Calculus in Economics

Introduction

Calculus is used throughout economics. Whether to minimize cost, maximize utility, or find consumer and producer surplus, the derivative and integral are used to solve real-world economics problems.

Derivatives

As we discussed in the last post, marginal values are derivatives. For example, the marginal cost per good produced, C’(x), is the derivative of the total cost function, C(x). The function C(x) can then be used to analyze other behaviors, which we will go into in the optimization section.

Integrals

Integration, also called antidifferentiation, is the opposite of differentiation. Since integrals work opposite to derivatives, we can integrate marginal functions to find the total function. For example, if we integrate C’(x), the marginal cost function, we get C(x), the total cost function, plus a constant (C is the generic label that mathematicians use), which represents the fixed, or “sunk” cost.

Optimization

Optimization uses derivatives to make “optimized” decisions, which include maximizing and minimizing functions. First, we set the derivative function equal to 0. Then, by checking nearby values or by using the second derivative test, we can determine whether the point is a local minimum, local maximum, or point of inflection. To minimize cost, we set C′(x)=0 to find critical points. Then, by applying the Extreme Value Theorem—checking values at endpoints and near critical points—we determine whether the function has a local or absolute minimum. Businesses often use optimization to ensure that their costs are low and that their products deliver maximum utility to their consumer. 

The Average Total Cost, abbreviated ATC, is the function that models the total cost of a good per quantity produced. We can take the derivative of this function and set it equal to zero to find when the ATC is minimized.

To What Extent Can Calculus Answer Economic Questions?

One of the first problems using Calculus for economics is the discrepancy between mathematical functions and real-world trends. Often, it is difficult to model real-world behaviors as continuous and differentiable functions. With mathematical models, complex patterns and trends can be oversimplified to simple functions. Statistics attempts to solve this problem; using experimentation and analyzing data instead of mathematical functions, we can determine the causal effects of economic decisions. With that being said, modern economists extensively use Calculus and Statistics to solve pressing economics issues.

Bibliography

1. Pettinger, T. (2019, January 11). Diagrams of cost curves [Diagram]. Economics Help. https://www.economicshelp.org/blog/189/economics/diagrams-of-cost-curves/

2. Marsitin, R. (2019). Analysis of differential calculus in economics. Journal of Physics: Conference Series, 1381(1), 012003. https://doi.org/10.1088/1742-6596/1381/1/012003