Math You Should Definitely Know About To Study Economics
This is just a quick note here. To study economics effectively, you at least have to know accounting math and probably basic calculus/analysis. Even Marx learned some calculus, so there's not much excuse to not master that level. But this article is not "Math you should definitely know", it is "Math you should definitely know about", and there is an important difference. In this case, I am talking about the basics of algorithms and computer science, complexity theory, differential equations and dynamical systems, information theory and control theory.
Knowing vs Knowing About
Much of our world today runs on information systems, and accounting itself is an information system. Economists tend to focus on statistics and analysis, which I think is probably the most useful thing for that discipline, but if you don't know about what computer scientists and IT people do, then you will have trouble analyzing how the modern world works.
The mathematics of computation
Much of the modern world is built by engineers using what I call "the mathematics of computation", this includes basic algorithms(sorting lists), discrete math(sets, graphs, boolean algebra), computational theory(turing machines, computational classes: ie P vs NP, problem reductions and mappings, halting problem), control theory and feed back, differential equations, mathematical optimization or what was previously called linear and non-linear programming, and information theory.
We use this mathematics to understand everything from language, to signal processing used for radios, to engineering machines that need to operate effectively in gritty environments, to planning logistics to deal with costs. If you don't know what "operations research" refers to, you will have difficulty knowing the issues that modern supply chains face, no matter how many times you can draw the intersecting supply and demand curves.
In this case, here are the mathematical disciplines you should know about, learn who studies them and what problems they are trying to solve:
- Algorithms and Asymptotic Analysis
- Theory of Computation
- Control Theory and Feedback Systems
- Dynamical Systems/differential equations
- Mathematical Optimization: Linear/Non-Linear Programming
- Operations Research
- Risk analysis and Actuarial Science
- Information theory and Signal Processing
It is just not possible for one person to be an expert on all of these, but appreciating all of these is critical for understanding how the economic system works. In particular, the basics of algorithms studies what is called "asymptotic analysis", which can serve to inform the economic concept of growth. Asymptotic analysis evaluates what resources are required to solve particular computational problems, but importantly, it is not about the algorithms themselves, but building up a mathematical toolkit to describe resource utilization of algorithms. This same toolkit can be extended to help evaluate resource usage of the total economic system, to help us understand growth. This topic is typically studied late in the first year of a typical computer science program, or early in the second year. Compared to the economic concept of price curves, asymptotic analysis evaluates specifically how costs scale when the size of a problem scales, and how the shape of these costs changes as the size approaches infinity.
While studying algorithmic resource usage is not the same thing as studying capital development and economic growth, it is concerning to me when the first or second year computer science course, presents a much greater level of nuance and complexity in how resource costs scale, than what I hear from economists talking about growth models who have completed a graduate degree. Of course the discipline is different and the focus should be different, but economists need these mathematical tools just as much as computer scientists.
I would not necessarily expect an economist to be able to write a quicksort algorithm or reduce a graph coloring problem to 3SAT, but they should be able to discuss the basics of big oh notation. While there are conventional economists who understand computer science, sometimes it is not clear how they justify the application of economic growth models. Certainly for a system with more uncertainty you may be forced to use a simpler mathematical framework, but at least the basic principle that costs can vary across scale in complex ways should be an essential part of economic analysis of growth.
Economic growth, seems to face the "uncanny valley", where it is either described with trivial and simplistic exponentials, or insanely difficult and incomprehensible statistical and empirical fitting techniques. The seems to be a deliberate avoidance of intermediate techniques focused on how specific resource constraints affect growth and scaling. Unfortunately, the topic of logarithmic growth, that as we grow larger we should grow slower, seems to get very little attention.
Much of this difficulty seems to be based on the tension between economies of scale and the global environmental costs of growth. Economies of scale means that production becomes more and more efficient as operations become larger, but the ability to produce rapidly is a double edged sword, as it means we can more easily stress the environmental limits in shorter and shorter time frames. Most conventional discussion of economic growth, which I am aware of, focuses on the benefits of the economic machine, and not its dangers, whereas other academics and scientists end up sounding the alarm bell on the global perils of growth. Growth is dangerous precisely because it is so effective.
You can never grow your way out of the need to care about the environment. Technology transforms simple easy problems, like the agricultural process of growing food as managing energy, to complex systemic issues, such as oil and shipping supply chains and global warming and natural disasters.
I am reminded of the biblical parable attributed to Jesus about casting out demons.
Technology often has a potential to create more problems as it is solving old ones, and the ability to regress to a previous mode of resource use or technology, when necessary, becomes incredibly important for dealing with this. I am the last person who wants to ban cars. I am more worried about what will happen if we can't continue to rely on our current energy systems. Do we have adequate rail and river networks to fall back on if our road networks face a significant threat, whether that is with the infrastructure themselves or a problem with energy networks?
In the rush to emphasize formal mathematical economic models, economists often appear to run straight past the mathematical toolkits and applications developed and utilized in other disciplines. Machine learning, statistical regression, and econometric experiments are center stage, while engineering, computer science fundamentals, and primitive skills are sideshows. In my opinion, the application of mathematics to the economic system is so complex and difficult, that it is essential to train our intuitions and skills on these more controlled engineering disciplines first. Econometric work without being able to spot an infinite loop bug in a computer program may be the worst case of "premature optimization" that we see in modern academics.
It is not that I think statistics, vector autoregressions, machine learning, and other advanced econometric principles are useless, I just think to focus on them to the exclusion of other skills, increases the chance of missing the perils that come with all technological adoption.
Much attention is given to the dangers of artificial intelligence. I am much less afraid about a robot takeover and enslavement of humanity, than I am that humans will neglect basic skills and lose self reliance. When computer brain interfaces come out, will the next generation never learn the ability to type. I am not one to disparage gen z people for not knowing cursive, they should learn at least one system of handwriting so they can make a shopping list when their phone battery dies. (Check out my project ["Voyager Keyboard"]: a chorded typing system that works well on both touchscreens, physical keyboards, controllers, and small 10 button bluetooth number pads).
How much math should economists be expected to know, and how much should they at least know about? I'm not entirely sure, but let me use an analogy. We should probably not expect a "Gen-Z"er to know cursive handwriting, but they should at least know what it is. For hundreds of years cursive was the most advanced handwriting system, and while we disfavor it now, because it is particularly hard to read, and takes more work to learn, it is still an essential skill for historians and other domain experts. Similarly, economists probably don't need to be able to compute the Shannon distance between two strings of text, but they should at least know what information theory and signal processing are, and how much effort engineers put in to master this discipline so that we can talk to each other over the internet.