The title pretty much explains the question. I've never taken a course in economics, so it's hard for me to judge what are the most important applications, the most interesting ones, or the ones that show the calculus techniques to best advantage.

What I have so far:

  • The marginal rate of substitution is minus the derivative of the indifference curve.

  • Various business-y applications of optimization, including economic order quantity.

  • The Laffer-curve argument related to Rolle's theorem.

  • Free-market equilibrium occurs where the supply and demand curves intersect (intermediate value theorem).

  • Consumer's and producer's surplus as applications of the definite integral.

  • $\begingroup$ You'll note, the Laffer Curve has no actual points aside from the zeroes that can actually be calculated. Legend has it, the curve was written on a napkin, and only meant to prove a point. $\endgroup$ – JoeTaxpayer May 29 '14 at 10:04
  • $\begingroup$ I am an economist, but not a mathematician. So I do not have the taxonomy of mathematics clearly in my head. Could you describe, using some labels, what "freshman calculus" includes? $\endgroup$ – Alecos Papadopoulos May 29 '14 at 21:44
  • $\begingroup$ @AlecosPapadopoulos: The first semester would be differentiation, including the chain rule, product rule, and transcendental functions. Definite and indefinite integrals, including substitutions. The second semester is mostly infinite series and methods of integration. $\endgroup$ – Ben Crowell May 30 '14 at 14:53

Present and future value of a continuous income stream $f(t)$, continuously compounded at a rate $r$, over $0 \le t \le T$:

  • Total income: $TI = \int_0^T f(t) \; dt$

  • Present value of the stream: $PV = \int_0^T f(t) e^{-rt} \; dt$

  • Future value over $0 \le t \le T$: $FT = e^{rT} \int_0^T f(t) e^{-rt} \; dt = e^{rT} PV$

  • $\begingroup$ What is your favorite example of $\int f(t) e^{-rt}dt$ as opposed to $\sum f(t)e^{-rt}$? In my experience, most discounting in finance and economics uses discrete time increments. Putting the integral on the greatest hits album would make more sense with a particular example from the economics discography. $\endgroup$ – user173 May 29 '14 at 13:41
  • $\begingroup$ This is a nice example because it's something practical that could come up in personal finance. $\endgroup$ – Ben Crowell May 29 '14 at 20:40

I'm fond of the Gini index, a useful and interesting measure of the "fairness" of income distribution and requires the ability to integrate.

  • 1
    $\begingroup$ Note the backlash against this: partially because the Gini index requires calculus, many economists now recommend using the Palma index instead. That index is the income of the top 10% divided by the income of the bottom 40%. I agree with this: the Palma index is more transparent, and a better target. See inequality.org/yardstick-measuring-inequality, especially the link about the 90 noted social scientists. $\endgroup$ – user173 May 29 '14 at 9:23
  • $\begingroup$ @MattF. Thanks for the link -- I wasn't aware of the Palma index. I think, though, the existence of the Palma index makes a project on the Gini index even more worthwhile since a student could be asked to compare them and analyze the differences/similarities between the models (both empirically and theoretically). $\endgroup$ – ncr May 29 '14 at 12:38

Expected utility is a key concept, a paradigm for rationality, which uses calculus in several ways:

  • Expected utility is an integral, $E[U]=\int U(x)\,p(x)\,dx$, where $U$ is utility and $p$ is probability.

  • The second derivative $U''(x)$ is typically required to be negative for risk aversion

  • Expected utility is to be maximized (e.g. by varying the amount of risky assets in a portfolio)

  • $\begingroup$ Wish I could accept more than one answer. This one is nice because it gets at rationality, which seems to be one of the central concepts of economics. $\endgroup$ – Ben Crowell May 30 '14 at 21:01

By far, the "freshman calculus" Superstar in economics is differentiation, because it is pervasive in all branches of the discipline, and at all levels, undergraduate post-graduate, PhD, professional academic.

And this is because, the core behavioral assumption in economics at the micro-level is that economic agents (persons, companies organizations, whatever), are all the time consciously and purposefully trying to optimize some measure, metric, whatever: utility, profit, value (maximization), cost, risk, loss (minimization)... in the world of economics, there is no such thing as "sit back and enjoy life", you are always on the look out (our agents, that is, inside our models), to ameliorate your position, in whatever way this "position" is represented -this is how you do enjoy life.

So we take derivatives as we breath, since, apart for special cases where the objective function may be constructed to be a bit weird, our functions are always differentiable -and even we have a model in discrete time, we find ways to do things similar to differentiation w.r.t time...

But I guess, what everybody knows about Economics is that the discipline has an obsession with the concept of "equilibrium", right? Well, watch us differentiate the equilibiurm condition of a model to obtain the relation between two variables of interest at the equilibrium point -and we are so used to it, some don't even know this is called the implicit function theorem.

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    $\begingroup$ The word "marginal" is also pervasive in economics, and its mathematical representation is usually a derivative: marginal utility, marginal cost, marginal profit are all derivatives with respect to quantity. $\endgroup$ – user173 May 31 '14 at 1:26

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