Weak Axiom of Revealed Preference and the Law of Demand

Weak Axiom of Revealed Preference and the Law of Demand

• The weak axiom of revealed preference (WA) places restrictions on consumer demand behavior. It is stated as follows:

  • For two price-wealth situations
  • (p, w) and (p', w')
  • If the consumer chooses a bundle x(p, w) over x(p', w') when both are affordable, then:
  • If prices change from p to p', where a new wealth w' makes the original bundle x(p, w) still affordable, then:
    • The condition that ensures revealed preference is:
      p' ullet x(p, w) > w'

• Definition 2.F.1: The Wa is satisfied if:

  • If p ullet x(p', w') ≤ w and x(p', w') ≠ x(p, w) , then p' ullet x(p, w) > w' .

• Implications of the Weak Axiom:

  • The weak axiom implies consistency in consumer choice. If a consumer prefers Bundle A over Bundle B when both are affordable, the consumer should prefer Bundle A over Bundle B in all situations where both are still affordable.
  • A violation leads to contradictions regarding preference consistency.

Slutsky Wealth Compensation

• Slutsky Wealth Compensation:
  • When there is a price change, the consumer's budget is adjusted so that the original consumption bundle is just affordable again at the new prices.
  • This condition helps isolate the effect of price changes on the consumer's real wealth.
• Proposition 2.F.1 states: The WA can also be understood through compensated changes. Specifically:
  • (p' - p) [x(p', w') - x(p, w)] ≤ 0 ,
    • If strict inequality occurs whenever x(p, w)
      eq x(p', w') .

Law of Demand

• The weak axiom leads to the law of demand under compensated price changes:
  • Demand and price move in opposite directions. As prices rise, the quantity demanded falls, and vice versa.
  • This can be analytically framed in terms of properties relating to the Slutsky matrix, denoted as S(p, w) , which measures substitution effects.

Differentiable Demand Functions

• If the Walrasian demand function x(p, w) is differentiable, the change in demand from a compensated price change leads to:
  • dp ullet S(p,w) ullet dX ≤ 0 ,
• This indicates that marginal changes in price lead to residual effects analyzed through the Slutsky matrix, which explores optimal consumption responses.

Properties of the Slutsky Matrix

• The Slutsky matrix has properties that relate closely to the weak axiom:
  • It is negative semi-definite, therefore ensuring non-positive substitution effects.
  • A non-positive substitution effect results indicates that as the price of a good rises, the quantity demanded does not increase.

Summary of Key Conclusions

• The weak axiom's implications suggest a deeper structure of consumer choice over independent price changes.
• Three Conclusive Properties:
  1. The consistency underpinning consumer choices aligns with the compensated law of demand.
  2. This law leads to the negation of the Slutsky matrix's definiteness, enhancing understanding of demand response behavior to price changes.
  3. The properties of the demand function derived under these constraints play a fundamental role in economic theories of consumer behavior.