Utility Functions A utility function U(x) represents a preference relation if and only if: x’ x” U(x’) > U(x”) x’ x” 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. 

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 1 2 x 2 2 ?

Utility Functions U(x 1,x 2 ) = x 1 x 2 (2,3) (4,1)  (2,2). How about W = 2U + 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! 

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 2 1-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-of- change 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?