Love it or hate it, economics is a discipline that becomes increasingly mathematical the deeper you get into it. It wasn’t always this way; many contributions to the field have been made through logical arguments and even philosophy.

However, as our mathematical prowess and computing capabilities grew, economics naturally grew more quantitative as well. Nowadays, even Bachelor’s courses in economics are relatively math-heavy, requiring basic knowledge of calculus from early on.

That’s not to say that economics was ever a purely qualitative field — it’s always been in a comfortable relationship with mathematics. Adam Smith, one of the earliest economists, was an accomplished mathematician himself. In fact, many economists were originally mathematicians who became interested in economics at some point in their careers. And it’s noteworthy that Joseph Schumpeter — the economist who coined the term “creative destruction” — was a strong advocate of marrying math and economics, despite seldom using math himself (because, as other economists have retold it, he allegedly regretted that he wasn’t very good at it despite recognizing its value).

While it is possible to pursue a career in economics without strong quantitative skills, refusing to partake in them at all is certainly an uphill battle. And so, we’re providing the following list of quantitative subjects that are increasingly integral (ha, ha) to becoming an economist in today’s world. In fact, they’re also arguably necessary if you want to become a data scientist or natural scientist as well!

Calculus

This subject should be thought of as a prerequisite for serious study in economics, computer science, and data science. Almost every single course in economics will feature several of the following concepts: derivatives, limits, integrals, and/or algebraic manipulation of complicated expressions including logarithms and exponentials.

For a brief explanation as to why, consider that optimizing a function — or in other words, finding a maximum or a minimum — requires taking derivatives. When we consider how consumers maximize their utility or how firms minimize their factor costs, we must use derivatives and other mathematical tools. Thus, optimization is a core building block of both macro- and microeconomics, which deal with the maximizing behavior of consumers and firms.

Calculus is perhaps the first challenging math subject that students take after learning algebra in their early years, and it can often leave a sour aftertaste when first encountered. Still, it’s one of the fundamental building blocks of mathematical literacy, and it remains an indispensable tool in any scientist’s toolkit.

Linear Algebra

Linear algebra is a branch of math that deals with systems of linear equations, which can be written in matrix form. Matrices can thus be thought of as big systems of equations (or a group of interrelated vectors). When these equations are all linear, they can be solved just like we learned in algebra classes.

But when matrices grow to become massive — think of when an economist gathers thousands of rows of data about (say) tax revenue, which are all related to each other and thus form one massive system of equations — finding a solution becomes difficult. A system of two equations is simple enough to solve, but two thousand? (Nevermind the fact that often, systems of equations based on real-world data don’t have an algebraic solution at all!) Enter linear algebra, which fortunately allows us to manipulate these matrices and solve the systems they represent.

And, yes, it’s extremely important to economics; most economics courses default to using matrix notation for lecture notes, model descriptions, and more. That’s because economics data (and most datasets in general) are matrices! Anytime an economist or data scientist collects quantitative (and many times qualitative) data about the economy, it gets stored in a matrix. This is a natural result of working with data, not an arbitrary decision by math-obsessed economists or data scientists. Large amounts of data inevitably get stored in some sort of “list”, and a matrix is essentially a large list.

Vectors and matrices are also the language of econometrics, which is a core tool for any modern economist. Econometricians and macroeconomists especially must become very comfortable with linear algebra, though it’s a necessary skill for essentially all economists.

Differential Equations

If algebra problems take a bit of unraveling, differential equations are like the thick sailor’s knots that hold cruise ships to their moorings. Differential equations are probably the single most important quantitative subject for natural scientists to know, as they govern an incredible amount of real-world relationships in the sciences. They’re also, unfortunately, frequently impossible to solve with a pencil and paper.

A basic differential equation defines a relationship between an equation and any number of its own derivatives. The solution to a differential equation is not a number or variable, but an equation that solves the system.

A simple example is the size of a population. The current population of human beings is based on the number of humans alive last year, which in turn is based on the number of humans alive the year before it, and so on. It also depends on the rate at which people reproduce. So, in order to determine (or predict) the size of a population at a point in time, we must know the size of the population at some point beforehand, the rate at which the population grows (which is defined by the first derivative), and the rate at which that growth changes (second derivative). This can be modeled by a differential equation.

In economics, differential equations come up most often in macroeconomics, since most real-world macroeconomic data depend on their past values and growth rates of myriad interrelated factors. However, they’re still useful for all economists — and for many other data scientists and natural scientists — to know.

Real Analysis

Real analysis can be considered the first genuine course in mathematics that students ever take. Everything else on this list can be considered subjects that merely teach how to do calculations. In contrast, real analysis uses logic to show how mathematical calculations can be true in the first place, and prove how they work. It is the basic language of professional mathematicians.

A typical real analysis course uses logic and set theory to prove how real numbers work and how we know the math of calculus is true. Rather than calculate answers, typical real analysis problems ask students to prove mathematical statements. As such, learning how to write arguments in mathematical logic is a core skill that this course teaches — and it’s often the first time students ever encounter something like it.

This might sound arcane and unnecessary for a student of economics. Yet, since real analysis teaches students how to reason in math terms, and since nearly all economic theory is either based upon or translated into math, real analysis is one of the best ways to become highly proficient at PhD-level economics.

In my personal experience, Master’s-level courses in economics taught calculations, and were difficult and interesting in their own right. But graduate/PhD-level economics goes further, forcing students to understand why things work and what quantitative facts about economic behavior mean (and imply about other economic phenomena).

Graduate microeconomics courses in particular could be described as real analysis applications masquerading as economics. And PhD-level macroeconomics courses frequently refer to and depend on theorems, proofs, and concepts that real analysis courses prove.

Take it from my personal experience: in my class of 12 PhD students, I was one of 4 or 5 who had never studied real analysis before. As a result, I had to work harder than most of my peers to understand certain topics, so much so that I actually decided to cram in an entire online course of real analysis in between my last week of classes and my first exam. It was a painful but incredibly enlightening experience.

Learn from my struggles and take note: the time you spend learning (even just basic) real analysis will pay itself off in spades. It is highly recommended, most especially for aspiring microeconomists.

Statistical Inference

Statistics is deeply related to math, but can be considered its own field. Still, we’d be remiss not to include it here, as understanding statistics is essential for economists. Basic statistics courses often introduce students to fundamentals like the normal distribution and basic methods like constructing confidence intervals.

Every applied economist absolutely must learn how to wield statistical methods effectively. That’s because real-world data is messy and contains noise. When collecting data and attempting to disentangle economic truths from them, likely by running a regression, it’s the tools and methods of statistics that allow an economist to make statements and draw conclusions. Research paper conclusions are built off of proper statistical analysis!

Statistical inference classes are therefore very important for an aspiring economist, and any data-minded scientist. Econometrics courses often teach statistical techniques and analyses side-by-side with linear algebra concepts, but seldom have time to explain the underpinnings of statistics (or linear algebra, for that matter). Fortunately, many economics programs do include statistical inference classes as part of the curriculum, and it is highly recommended that students take these courses if given the chance.

You owe it to yourself

“Painful but enlightening” could probably describe most courses in mathematics. Still, if you’re serious about becoming an economist, a data scientist, or even a natural scientist, these subjects will come up in your field. As such, you owe it to yourself to give them a shot as soon as possible.

And don’t be discouraged when things are difficult. Often, learning a new subject — especially a quantitative one — can be challenging. It can take a long time for things to “click”, but repetition and genuine effort will reward you in the long run. And, it’s much easier to attempt to learn them beforehand, rather than try to learn foundational concepts alongside the economic applications that are built off of them!

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