Choices/Preference/Utility over Finite Outcomes
* in conjunction with §1 of Microeconomics Foundation I (Kreps, 2013)
yedlu, Fall 2024
Key Concepts Overview
1. Choices
Choices are observable actions taken when confronted with alternatives. Economists model choices using the following notations:
Notation | Definition | Example |
---|---|---|
\(X\) | Finite set of alternatives | Colleges someone can apply: \(X = \{a,b,c\}\) |
\(P(X)\) | All non-empty subsets of \(X\) | \[ P(X) = \{\{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\} \] |
\(B \subseteq P(X)\) | Set of feasible choice menus | \(B = \{\{a\}, \{a,b\}, \{a,b,c\}\}\) |
\(c: B \to P(X)\) | Choice function: \(c(A) \subset A\) | \(c(\{a,b,c\}) = \{a,b\}\) |
2. Preferences
Preferences reflect internal evaluations of choices, modeled abstractly:
Notation | Definition | Example |
---|---|---|
\(X\) | Set of alternatives | \(X = \{a,b,c\}\) |
\(X \times X\) | Cartesian product: pairs of alternatives | \((a,b), (b,c), \dots\) |
\(R \subseteq X \times X\) | Binary relation: \((a,b) \in R \iff a \succsim b\) | Preferences: \(a \succsim b, b \succsim c\) |
Properties of \(R\):
- Complete: For all \(x, y \in X\), \(x \succsim y\) or \(y \succsim x\).
- Transitive: \(x \succsim y\) and \(y \succsim z \implies x \succsim z\).
- Rational: \(R\) is complete and transitive.
3. Choices from Preferences
A rational binary relation \(R\) generates a choice function \(c_R\):
\[ c_R(A) := \{x \in A \mid (\forall y \in A) \, (x,y) \in R\} \]
Proofs
Proposition 1: If \(R\) is rational, then \(c_R\) is a choice function.
Proof (Click to Expand)
By induction. Need to show that \(c_R (A) \neq \emptyset\) for all \(A \subseteq X\).
Base Case (\(|A| = 1\)):
Let \(A = \{x\}\). A complete \(R\) implies that \((x,x) \in R\), which in turn implies that \(c_R (A) = \{x\} \neq \emptyset\).Inductive Step (\(|A| = n + 1\)):
Assume true for \(|A| = n\). Pick any \(x \in A\). By assumption, \(c_R (A \setminus \{x\}) \neq \emptyset\).- By definition, there exists \(y \in A \setminus \{x\}\) such that \((y,z) \in R\) for all \(z \in A \setminus \{x\}\).
- Since \(R\) is complete, either \((x,y) \in R\) or \((y,x) \in R\).
- If \((x,y) \in R\), then transitivity implies \((x,z) \in R\) for all \(z \in A\), so \(x \in c_R(A)\).
- If \((y,x) \in R\), then \(y \in c_R(A)\).
\(\blacksquare\)
Proposition 2: If \(c\) satisfies WARP, then \(R_c\) is a rational preference relation.
Proof (Click to Expand)
- Completeness:
Take any \(x, y \in X\).- If \((x,y) \notin R_c\), then \(x \notin c(\{x,y\})\).
- By definition, \(y \in c(\{x,y\})\), so \((y,x) \in R_c\).
Therefore, \(R_c\) is complete.
- If \((x,y) \notin R_c\), then \(x \notin c(\{x,y\})\).
- Transitivity:
Suppose \((x,y) \in R_c\) and \((y,z) \in R_c\).- This implies \(x \in c(\{x,y\})\) and \(y \in c(\{y,z\})\).
- Assume \((x,z) \notin R_c\). Then \(z \in c(\{x,z\})\) contradicts WARP.
Hence, \((x,z) \in R_c\), and \(R_c\) is transitive.
\(\blacksquare\)
- This implies \(x \in c(\{x,y\})\) and \(y \in c(\{y,z\})\).
4. Preferences Revealed by Choices
Revealed preference relation \(R_c\):
\[ R_c := \{(x,y) \in X \times X \mid x \in c(\{x,y\})\} \]
Weak Axiom of Revealed Preferences (WARP):
If \(x, y \in A \cap B\), \(x \in c(A)\), and \(y \in c(B)\), then \(x \in c(B)\).