C1-1: Consumer Theory-Primitive Notions, Preference Relations and categories.

Chapter1-Consumer Theory

Primitive Notions

4 building blocks in any model of consumer choice

• consumption set, $\mathbf{X}:$ SET, all alternatives or complete consumption plans.

  • $\mathbf{X}\subseteq {\mathbb{R}}_{+}^n$, that is to say, set $\mathbf{X}$ is a subset of the non-negative n-dimension set $\mathbb{R}$.

  

  • $\mathbf{X}$ is closed. A closed set is a set that contains all its limit points. This implies that the complement of the closed set is an open set.

• $\mathbf{X}$ is convex

  A set $\mathbf{X}$ is convex if, for any two points within the set, the line segment connecting them is also contained within the set. Mathematically, this can be represented as:

  For any $x,y\in \mathbf{X}$ and any $t\in [0,1],tx+(1-t)y\in \mathbf{X}$.

• $0\in \mathbf{X}$

$\mathbf{x}=(x_1,...,x_n),x_i\in \mathbb{R},\mathbf{x}\in \mathbf{X},\mathbf{x}\in {\mathbb{R}}_{+}^n:$ Vector, consumption bundle/plan

• feasible set, $\mathbf{B}\in \mathbf{X}$

  • subset of the consumption set $\mathbf{X}$
  • satisfy the constraints
    

• preference relation,

  • represents the subjective ranking of options from the most preferred to the least preferred

  • The preference relation $\succ$ on a set $\mathbf{X}$ is a binary relation that satisfies the following properties:

    Axiom1: Completeness: For any ${\mathbf{x}}^1,{\mathbf{x}}^2\in \mathbf{X}$, either $x⪰y$ or $y⪰x$.

    Axiom2: Transitivity: For any ${\mathbf{x}}^1,{\mathbf{x}}^2,{\mathbf{x}}^3,\in \mathbf{X}$, if ${\mathbf{x}}^1⪰{\mathbf{x}}^2$ and ${\mathbf{x}}^2⪰{\mathbf{x}}^3$, then ${\mathbf{x}}^1⪰{\mathbf{x}}^3$.

• behavioural assumption

  • the consumer seeks to identify and select an available alternative that is most preferred in the light of his personal tastes.

Preference and Utility

preference relations

• preference relation $⪰$ :

  A preference relation $⪰$ on a set $\mathbf{X}$ is a binary relation that represents the way in which an individual ranks different alternatives or choices.

  • read as: is at least as good as
  • defined on the consumption set, $\mathbf{X}$.
  • If $x^1⪰x^2$, we say that ‘$x^1$ is at least as good as $x^2$’, for this specific consumer.
  • It satisfies the following axioms:
    1. Completeness: For any ${\mathbf{x}}^1,{\mathbf{x}}^2\in \mathbf{X}$, either ${\mathbf{x}}^1⪰{\mathbf{x}}^2$ or ${\mathbf{x}}^2⪰{\mathbf{x}}^1$ or both.
    2. Transitivity: For any ${\mathbf{x}}^1,{\mathbf{x}}^2,{\mathbf{x}}^3\in \mathbf{X}$, if ${\mathbf{x}}^1⪰{\mathbf{x}}^2$ and ${\mathbf{x}}^2⪰{\mathbf{x}}^3$, then ${\mathbf{x}}^1⪰{\mathbf{x}}^3$.

• strict preference relation $\succ$

  A strict preference relation indicates that one option is strictly preferred over another, without considering them as equally desirable.

  • read as: is strictly preferred to
  • The relation $x^1\succ x^2$ holds if and only if $x^1⪰x^2$ and $x^2/⪰x^1$.
  • It satisfies the following axioms:
    1. Irreflexivity: No element is strictly preferred to itself, formally represented as $x\nsucc x$.
    2. Asymmetry: If $x\succ y$, then $y\nsucc x$ for any elements $x$ and $y$.
    3. Transitivity: For any elements $x$, $y$, and $z$, if $x\succ y$ and $y\succ z$, then $x\succ z$.

• indifference relation $\sim$

  • read as: is indifferent to

  • The relation $x^1\sim x^2$ holds if and only if $x^1⪰x^2$ and $x^2⪰x^1$.

• Conclusion:

  For any pair $x_1$ and $x_2$, exactly one of the following three mutually exclusive possibilities holds: $x^1\succ x^2$, or $x^2\succ x^1$, or $x^1\sim x^2$.

What we get for the above assumptions:

• For example:

  For $\mathbf{X}={\mathbb{R}}_{+}^2$, Any point in the consumption set, such as $x^0=(x_1^0,x_2^0)$, represents a consumption plan consisting of a certain amount $x_1^0$ of commodity 1, together with a certain amount $x_2^0$ of commodity 2.

Based on Axioms 1 and 2, the consumer’s preference relation with respect to any given point $x_0$ can be categorized into three mutually exclusive sets relative to $x^0$:

• the set of points worse than $x^0$ ($ \prec (x^0) $),

• the set of points indifferent to $x^0$ ($ \sim (x^0) $),

• and the set of points preferred to $x^0$ ($ \succ (x^0) $).

  Therefore, for any bundle $x_0$, the three sets $ \prec (x^0) $, $ \sim (x^0) $, and $ \succ (x^0) $ partition the consumption set $X$.

References

[1] Geoffrey A. Jehle, Philip J. Reny Advanced Microeconomic Theory, 3rd Edition ( 2011, Prentice Hall).

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