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I'm currently studying economics and am interested in getting into economic research. I'm aware of how math based economics research is. I have done some 1st math courses (probability, linear algebra, univariate calculus and discrete math), a 2nd year course in probability and have done/am going to do multiple courses in econometrics. However, I would like to learn the remaining math needed in my own time. When looking at advice pages on this topic people usually say some combination of needing: linear algebra, multivariable calculus, real analysis and probability. However, I'm hesitant to learn all of these topics from text books as I feel like I would learn a large number of topics that aren't used in economics. For example, in multivariable calculus divergence doesn't seem that applicable to economics. I feel that my time could be spend more productively going deep in the concepts that come up a lot in economics. I was wondering whether someone would be able to collate a list of all the useful concepts needed in each of these subjects in the format:
Multivariable calculus:

  • Langrangian
  • Hessian matrix
  • etc

Thanks in advance.

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Given the level you say you are at I would recommend you to go through this book

Simon and Blume: Mathematics for Economists.

it will give you the core of math skills you need for undergraduate and graduate economics at most universities I would guess. In graduate courses you will probably have to learn more depending on the course and university but it is standard to learn some of the math on the fly when you do economics.

I could recommend other books but they would then be more specialized in certain topics and anyway Simon and Blume is the book that I have most often seen references to on various university courses.

Another virtue of the book is that the examples are to a large extent taken from economics. So you do not "waste your time" on nice mathematical examples that illustrate the mathematical methods but will never appear in economics.

For some graduate courses of course there will be special math skills that are required. Econ departments of some universities specialize in different areas so what you need can also vary from university to university. However, as an economist I would dare say you are very often learning the mathematics as you learn new areas of economics.

Choose a programming language

Although you did not as for a recommendation of this kind, I would strongly encourage you to start programming. Very often being able to simulate some problem in optimization, probability theory, dynamic programming etc. is an invaluable resource to get a deeper understanding of complicated math/economic models.

Currently Matlab is used a lot especially for structural economics. R is very versatile better than Matlab when it comes to econometrics. Python is strong for Machine Learning which is also finding its way into econometrics/economics these days.

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If you want a comprehensive list of the fundamentals, it may be useful to reference the math camp materials for PhD programs, many of which are available online. After all, these are intended to get students up to speed on the necessary math.

In addition, most first year PhD students will take a class along the lines of "Math for Economists" which cover such material more in depth.

Note that most of these classes are tailored towards the tools you need in microeconomics, not necessarily in macroeconomics or econometrics. There, you may want to look into dynamic programming and measure theory.

Here is a nice page with different contents from Osborne at Toronto to get you started: Mathematical methods for economic theory

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