Chapter Introduction

Whether you’re reading about economics in the news or in your economics textbook, you will see many graphs. Visual presentations can make it easier to understand verbal descriptions, numerical information, or ideas. In economics, graphs are used to facilitate understanding. But to get the full benefit from these visual aids, you need to know how to interpret them. This appendix explains how graphs are constructed, interpreted, and used in economics.

Graphs, Variables, and Economic Models

If you were reading an article about the relationship between educational attainment and income, you would probably see a graph showing the income levels for workers with different levels of education. This graph would depict the idea that, in general, having more education increases a person’s income. This graph, like most graphs in economics, would depict the relationship between two economic variables. A variable is a quantity that can take on more than one value, such as the number of years of education a person has, the price of a can of soda, or a household’s income.

Earlier we discussed how economic analysis relies heavily on models, simplified descriptions of real situations. Most economic models describe the relationship between two variables, simplified by holding constant other variables that may affect the relationship. For example, an economic model might describe the relationship between the price of a can of soda and the number of cans of soda that consumers will buy, assuming that everything else that affects consumers’ purchases of soda stays constant. This type of model can be described mathematically or verbally, but illustrating the relationship in a graph makes it easier to understand. Next we show how graphs that depict economic models are constructed and interpreted.

How Graphs Work

Most graphs in economics are based on a grid built around two perpendicular lines that show the values of two variables, helping you visualize the relationship between them. So a first step in understanding the use of such graphs is to see how this system works.

Two-Variable Graphs

Figure A-1 shows a typical two-variable graph. It illustrates the data in the accompanying table on outside temperature and the number of sodas a typical vendor can expect to sell at a baseball stadium during one game. The first column shows the values of outside temperature (the first variable) and the second column shows the values of the number of sodas sold (the second variable). Five combinations or pairs of the two variables are shown, denoted by points A through E in the third column.

The data from the table are plotted where outside temperature (the independent variable) is measured along the horizontal axis and number of sodas sold (the dependent variable) is measured along the vertical axis. Each of the five combinations of temperature and sodas sold is represented by a point: A, B, C, D, and E. Each point in the graph is identified by a pair of values. For example, point C corresponds to the pair (40, 30)—an outside temperature of 40°F (the value of the x-variable) and 30 sodas sold (the value of the y-variable).

Now let’s turn to graphing the data in this table. In any two-variable graph, one variable is called the x-variable and the other is called the y-variable. Here we have made outside temperature the x-variable and number of sodas sold the y-variable.

The solid horizontal line in the graph is called the horizontal axis or x-axis, and values of the x-variable—outside temperature—are measured along it. Similarly, the solid vertical line in the graph is called the vertical axis or y-axis, and values of the y-variable—number of sodas sold—are measured along it. At the origin, the point where the two axes meet, each variable is equal to zero. As you move rightward from the origin along the x-axis, values of the x-variable are positive and increasing. As you move up from the origin along the y-axis, values of the y-variable are positive and increasing.

You can plot each of the five points A through E on this graph by using a pair of numbers—the values that the x-variable and the y-variable take on for a given point. In Figure A-1, at point C, the x-variable takes on the value 40 and the y-variable takes on the value 30. You plot point C by drawing a line straight up from 40 on the x-axis and a horizontal line across from 30 on the y-axis. We write point C as (40, 30). We write the origin as (0, 0).

Looking at point A and point B in Figure A-1, you can see that when one of the variables for a point has a value of zero, it will lie on one of the axes. If the value of the x-variable is zero, the point will lie on the vertical axis, like point A. If the value of the y-variable is zero, the point will lie on the horizontal axis, like point B.

Most graphs that depict relationships between two economic variables represent a causal relationship, a relationship in which the value taken by one variable directly influences or determines the value taken by the other variable. In a causal relationship, the determining variable is called the independent variable; the variable it determines is called the dependent variable. In our example of soda sales, the outside temperature is the independent variable. It directly influences the number of sodas that are sold, which is the dependent variable in this case.

By convention, we put the independent variable on the horizontal axis and the dependent variable on the vertical axis. Figure A-1 follows this convention: the independent variable (outside temperature) is on the horizontal axis and the dependent variable (number of sodas sold) is on the vertical axis.

An important exception to this convention is in graphs showing the economic relationship between the price of a product and quantity of the product: although price is generally the independent variable that determines quantity, it is always measured on the vertical axis.

Curves on a Graph

Panel (a) of Figure A-2 contains some of the same information as Figure A-1, with a line drawn through the points B, C, D, and E. Such a line on a graph is called a curve, regardless of whether it is a straight line or a curved line. If the curve that shows the relationship between two variables is a straight line, or linear, the variables have a linear relationship. When the curve is not a straight line, or nonlinear, the variables have a nonlinear relationship.

