumer theory do not need to assign precise numerical values to different indifference curves Ordinal Utility • Often convenient to do so especially if dealing with more than 2 goods • To do this, it is convenient to represent the ECON 370: Microeconomic Theory preference relation by a utility function U(x1,x2,…) Summer 2004 – Rice University Stanley Gilbert Lecture 2 Econ 370 - Ordinal Utility 2 Conditions Construction • U(x) is a utility function representing the • Construction of a utility function: preference relation ‘≿’: – Draw the indifference curves – U(x) ≥ U(y) if and only if x y ≿ – Drawing the diagonal (i.e. the line x1 = x2) – For preferences that satisfy A1 – A4 – Labeling each indifference curve with how far from the origin the intersection of that indifference curve and the diagonal it is. Econ 370 - Ordinal Utility 3 Econ 370 - Ordinal Utility 4 1 Identical Representations Equivalence • Many different utility functions can describe same • Our model of consumer behavior would be exactly set of preferences the same whichever of these utility functions were assumed. • Any monotonic transformation of a utility function is still a utility function • See that: U a (x) = [U(x)]2 + 42 • For Example: U ()x = x1x2 a 2 2 b 1 2 U ()x = 4x1 x2 + 42 U (x) = [U(x)] U b ()x = (x x )1 2 1 2 U c (x) = ln[U(x)] c U ()x = ln x1 + ln x2 All three are monotonic transformations of U(x). Econ 370 - Ordinal Utility 5 Econ 370 - Ordinal Utility 6 Characteristics Interpersonal Comparisons • Consumer theory depends only on the shapes of • An associated issue is that of interpersonal indifference curves and the direction in which comparability: utility is increasing (normally northeast). Most – If Jack consumes cans of soda and pizzas and prefers more of each good to less, we can conclude that he economists argue that precise numerical values would prefer 3 cans of soda and 2 pizzas per day to 1 attached to individual indifference curves have no can of soda and 1 pizza per day economic meaning – Suppose, however, that we have two consumers Jack and Jill and Jack consumes 3 cans of soda and 2 pizzas • Because the ordering of indifference curves is all per day while Jill consumes 1 can of soda and 1 pizza that is important, this is described as an ordinal per day. Can we say Jack is happier? theory of utility • There is no objective (i.e. value-free) way to • That is, it can only be used to rank choices for a compare the happiness of the two people single individual • In our analysis we treat utility functions as ordinal Econ 370 - Ordinal Utility 7 Econ 370 - Ordinal Utility 8 2 Common Utility Functions Indifference Curves • Perfect Substitutes: • If we have a utility function u = f(x1, x2): – U(x) = ax + bx a, b > 0 1 2 • Indifference curves are represented by • Perfect Complements: – c = f(x1, x2) – U(x) = min{ ax1, bx2} a, b > 0 – Where ‘c’ is some constant • Cobb-Douglas Preferences α 1 - α – U(x) = x1 x2 0 ≤ α≤ 1 – U(x) = αln(x1) + (1 – α)ln(x2) 0 ≤ α≤ 1 – (Why are these equivalent?) • Quasilinear Preferences – U(x) = x1 + f(x2) Econ 370 - Ordinal Utility 9 Econ 370 - Ordinal Utility 10 Marginal Utility Marginal Utilities and the MRS • Marginal means “incremental”. • The equation of an indifference curve is U(x1,x2) ≡ k, a constant. • MUi = change in U with small change in xi (holding constant consumption of all other goods) • Totally differentiating this identity gives ∂ U MU i = ∂ xi ∂U ∂U dx1 + dx2 = 0 or ∂x1 ∂x2 • For example, if U(x , x ) = x 1/2x 2 1 2 1 2 dx ∂U / ∂x MU (x ) MRS = 2 = − 1 = − 1 • Then, MU(x ) = (½)x -1/2x 2 1 1 2 dx1 ∂U /∂x2 MU (x2 ) 1/2 • And, MU(x2) = 2x1 x2 Econ 370 - Ordinal Utility 11 Econ 370 - Ordinal Utility 12 3 Example of Marginal Utilities and MRS Monotonic Transformations and MRS • Suppose U(x1,x2) = x1x2. Then • Monotonic transformation of U does not change underlying preference structure • Example: U(x1, x2) = x1x2 with MRS = - x2/x1. ∂ U = (1)(x ) = x 2, 2 2 2 2 • Consider V = U or V(x1, x2) = x1 x2 : ∂ x1 2 ∂ V /∂ x1 2x1x2 x2 ∂ U MRS = − = − 2 = − = (x1)(1) = x1 ∂ V /∂ x2 2x1 x2 x1 ∂ x2 d x ∂ U /∂ x x • This is the same as the MRS for U. so MRS = 2 = − 1 = − 2 d x1 ∂ U /∂ x2 x1 • Monotonic transformation does not change MRS. Econ 370 - Ordinal Utility 13 Econ 370 - Ordinal Utility 14 4.