All Terrain Thinking

A Compendium of things I think are Important

Algebra: Calculating why it's not in your wallet

Introduction

Algebra is an extremely powerful tool, and one that you should make an effort to master. First, a working knowledge of it is essential if you are to realize the power of spreadsheet software, one of the most popular software programs on the market and one which employers are increasingly looking for mastery of when they hire college grads. It is also a valuable tool in economics-no surprise given that our main concern is with the study of relationships between observed quantities, and algebra/ mathematics offers us a way to express the existence of such relationships. Algebra can be thought of as an alternative to the graphs which fill nearly all economics texts.

At the heart of algebra is the function-the technical term used to symbolize a relationship between variables. It is an abstract but essentially simple concept. A function is merely a mathematical statement that two or more variables are related. For example, if we believe that two variables, x and y, are related to one another, we may write: y = f(x). This expression is read, 'y' is a function of 'x', or 'y' is dependent upon 'x'. The choice of the letter f was purely arbitrary. An alternative approach would be to repeat the letter used on the left hand side of the equation (ex. y = y(x)). How we specify the relationship is fairly unimportant. What is important is that when you see this notation y = h(x), it should be interpreted as a shorthand for the value of y depends on the value of x.

Oftentimes in economics, however, we are called upon to specify the exact nature of the relationship between the variables. When we do this we are specifying the relationship in the form of equations. Examples of equations are presented below. These equations are the inputs into models, collections of equations that describe the interrelationship between a set of variables.

• y = 4x + 3
• y = 4x^1/2 + 3
• y = 4x - 2z +3
• y = 10/x
• y = 2^x
• lny = 4lnx
• y = 3 + 4x²
• lnQ = -2lnP +1.5lnY

At this point do not be concerned by the fact that many of these equations look 'new' to you. Most of them you will never have to deal with, but you should know that they are all just simply ways of expressing relationships between the variables y and x. Now move on to a discussion of those that you are likely to run into, beginning with a discussion of linear equations and then moving to some nonlinear equations and then finishing with a discussion of some important equations of change.

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