L05 Choice Problem: We know Preferences U ( x 1 , x 2 ) ln x 1 ln x 2 Prices and income p 1 1 , p 2 1 , m We want optimal choice
* 1 * 2 (x , x ) 1 0 Secrets of Happiness (SOH): x2
x1 Abstract approach In the example we were given U ( x 1 , x 2 ) ln x 1 ln x 2 p 1 1, p 2 1, m 1 0 we found demands - two numbers
x1 5 , x 2 5 Now we use abstract parameters p1 , p 2 , m U ( x , x
) 1 2 we find demand functionsNow we 4 types xof 1 ( ppreferences 1, p2 , m ) x 2 ( p1 , p 2 , m ) Abstract Cobb Douglass Function
Cobb Douglass utility functions and a 1 U ( x1 , x 2 ) x x b 2 V ( x 1 , x 2 ) ln U ( x 1 , x 2 )
are equivalent in terms of preferences Magic (Cobb-Douglass) formula U ( x 1 , x 2 ) a ln x 1 b ln x 2 Parameters: a , b , p 1 , p 2 , m p1 , p 2 , m Cobb-Douglas: Summary a b V a
ln x b ln x U x Utility function: 1 2 or 1 x2
Solution: a m b m * * x1 , x2 a b p1 a b p2 Shares of income A) Let U x
0 .5 1 x * 1 1 0 .5 2 and p 1 2 , p 2 4 , m 4 0
* 2 p x , p2x * 1 * 2
x ,x 10 1 B) Let U x x 20 2 and p 1 1 0 , p 2 1 0 , m 9 0 0
p 1 x 1* , p 2 x 2* x 1* , x 2* Interior and corner solution Interiority Cobb Douglass (always interior solution)
a m x , a b p1 * 1 b m x a b p2 * 2
SOH: Extreme preferences Perfect Complements (shoes) Perfect substitutes (cheese) Troublemakers
Perfect complements Right and left shoe Perfect complements Coffee and 2 teaspoons of sugar Perfect Complements: Problem U ( x 1 , x 2 ) m in( x 1 , x 2 ) p 1 1, p 2 1, m 1 0
SOH (Perfect Complements) U ( x 1 , x 2 ) min( 2 x 1 , x 2 ) p 1 1, p 2 1, m 1 0 Perfect Complements (SOH) U ( x 1 , x 2 ) m in( a x 1 , b x 2 ) nterior or corner solution? p1 , p 2 , m Is solution always interior?
Not necessarily Perfect Substitutes Quasilinear Perfect Substitutes:Problem U ( x1 , x 2 ) x1 x 2
p 1 1, p 2 2 , m 1 0 x2 x1 Magic Formula (Substitutes) U ( x1 , x 2 ) a x1 b x 2 p1 , p 2 , m