Afriat’s Theorem

* in conjunction with §4 of Microeconomics Foundation I (Kreps 2013)

Y. Eddie Lu, Fall 2024

Recap

Notations

• \(X = \mathbb{R}^{n}_{+}\) be consumption bundles. A vector with \(n\) non-negative elements.
• \(B(p,m) = \{x \in X: p \cdot x \leq m\}\) be the budget set.
• \(c(B(p,m))\) be the demand chosen by the consumer.

Key: what to infer from finite number of actual choices?

Concepts and Assumptions

Locally Insatiable

Dfn: Locally Insatiable

For every bundle \(x \in X\), for every \(\epsilon > 0\), there exists \(y \in X\) with \(|x-y| < \epsilon\) and \(y \succ x\).

Intuition: a consumer would always accept if they are provided a bundle with slightly more stuff.

Lemma

If \(\succsim\) is complete, transitive, locally insatiable; \(x^*\) is chosen from the budget set \(B(p,m)\), then

• \(x^* \succsim x\) for all \(x\) with \(p \cdot x = m\)
• \(x^* \succ x\) for all \(x\) with \(p \cdot x < m\).

GARP (Generalized Axioms of Revealed Preferences)

Dfn: GARP

Given a finite dataset:

• p = (\(p^1, ..., p^n\)) as the price vector
• m = (\(m^1, ..., m^n\)) as the income/budget vector
• x = (\(x^1, ..., x^n\)) as the choice vector

with \(p^i \cdot x^i \leq m^i\) for each \(i\), we say that:

• \(x^i \succsim^d x^j\) when \(p^i \cdot x^j \leq m^i\)
• \(x^i \succ^d x^j\) when \(p^i \cdot x^j < m^i\)
• \(x^i \succsim^R x^j\) when \(x^i \succsim^d x \succsim^d ... \succsim^d x^j\)
• \(x^i \succ^R x^j\) when \(x^i \succsim^d ... \succ^d x \succsim^d ... \succsim^d x^j\)

The dataset satistify GARP if \(x^i \not\succ^R x^i\) for every \(i\).

Strictly Increasing Preference

Dfn: Strictly Increasing Preference

The preference \(\succsim \subseteq \mathbb{R}^n_+ \times \mathbb{R}^n_+\) is strictly increasing if

• \(x_i \geq y_i\) for all \(i\)
• \(x \neq y\)

implies \(x \succ y\). This \(\succsim\) is automatically locally insatiable.

Convex Preference

Dfn: Convex Preference

The preference \(\succsim \subseteq \mathbb{R}^n_+ \times \mathbb{R}^n_+\) is convex if

• \(x \succsim y\)
• \(\alpha \in [0,1]\)

implies \(\alpha x + (1-\alpha) y \succsim y\).

Afriat’s Theorem

Afriat’s Theorem

1. If dataset violates GARP, then there cannot exist a complete, transitive, and locally insatiable \(\succsim\) with \(x^i \in c_{\succsim}(B(p^i, m^i))\).
2. If dataset satisfies GARP, then there exists a complete, transitive, locally insatiable, strictly increasing, continuous, convex \(\succsim\) with \(x^i \in c_{\succsim}(B(p^i, m^i))\).

Proof of (1)

Suppose we have a complete, transitive, and locally insatiable \(\succsim\) with \(x^i \in c_{\succsim}(B(p^i, m^i))\) for each \(i = 1, ..., n\).

If we had a GARP violation, then \(x^i \succ^R x^i\). This implies that:

\[x^i \succsim^d ... \succsim^d x^j \succ^d x^k \succsim^d ... \succsim^d x^i\]

\[\Rightarrow x^i \succsim ... \succsim x^j \succ x^k \succsim ... \succsim x^i\]

\(\Rightarrow x^i \succ x^i\), which leads to a contradiction.

\(\blacksquare\)

Proof of 2 (Afriat’s Approach to Constructing Utility)

Suppose data \((p, m, x)\) satisfies \(p^i \cdot x^i \leq m^i\) and GARP.

1. \(p^i \cdot x^i = m^i\) for all \(i\).

Intuition

Every choice must be preference-maximizing under budget constraint. If not, then there must be a better choice under this price \(p\) and budget constraint \(m\).

Proof

Assume to contrary that \(p^i \cdot x^i < m^i\). Then by GARP, \(x^i \succ^d x^i\), which implies that \(x^i \succ^R x^i\), which creates a contradiction by violates GARP.

2. \(|\{x^k: x^i \succ^R x^k\}| < |\{x^k: x^j \succ^R x^k\}| \Rightarrow p^i \cdot x^j > m^i\).

Intuition

A choice \(x^j\) strictly more superior than the other choice \(x^i\) should never be feasible under the condition where \(x^i\) was chosen. If not, then the choice should be \(x^j\) instead of \(x^i\).

