Transcript Utility Functions • A utility function U(x) represents a preference relation f if and only if: ~ p x’ x” U(x’) > U(x”) x’ p x” U(x’) x’ ~

Utility Functions
• A utility function U(x) represents a
preference relation f if and only if:
~
p
x’
x”
U(x’) > U(x”)
x’ p 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.
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(x1,x2) = x1x2 represents a
preference relation.
• Does it work for (2,3) > (4,1) ~ (2,2)?
• How about V(x1,x2) =U2 = x12x22 ?
Utility Functions
• U(x1,x2) = x1x2
(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!
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My favorite utility functions.
•
•
•
•
Perfect substitutes: u(x1,x2)=x1+x2
Perfect complements: u(x1,x2)=min{x1,x2}
Quasilinear: u(x1,x2)= g(x1) + x2
Cobb-Douglas: U(x1,x2) = x1a x2b
where a>0 and b>0.
What do the indifference curves look like of all
four?
Find a function that changes C-D into x1c x21-c
Lexicographic preferences.
• Look first at good x1 and then look at x2.
• 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.
U
MU i 
 xi
• What is the marginal utilities of perfect
substitutes, perfect complements, quasi-linear, and
Cobb-Douglas?
Marginal Rate of Substitution
• MRS= _ MU1/MU2
• Why??
• MRS is not affected by f. Why? (hint: chain
rule).
• What is MRS of Cobb-Douglas, QuasiLinear, Perfect Substitutes and Perfect
Complements?