Lecture 1: Welcome to Econ 50

Welcome to Econ 50! I’m excited to get to see you in person soon.

How to use this textbook

The primary purpose of this textbook is to prepare you for lectures. There is one (long) page for each lecture; there is a quiz on that reading due before class, which you’ll find in Canvas or linked at the top and bottom of the reading. Please note that I do not expect you to master the material by doing the reading. The quiz is meant to (a) give you an incentive to read before coming to class, which will make class much more productive to your learning; and (b) give you a heads-up about the tricky bits that trip up most students. But do NOT feel you have to memorize the readings or fully understand the material just from having read the textbook!

You’ll notice that the diagrams in the textbook are interactive; this is meant to encourage exploration, especially of how changes in underlying parameters of a graph affect the graph’s appearance. Because of the interactivity, if you print the textbook, instead of the graphs printing, you’ll find a link to the interactive version.

This past Spring we embarked on a substantial redesign of Econ 50: based on several years of student feedback, I’m including more real-life applications, and also moving some topics from 51 into 50. As a result, this textbook has been substantially rewritten. We ironed out a lot of the kinks in the Spring, but if you spot a typo, or find an explanation confusing, or even just want to make a suggestion, please submit a note using the link at the top of each page; we’ll work to fix it ASAP. (If you have a question about content, though, please post that to Ed Discussion.)

Supplemental resources

Beyond this textbook, you might want to use Hal Varian’s Intermediate Microeconomics: A Modern Approach. Any edition will do. I used to teach the course out of that textbook, so it covers much of the same material and uses much the same approach as we will. I’ll post which chapter is most relevant for each lecture.

I’ve created this course on Khan Academy to help you review univariate calculus and learn some basics of multivariate calculus.

UCSD has graciously allowed Stanford students to use the Intermediate Microeconomics Video Handbook (IMVH) for free. Just go to this page and click the yellow “Add to Cart” button, then “buy” the product; the price will be zero if you use your Stanford email address. I will link to appropriate IMVH videos in each page of this textbook. Except for the IMVH, I would caution against using YouTube videos on the economic content of this course. People cover this stuff very differently, and students have run into problems in the past by using approaches which differed from the ones we use in this class.

About the math in this course

The official math prerequisite for Econ 50 is Math 20. Stanford gives a number of ways to waive this requirement, most commonly with a 5 on the AB Calculus exam. However, there is a difference between being permitted to take Econ 50, and being prepared to take Econ 50. Feedback from the last year suggests that students who had previously taken a multivariable calculus like Math 51 felt much better prepared to take Econ 50 than those who hadn’t. Furthermore, students who had taken any math at Stanford also felt more prepared than those who had only had math in high school. Here’s why.

Economists use three “languages” to describe their models:

• Verbal: a description of who is in the model, what they care about, what external stimuli they are responding to, and how changes in the parameters of the model affect the decisions they make.
• Mathematical: for example, functions describing a consumer’s preferences, or a firm’s technology, or equations characterizing equilibria.
• Visual: graphical representations of the behavior, meant to succinctly and intuitively convey the key elements of the model.

Success in economics requires fluency in all three of these “languages.” Just as in learning a language, fluency is different from the ability to receive a passing grade on a test, or to work through a problem with unlimited time and reference materials. It definitely doesn’t mean being able to “plug and chug” your way to a numerical solution. Fluency means an intuitive understanding of how words, math, and graphs are related.

For example, consider the core economic concept of diminishing marginal returns, which means that as you do more of something (e.g. study), you get fewer and fewer results (e.g. more points on an exam). If $x$ is hours of studying and $y$ is your grade, we might express the grade as a function of studying: \(y = f(x)\) Suppose we go further, and say that this relationship takes on the form \(y = x^a\) where $a$ is some parameter that’s greater than zero.

If studying exhibits diminishing marginal returns (a verbal concept), what does that imply about the value of $a$ (a mathematical concept), and the graph of $f(x)$ ( a visual representation)?

The graph below shows the function $y = x^a$. The actual numbers aren’t that important (indeed, the scale of the $y$ axis changes to keep the graph in the field of view); but the shape of the graph tells an important story.

Now think about how the slope of the graph changes as you study more. Verbally, this is the additional improvement to your grade from another hour studying. Mathematically, by the power rule, this is given by \(f^\prime(x) = {d \over dx}\left(x^a\right) = ax^{a - 1}\) Visually, we can plot this below the graph of $f(x)$:

By now you’ve probably figured out what’s going on. The phrase “diminishing marginal returns” means that the marginal returns (i.e., the first derivative of the function) is diminishing (getting smaller, though still remaining positive). Visually, this means that the curve representing $f(x)$ gets flatter as you move to the right, and that the graph of $f^\prime(x)$ is downward sloping. How does this relate to $a$, though?

To answer this question rigorously, we can look at the second derivative of $f(x)$, which tells us how the first derivative changes: \(\begin{aligned} y &= f(x) = x^a \\ {dy \over dx} &= f'(x) = ax^{a-1}\\ {d^2y \over dy^2} &= f''(x) = (a-1)ax^{a-2} \end{aligned}\) In order to have the first derivative be decreasing, we need the second derivative to be negative. Note that since $a \ge 0$ and $x \ge 0$ (you can’t study a negative number of hours), it must be the case that $ax^{a-2} \ge 0$; so the sign of $f^{\prime\prime}(x)$ must be the same sign as $a - 1$: positive when $a > 1$, zero when $a = 1$, and negative when $a < 1$. Therefore, in order to model “diminishing marginal returns to studying” with this functional form, we need $a < 1$.