Proof

If \(p^i \cdot x^j \leq m^i\), then according to GARP, \(x^i \succsim^d x^j\). Then, \(\{x^k: x^j \succ^R x^k\} \subseteq \{x^k: x^i \succ^R x^k\}\), which contradicts \(\{x^k: x^i \succ^R x^k\}| < |\{x^k: x^j \succ^R x^k\}|\).

3. \(|\{x^k: x^i \succ^R x^k\}| \leq |\{x^k: x^j \succ^R x^k\}| \Rightarrow p^i \cdot x^j \geq m^i\).

Intuition

A choice \(x^j\) at least as good as the other choice \(x^i\) should never be strictly inferior when under price and budget constraint of \(x^i\).

Proof

Assume to contrary that \(p^i \cdot x^j < m^i\). Then it implies that \(x^i \succ^d x^j\). Also by GARP, \(x^j \in \{x^k: x^i \succ^R x^k\}\) and \(x^j \in \{x^k: x^i \succ^R x^k\}\). Thus, \(\{x^k: x^j \succ^R x^k\} \subsetneq \{x^k: x^i \succ^R x^k\}\), contradicting the assumption.

4. \(\exists x^k: \forall i \quad x^k \not\succ^R x^i\)

Intuition

There must be a lowest-ranking choice that is never strictly prefered to any other choices. If not, there will exist a loop that violates transitivity.

Proof

Suppoose for all \(x^k\), there exists \(x^i\) such that \(x^k \succ^R x^i\). Then it must be the case that the following chain with \(2n+1\) elements: \(x^1 \succ x^{j1} \succ x^{j2} \succ ... \succ x^{j2n}\). Since we are in a finite set with \(n\) choice bundles, there must be somewhere, in between, that \(x^k \succ x^k\) happens, which directly violates GARP.

5. \(\exists \, v^1, ..., v^n \in \mathbb{R} \quad \exists \, \alpha^1, ..., \alpha^n > 0 \quad \forall i, j \quad v^i \leq v^j + \alpha^j [p^j x^i - p^j x^j]\)

Intuition

We want to use \(v^j\) as the baseline of utility. \(\alpha^j\) controls the speed of moving the baseline.

Proof

By induction. When \(n=1\), let \(v^1 = 0\) and \(\alpha^1 = 1\). Suppose this is true when we have \(n\) choices. Now, denote \(x^{n+1}\) as \(x^{n+1} \not\succ^R x^i\) for every \(i\). We can imply from the assumption that this statement is true for observation \(1, ..., n\) that:

\[\exists \, v^1, ..., v^n \in \mathbb{R} \quad \exists \, \alpha^1, ..., \alpha^n > 0 \quad \forall i, j \quad v^i \leq v^j + \alpha^j [p^j x^i - p^j x^j].\]

Define: \(v^{n+1} = \min_{1 \leq j \leq n} v^j + \alpha^j [p^j x^{n+1} - p^j x^j]\). Since \(|\{x^k: x^{n+1} \succ^R x^k\}| = 0\), we can say that \(p^{n+1} x^j \geq m^{n+1} = p^{n+1} x^{n+1}\) for every \(1 \leq j \leq n\).

We assert that for \(1 \leq j \leq n\), either \(p^{n+1} x^j > m^{n+1} = p^{n+1} x^{n+1}\) or \(p^{n+1} x^j = m^{n+1} = p^{n+1} x^{n+1}\) holds.

• If \(p^{n+1} x^j > m^{n+1} = p^{n+1} x^{n+1}\), \(p^{n+1} x^j - p^{n+1} x^{n+1} > 0\). Then there exists \(\alpha^{n+1} > 0\) large enough such that \(v^j \leq v^{n+1} + \alpha^{n+1} [p^{n+1} x^j - p^{n+1} x^{n+1}]\) holds.
  
• If \(p^{n+1} x^j = m^{n+1} = p^{n+1} x^{n+1}\), \(p^{n+1} x^j - p^{n+1} x^{n+1} = 0\).

Then, as long as \(\alpha^{n+1} \in \mathbb{R}_{++}\), \(v^j \leq v^{n+1} + \alpha^{n+1} [p^{n+1} x^j - p^{n+1} x^{n+1}] = v^{n+1}\) holds.

6. Let \(u(x) := \min_{1 \leq j \leq n} v^i + \alpha^i [p^i x - p^i x^i]\).

Intuition

This function yields a concave form.

Application: Giffin Goods

• Normal goods: \(p_i \uparrow \quad \Rightarrow x^i \downarrow\)
• Giffin goods: \(p_i \uparrow \quad \Rightarrow x^i \uparrow\)

One application of Afriat’s theorem is prooving the existance of Giffin goods. Since an individual with observable Giffin goods purchasing behavior does not necessarily violate GARP, then there exists a preference relation that is complete, transitive, locally insatiable, continuous, strictly increasing, and convex that generates it.