# Utility Functions A utility function Ux represents a

• Slides: 8

Utility Functions • A utility function U(x) represents a preference relation fif and only if: ~ p x’ x” U(x’) > U(x”) x’ px” U(x’) < U(x”) x’ ~ x” U(x’) = U(x”).

Utility Functions & Indiff. Curves • 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. Call these numbers utility levels. • Note: All bundles in an indifference curve have the same utility level. p

Utility Functions • A preference relation that is complete, reflexive, transitive and (continuous) can be represented by a utility function • There is no unique utility function representation of a preference relation. • Suppose U(x 1, x 2) = x 1 x 2 represents a preference relation. • Does it work for (2, 3) > (4, 1) ~ (2, 2)? • How about V(x 1, x 2) =U 2 = x 12 x 22 ?

Utility Functions • U(x 1, x 2) = x 1 x 2 (2, 3) (4, 1) ~ (2, 2). • How about W = 2 U + 10? • W preserves the same order as U and V and so represents the same preferences. • If U is a utility function that represents a preference relation and f is a strictly increasing function, • then V = f(U) is also a utility function representing those same preferences! • WARNING: This does not include risk! p

My favorite utility functions. • • Perfect substitutes: u(x 1, x 2)=x 1+x 2 Perfect complements: u(x 1, x 2)=min{x 1, x 2} Quasilinear: u(x 1, x 2)= g(x 1) + x 2 Cobb-Douglas: U(x 1, x 2) = x 1 a x 2 b where a>0 and b>0. What do the indifference curves look like of all four? Find a function that changes C-D into x 1 c x 21 -c

Lexicographic preferences. • Look first at good x 1 and then look at x 2. • For instance, Tall Tom likes to date tall women. His first choice in a partner is by height. If two women are of the same height, he chooses the one with the highest IQ. • Does such preferences satisfy transitivity? • Can one represent one by a utility function?

Marginal Utility • The marginal utility of commodity i is the rate-ofchange of total utility as the quantity of commodity i consumed changes. • What is the marginal utilities of perfect substitutes, perfect complements, quasi-linear, and Cobb. Douglas?

Marginal Rate of Substitution • MRS= _ MU 1/MU 2 • Why? ? • MRS is not affected by f. Why? (hint: chain rule). • What is MRS of Cobb-Douglas, Quasi. Linear, Perfect Substitutes and Perfect Complements?