Definitions of Economics Terms

These are taken from the book [#!MWG95!#].

Preference Relations:

A relation that captures the decision makers taste! Its a usual binary relation, denoted by $\ge$ (greater than equal to) on the set X, where X denotes the set of alternatives. A rational preference relation is the one which is complete and transitive. By complete we mean that any two elements in the set X are comparable.

Utility Functions:

A utility function assigns a numerical value to each element of the set X (i.e. the set of choices/alternatives). If $x \ge y$, then $u(x) \ge u(y)$, where $x, y \in X$. Here it is assumed that there is a rational preference relation on the set X, and the utility function preserves that preference relation.

Commodities:

The objects of choice by consumers in a market economy. Commodities are finite in number. The commodity vector represents amounts of different commodities, usually referred to as a consumption bundle/vector. Let us assume that there are in all L different commodities. Consumption set is a subset of the commodity space R^L, where each element of the subspace is a consumption bundle.

Who is Walras?

Here is a great web-site about Who is Walras and on his Contributions to Economics.

Walrasian Set/ Competitive Budget Set:

Assume that each commodity has a given price (per unit). The consumption bundle $x \in R^L$ is affordable for a consumer if its total cost does not exceed the wealth of the consumer w (i.e., $p.x \le w$). The Walrasian set or the competitive budget set refers to the set of affordable consumption bundles (i.e., $x \in {R_+}^L : p.x \le w$).

Walrasian demand function:

For each price wealth pair, (p,w), x(p,w) refers to the set of consumption bundles satisfying (p,w). In general x(p,w) is a relation, but in the case it is single valued, it is called a demand function, otherwise demand correspondence. Walrasian demand correspondence is homogeneous of degree zero if for any constant r >0, x(rp, rw)=x(p,w) (i.e. scaling of price and wealth by the same proportion does not alter the choices for the consumers).

Walras's Law:

Walrasian demand correspondence satisfies Walras's Law if for every p and w >0, p.x = w, for all $x \in x(p,w)$. Essentially it says that consumers fully expends their wealth!

Monotone preference relation:

A preference relation on X is monotone, if $x \in X$, and $y \ge x$, then $y \in X$ (consumption of larger amounts of goods is always feasible).

Non-satiated preference relation:

For every $x \in X$, and every $\epsilon > 0$, there is a $y \in X$, such that $\vert\vert y-x\vert\vert \le \epsilon$ and $y \ge x$. (Always in the neighborhood of x, we can find another consumption bundle y, that is preferred over x, and y is not same as x.)

Convex preference relation:

A preference relation is convex, if for every $x \in X$, the set $\{y\in X: y \ge x\}$ is convex (upper contour set is convex).

Continuous preference relation:

It is continuous if it is preserved under limits. For any sequence of pairs $\{x^n,y^n\}, n=1,2,...,$ with $x^n \ge y^n$, and $x=\lim x^n$ and $y= \lim y^n$, when $n \rightarrow \infty$, then $x \ge y$. Alternatively, the upper contour set $\{y\in X: y \ge x\}$, and the lower contour set $\{y \in x: y \le x\}$ are closed. Consumer preferences do not exhibit jumps!! It turns out that for every continuous preference relation, one can find a continuous utility function!

Utility maximization problem:

Assume that consumers have rational, continuous, and locally non-satiated preference relation over the set X=R₊^L and the corresponding continuous utility function u(x). Then the maximization problem is to max u(x), subject to $p.x \le w$ and $x \ge 0$. Why does utility maximization problem has a solution? Note that continuous functions always have an optimum value on a compact set. Utility function is continuous. The set of feasible solutions, namely the budget set, $\{x\in {R_+}^L: p.x\le w\}$ is compact, since it is bounded and closed.