Getting Acquainted with Economics

This chapter introduces some of the techniques of graphic analysis - tools you will use throughout the book and, more important, very likely throughout your careers. The second part of the chapter examines some pitfalls in graphic analysis - common ways in which graphs can be misleading if not drawn and interpreted with care. 

Graphs Used in Economic Analysis

Two-variable graphs

A graph, or diagram, expresses the relationship between two variables. There are only three things that a variable can do: increase, decrease or remain constant. Graphs, though a math tool, play important roles in economics by representing geometrically economic variables such as prices, quantity, interest rates, income, consumption, etc. Because the surface of a piece of paper is two-dimensional, a graph may be readily used to represent any relationship between two variables.

In order to understand and interpret graphs, you must know the rules for how to read and construct them. Graphs are not really difficult once you understand their components. Time spent in becoming comfortable with graphs will pay dividends not only in this course but in other economics and business courses.

A rectangular coordinate system is formed by two number lines, one horizontal and one vertical, that intersect at the zero point of each line. The number lines are called axes.The vertical line is called the y-axis and the horizontal line is called the x-axis. Each axis has a zero point which is shared by the two axes. This zero point, which is common to both axes, is called the origin.

The two axes divide the plane - which can be thought of as a large, flat sheet of paper - into four regions called quadrants, which are numbered counterclockwise from I to IV. The variables X and Y assume positive or negative values depending on their position in the plane:

Economics is often concerned only with the positive values of variables and graphs are usually confined to the positive quadrant (Quadrant I). Whenever a variable has a negative value, one or more of the other quadrants must be included.

Each point in the plane is identified by a pair of numbers called an ordered pair. The first number of the pair (the x-coordinate) measures the horizontal distance and the second number (the y-coordinate) measures the vertical distance. The coordinates of the point are the numbers associated with the point. And remember: graphs are always read from left to right. The above concepts are represented graphically in Figure 1 below:

Figure 1

To graph or plot a point, we place a dot at the location given by the ordered pair. The graph of an ordered pair is the dot drawn at the coordinates of the point in the plane. The points whose coordinates are (3,4) and (-2,-3) are graphed in the Figure 2 below:

Figure 2

A two-dimensional graph, like Figure 2-1 in the textbook, can show how two variables are related. Note that the book shows price on the vertical axis and quantity demanded on the horizontal axis. It is customary to place on the horizontal axis the variable that changes first. This variable is called independent variable. The variable that changes in response to a change in the independent variable is called dependent variable and is placed on the vertical axis. This convention is not followed in the case of price and quantity: Price (the independent variable) is plotted on the vertical axis and quantity (the dependent variable) is plotted on the horizontal axis. This practice dates back to Alfred Marshall's classic book Principles of Economics (1890) that laid the foundations of modern price theory. The two variables, the price of natural gas and the quantity people wish to purchase are shown by the downward sloping curve labelled DD in Fig 2-1 in the textbook. Particular points on the curve are labelled a and b. For example, point a shows that a price of $3, the demand to purchase natural gas is 80 billion cubic feet per year. The ordered pair at point a is (80, 3), that is, the x-coordinate is 80 and the y-coordinate is 3.

The definition and measurement of slope

Slope is both a qualitative and quantitative measure. It is one of the most important elements of economic graphs and measures the influence of one variable on another. Slope is defined as the ratio of the vertical change (change in the value of the variable measured in the y-axis) to the corresponding horizontal change (change in the value of the variable measured in the x-axis). We use the Greek letter Δ (delta) as a shorthand method of saying "a change in". Thus, Δy means "a change in y" and Δx means "a change in x". The formula for calculating the slope is

 Slope = Δy / Δx

For example, in measuring the slopes in Figure 2-3 of the textbook, we make the following calculations:

Figure 2-3 (a): Slope = Δy / Δx = 9 - 8 / 13 - 3 = 1/10

Figure 2-3 (b): Slope = Δy / Δx = 11 - 8 / 13 - 3 = 3/10

If a large change in y is associated with a smaller change in x, the slope is large and the curve (or straight line) is steeper (Figure 2-3b). If a small change in y is associated with a larger change in x, the slope is small and the curve (or straight line) is flatter (Figure 2-3a).

