As economists, we all know the value of being precise in our research. Imprecise assumptions inevitably lead to results which aren't terribly robust. Hand-waving your way through a proof only leads to disaster.
The same principle applies when asking your students to solve problems. As professors, the fall offers a chance to rewrite problem sets, create new ways to challenge your students, and improve on the previous year. When writing a problem, we have a specific idea in mind about what we want a student to understand after answering the question. But in a question's first draft we often inadvertently include language with, from some perspective or another, could lead students down the wrong path. Writing more precise questions allows us to pinpoint the specific concept we want students to be analyzing.
In the classroom, too, precision is important in helping students to understand the exact point you're trying to get across. This is especially true of principles classes, where we don't have the luxury of using mathematics as a "precision safety net."
For example, consider the problem of how to teach marginal analysis—a simple and powerful concept, but one of the most difficult to teach well. If we could simply rely on math, we could say something like the following.
Say we have two functions, R(x) and C(x), which represent the revenue and cost from producing x units of a good. Assume these functions are continuous and differentiable, that R(x) is increasing and strictly concave on support of x, that C(x) is strictly convex on the support of x, and that the support of x is a closed interval on the real line. We're trying to maximize profit, which we may write as П(x) = R(x) - C(x). Because R(x) is strictly concave and C(x) is strictly convex, we know that П(x) must be strictly concave, and that therefore the first-order condition for maximization is R'(x) - C'(x) ≥ 0, with strict equality if x is on the interior of its support.
Even this paragraph is surely not precise enough for a prelim, but let's consider it as a benchmark of how you might explain optimization to a math-savvy audience.
Now suppose you tried to simplify this for a principles class. You might be tempted to say something like the following:
You own a business that makes chairs, and you're trying to figure out how many chairs to make. In order to maximize your profit, you should to keep producing chairs as long as your additional revenue from producing another chair is equal to the additional cost you incur from making that chair; that is, until the point when your marginal benefit is equal to your marginal cost.
You've essentially said "the function R(x) - C(x) is maximized where R'(x) = C'(x)," which is true, but only under a host of other assumptions. Now, clearly, you don't need to go into all of those assumptions, and indeed some of them you'll need to violate in order to simplify what you're trying to say. (For example, if you're dealing with a discrete number of chairs, the revenue and costs functions are neither continuous nor differentiable!) But in order to present the material clearly to your class, you do need to think clearly about which assumptions need to be retained, which ones need to be addressed, and which ones need to be glossed over.
In this example, a better way to demonstrate the principle of marginal analysis might be something like the following:
You own a business that makes chairs, and you're trying to figure out how many chairs to make. The more chairs you make, the more revenue you'll receive, and the more production costs you'll incur. Your problem is therefore to choose the number of chairs that maximizes your profits—that is, your total revenues minus your total costs.
Let's suppose that the more chairs you make, the less additional revenue you get from each additional chair sold. Furthermore, let's assume that the more chairs you make, the more each additional chair costs you to produce. These two assumptions together mean that each additional chair you produce and sell brings you less additional profit than the one before. Eventually, the cost of producing an additional chair will be greater than the additional revenue that chair brings in.
Your optimal choice, therefore, is to stop producing at the last chair at which your additional profit from that chair is positive. That is, you should stop producing at the last chair for which your additional revenue is from that chair is at least as much as the additional cost you incur from making that chair.
Notice that this explanation implicitly states that R(x) is strictly concave and C(x) is strictly convex; that П(x) is therefore strictly concave; and that the therefore the maximum of П(x) may be found by comparing approximations of R'(x) with approximations of C'(x), and choosing the last point for which R'(x) - C'(x) ≥ 0.
It may seem as if this is far too much information for a student to grasp. But it's important to realize that students crave structure, and if you don't provide a precise structure within which to think about problems, they will generate a (probably incorrect) structure on their own. By being precise about your assumptions, and exactly why your assumptions lead to your results, you provide them a concrete storyline that they will be sure to remember.
