<p>A textbook on dynamic optimization which treats transversality conditions at some length is <a href="https://www.amazon.co.uk/Elements-Dynamic-Optimization-Alpha-Chiang/dp/157766096X" rel="nofollow noreferrer">Chiang <em>Elements of Dynamic Optimization</em></a>.  See especially sections 7.2 - 7.4 and 9.1.</p>

CC BY-SA 4.0
2022-10-06T16:26:47.997Z
2022-10-06T16:26:47.997Z
Adam Bailey
<p>I think that references in this field can be of interest and useful also at distance of time from the original question.</p>
<p>Transversality conditions are a complex subject in calculus of variations and optimal control theory.</p>
<p>From a (mathematical) economics point of view, classical references are, of course, the books of Takayma, <em>Analitycal Methods in Economics</em>, and <em>Mathematical</em> <em>Economics</em>, where optimal control and calculus of variations are exposed in detail, of course with applications to economics.</p>
<p>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.</p>
<p>A classical reference in calculus of variations is Gelfand and Fomin, <em>Calculus of Variations</em>, Dover Publications Inc., 1991.
A good introduction to optimal control theory is Macki-Strauss, <em>Introduction</em> <em>to Optimal Control Theory</em>, Springer-Verlag, 1982.</p>
<p>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, <em>The mathematical Theory of</em> <em>Optimal Processes</em>, Wiley (1962), where the 'Pontryagin's Principles of Maximum 'are stated.</p>
<p>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.</p>
<p>But I warn you: the proof of the <strong>first</strong> Maximum Theorem in Pontryagin's book is about forty (40) pages!</p>
<p>That's to say that it is not a simple subject.</p>

CC BY-SA 4.0
2022-10-06T14:59:04.733Z
2022-10-06T14:59:04.733Z
BakerStreet
<p>I know how to derive the transversality condition in simple models like the Ramsey model. </p>

<p>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. </p>

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

CC BY-SA 3.0
2017-11-18T19:08:22.340Z
@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.
user56834
CC BY-SA 3.0
2017-11-18T19:08:42.640Z
Again, that would be an **example** of what I'm talking about
user56834
<macroeconomics><mathematical-economics><dynamic-optimization>
Good book/article that goes into depth about transversality conditions?
user56834
# Good book/article that goes into depth about transversality conditions?

(Asked by: username_0 on 2017-11-18)

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?

**Tags:** macroeconomics, mathematical-economics, dynamic-optimization
> **Comments:**
> **username_0**: @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.  
> **username_0**: Again, that would be an **example** of what I'm talking about  

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## 2 answers

### Answer by username_1 on 2022-10-06. Score: 3

A textbook on dynamic optimization which treats transversality conditions at some length is [Chiang *Elements of Dynamic Optimization*](https://www.amazon.co.uk/Elements-Dynamic-Optimization-Alpha-Chiang/dp/157766096X). See especially sections 7.2 - 7.4 and 9.1.


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### Answer by username_2 on 2022-10-06. Score: 2

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.


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