# Variables, Functions and Equations

Economists are interested in examining types of relationships. For example an economist may look at the amount of money a person earns and the amount that person chooses to spend. This is a consumption relationship or function. As another example an economist may look at the amount of money a business firm has and the amount it chooses to spend on new equipment. This is an investment relationship or investment function.

A function tries to define these relationsips. It tries to give the relationship a mathematical form. An equation is a mathematical way of looking at the relationship between concepts or items. These concepts or items ar represented by what are called variables.

A variable represents a concept or an item whose magnitude can be represented by a number, i.e. measured quantitatively. Variables are called variables because they vary, i.e. they can have a variety of values. Thus a variable can be considered as a quantity which assumes a variety of values in a particular problem. Many items in economics can take on different values. Mathematics usually uses letters from the end of the alphabet to represent variables. Economics however often uses the first letter of the item which varies to represent variables. Thus p is used for the variable price and q is used for the variable quantity.

An expression such as 4x3 is a variable. It can assume different values because x can assume different values. In this expression x is the variable and 4 is the coefficient of x. Coefficient means 4 works together with x. Expressions such as 4x3 which consists of a coefficient times a variable raised to a power are called monomials.

A monomial is an algebraic expression that is either a numeral, a variable, or the product of numerals and variables. (Monomial comes from the Greek word, monos, which means one.) Real numbers such as 5 which are not multiplied by a variable are also called monomials. Monomials may also have more than one variable. 4x3y2 is such an example. In this expression both x and y are variables and 4 is their coefficient.

The following are examples of monomials:

x, 4x2, -6xy2z, 7

One or more monomials can be combined by addition or subtraction to form what are called polynomials. (Polynomial comes from the Greek word, poly, which means many.) A polynomial has two or more terms i.e. two or more monomials. If there are only two terms in the polynomial, the polynomial is called a binomial.

The expression 4x3y2 - 2xy2 +3 is a polynomial with three terms.

These terms are 4x3y2, - 2xy2, and 3. The coefficients of the terms are 4, -2, and 3.

The degree of a term or monomial is the sum of the exponents of the variables. The degree of a polynomial is the degree of the term of highest degree. In the above example the degrees of the terms are 5, 3, and 0. The degree of the polynomial is 5.

Remember that variables are items which can assume different values. A function tries to explain one variable in terms of another.

Consider the above example where the amount you choose to spend depends on your salary. Here there are two variables: your salary and the amount you spend.

Independent variables are those which do not depend on other variables. Dependent variables are those which are changed by the independent variables. The change is caused by the independent variable. In our example salary is the independent variable and the amount you spend is the dependent variable.

To continue with the same example what if the amount you choose to spend depends not only on your salary but also on the income you receive from investments in the stock market. Now there are three variables: your salary and your investment income are independent variables and the amount you spend is the dependent variable.

Definition: A function is a mathematical relationship in which the values of a single dependent variable are determined by the values of one or more independent variables. Function means the dependent variable is determined by the independent variable(s).

A goal of economic analysis is to determine the independent variable(s) which explain certain dependent variables. For example what explains changes in employment, in consumer spending, in business investment etc.?

Functions with a single independent variable are called univariate functions. There is a one to one correspondence. Functions with more than one independent variable are called multivariate functions.

The independent variable is often designated by x. The dependent variable is often designated by y.

We say y is a function of x. This means y depends on or is determined by x.

Mathematically we write y = f(x)

It means that mathematically y depends on x. If we know the value of x, then we can find the value of y.

In pronunciation we say " y is f of x." This does not mean that y is the product of two separate quantities, f and x but rather that f is used to indicate the idea of a function. In other words the parenthesis does not mean that f is multiplied by x.

It is not necessary to use the letter f. For example we could say
y = g(x) which also means that y is a function of x or we could say y = h(x) which too means that y is a function of x.

We may look at functions algebraically or graphically. If we use algebra we look at equations. If we use geometry we use graphs.

A simple example of functional notation

Qd = the number of pizzas (quantity) demanded

Pp = the price of a pizza

Pt = the price of tomato sauce

Pc = the price of cheese

Pd = the price of pizza dough

N = the number of potential pizza eaters

Pp = f(Pt, Pc, Pd)

This is an example of a function that says the price of pizza depends on the prices of tomato sauce, cheese, and pizza dough. There is one dependent variable, the price of pizza and there are three independent variables, the prices of tomato sauce, cheese, and pizza dough.

Qd = f(Pp, N)

This is another example of a function. It says that the quantity of pizza demanded depends on the price of pizza and the number of potential pizza eaters. There is one dependent variable, the quantity of pizza demanded, and there are two independent variables, the price of pizza and the number of potential pizza eaters.

A common economic example of functional notation

C = consumption, the amount spent on goods and services

Y = income, the amount available to spend

C = C(Y)

This is an example of a function that says the amount spent on consumption depends on income. This is a very general form of the consumption function. In order to use it economists must put it into a more precise mathematical form. For example

C = 25 + .75Y

This is a function which says that consumption is 25 regardless of the level of income and that for every extra dollar of income 75 cents are spent on consumption.

The use of functional notation: some examples

Example 1

y = f(x) = 3x + 4

This is a function that says that, y, a dependent variable, depends on x, an independent variable. The independent variable, x, can have different values. When x changes y also changes.

Find f(0). This means find the value of y when x equals 0.

f(0) = 3 times 0 plus 4

f(0) = 3(0) + 4 = 4

Find f(1). This means find the value of y when x equals 1.

f(1) = 3 times 1 plus 4

f(1) = 3(1) + 4 = 7

Find f(-1). This means find the value of y when x equals -1.

f(-1) = 3 times (-1) plus 4

f(1) = 3(-1) + 4 = 1

Example 2

d(p) = p2 -20p + 125

This is a function that describes the demand for an item where p is the dollar price per item. It says that demand depends on price.

Find the demand when one item costs \$2.

d(2) = 22 - 20(2) + 125 = 89

Find the demand when one item costs \$5.

d(5) = 52 - 20(5) + 125 = 50

Notice that, as we might expect, the demand declines as the price rises.

Example 3

Two or more functions can be added, subtracted, multiplied or divided.

g(x) = x - 3                   h(x) = x2 + 2

Find g(0) + h(0)

g(0) = 0 - 3 = -3

h(0) = 02 + 2 = 2

g(0) + h(0) = -3 + 2 = -1

Find g(1) h(2)

g(1) = 1 - 3 = -2

h(2) = 22 + 2 = 6

g(1) h(2) = (-2) ( 6) = -12

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