At this point, we know a lot about straight lines. What we need to do now is to use this information to build models of the data. These models will allow us to explore what happens to the dependent variable for different values of the independent variable. Right now, we’re after models that are proportional. A proportion is simply a ratio (a fraction) between two quantities. Most people use the term proportional model interchangeably with linear model. These are models in which the change in the y-variable is in a fixed proportion to the change in the x-variable. What this means is that no matter what x-value you are looking at, if you increase it by a fixed amount, any fixed amount, then the change in y is fixed by the constant of proportionality. In this case, the constant of proportionality is the slope of the line.
Consider the cost of manufacturing widgets. (Widget is simply a word to describe something non-specific; a widget could be anything: a baseball bat, an engine part, or even a sandwich.) Normally there are fixed costs associated with manufacturing. These costs are constant, regardless of how many widgets you make. The fixed costs include things like payment for the production facilities, coverage of salaried employees, electricity, and other costs that are reasonably constant. Fixed costs look pretty much like a y-intercept on a graph. There are also variable costs in production. These include the cost of the materials to make the widgets and the wages of the employees who make the widgets. They may also include costs for quality control. Clearly, the more widgets you make, the more materials and labor you will use. It is easiest to assume that these variable costs are like the slope of a linear model, so each additional widget adds a certain amount of cost to the total manufacturing costs. Thus, we have a linear model:
Total cost of producing widgets = Fixed Costs + (Variable Costs) * (Number of Widgets)
(There are certainly other ways of modeling cost, but this is the easiest to understand, so it makes a nice starting point.) Suppose your fixed costs are $1,000 and the variable costs are $3.50 per widget. If you make 10 widgets, it will cost you $1,000 + $3.50 per widget* 10 widgets = $1,000 + $35 = $1,035. If you make five more widgets, for a total of 15 widgets, it will cost $1,000 + $3.50 (15) = $1,052.50, exactly $17.50 more. Notice that $17.50 = $3.50 * 5. In a proportional model, no matter what the current production level is (how many widgets you are making), the model always predicts the same change in y for a fixed change in x. Such models are sometimes called level independent models.
What this really means is that whether you are making 10 widgets or 20,000 widgets, if you make 5 more widgets it will cost an additional $17.50. This is the reason that linear models are so useful; they are easy to interpret. Each coefficient in the equation of the model has a meaning that is easily understood in terms of the problem context. Just to keep you even more confused, economists often refer to marginal costs, the additional cost you will pay in order to make one more widget. For a linear model, the marginal cost is simply the slope.
For more information, see the interactive Excel workbook C08 StepByStep.xls [.rda].