### Learning Objectives

- Understand how graphs show the relationship between two or more variables and explain how a graph elucidates the nature of the relationship.
- Define the slope of a curve.
- Distinguish between a movement along a curve, a shift in a curve, and a rotation in a curve.

Much of the analysis in economics deals with relationships between variables. A variable is simply a quantity whose value can change. A graph is a pictorial representation of the relationship between two or more variables. The key to understanding graphs is knowing the rules that apply to their construction and interpretation. This section defines those rules and explains how to draw a graph.

## Drawing a Graph

To see how a graph is constructed from numerical data, we will consider a hypothetical example. Suppose a college campus has a ski club that organizes day-long bus trips to a ski area about 100 miles from the campus. The club leases the bus and charges $10 per passenger for a round trip to the ski area. In addition to the revenue the club collects from passengers, it also receives a grant of $200 from the school’s student government for each day the bus trip is available. The club thus would receive $200 even if no passengers wanted to ride on a particular day.

The table in Figure 35.1 “Ski Club Revenues” shows the relationship between two variables: the number of students who ride the bus on a particular day and the revenue the club receives from a trip. In the table, each combination is assigned a letter (A, B, etc.); we will use these letters when we transfer the information from the table to a graph.

Figure 35.1 Ski Club Revenues

Combination | Number of passengers | Club revenue |
---|---|---|

A | 0 | $200 |

B | 10 | 300 |

C | 20 | 400 |

D | 30 | 500 |

E | 40 | 600 |

The ski club receives $10 from each passenger riding its bus for a trip to and from the ski area plus a payment of $200 from the student government for each day the bus is available for these trips. The club’s revenues from any single day thus equal $200 plus $10 times the number of passengers. The table relates various combinations of the number of passengers and club revenues.

We can illustrate the relationship shown in the table with a graph. The procedure for showing the relationship between two variables, like the ones in Figure 35.1 “Ski Club Revenues”, on a graph is illustrated in Figure 35.2 “Plotting a Graph”. Let us look at the steps involved.

Figure 35.2 Plotting a Graph

Here we see how to show the information given in Figure 35.1 “Ski Club Revenues” in a graph.

## Step 1. Draw and Label the Axes

The two variables shown in the table are the number of passengers taking the bus on a particular day and the club’s revenue from that trip. We begin our graph in Panel (a) of Figure 35.2 “Plotting a Graph” by drawing two axes to form a right angle. Each axis will represent a variable. The axes should be carefully labeled to reflect what is being measured on each axis.

It is customary to place the independent variable on the horizontal axis and the dependent variable on the vertical axis. Recall that, when two variables are related, the dependent variable is the one that changes in response to changes in the independent variable. Passengers generate revenue, so we can consider the number of passengers as the independent variable and the club’s revenue as the dependent variable. The number of passengers thus goes on the horizontal axis; the club’s revenue from a trip goes on the vertical axis. In some cases, the variables in a graph cannot be considered independent or dependent. In those cases, the variables may be placed on either axis; we will encounter such a case in the chapter that introduces the production possibilities model. In other cases, economists simply ignore the rule; we will encounter that case in the chapter that introduces the model of demand and supply. The rule that the independent variable goes on the horizontal axis and the dependent variable goes on the vertical usually holds, but not always.

The point at which the axes intersect is called the origin of the graph. Notice that in Figure 35.2 “Plotting a Graph” the origin has a value of zero for each variable.

In drawing a graph showing numeric values, we also need to put numbers on the axes. For the axes in Panel (a), we have chosen numbers that correspond to the values in the table. The number of passengers ranges up to 40 for a trip; club revenues from a trip range from $200 (the payment the club receives from student government) to $600. We have extended the vertical axis to $800 to allow some changes we will consider below. We have chosen intervals of 10 passengers on the horizontal axis and $100 on the vertical axis. The choice of particular intervals is mainly a matter of convenience in drawing and reading the graph; we have chosen the ones here because they correspond to the intervals given in the table.

We have drawn vertical lines from each of the values on the horizontal axis and horizontal lines from each of the values on the vertical axis. These lines, called gridlines, will help us in Step 2.

## Step 2. Plot the Points

Each of the rows in the table in Figure 35.1 “Ski Club Revenues” gives a combination of the number of passengers on the bus and club revenue from a particular trip. We can plot these values in our graph.

We begin with the first row, A, corresponding to zero passengers and club revenue of $200, the payment from student government. We read up from zero passengers on the horizontal axis to $200 on the vertical axis and mark point A. This point shows that zero passengers result in club revenues of $200.

The second combination, B, tells us that if 10 passengers ride the bus, the club receives $300 in revenue from the trip—$100 from the $10-per-passenger charge plus the $200 from student government. We start at 10 passengers on the horizontal axis and follow the gridline up. When we travel up in a graph, we are traveling with respect to values on the vertical axis. We travel up by $300 and mark point B.

