L04 Choice

Big picture u Behavioral Postulate: A decisionmaker chooses its most preferred alternative from the set of affordable alternatives. u Budget set = affordable alternatives u To model choice we must have decisionmaker’s preferences.

MRS and Utility Function x2x2x2x2 x1x1x1x1 RS is a slope of the indifference curve at x MRS is a slope of the indifference curve at x x Q: Can we recover MRS from knowledge of U(x)?

Utility and Marginal Utility

MRS and Marginal Utility u Yes, but we have to find marginal utilities for both goods first (MU) u MU: How much utility we gain by adding an extra unit of good i

Example u U(x 1,x 2 ) = x 1 x 2

Example u V(x 1,x 2 ) = ln(x 1 )+ln(x 2 )

u Cobb Douglass utility function u Log function Cobb-Douglass utility u Conclusion: preferences are Cobb-Douglass

Problem: u We know preferences (utility function) and u We want to know optimal choice

MRS of Monotonic transformations u V=f(U) and f strictly increasing u Can we say something about MRS?

Choice $ $ $$$ $ $$ $$

u How can we modify our argument if u Marginal utility of a dollar Different prices u We should equalize MU of a $!

Choice $ $ $$$ $ $$ $$

Problem (thinking on the margin!) Two secrets of happiness: 1. Spend your total income 2. Equalize marginal utility of a $ Rearranging:

Choice: Calculation

Choice: geometric solution x1x1 x2x2

u SOH for Well-behaved preferences But: u Perfect Complements (Right and Left shoe) u Perfect Substitutes (Example: French and Dutch Cheese) SOH for other preferences

Perfect Complements (Shoes) L R

More generally L R

Choice: Calculation

Perfect Substitutes (F & D cheese) F D

More generally X1X1 X2X2

Perfect Substitutes