More generally, when we analyze the curvature of a function $f(x)$ by analyzing the sign of its second derivative, we will say that the function is:

• convex (or “concave up”) if its second derivative is positive: that is, if its slope is increasing (getting more positive or less negative) as $x$ increases.
• concave (or “concave down”) if its second derivative is negative: that is, if its slope is decreasing (getting less positive or more negative) as $x$ increases.
• linear if its second derivative is zero: that is, if its slope is not changing as $x$ increases.

Note that a function, like the one we analyzed above, may be concave for some values and convex for others. Others, like the function we analyzed at the beginning of this lecture, are either concave, convex, or linear over their entire domain.

One way to test if a function is always concave or convex is to take any two points and draw a line between them. If no matter which two points you choose, the segment connecting them lies above the function, then the function is convex. Conversely, if the segment always lies below the function, then the function is concave. (And obviously, if the segment always lies along the function, then the function must be linear.)

Sketching a function: communicating ideas visually

You will often be asked to “sketch” functions in this class. One of the most famous stories in economics is that economist Art Laffer sketched what would become known as the “Laffer Curve” on a cocktail napkin. The napkin is now in the Smithsonian National Museum of American History; you can see it here. There is nothing precise about his diagram; but the story it told was so powerful (that taxes not only have diminishing marginal revenue, but high enough taxes have negative marginal revenue) that it animated the conservative movement in the United States for over a generation.

Key to a good sketch is being faithful to its general shape: where it’s increasing, where it’s decreasing, where it’s concave, where it’s convex. For example, many economic problems are concerned with optimizing something. This can mean maximizing something desirable like utility, or social welfare, or minimizing something undesirable like expense or risk. It can also mean balancing costs and benefits. To round out this lecture, let’s quickly review the mathematics of finding local maxima and minima of a function using calculus, and how we can use this information to sketch the graph of a function.

For example, if we plot the function \(y = f(x) = 24x-6x^{2}-8x^{3}+3x^{4}\) we get the following graph:

As you can see, there are two local minima, at $x = -1$ and $x = 2$. Of these, the one at $x = -1$ is also a global minimum, because there is no value of $x$ for which $f(x) < f(-1)$. There is no global maximum, because the function increases as $x$ becomes very large or very small.

In the case of a continuous, smooth function (one which is both continuous and continuously differentiable), a critical point — that is, a local maximum or minimum — occurs at a point where the function is “flat”. For a univariate function $y = f(x)$, this occurs where the derivative $dy/dx$ is equal to zero. If we plot the function above as well as its derivative, we can see that when the derivative is positive, the function is rising; when the derivative is negative, the functions is falling; and when the derivative is zero, the function is at a local maximum or minimum:

We can use this observation to sketch a function from its formula. In this case, the derivative of the function is \(f^\prime(x) = 24 - 12x - 24x^2 + 12x^3\) If we factor this ⊕ Don’t panic. You won’t ever have to factor a cubic in this course! You might have to factor a quadratic, though…including on the homework for this week…, we get \(f^\prime(x) = 12(2 - x - 2x^2 + x^3) = 12(x+1)(x-1)(x-2)\) This is zero when $x = -1$, $x = 1$, or $x = 2$; so those are the $x$-values of its critical points. If we plug those back into the function, we see that this means there are critical points at $(-1, -19)$, $(1, 13)$, and $(2,8)$.

We can also find out whether the function is increasing or decreasing between those points, by evaluating the signs of the factors $(x+1)$, $(x-1)$, and $(x-2)$ between the critical points:

Try sketching the graph on a piece of paper using just that information. Then see how close your drawing is to the function shown above.

Asymptotes and Restricted Domains

It’s important to note that not every function is defined for every possible value of $x$. Let’s think of two examples: square root functions, and functions with $x$ in the denominator.

First, (real) square roots are only defined for positive values. Therefore, the function $f(x) = \sqrt{x - c}$ is restricted to domain in which $x \ge c$. In the graph below, there are no values plotted for $x < c$:

Second, rational functions are not defined for values when their denominator is equal to zero. Therefore, the function $f(x) = {1 \over x - c}$ is not defined when $x = c$. As $x$ approaches $c$ from below, the function approaches $-\infty$, while as $x$ approaches $c$ from above, the function approaches $\infty$. When a function approaches negative or positive infinity at a point on its domain, but is not defined at that point, it’s called a vertical asymptote. The best way to indicate this on a graph is to draw a vertical dashed line that the function approaches, but never reaches:

In conclusion, when you’re sketching a function, understand its domain; get its critical points right; understand its curvature (convexity and concavity); and then draw something which tells the story you need it to tell.

What to do before the first class

Before the first class, please read the course syllabus take the quiz on Canvas using the link below.

Please also fill out the section preferences form as soon as possible, and definitely before the first class; sections start in Week 1, and we’ll assign people to sections no later than Tuesday. Don’t worry, you can easily change sections!

Reading Quiz

That's it for today! Click here to take the quiz on this reading.