I am a high school student with keen interest in mathematics and economics. I wish to study mathematical economics, but most of the books I have encountered begin with quite advanced mathematics. In particular, Mathematical Economics by Akira Takayama begins with Nonlinear Programming [Concave Programming, Differentiation and Unconstrained Maximum Problem and Quasi Saddle Point Charecterization]. Frankly, I find myself at sea while trying to understand this content. Could someone please recommend what prerequisites one must have in order to fully understand books of such caliber.
In the preface, Takayama writes that the book was written with the intention to keep the prerequisites to a minimum: elementary calculus and matrix algebra.
Perhaps he was exaggerating a little, but I suspect, after skimming the table of contents, that knowledge of the aforementioned subjects and experience working with (i.e. reading/understanding and independently constructing) mathematical proofs should be enough.
I'm assuming you've taken calculus and linear algebra -- in which case you might want to get more experience with working with proofs. I'm quite fond of Axler's Linear Algebra Done Right, and I've heard good things about Abbott's Understanding Analysis.
If you haven't studied those two subjects, work on learning those first. You can try the two textbooks I suggested to do so, but that might be a bit rough-going. I'm sure your math teacher might have more helpful suggestions for a first reference to learn calculus and matrix algebra from.
I think with a bit of what mathematicians like to call 'mathematical maturity', you should be able to manage Takayama just fine.
Economics in undergraduate and graduate programs already contain a non negliglible amount of maths. Mathematics are at the core of modern orthodox economics.
I would propose you to have a look at the general textbooks for undergraduates/graduates in economics. You will find an extensive use of maths in some of them. If you are interested in theory Microeconomic Theory (by Mas-Collel, Whinston and Green) is a masterpiece. Introductory Econometrics (by Woolridge) may be relevant if you are interested in applied econometrics.
This is not an exhaustive list of course.