Points in a graph have a special significance. They relate the values of the variables on the two axes to each other. Reading to the left from point B, we see that it shows $300 in club revenue. Reading down from point B, we see that it shows 10 passengers. Those values are, of course, the values given for combination B in the table.

We repeat this process to obtain points C, D, and E. Check to be sure that you see that each point corresponds to the values of the two variables given in the corresponding row of the table.

The graph in Panel (b) is called a scatter diagram. A scatter diagram shows individual points relating values of the variable on one axis to values of the variable on the other.

## Step 3. Draw the Curve

The final step is to draw the curve that shows the relationship between the number of passengers who ride the bus and the club’s revenues from the trip. The term “curve” is used for any line in a graph that shows a relationship between two variables.

We draw a line that passes through points A through E. Our curve shows club revenues; we shall call it *R*_{1}. Notice that *R*_{1} is an upward-sloping straight line. Notice also that *R*_{1} intersects the vertical axis at $200 (point A). The point at which a curve intersects an axis is called the intercept of the curve. We often refer to the vertical or horizontal intercept of a curve; such intercepts can play a special role in economic analysis. The vertical intercept in this case shows the revenue the club would receive on a day it offered the trip and no one rode the bus.

To check your understanding of these steps, we recommend that you try plotting the points and drawing *R*_{1} for yourself in Panel (a). Better yet, draw the axes for yourself on a sheet of graph paper and plot the curve.

## The Slope of a Curve

In this section, we will see how to compute the slope of a curve. The slopes of curves tell an important story: they show the rate at which one variable changes with respect to another.

The slope of a curve equals the ratio of the change in the value of the variable on the vertical axis to the change in the value of the variable on the horizontal axis, measured between two points on the curve. You may have heard this called “the rise over the run.” In equation form, we can write the definition of the slope as

Equation 35.1

Equation 35.1 is the first equation in this text. Figure 35.3 “Reading and Using Equations” provides a short review of working with equations. The material in this text relies much more heavily on graphs than on equations, but we will use equations from time to time. It is important that you understand how to use them.

Figure 35.3 Reading and Using Equations

Many equations in economics begin in the form of Equation 35.1, with the statement that one thing (in this case the slope) equals another (the vertical change divided by the horizontal change). In this example, the equation is written in words. Sometimes we use symbols in place of words. The basic idea though, is always the same: the term represented on the left side of the equals sign equals the term on the right side. In Equation 35.1 there are three variables: the slope, the vertical change, and the horizontal change. If we know the values of two of the three, we can compute the third. In the computation of slopes that follow, for example, we will use values for the two variables on the right side of the equation to compute the slope.

Figure 35.4 “Computing the Slope of a Curve” shows *R*_{1} and the computation of its slope between points B and D. Point B corresponds to 10 passengers on the bus; point D corresponds to 30. The change in the horizontal axis when we go from B to D thus equals 20 passengers. Point B corresponds to club revenues of $300; point D corresponds to club revenues of $500. The change in the vertical axis equals $200. The slope thus equals $200/20 passengers, or $10/passenger.

- Select two points; we have selected points B and D.
- The slope equals the vertical change divided by the horizontal change between the two points.
- Between points B and D, the slope equals $200/20 passengers = $10/passenger.
- The slope of this curve is the price per passenger. The fact that it is positive suggests a positive relationship between revenue per trip and the number of passengers riding the bus. Because the slope of this curve is $10/passenger between any two points on the curve, the relationship between club revenue per trip and the number of passengers is linear.

We have applied the definition of the slope of a curve to compute the slope of *R*_{1} between points B and D. That same definition is given in Equation 35.1. Applying the equation, we have:

The slope of this curve tells us the amount by which revenues rise with an increase in the number of passengers. It should come as no surprise that this amount equals the price per passenger. Adding a passenger adds $10 to the club’s revenues.

Notice that we can compute the slope of *R*_{1} between any two points on the curve and get the same value; the slope is constant. Consider, for example, points A and E. The vertical change between these points is $400 (we go from revenues of $200 at A to revenues of $600 at E). The horizontal change is 40 passengers (from zero passengers at A to 40 at E). The slope between A and E thus equals $400/(40 passengers) = $10/passenger. We get the same slope regardless of which pair of points we pick on *R*_{1} to compute the slope. The slope of *R*_{1} can be considered a constant, which suggests that it is a straight line. When the curve showing the relationship between two variables has a constant slope, we say there is a linear relationship between the variables. A linear curve is a curve with constant slope.

The fact that the slope of our curve equals $10/passenger tells us something else about the curve—$10/passenger is a positive, not a negative, value. A curve whose slope is positive is upward sloping. As we travel up and to the right along *R*_{1}, we travel in the direction of increasing values for both variables. A positive relationship between two variables is one in which both variables move in the same direction. Positive relationships are sometimes called direct relationships. There is a positive relationship between club revenues and passengers on the bus. We will look at a graph showing a negative relationship between two variables in the next section.