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When I say "theories of the firm", I'm referring, in particular, to the theories exposed in the next works: 1) Transaction Cost Economics exposed in "Transaction Cost Economics" by Tadelis and Williamson (2013) (chapter 4 of "The Handbook of Organizational Economics" by Gibbons and Roberts); 2) Property Rights Theories exposed in "The Costs and Benefits of Ownership" by Grossman and Hart (1986), "Property Rights and The Nature of The Firm" by Hart and Moore (1990), and "Firms, Contracts, and Financial Structure" by Hart (1995); and 3) Incentive System Theories exposed in "The Firm as an Incentive System" by Holmstrom and Milgrom (1994), and "Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design" by Holmström and Milgrom (1991).

When I say "Which mathematics", I would appreciate if a list of chapters of a highly recommended math book is given, due to I need to be prepared for analyzing those theories as soon as possible. In this regard, it should be noted that I have studied the first eight chapters of the first volume of Apostol's Calculus. In advance, thanks so much.

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  • $\begingroup$ Considering that I received a bad vote, I would appreciate any feedback that allows me to improve the question. $\endgroup$ – David Fernando Jiménez Sep 29 at 2:33
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    $\begingroup$ I am not the downvoter- but this question is too broad. You ask "I would appreciate if a detailed list of the required mathematical topics is given". That is too broad. For example, in mathematics basic operations summation, addition, multiplication and division can be considered a 'topic'. I guess a detailed list of topics would have over 100 entries easily. If you want a comprehensive list of all mathematics you should know then you can have a look at table of contents for Essential Mathematics for Economic Analysis and Further Mathematics for Economic Analysis by Hammond et al. $\endgroup$ – 1muflon1 Sep 29 at 13:03
  • $\begingroup$ @1muflon1 Thanks for your recommendation. I edited the post taking into account what you said. I hope it is ok now. $\endgroup$ – David Fernando Jiménez Sep 29 at 14:58
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I haven't gone through all the papers so I just sampled 'The Firm as an Incentive System'. Browsing through it, it relies on Linear Algebra and Real Analysis. Again, I warn that I have not gone through the entire paper but seeing some of the terminology used there, I could guess that these two a clearly involved (maybe they use some bit of topology as well). They use a concept of sublattice somewhere and that pertains to group theory.
A good starting place for real analysis could be Steven R. Lay's 'Analysis'. There you can learn some of the basic terminologies that they have used like supremum and infimum. For Linear Algebra, it has got to be Gilbert Strang's text 'Introduction to Linear Algebra'. Linear Algebra is used extensively in Econometrics as well. You can find his lectures at: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ .
Hope this helps!

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    $\begingroup$ Thanks so much. I am going to check the suggested material. On the other hand, do you think that Apostol's Calculus is a good book for studying Linear Algebra? $\endgroup$ – David Fernando Jiménez Oct 2 at 23:11
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    $\begingroup$ Before I comment I must let you know that my knowledge of math is not extensive, having studied only two years of college courses. I browsed through Apstol's book and in my opinion, it is not that good a book if you want to study Linear Algebra. It doesn't add any specific linkage between Algebra and Calculus. While that happens in Differential Equations and he covers that part in Volume II, I think you'd be better off learning Linear Algebra from a specific text devoted to it. If you want to learn Multi-variable calculus which is the other thing he covers, you can refer to Thomas' Calculus $\endgroup$ – Vedant Monger Oct 3 at 17:32
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    $\begingroup$ Secondly, let me briefly describe the major branches of mathematics. Broadly, mathematics has three branches: Analysis, Algebra and Geometry. Analysis is the study of limit processes and that is where Calculus comes as well. Algebra is the study of operators on algebraic structures like Vector Spaces, groups and fields. I do not know much of geometry as I haven't taken any course on it.. Algebra and Analysis merge in a field called Number Theory. As you pursue math further you will realize that they are closely intertwined. But for economics I think studying Analysis and Algebra will suffice. $\endgroup$ – Vedant Monger Oct 3 at 17:37
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    $\begingroup$ I went through the paper but not in its entirety. The mathematics here is not very involved. If you understand multi-variable calculus and some algebraic manipulations, it will suffice. But if you want to keenly understand the logic of the proofs for the lemmas, you could take a crash course in mathematical logic. l again reiterate that Analysis is used as a tool. A proof is a solid argument that you use to prove your point. Various proof techniques can be found in mathematical logic. The key here is understanding the chronology and the line of argument. Tell me if you want to learn more. $\endgroup$ – Vedant Monger Oct 5 at 17:56
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    $\begingroup$ There is no need to understand Linear Algebra. For the article that you shared, you need not know linear algebra. However, you need to follow and understand the algebraic manipulations they are using. Specifically, you need to know multi-variable optimization. That should suffice. Try having a look at Walter Nicholson and Christopher Snyder's 'Microeconomic Theory'. It has a good introduction to math. If you want a PDF version of it, give me your email id. I can forward it to you. $\endgroup$ – Vedant Monger Oct 6 at 12:07

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