Weak Axiom of Revealed Preference

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Weak Axiom of Revealed Preference (WARP) WARP is a fundamental concept in consumer theory that states if a consumer chooses bundle $A$ over bundle $B$ when both are affordable, then they should not choose bundle $B$ over bundle $A$ when both are affordable under a different price set. Formal Definition Let $x_1$ and $x_2$ be two consumption bundles. Let $p_1$ and $p_2$ be two price vectors. If bundle $x_1$ is chosen at prices $p_1$, and bundle $x_2$ is affordable at prices $p_1$ (i.e., $p_1 \cdot x_1 \ge p_1 \cdot x_2$), then $x_1$ is revealed preferred to $x_2$. WARP states that if $x_1$ is directly revealed preferred to $x_2$ (i.e., $p_1 \cdot x_1 \ge p_1 \cdot x_2$ and $x_1$ is chosen at $p_1$), then it cannot be the case that $x_2$ is directly revealed preferred to $x_1$ (i.e., $p_2 \cdot x_2 \ge p_2 \cdot x_1$ and $x_2$ is chosen at $p_2$). Mathematically: If $x_1$ is chosen when $x_2$ is affordable (i.e., $p_1 \cdot x_1 \ge p_1 \cdot x_2$), Then, if $x_2$ is chosen at some prices $p_2$, $x_1$ must not be affordable at those prices (i.e., $p_2 \cdot x_2 Implications of WARP Consistency: WARP implies a basic level of consistency in consumer choices. If a consumer makes a choice, they should not contradict that choice when faced with a similar or less restrictive budget constraint. Downward Sloping Demand: For normal goods, WARP implies that demand curves are downward sloping. If the price of a good increases, a rational consumer (satisfying WARP) will demand less of that good (assuming income is adjusted to keep the original bundle affordable). No Giffen Goods: WARP rules out the existence of Giffen goods where demand increases as price increases. However, it does not rule out inferior goods. Revealed Preference Theory: WARP is the cornerstone of revealed preference theory, which aims to deduce consumer preferences from observed choices rather than assuming utility functions. Relationship to Other Axioms Strong Axiom of Revealed Preference (SARP): SARP is a stronger condition than WARP. It says that if $x_1$ is revealed preferred (directly or indirectly) to $x_2$, then $x_2$ cannot be revealed preferred (directly or indirectly) to $x_1$. SARP is equivalent to the existence of a continuous, strictly quasi-concave utility function that rationalizes the observed choices. WARP is a necessary but not sufficient condition for SARP. Generalized Axiom of Revealed Preference (GARP): GARP is the most general axiom. It states that if $x_0$ is revealed preferred (directly or indirectly) to $x_k$, then $x_k$ cannot be strictly revealed preferred to $x_0$. GARP is equivalent to the existence of a non-satiated, continuous, and quasi-concave utility function. If choices are single-valued, GARP collapses to SARP. Example Consider a consumer with income $M$. Scenario 1: Prices are $p_A = (2, 3)$. Consumer chooses bundle $A = (5, 5)$. The cost of bundle $A$ is $2 \cdot 5 + 3 \cdot 5 = 25$. Suppose bundle $B = (6, 2)$ is also affordable at these prices: $2 \cdot 6 + 3 \cdot 2 = 12 + 6 = 18 \le 25$. Since the consumer chose $A$ when $B$ was affordable, $A$ is revealed preferred to $B$. ($A \succ^R B$) Scenario 2: Prices change to $p_B = (4, 1)$. Consumer chooses bundle $B = (6, 2)$. The cost of bundle $B$ is $4 \cdot 6 + 1 \cdot 2 = 24 + 2 = 26$. For WARP to hold, bundle $A = (5, 5)$ must NOT be affordable at these new prices. Cost of bundle $A$ at $p_B$: $4 \cdot 5 + 1 \cdot 5 = 20 + 5 = 25$. If the consumer chose $B$ when its cost is $26$, and $A$ costs $25$, then $A$ is affordable. In this case, $p_B \cdot B = 26 \ge p_B \cdot A = 25$. This violates WARP. The consumer chose $A$ over $B$ in Scenario 1, but then chose $B$ over $A$ in Scenario 2 when $A$ was also affordable. This is inconsistent. For WARP to hold, if $A \succ^R B$ (as in Scenario 1), then it must be that if $B$ is chosen at prices $p_B$, bundle $A$ is not affordable (i.e., $p_B \cdot x_A > p_B \cdot x_B$).