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Essential mathematics for economic analysis course

Essential mathematics for economic analysis

ESSENTIALS OF LOGIC AND SET THEORY

Arguments in mathematics require tight logical reasoning; arguments in economic analysis are no exception to this rule. We therefore present some basic concepts from logic. A brief section on mathematical proofs might be useful for more ambitious students.

A short introduction to set theory precedes this. This is useful not just for its importance in mathematics, but also because of the role sets play in economics: in most economics models, it is assumed that, following some specific criterion, economic agents are to choose, optimally, from a feasible set of alternatives.

The chapter winds up with a discussion of mathematical induction. Very occasionally, this is used directly in economic arguments; more often, it is needed to understand mathematical results which economists often use.

 

1.1 Essentials of Set Theory

In daily life, we constantly group together objects of the same kind. For instance, we refer to the faculty of a university to signify all the members of the academic staff. A garden refers to all the plants that are growing in it.

We talk about all Scottish firms with more than 300 employees, all taxpayers in Germany who earned between €50 000 and €100 000 in 2004. In all these cases, we have a collection of objects viewed as a whole. In mathematics, such a collection is called a set, and its objects are called its elements, or its members.

How is a set specified? The simplest method is to list its members, in any order, between the two braces { and }. An example is the set S = {a, b, c} whose members are the first three letters in the English alphabet. Or it might be a set consisting of three members represented by the letters a, b, and c. For example, if a = 0, b = 1, and c = 2, then S = {0, 1, 2}. Also, S = {a, b, c} denotes the set of roots of the cubic equation (x − a)(x − b)(x − c) = 0 in the unknown x, where a, b, and c are any three real numbers.

Two sets A and B are considered equal if each element of A is an element of B and each element of B is an element of A. In this case, we write A = B. This means that the two sets consist of exactly the same elements. Consequently, {1, 2, 3} = {3, 2, 1}, because the order in which the elements are listed has no significance; and {1, 1, 2, 3} = {1, 2, 3}, because a set is not changed if some elements are listed more than once.

Alternatively, suppose that you are to eat a meal at a restaurant that offers a choice of several main dishes. Four choices might be feasible—fish, pasta, omelette, and chicken. Then the feasible set, F, has these four members, and is fully specified as

F = {fish, pasta, omelette, chicken}

Notice that the order in which the dishes are listed does not matter. The feasible set remains the same even if the order of the items on the menu is changed.
The symbol “∅” denotes the set that has no elements. It is called the empty set.

 

Specifying a Property

Not every set can be defined by listing all its members, however. For one thing, some sets are infinite—that is, they contain infinitely many members. Such infinite sets are rather common in economics. Take, for instance, the budget set that arises in consumer theory.

Suppose there are two goods with quantities denoted by x and y. Suppose one unit of these goods can be bought at prices p and q, respectively. A consumption bundle (x, y) is a pair of quantities of the two goods. Its value at prices p and q is px + qy. Suppose that a consumer has an amount m to spend on the two goods. Then the budget constraint is px + qy ≤ m (assuming that the consumer is free to underspend).

If one also accepts that the quantity consumed of each good must be nonnegative, then the budget set, which will be denoted by B, consists of those consumption bundles (x, y) satisfying the three inequalities px + qy ≤ m, x ≥ 0, and y ≥ 0. (The set B is shown in Fig. 4.4.12.) Standard notation for such a set is:

B = {(x,y) : px + qy m, x ≥ 0, y ≥ 0}

The braces { } are still used to denote “the set consisting of”. However, instead of listing all the members, which is impossible for the infinite set of points in the triangular budget set B, the specification of the set B is given in two parts.

To the left of the colon, (x, y) is used to denote the typical member of B, here a consumption bundle that is specified by listing the respective quantities of the two goods. To the right of the colon, the three properties that these typical members must satisfy are all listed, and the set thereby specified. This is an example of the general specification:

S = {typical member : defining properties}

Note that it is not just infinite sets that can be specified by properties—finite sets can also be specified in this way. Indeed, some finite sets almost have to be specified in this way, such as the set of all human beings currently alive.

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