# Good book/article that goes into depth about transversality conditions?

I know how to derive the transversality condition in simple models like the Ramsey model.

However, I am looking to develop a deeper understanding of transversality conditions in more complex models. However most macroeconomics books I've seen only have about 1 page about the transversality condition, namely its simplest form.

Is there a good book that shows how to deal with more complex trannversality conditions than the one in the ramsey model? e.g. if the utility function contains a term for assets?

• @HerrK. No those questions are all very simple for me. As an example of what I'm talking about is the following (mind you I'm not looking for a specific answer to that example, but for a book that delves into the theory to understand things like that example). Take the standard Ramsey model, but include capital in the instantaneous utility function. I.e. instead of $u(c_t)$ assume $u(c_t, k_t)$. If you solve this using the Euler Lagrange equations or Hamiltonian for a free endpoint problem, you will get a different (more complicated) transversality condition. Try it out if you want. Nov 18, 2017 at 19:08
• Again, that would be an example of what I'm talking about Nov 18, 2017 at 19:08

A textbook on dynamic optimization which treats transversality conditions at some length is Chiang Elements of Dynamic Optimization. See especially sections 7.2 - 7.4 and 9.1.

I think that references in this field can be of interest and useful also at distance of time from the original question.

Transversality conditions are a complex subject in calculus of variations and optimal control theory.

From a (mathematical) economics point of view, classical references are, of course, the books of Takayma, Analitycal Methods in Economics, and Mathematical Economics, where optimal control and calculus of variations are exposed in detail, of course with applications to economics.

If you want a deeper insight from a mathematical point of view, there are a lot of books in calculus of variations and optimal control theory.

A classical reference in calculus of variations is Gelfand and Fomin, Calculus of Variations, Dover Publications Inc., 1991. A good introduction to optimal control theory is Macki-Strauss, Introduction to Optimal Control Theory, Springer-Verlag, 1982.

For optimal control theory, if you have the curiosity of seeing original sources, there is the original text by Pontryagin, L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, The mathematical Theory of Optimal Processes, Wiley (1962), where the 'Pontryagin's Principles of Maximum 'are stated.

Actually, the 'Principle of Maximum', is not a single theorem, but a collection of theorems, and the question of transversality conditions is analyzed in each case.

But I warn you: the proof of the first Maximum Theorem in Pontryagin's book is about forty (40) pages!

That's to say that it is not a simple subject.