MR-1

Math Review Appendix

There is no doubt that economics is about ideas. These economic ideas can be conveyed in a variety of languages—from words to the use of algebraic equations and graphs. We’ve used all these methods in the text, and the appendices scattered throughout the book and online add yet another language: calculus.

Sometimes it’s easier to express and understand economic ideas using mathematical symbols rather than words. The material we present here reviews the algebra and geometry concepts we use in the text, as well as the calculus that the material in the appendices relies on. You have likely encountered most of these concepts and techniques before in your high school or college algebra or calculus classes, but we compile them here in a way that is most relevant to your study of economics. We will begin with a discussion of lines and curves, which are crucial to economic ideas such as utility and income, among others, and will set the foundation for the rest of our review.

Lines and Curves Functions describe the relationships between input variables and outputs, and may be written generically as y = f(x), where x is some input and y the output. One function crucial to our study is that of the line, which we commonly write in the form y=mx + b, and which we’ve graphed on the x-y plane—the Cartesian plane—in Figure A.1 below. As in our generic function, x and y are inputs and outputs, respectively.

We can learn a lot from this functional form of the line, which is known as the slope-intercept form for reasons that will soon become clear. The line’s slope, m, describes the change in y from a given change in x, and can be written mathematically as

where Δ represents the change in a variable, and (x₁, y₁) and (x₂, y₂) are two points on the line.

MR-2

The slope tells us two important pieces of information about the line. First, it describes how flat or steep the line is. In other words, we can answer the question: If x increases by one unit, by how much does y change? The slope also tells us whether the relationship between x and y is negative or positive. An upward-sloping line, as in Figure A.1, has a positive slope, meaning that when x increases, so does y. A negative slope indicates that an increase in x results in a decrease in y, creating a downward-sloping line. A flat, horizontal line has a slope equal to zero; as x changes, there is no corresponding change in y. On a vertical line, x doesn’t change at all, even as y increases or decreases. As a result, we describe a vertical line’s slope—which is equal to infinity—as undefined.

The form y = mx + b holds one final piece of information about the line: b, or the y-intercept. You can clearly see this in Figure A.1—it’s where the line intersects the y-axis. In this case, the y-intercept is positive, but the line could also have a negative intercept and intersect the y-axis below the x-axis, or have a y-intercept equal to zero.1

1 A line may also have no y-intercept, although we won’t see these much in our study of intermediate microeconomics. This occurs when the line is vertical. Vertical lines will either intersect the y-axis at all points (if the line is equal to the y-axis, or x = 0) or, more commonly, not intersect the y-axis at all (say, for the line x = 2 or x = –4).

We have described the equation for the line using the slope-intercept form, but a line can be expressed in any way that demonstrates the relationship between the input x and its output y. You might also see a line written in the standard form of a line, mx – y = –b, or even as . In fact, we’ll see that last form a lot throughout our study of economics: It’s how we typically write supply and demand functions.

By definition, the slope of a straight line is constant; the slope will be the same no matter where you measure the change in x and y. But, the relationship between x and y may be more complicated. In contrast to the line, a curve can have different slopes at different points along it. Because a curve can have an almost infinite number of shapes and curvatures, there are no standard functional forms for curves.

Tangency The point at which a given line and curve just touch—without intersecting or overlapping—is the tangent point. It describes the point at which the slope of the line is equal to the slope at the particular point on the curve. This concept of tangency is particularly useful in the study of microeconomics, which relies on tangencies in optimization problems such as profit and utility maximization.

Graphically, a tangent point is easy to see. Point A in panel a of Figure A.2 is a tangent point between the curve and a line. How can we tell this is a tangency? At A, the curve just touches the line without crossing it. In this case, this is the only point at which the slope of the curve equals the slope of the line, and is therefore tangent to the line.

MR-3

Not all lines and curves are tangent to each other. Panel b illustrates an example of a curve and a line that, while intersecting, have no tangencies.