A positive slope means that the two variables have a positive or direct relationship (i.e., move together - the value of y increases as the value of x increases); in that case the line or curve slopes upward to the right. A negative slope indicates that the two variables have a negative or inverse relationship (i.e., move in opposite directions - the value of y decreases as x increases and vice versa); in that case, the line or curve slopes downward to the right. A relationship shown by a straight line is called a linear relationship. See Figure 2-2a, 2-2b and Figure 2-4a, 2-4b in the textbook. Also, there are many situations where there is no relationship between the variables. In Figure 2-2c in the textbook, a zero slope (because Δy = 0) means that the y variable remains constant no matter what happens to the x variable. In Figure 2-2d, an infinite slope (because Δx = 0 and, thus, division by zero is undefined) means that the x variable remains constant irrespective of the changes in the y variable. These cases are directly related with the concept of elasticity which is studied in ECON 203.

The slope of a straight line is constant, that is, the slope is the same no matter where on the line we calculate it. However, the slope of a curved line is not constant and depends on where on the line we calculate it. We can calculate the slope of a curved line either at a point or across an arc. The former case is illustrated in Figure 2-5 in the textbook. The slope at a point R or T is equal to the slope of a straight line that is tangent to the curved line at those  points. The slope of a curved line across an arc equals the slope of the straight line between two points on the curved line. This case is illustrated in Figure 3 below:

Figure 3

To calculate the average slope of the curve along the arc ab, we draw a straight line from a to b. The slope of the line ab is calculated by dividing the change in y, Δy, by the change in x, Δx. In moving from a to b, Δx equals 3 (4 -1) and Δy equals 2 (4.5 - 2.5). The slope of the line ab is 2/3 and, thus, the slope of the curved line across the arc ab is also 2/3.

Curved lines which are shaped like hills or valleys as in Fig 2-4c and 2-4d in the text have maximum or minimum points. In Figure 2-4c, the maximum point is where the slope changes from positive to negative whereas in Figure 2-4d, the minimum point is where the slope changes from negative to positive. The slope is always equal to zero at maximum or minimum points of curved lines because the line tangent to those points is horizontal and equal to zero.

45-degree lines

Straight lines through the origin, whose y-intercept is zero, have special application in economics. A line through the origin whose slope is equal to one is called a 45-degree line or a ray through the origin. Any point on the 45-degree line is equidistant from the horizontal and vertical axes. If a point is above that line, the value of y is greater than the value of x. In contrast, if a point is below that line, the value of x exceeds that of y. For example, if a point is on the line whose slope is equal to 2 in Figure 2-6, the value of y ( the vertical distance) is twice the value of x. Conversely, if a point is on the line whose slope is equal to 1/2, the value of x ( the horizontal distance) is twice the value of y.

Economic interpretation of the slope

It is important to understand the economic interpretation of the slope. In general, any quantity depicting rate of change and is expressed by using the word "per" (per year, per month, per week, per second, per minute, per hour etc) is represented mathematically as a slope.

Many relationships in economics are linear, that is, the data approximates a straight line. Such relationships are represented by the slope-intercept equation of a line of the form y = ax + b where y is the dependent variable, x is the independent variable, a is the coefficient of x and slope of the line, b the y-intercept (the point at which the line cuts the y-axis). In many cases, this equation is used to approximate empirical data. The data are then graphed as points in the coordinate plane and a line is drawn approximating the data. The graph of the points is called a scatter diagram and the line is called the line of best fit. A thorough analysis of lines of best fit is undertaken in statistics courses. However, it is important to understand the interpretation of the slope in a line of best fit. A couple examples will help:

Often we want to show graphically more than two dimensions. For example, a topographic map seeks to show latitude, longitude and altitude in a two dimensional page. This is done by using contour lines as in Figure 2-7 in the text. Now consider the function Z in Figure 2-8 where X, Y and Z are variables. Figure 2-8 plots this function for four different values of Z. The variable X and Y are represented on the two axes. The variable Z is represented by the labels on the curves. Point A represents Y = 40, X = 30 and Z = 20. By the same reasoning, point B represents Y = 28 (approximately), X = 40 and Z = 20. In a nutshell, a contour map or a production indifference map shows three variables is two-dimensional scale.

 Perils in the Interpretation of Graphs

A time-series graph is a particular type of two-variable diagram that is useful in depicting statistical data regarding the behavior of specific economic variables. Time (years, months, weeks, days, etc), the independent variable, is measured along the x-axis and some variable of interest (prices, interest rates, income, consumption, etc), the dependent variable, is measured along the vertical axis.

A time-series graph can give us at a glance a great amount of information. For example, Figure 2-11 in the text shows:

Time-series graphs can be misleading if they are not drawn and interpreted with care.

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