A point on a curve indicates the value of the y-variable for a specific value of the x-variable. For example, point D indicates that at a temperature of 60°F, a vendor can expect to sell 50 sodas. The shape and orientation of a curve reveal the general nature of the relationship between the two variables. The upward tilt of the curve in panel (a) of Figure A-2 suggests that vendors can expect to sell more sodas at higher outside temperatures.

When variables are related in this way—that is, when an increase in one variable is associated with an increase in the other variable—the variables are said to have a positive relationship. It is illustrated by a curve that slopes upward from left to right. Because this curve is also linear, the relationship between outside temperature and number of sodas sold illustrated by the curve in panel (a) of Figure A-2 is a positive linear relationship.

When an increase in one variable is associated with a decrease in the other variable, the two variables are said to have a negative relationship. It is illustrated by a curve that slopes downward from left to right, like the curve in panel (b) of Figure A-2. Because this curve is also linear, the relationship it depicts is a negative linear relationship. Two variables that might have such a relationship are the outside temperature and the number of hot drinks a vendor can expect to sell at a baseball stadium.

Return for a moment to the curve in panel (a) of Figure A-2, and you can see that it hits the horizontal axis at point B. This point, known as the horizontal intercept, shows the value of the x-variable when the value of the y-variable is zero. In panel (b) of Figure A-2, the curve hits the vertical axis at point J. This point, called the vertical intercept, indicates the value of the y-variable when the value of the x-variable is zero.

The curve in panel (a) illustrates the relationship between the two variables, outside temperature and number of sodas sold. The two variables have a positive linear relationship: positive because the curve has an upward tilt, and linear because it is a straight line. The curve implies that an increase in the x-variable (outside temperature) leads to an increase in the y-variable (number of sodas sold). The curve in panel (b) is also a straight line, but it tilts downward. The two variables here, outside temperature and number of hot drinks sold, have a negative linear relationship: an increase in the x-variable (outside temperature) leads to a decrease in the y-variable (number of hot drinks sold). The curve in panel (a) has a horizontal intercept at point B, where it hits the horizontal axis. The curve in panel (b) has a vertical intercept at point J, where it hits the vertical axis, and a horizontal intercept at point M, where it hits the horizontal axis.

A Key Concept: The Slope of a Curve

The slope of a curve is a measure of how steep it is; the slope indicates how sensitive the y-variable is to a change in the x-variable. In our example of outside temperature and the number of cans of soda a vendor can expect to sell, the slope of the curve would indicate how many more cans of soda the vendor could expect to sell with each 1° increase in temperature. Interpreted this way, the slope gives meaningful information.

Even without numbers for x and y, it is possible to arrive at important conclusions about the relationship between the two variables by examining the slope of a curve at various points.

The Slope of a Linear Curve

Along a linear curve the slope, or steepness, is measured by dividing the “rise” between two points on the curve by the “run” between those same two points. The rise is the amount that y changes, and the run is the amount that x changes. Here is the formula:

In the formula, the symbol Δ (the Greek uppercase delta) stands for “change in.” When a variable increases, the change in that variable is positive; when a variable decreases, the change in that variable is negative.

The slope of a curve is positive when the rise (the change in the y-variable) has the same sign as the run (the change in the x-variable). That’s because when two numbers have the same sign, the ratio of those two numbers is positive.

The curve in panel (a) of Figure A-2 has a positive slope: along the curve, both the y-variable and the x-variable increase. The slope of a curve is negative when the rise and the run have different signs. That’s because when two numbers have different signs, the ratio of those two numbers is negative. The curve in panel (b) of Figure A-2 has a negative slope: along the curve, an increase in the x-variable is associated with a decrease in the y-variable.

Figure A-3 illustrates how to calculate the slope of a linear curve. Let’s focus first on panel (a). From point A to point B the value of the y-variable changes from 25 to 20 and the value of the x-variable changes from 10 to 20. So the slope of the line between these two points is

Because a straight line is equally steep at all points, the slope of a straight line is the same at all points. In other words, a straight line has a constant slope. You can check this by calculating the slope of the linear curve between points A and B and between points C and D in panel (b) of Figure A-3.

Panels (a) and (b) show two linear curves. Between points A and B on the curve in panel (a), the change in y (the rise) is −5 and the change in x (the run) is 10. So the slope from A to B is , where the negative sign indicates that the curve is downward sloping. In panel (b), the curve has a slope from A to B of . The slope from C to D is . The slope is positive, indicating that the curve is upward sloping. Furthermore, the slope between A and B is the same as the slope between C and D, making this a linear curve. The slope of a linear curve is constant: it is the same regardless of where it is calculated along the curve.

Horizontal and Vertical Curves and Their Slopes

When a curve is horizontal, the value of y along that curve never changes—it is constant. Everywhere along the curve, the change in y is zero. Now, zero divided by any number is zero. So regardless of the value of the change in x, the slope of a horizontal curve is always zero.

