Advanced Microeconomics/Preference Relations

Preference relations
The preference relation provides a foundation upon which classical microeconomics erects a theory of rational choice. This section describes preference relations and their properties. The rational preferences approach to studying human decision making treats preferences as given, imposing axiomatic assumptions intended to represent rational choice. We begin by envisioning a set of mutually exclusive choices facing a decision maker, X. The binary relation $$\succeq$$ represents a preference over the elements of X. In the natural language $$x \succsim y$$ is read as "x is at least as good as y" or "x is weakly preferred to y".

Two useful derivative relations are

Rational Preferences
The preference relation $$\succsim $$ on X is called rational if it is both complete and transitive. Completeness requires all pairs $$(x,y)$$ can be compared.

Transitivity imposes a 'consistency' requirement, enabling a ranking or ordinal mapping onto the elements of X. For example, if a shopper strictly prefers apples to oranges and prefers oranges to bananas, the transitivity assumption requires him/her to also prefer apples to bananas.

properties of rational preferences:
 * 1) $$\succeq$$ is reflexive $$\Leftrightarrow \forall x \in \mathbf{X}\, x \succeq x$$. This is implied by completeness.
 * 2) $$\succ$$ is irreflexive $$\Leftrightarrow x \succ x$$ never holds
 * 3) $$\succ$$ is transitive $$\Leftrightarrow x \sim y \land y \sim z \Leftrightarrow z\sim z$$
 * 4) $$\sim$$ is reflexive.
 * 5) $$\sim$$ is transitive $$\Leftrightarrow x\succ y \succsim z \Leftrightarrow x\succ z$$
 * 6) $$\sim$$ is symmetric $$\Leftrightarrow x \sim y \Rightarrow y \sim x$$.
 * 7) If $$x \succ y$$ and $$y \succeq z$$ then $$x \succ z$$.

A moment of reflection reveals the gravity of the rationality assumption. In colloquial terms, rationality implies individuals fully know and understand their preferences. In any moment the decision maker could be called upon to act it must be able to provide a complete, consistent ranking of all elements in the choice set. Note, however, the rationality assumption places no restrictions on the subjective quality of preferences, nor has this framework imposed any requirements on the information or computational ability available to a decision maker. Simply, the subjectivity of preferences survives an assumption of rationality.

Additional Assumptions on Preferences
A number of assumptions often facilitate formal analysis. The partial list included below serves as reference.

Desirability assumptions

 * Monotonicity: The relation $$\succsim$$ is (weakly) monotone if $$x \in \mathbf{X} \land y \geq x \Rightarrow y \succsim x$$ and strongly monotone if $$x \geq y \land y \neq x \Rightarrow y \sim x$$ The monotonicity assumption translates roughly to "more is better". Though trivial in some instances, choice sets including economic 'bads' such as pollution may violate monotonicity.
 * Local nonsatiation: $$\succsim$$ is nonsatiated if $$ \forall x \in \mathbf{X},\; \forall \varepsilon > 0\; \exists y \in \mathbf{X} \mbox{ such that } |y-x| \leq \varepsilon \land \; y\sim x$$ Notice that monotonicity, translating to "more is better, even for infinitesimal deviations", implies local non-satiation, but not vice-versa. The assumption of non-satiation plays a key role in the discussion of indifference sets and indifference curves, below.

Convexity

 * $$\succsim$$ on X is convex if $$\forall x \in \mathbf{X} \mbox{ the upper contour } y \in \mathbf{X}: y\succsim x \mbox{ is convex; if} y\succsim x \land z \succsim x,\;(\alpha y + (1-\alpha)z) \succsim x\forall \alpha\in[0,1]$$


 * Convex preferences imply diminishing marginal rates of substitution. For any two goods, increasingly large amounts of one good are required to compensate for marginal losses of the other.
 * Convex preferences may be interpreted as a desire for diversification in consumption

Homothetic Preferences
Preferences are said to be homothetic if all indifference sets are related by proportional expansion along rays, $$x\sim y, \; \alpha x \sim \alpha y \; \forall \alpha \geq 0$$

Quasilinear Preferences
$$\succsim \mbox{ on } \mathbf{X} $$ is quasilinear with respect to its numeraire if:
 * 1) indifference sets are parallel displacements along the axis of commodity 1; $$x\sim y \Rightarrow (x+\alpha e_1) \sim (y+\alpha e_1), \mbox{ where } e_1 = (1,0,\dots,0) \mbox{ and } \alpha \in \mathbb{R}$$
 * 2) the numéraire is desirable,