Econ D10-1: Lecture 3

1. Econ D10-1: Lecture 3 Individual Demand and Revealed Preference: Choice in an Economic Environment (MWG 2)

2. The Market Choice Structure: Consumption and Budget Sets • Consumers choose a commodity vector x = (x1, …, xL) from the consumption set X+L • Feasible choices for the consumer are determined by his budget set, which, in turn, is determined by his wealth w>0 and the vector of commodity prices p>>0 that he faces. • The consumer’s competitive or Walrasian budget set is given by Bpw = {x ≥ 0: p.x ≤ w} • Walrasian budget family:B={Bpw: p>>0, w>0}

3. The Walrasian Demand Correspondence • x(p,w) is a choice rule defined on the Walrasian budget family B. • MWG make the following assumptions. • x is a continuous, single valued function x: ++L+1  +L • x is homogeneous of degree zero:  • (Walras Law) The consumer exhausts his budget: p.x(p,w)=w. 

4. WARP for Walrasian Demand Functions • Consistency of demand: If bundle x is chosen when (a different) bundlex is affordable, then, if x is ever chosen, bundle x must be unaffordable. • (Samuelsonian) WARP: Given (p,w) and (p,w), if p.x(p,w)≤w and x(p,w)≠x(p,w), then p.x(p,w)>w. • Exercise: Assume that the demand correspondence satisfies Samuelsonian WARP and Walras Law. Prove that it is single valued and homogeneous of degree zero.

5. Compensated Law of Demand • If x(p,w) satisfies WARP for all (p,w), then compensated demand curves are downward sloping. Proof: Choose (p,w) so that p.x=w=p.x; i.e., x is exactly affordable when x is chosen. Then, p.(x-x) = 0. For x≠x, WARP requires that x be unaffordable when x is chosen, so that p.x>p.x=w or p.(x-x)>0. Subtracting the former from the later yields (p-p).(x-x) = p.x < 0.For pk=0 for all k≠j, this becomes pjxj < 0 or (xj /pj)<0 • (Q: What is the significance of MWG’s Prop. 2.F.1?)

6. If Walrasian demand function is continuously differentiable: For compensated changes: Substituting yields: The Slutsky matrix of terms involving the cross partial derivatives is negative definite, but not necessarily symmetric. Differential Compensated Law of Demand and the Slutsky Matrix