Preference Axioms, Indifference Sets & Technical Conditions

Indifference Relations

• We work with a universal choice set $X$ (also called the “consumption set”).
• Weak preference ( (\succsim) ) is the basic relation. All other relations are defined from it.
• Indifference ( (\sim) )
  • Definition: $x \sim x' \iff (x \succsim x') \land (x' \succsim x).$
  • Properties that can be proved directly from weak‐preference axioms:
  • Reflexive: $x \sim x \; \forall x \in X.$
  • Symmetric: $x \sim x' \implies x' \sim x.$
  • Transitive: $x \sim y \land y \sim z \implies x \sim z.$
  • Therefore (\sim) is an equivalence relation.
• Strict preference ( (\succ) )
  • Definition: $x \succ x' \iff (x \succsim x') \land \neg(x' \succsim x).$
  • Consequences:
  • Irreflexive: never true that $x \succ x.$
  • Transitive: $x \succ y \land y \succ z \implies x \succ z.$
  • Generates the upper contour set at a bundle $x$: ${y \in X : y \succ x}.$

Indifference Classes / Curves

• For a given bundle $x'$ the indifference class is $I(x') = {x \in X : x \sim x'}.$
• Language:
  • “Classes” (Villar’s textbook) ≈ “Indifference sets” (Mas-Colell et al.).
  • In 2-goods diagrams the class appears as a curve, hence the textbook phrase “indifference curve.”
• Key facts:
  • Non-empty because $x' \in I(x')$ (reflexivity).
  • The union $\bigcup_{x' \in X} I(x') = X$ – plotting all curves reproduces the whole choice set.
  • Two classes intersect non-trivially iff they are the same class (any overlap forces indifference of the two anchor bundles).

Why Set Theory Matters

• Consumer theory manipulates sets: choice set, budget set, indifference sets, etc.
• Formal set language allows rigorous proofs & comparative-statics.

Rationality Axioms

• Completeness: every pair of bundles is comparable.
• Transitivity: rankings are consistent (no cycles).
• Together these axioms justify treating the consumer as rational and permit falsifiable predictions.
• Debate: realism questioned by philosophy & behavioral economics, yet the axioms are pragmatic foundations for modelling.

Additional Axioms Needed for Useful Models

We “hammer” the hot iron of preferences with more axioms to obtain tractable shapes.

Continuity

• Formal statement: For any $x, x' \in X$ with $x \succ x'$ there exist \varepsilon, \delta > 0 such that
  $z \in B<em>\varepsilon(x) \implies z \succ x', \quad s \in B</em>\delta(x') \implies x \succ s,$
  where $B_\varepsilon(x)$ is an $\varepsilon$-radius ball around $x$.
• Intuition: small changes in bundles cannot flip the ranking catastrophically; avoids “jumps.”
• Role: guarantees existence of a continuous utility representation, which is crucial for solving optimisation.
• Graphical idea: around a preferred point A all neighbouring points are still preferred to a dominated point B; no broken indifference curves.
• Counter-example: Lexicographic preferences.
  • Definition for two goods $x=(x<em>1,x</em>2),\; z=(z<em>1,z</em>2)$:
    $z \succ x \iff \big[z<em>1 > x</em>1\big] \;\text{or}\; \big(z<em>1 = x</em>1 \land z<em>2 > x</em>2\big).$
  • Properties: complete & transitive, not continuous → no utility function, optimisation impossible.
  • Behavioural story: small children or obsession with one good; discontinuous “jumps.”
  • Lexicographic preferences cannot be represented by any real-valued utility despite rationality in the basic sense.
• Debreu’s theorem (1950s): preferences are representable by a utility function iff they are complete, transitive, and continuous.

Convexity

• Weak/strict versions; focus on strict convexity:
  • If $x \neq y$ and $\lambda \in (0,1)$ then $z = \lambda x + (1-\lambda) y \succ x, y.$
• Economic intuition: consumers like diversification; balanced bundles trump extreme ones.
• Implications:
  • Indifference curves are smooth, bowed inward, never linear or crossing.
  • Strict convexity ↔ utility strictly quasi-concave.
  • Mathematical link to convex sets: any line segment between two points in a convex set remains inside the set.

Local Non-Satiation (Monotonicity)

• Formal: For every $x$ and \alpha>0 there exists $x' \in B_\alpha(x)$ such that $x' \succ x.$
• Meaning: “More is better” locally; can be satiated in some goods but not in all simultaneously.
• Consequences: indifference curves have negative slope and zero thickness.

Metaphor: Axioms as Hammers

• Completeness & transitivity = first two hammers forging basic rationality.
• Continuity, convexity, local non-satiation = additional hammers shaping the iron so that optimisation, comparative statics, and empirical testing become possible.

Slide 18 Discussion (Preview for Exercise)

• Two example decision rules to test for completeness/transitivity:
  1. Compare products of quantities in each bundle: x \succ y \iff \prodi xi > \prodi yi.
  2. Other (unspecified) rule with fixed number of goods.
• Hint: Since the rule maps bundles to real numbers, comparison reduces to the natural order on $\mathbb R$, making completeness & transitivity likely. Students are asked to prove it formally.

Logistics / Course Notes

• Instructor will upload slides & readings (Villar, Mas-Colell et al.).
• Current EVA platform contains 2023 material; updates coming.
• Next class starts at 09:30.
• Upcoming topics: choice set, budget set, conditions for utility maximisation, comparative statics, empirical tests.