2.1 Expectations of Fluency
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.
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 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.
[ See interactive graph online at https://www.econgraphs.org/graphs/math/univariate_calculus/returns_to_studying ]
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 \(f'(x) = {d \over dx}\left(x^a\right) = ax^{a - 1}\) Visually, we can plot this below the graph of $f(x)$:
[ See interactive graph online at https://www.econgraphs.org/graphs/math/univariate_calculus/marginal_returns_to_studying ]
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 $f’‘(x)$ 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 $a < 1$.