Weak Axiom of Revealed Preference and the Law of Demand

This set of flashcards covers key concepts related to the weak axiom of revealed preference and the law of demand as discussed in the lecture notes.

What does the weak axiom of revealed preference state in the context of consumer choices?

If a consumer chooses consumption bundle x(p, w) over x(p', w'), then bundle x(p, w) must not be affordable at prices p' and wealth w'.

How is the weak axiom of revealed preference mathematically expressed with respect to Walrasian demand?

The Walrasian demand function x(p, w) satisfies the weak axiom if, for any price-wealth pair (p, w) and (p', w'), if p'•x(p', w') ≤ w and x(p', w') ≠ x(p, w), then p•x(p, w) > w'.

What is Slutsky wealth compensation?

It is the adjustment of wealth to ensure that the initial consumption bundle remains affordable at new prices, expressed as w' = p'•x(p, w).

What implication does the weak axiom have for the law of demand?

The weak axiom implies that demand responds negatively to price changes, confirming that higher prices lead to lower demand for a good.

Describe the relationship between the weak axiom and compensated price changes.

The weak axiom can be tested with compensated price changes; it holds if the demand response to such price changes satisfies certain inequalities.

What does Proposition 2.F.1 state about compensated price changes?

If demand x(p, w) is homogeneous of degree zero and satisfies Walras' law, then demand satisfies the weak axiom if the inequality (p' - p)•[x(p', w') - x(p, w)] ≤ 0 holds.

What does it mean if a function or matrix is negative semidefinite?

It indicates that the function does not increase for any permissible direction, implying that the substitution matrix for demand is not positive.

What is the significance of a symmetric Slutsky matrix in demand theory?

A symmetric Slutsky matrix implies that demand behavior can be derived from rational preferences, involving consistency between choices.

How can the weak axiom be violated in the context of consumer choice?

Violation occurs when a consumer chooses a less preferred bundle when a higher preferred option is affordable, conflicting with transitive preferences.

What three conclusions can be drawn from Section 2.F on consumer choice and preferences?

1. The weak axiom combined with Walras' law is equivalent to the compensated law of demand. 2. The compensated law implies negative semidefiniteness of the substitution matrix. 3. These axioms do not guarantee symmetry of the substitution matrix except when there are two goods.