1. Utility Functions f ~ p p
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”). ~ f p p

2. 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

3. 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 ?

4. Utility Functions p 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! p

5. 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

6. 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?

7. 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?

8. Marginal Rate of Substitution
MRS= _ MU1/MU2
Why??
MRS is not affected by f. Why? (hint: chain rule).
What is MRS of Cobb-Douglas, Quasi-Linear, Perfect Substitutes and Perfect Complements?