All Terrain Thinking

A Compendium of things I think are Important

Algebra: Calculating why it's not in your wallet

At the heart of algebra is the equation-the technical term used to symbolize a relationship between variables. It is an abstract, but essentially simple concept. Equations are similar to graphs which we have already looked at in that they are both short-hand ways of conveying information concerning a relationship between phenomena. In the case of graphs we use a visual means to communicate the information, while with equations we use mathematical symbols.

For example, consider the problem of conveying information on sales at Max's Market, a local retail outlet. Max needs to put together some numbers and prepare a financial statement before he meets with his banker. Although we can count on Max to make a full accounting of his situation, we will only look at the revenue side of the books. We want to compare two revenue projections. The first is based on the assumption that Max's sales are currently $200,000 per month and will increase $2,000 per month for each of the next 48 months. The second is that from the same existing base of $200,000, sales will increase 1 percent a month. How can we convey this information in a graph and in an equation?

The graph, which appears below, provides a valuable visual representation of the sales which a good number of people should understand. Certainly Ross Perot believed this when he brought his graphs into the 1992 presidential campaign TV infomercials.

But how did we get this graph? With algebra. We needed algebra to produce this visual representation of future sales projections. To reproduce this graph with a spreadsheet, which is how it was produced, put the numbers 1 through 48 in the first column which could be labeled months. In the second column we would program Excel to give us a number that started at 200,000 and increased by 2,000 every time month increased. If the number 1 appeared in cell A2, then in cell B2 we would type in the following = 200000 + 2000*A2. For our second forecast in the third column you would type into cell C2 the following equation: = 200000*(1.01)^A2 [ Do not worry about this equation yet-we'll get to it soon].

Given that the picture is generated by an algebraic equation, it is no surprise that some people prefer the ignore the extra step of generating the graph and simply work with the algebraic version of the relationship. While the graph provides a useful visual representation of the sales which a good number of people could understand, we could also present the two relationship between Max's sales (S) and months (M) by the following equations.

If we want to know exactly what sales will be in 6 months, we simply plug in 6 for the value of M and we get 212,000. This precision is the primary advantage of the algebra and it is the reason why behind the majority of those forecasts you here each month are algebraic equations. The disadvantage of the algebraic representation is that not as many people can interpret the language of math necessary to translate the equation back into a picture (the graph) or a story (the text). Have you ever listened to a conversation in a foreign language and marveled at the speed with which they talked? If you have, you probably have a good idea of how many people feel when they listen to people telling stories using math. They simply do not know the basics of the language of math which would allow them to understand what they are seeing / hearing.

To help you with the translation, we will spend a little time learning some of the basics of the language of math. As a starting point you should recognize mathematics as a language and think of equations as sentences and the characters that appear in the equations as the words. In a later section we will talk about models which are the paragraphs.

So let's start with the words. Rather than talk about verbs, adjectives, or nouns, we will talk about variables and parameters. Let's return to our Max's market problem. The variables are the terms used to represent the phenomena that are being studied. In this equation there are two variables, S which we use to represent the volume of sales and M which represents the number of months. We can add more precision to our analysis if we recognize that there is a causality implied in the relationship - once we know the month we will know the value for sale. The variable that is the cause, M in this case, is an exogenous or independent variable which tells us something about the external environment. Once we know the value of M, the parameters in the equation allow us to determine the value of S, the endogenous or dependent variable.

In addition to the variables, we also see the numbers 200,000 and 2,000. These are called parameters, constants that specify the exact nature of the relationship between the variables. If we change one of the numbers it would change both the picture and the story.

When we combine the words with a variety of mathematical symbols (ex. +, -, /, and *) we have equations which provide the reader with some information regarding the nature of the relationship. Once you have the values for the parameters and exogenous variables, you will be able to calculate the value of the dependent variable. In our simple example, once I know the value of M (10 months from now), I can determine the value for S (220,000). The situation is described in the diagram below.

If you return to your English grammar books, you will note that there are many types of sentences. The same is true in mathematics where we can use a number of classification schemes for sorting equations. One way to classify them is by their mathematical structure. We will talk about linear and some nonlinear relationships that are popular with economists.

What do we do with our equations? Often times we will combine them to build a model which allows us to specify the interrelationship between a number of variables. For example, let's look at the model of supply and demand. You remember those supply and demand curves and the designation of the intersection as the equilibrium point giving us reason to believe price will be P* and output will be Q*.

Just as we saw that there was an algebraic skeleton to the graphs, we can now create the algebraic structure of the supply - demand model. We begin by recognizing that we need to identify the key pieces of the supply-demand model which we want to capture in equations. One piece of information is the behavior of suppliers where we would specify the quantity supplied (Qs) as being dependent upon price (P). The second piece of information would be the behavior of demanders where the quantity demanded (Qd) is specified as being dependent upon price (P). The last bit of information would be the equilibrium condition that specified the conditions under which there would be no tendency for change in the market-what you will remember as the supply equals demand condition. We can represent this algebraically with our three-equation, linear model where I have specified the values of the parameters.

If we take these three pieces of information together we have a simple economic model which will explain the market price (P), quantity supplied (Qs), and quantity demanded (Qd). The graphical analysis would lead us to look for the intersection of the supply and demand curves while the algebraic approach would lead us toward simultaneously solving the three equations for the equilibrium values for P (50), QS (200), and Qd (200).

It is now time to move on to look at some of the basics of algebra.

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