Policy economists analyze relationships between variables in order to understand, predict, plan for, and influence behavior, among other things. In order to understand the sophisticated, complex behavior of economic agents in the marketplace, then, we have to be able to model that behavior using complex, nonlinear functions, and then analyze our models appropriately.
More simply, real world behavior is not linear. Therefore, any attempt to understand real economic behavior requires nonlinear modeling and the appropriate use of calculus. Any other, simpler forms of analysis lack the detail and sophistication essential to creating and evaluating policy.
A nonlinear function by definition has a different slope at every point along the curve, unlike a linear function whose slope stays constant. In order to calculate a constantly changing slope, we will use the techniques developed in calculus. Since our focus is practical analysis, we'll review just enough theory to be confident that our economic models are mathematically well founded.
Recall that a circle is an example of a nonlinear function. A tangent line is a straight line that touches our circle only at a single point. As you move around a circle, imagine drawing a tangent line for each point; it's clear to see that the slope of that tangent line will constantly be changing, and can be used as an approximation for the slope of the (nonlinear) circle at that particular point. Then all we need to do is to correctly identify the tangent line associated with that point on the curve, and then determine the slope of that (linear) tangent line.
Basic calculus, briefly, is the determination of the slope of a point on a nonlinear function. The techniques are based on this idea of using a tangent line to approximate the slope of a nonlinear function at a point. Lucky for us, Newton did all the hard work proving the theory and developing the rules. For purposes of this review, we will skip the proofs and go straight to the mechanics of the method of differentiation.
One last caveat. While the upside of a "cookbook" review (i.e. just recalling the how without worrying too much about the what or why), is found in its simplicity, so is the downside. As you learn or review technique, or use technique in the majority of cases, keep in mind that there are theoretical underlying assumptions that are crucial to the integrity of your results. These assumptions are usually treated as axiomatic, but as you become more sophisticated in your economic analytic skills and/or use real world data, it is imperative that you increase your theoretical sophistication as well by justifying your use of the assumptions as well as properly proving your results.
The main assumption we need in order to use the techniques of calculus is continuity.
Informally, a continuous function is one with no breaks in it. A more formal definition will come later; for now, note that all polynomial functions are continuous. In addition, all rational functions [link], except where undefined, that is where their denominators are equal to zero, are also continuous.
If a function is continuous and the slope exists, we can find it using the techniques of differentiation, our main calculus tool.
So, let's begin!