In consumer theory, every consumer has their own preference for choosing consumption bundles. Based on Advanced Microeconomic Theory by Jehle and Reny (2011), consumer preferences are described by axioms to characterize the preferences’ properties with mathematical expression. These preferences are represented by a binary relation ≿ on the consumption set (for instance, 𝑥1 ≿ 𝑥2 means 𝑥1 is at least as good as 𝑥2). Overall, consumer preferences have five axioms.
Axiom 1. Completeness
This axiom states that for all 𝑥1 and 𝑥2 in 𝑋, the relation is either 𝑥1 ≿ 𝑥2 or 𝑥1 ≾ 𝑥2 (or both). This means consumers can always compare two consumption bundles and determine which one is preferable.
Axiom 2. Transitivity
This axiom says that for three bundles 𝑥1, 𝑥2 and 𝑥3 in 𝑋, if 𝑥1 ≿ 𝑥2 and 𝑥2 ≿ 𝑥3, then 𝑥1 ≿ 𝑥3. This axiom provides consistent pairwise comparisons throughout the consumption set 𝑋.
Axiom 3. Continuity
This axiom says that for all 𝑥 ∈ ℝ+ 𝑛 , the ‘at least as good as’ set (≿ (𝑥)) and ‘no better than’ set (≾ (𝒙)), are closed in ℝ+ 𝑛 . This axiom removes the open are in the indifference set and set a technical assumption of no sudden preference reversals.
Axiom 4. Strict Monotonicity
This axiom states that for all 𝑥0, 𝑥1 ∈ ℝ+ 𝑛 , if 𝑥0 ≥ 𝑥1, then 𝑥0 ≿ 𝑥1, while if 𝑥0 ≫ 𝑥1, then 𝑥0 ≻ 𝑥1. It means that consumers always prefer bundles with more items to one with less items. For instance, given two bundles consisting of 3 apples and 5 apples respectively. If these bundles have the same price, then consumers would prefer take the 5-apple bundle.
Axiom 5. Strict Convexity
This axiom says that if 𝑥0 ≠ 𝑥1 and 𝑥0 ≿ 𝑥1, then 𝑡𝑥1 + (1 − 𝑡)𝑥0 ≿ 𝑥0 for all 𝑡 ∈ (0,1). This axiom means that consumers prefer more balanced bundles to extreme ones. For example, given two bundles with the same price, consumers prefer the bundle with five apples and five oranges to the bundle with nine apples and one orange.
These axioms for consumer preferences could be classified into three different parts. The completeness and transitivity focus on the consistency in comparisons for available alternatives. Continuity is created for a mathematical purpose such that the ‘at least as good as’ and ‘no better than’ sets topologically (in mathematical context) exists. The last two axioms describe the characteristics of consumers’ taste for goods within the bundles
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