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 = (x 1, …, x L ) 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 B pw = {x ≥ 0: p. x ≤ w} Walrasian budget family: B={B pw : 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) bundle x 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  p k =0 for all k≠j, this becomes  p j  x j < 0 or (  x j /  p j )<0 (Q: What is the significance of MWG’s Prop. 2.F.1?)

6. Differential Compensated Law of Demand and the Slutsky Matrix 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.