If a curve is vertical, the value of x along the curve never changes—it is constant. Everywhere along the curve, the change in x is zero. This means that the slope of a vertical line is a ratio with zero in the denominator. A ratio with zero in the denominator is equal to infinity—that is, an infinitely large number. So the slope of a vertical line is equal to infinity.

A vertical or a horizontal curve has a special implication: it means that the x-variable and the y-variable are unrelated. Two variables are unrelated when a change in one variable (the independent variable) has no effect on the other variable (the dependent variable). To put it a slightly different way, two variables are unrelated when the dependent variable is constant regardless of the value of the independent variable. If, as is usual, the y-variable is the dependent variable, the curve is horizontal. If the dependent variable is the x-variable, the curve is vertical.

The Slope of a Nonlinear Curve

A nonlinear curve is one in which the slope changes as you move along it. Panels (a), (b), (c), and (d) of Figure A-4 show various nonlinear curves. Panels (a) and (b) show nonlinear curves whose slopes change as you follow the line’s progression, but the slopes always remain positive.

In panel (a) the slope of the curve from A to B is , and from C to D it is . The slope is positive and increasing; it gets steeper as it moves to the right. In panel (b) the slope of the curve from A to B is , and from C to D it is . The slope is positive and decreasing; it gets flatter as it moves to the right. In panel (c) the slope from A to B is , and from C to D it is . The slope is negative and increasing; it gets steeper as it moves to the right. And in panel (d) the slope from A to B is , and from C to D it is . The slope is negative and decreasing; it gets flatter as it moves to the right.
The slope in each case has been calculated by using the arc method—that is, by drawing a straight line connecting two points along a curve. The average slope between those two points is equal to the slope of the straight line between those two points.

Although both curves tilt upward, the curve in panel (a) gets steeper as the line moves from left to right in contrast to the curve in panel (b), which gets flatter. A curve that is upward sloping and gets steeper, as in panel (a), is said to have positive increasing slope. A curve that is upward sloping but gets flatter, as in panel (b), is said to have positive decreasing slope.

When we calculate the slope along these nonlinear curves, we obtain different values for the slope at different points. How the slope changes along the curve determines the curve’s shape. For example, in panel (a) of Figure A-4, the slope of the curve is a positive number that steadily increases as the line moves from left to right, whereas in panel (b), the slope is a positive number that steadily decreases.

ABSOLUTE VALUEThe slopes of the curves in panels (c) and (d) are negative numbers. Economists often prefer to express a negative number as its absolute value, which is the value of the negative number without the minus sign.

In general, we denote the absolute value of a number by two parallel bars around the number; for example, the absolute value of −4 is written as |−4| = 4. In panel (c), the absolute value of the slope steadily increases as the line moves from left to right. The curve therefore has negative increasing slope. And in panel (d), the absolute value of the slope of the curve steadily decreases along the curve. This curve therefore has negative decreasing slope.

Maximum and Minimum Points

The slope of a nonlinear curve can change from positive to negative or vice versa. When the slope of a curve changes from positive to negative, it creates what is called a maximum point of the curve. When the slope of a curve changes from negative to positive, it creates a minimum point.

Panel (a) of Figure A-5 illustrates a curve in which the slope changes from positive to negative as the line moves from left to right. When x is between 0 and 50, the slope of the curve is positive. At x equal to 50, the curve attains its highest point—the largest value of y along the curve. This point is called the maximum of the curve. When x exceeds 50, the slope becomes negative as the curve turns downward. Many important curves in economics, such as the curve that represents how the profit of a firm changes as it produces more output, are hill-shaped like this one.

Panel (a) shows a curve with a maximum point, the point at which the slope changes from positive to negative.
Panel (b) shows a curve with a minimum point, the point at which the slope changes from negative to positive.

In contrast, the curve shown in panel (b) of Figure A-5 is U-shaped: it has a slope that changes from negative to positive. At x equal to 50, the curve reaches its lowest point—the smallest value of y along the curve. This point is called the minimum of the curve. Various important curves in economics, such as the curve that represents how a firm’s cost per unit changes as output increases, are U–shaped like this one.

Calculating the Area Below or Above a Curve

Sometimes it is useful to be able to measure the size of the area below or above a curve. To keep things simple, we’ll only calculate the area below or above a linear curve.

How large is the shaded area below the linear curve in panel (a) of Figure A-6? First, note that this area has the shape of a right triangle. A right triangle is a triangle in which two adjacent sides form a 90° angle. We will refer to one of these sides as the height of the triangle and the other side as the base of the triangle. For our purposes, it doesn’t matter which of these two sides we refer to as the base and which as the height.

