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 wgt0 and the vector of commodity
  prices pgtgt0 that he faces.
• The consumers competitive or Walrasian budget
  set is given by Bpw x 0 p.x w
• Walrasian budget family BBpw pgtgt0, wgt0

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
  ?L1 ? ?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)gtw?.
• 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?.xw?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?gtp.xw or p.(x?-x)gt0. Subtracting the former
  from the later yields (p?-p).(x?-x) ?p.?x lt
  0.For ?pk0 for all k?j, this becomes ?pj?xj lt 0
  or (?xj /?pj)lt0
• (Q What is the significance of MWGs 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.