X RL {x R : xl 0 for l 1,..., L}
The economic constraint:
px p1x1 ... pL xL w
The Walrasian budget set (Definition 2.D.1)
Bp,w {x RL : px w}
or
u(x* ) xl
pl
px w
Solution: Walrasian demand function x*( p, w)
Utility Maximization -- Example
Example 3.D.1: the transformed Cobb-Douglas Utility Function
Expenditure Function
Expenditure function e( p,u) Min px s.t. u(x) u {x}
Properties: 1. Homogeneous of degree of one in p 2. Strictly increasing in u and nondecreasing in p 3. Concave in p 4. Continuous in p and u
Comparative Statics – Wealth Effects
The consumer’s Engel function x( p, w)
The wealth effect xl ( p, w) / w or Dwx( p, w) Normal goods and inferior goods
A choice rule C(B) B
The weak axiom of revealed preference (WARP): if x is revealed at least as good as y, then y cannot be revealed preferred to x