The area below or above a linear curve forms a right triangle. The area of a right triangle is calculated by multiplying the height of the triangle by the base of the triangle, and dividing the result by 2. In panel (a) the area of the shaded triangle is 9. In panel (b) the area of the shaded triangle is 12.

Calculating the area of a right triangle is straightforward: multiply the height of the triangle by the base of the triangle, and divide the result by 2. The height of the triangle in panel (a) of Figure A-6 is 10 − 4 = 6. And the base of the triangle is 3 − 0 = 3. So the area of that triangle is

How about the shaded area above the linear curve in panel (b) of Figure A-6? We can use the same formula to calculate the area of this right triangle. The height of the triangle is 8 − 2 = 6. And the base of the triangle is 4 − 0 = 4. So the area of that triangle is

Graphs That Depict Numerical Information

Graphs are also used as a convenient way to summarize and display data without assuming some underlying causal relationship. Graphs that simply display numerical information are called numerical graphs.

Here we will consider the four most widely used types of numerical graphs: time-series graphs, scatter diagrams, pie charts, and bar graphs. These graphs are used to display real, empirical data about different economic variables and help economists and policy makers identify patterns or trends in the economy.

TIME-SERIES GRAPHYou have probably seen graphs that show what happens over time to economic variables such as the unemployment rate or stock prices. A time-series graph has successive dates on the horizontal axis and the values of a variable that occurred on those dates on the vertical axis. For example, Figure A-7 shows real gross domestic product (GDP) per capita—a rough measure of a country’s standard of living—in the United States from 1947 to late 2010. A line connecting the points that correspond to real GDP per capita for each calendar quarter during those years gives a clear idea of the overall trend in the standard of living over these years.

Time-series graphs show successive dates on the x-axis and values for a variable on the y-axis. This time-series graph shows real gross domestic product per capita, a measure of a country’s standard of living, in the United States from 1947 to late 2010.

Source: Bureau of Economic Analysis.

SCATTER DIAGRAMFigure A-8 is an example of a different kind of numerical graph. It represents information from a sample of 184 countries on the standard of living, measured by GDP per capita, and the amount of carbon emissions per capita, a measure of environmental pollution. This type of graph is a scatter diagram. In these graphs, each point corresponds to an actual observation of the x-variable and the y-variable. Typically, a curve is fitted to the scatter of points; that is, a curve is drawn to approximate, as closely as possible, the general relationship between the variables. Each point in Figure A-8 indicates an average resident’s standard of living and his or her annual carbon emissions for a given country.

In a scatter diagram, each point represents the corresponding values of the x- and y-variables for a given observation. Here, each point indicates the GDP per capita and the amount of carbon emissions per capita for a given country for a sample of 184 countries. The upward-sloping fitted line here is the best approximation of the general relationship between the two variables.

Source: World Bank.

The points lying in the upper right of the graph, which show combinations of a high standard of living and high carbon emissions, represent economically advanced countries such as the United States. (The country with the highest carbon emissions, at the top of the graph, is Qatar.) Points lying in the bottom left of the graph, which show combinations of a low standard of living and low carbon emissions, represent economically less advanced countries such as Afghanistan and Sierra Leone.

The pattern of points indicates that there is a positive relationship between living standard and carbon emissions per capita: on the whole, people create more pollution in countries with a higher standard of living. As you can see, the fitted line in Figure A-8 is upward sloping, indicating the underlying positive relationship between the two variables.

FIGUREA-9Pie Chart

FIGUREA-9Pie Chart

A pie chart shows the percentages of a total amount that can be attributed to various components. This pie chart shows the percentages of workers with given education levels who were paid at or below the federal minimum wage in 2009.

Source: Bureau of Labor Statistics.

Scatter diagrams are often used to show how a general relationship can be inferred from a set of data.

PIE CHARTA pie chart shows the share of a total amount that is accounted for by various components, usually expressed in percentages. Figure A-9 is a pie chart that depicts the education levels of workers who in 2009 were paid the federal minimum wage or less. As you can see, the majority of workers paid at or below the minimum wage had no college degree. Only 8% of workers who were paid at or below the minimum wage had a bachelor’s degree or higher.

BAR GRAPHA graph that uses bars of various heights or lengths to indicate values of a variable is a bar graph. In the bar graph in Figure A-10, the bars show the percent change in the number of unemployed workers in the United States from 2009 to 2010, separately for White, Black or African-American, and Asian workers. Exact values of the variable that is being measured may be written at the end of the bar, as in this figure. For instance, the number of unemployed Black or African-American workers in the United States increased by 9.4% between 2009 and 2010. But even without the precise values, comparing the heights or lengths of the bars can give useful insight into the relative magnitudes of the different values of the variable.

A bar graph measures a variable by using bars of various heights or lengths. This bar graph shows the percent change in the number of unemployed workers between 2009 and 2010, separately for White, Black or African-American, and Asian workers.

Source: Bureau of Labor Statistics.