*Revised March 1, 2011*

This is an interactive platform for my introductory course on non-stochastic Dynamic Programming. Please click lectures to access the material to be used for the course. What you will find here is based mainly on “Introduction to Dynamic Programming Theory” by Tapan Mitra (in Optimization and Chaos, ed. by Mukul Majumdar, Tapan Mitra and Kazuo Nishimura, Springer, 2000). However, I shall also use parts of “Recursive Methods in Economic Dynamics” by Nancy Stokey, Robert Lucas and Edward Prescott (Harvard, 1989). The necessary background for the theory of convex sets is lifted out of the classic text “Nonlinear Programming” by O. L. Mangasarian (McGraw-Hill, 1969).

Readers are encouraged to raise questions and point out typos using the comment section below.

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About dipankardasgupta

Dipankar Dasgupta received his early education in Calcutta (now called Kolkata), India and moved on to the University of Rochester, NY, USA, where he was awarded a PhD degree in Economics. He did most of his academic research and teaching in the Delhi and Kolkata campuses of the Indian Statistical Institute, from where he retired in August, 2006 as Professor of Economics. He has also taught and researched in visiting capacities as well as a regular faculty member in different universities in Canada, Hong Kong, Japan and the USA.
His interests vary from Economic Theory to creative literature and vocal music. He writes stories, memoirs and poems in English and Bengali and sings semi-classical music, mostly in Bengali. He is also interested in foreign languages, Japanese being his favorite. He writes for the printed media and is a regular TV commentator on subjects of socio-economic interest. Dipankar and his wife, Sankari, live in Kolkata, India.

Dear Kaustav:

I was delighted to read your comment. Here are my initial reactions, though I shall think more about the matter.

1. Regarding “contraction mappings”. Remember that my purpose is to present things to a group that hasn’t had the good fortune of being exposed to any advanced course in math. So, I need to build up from basics. To do this, I find the Tapan Mitra approach more helpful than, say, Stokey et al.

2. I don’t want to start off with the recursive approach. If one does this, one needs to begin with the functional equation appealing to intuition alone. I will first derive this equation rigorously and then proceed. You will notice that the second of my theorems includes the derivation of the functional equation.

Keep commenting. We will discuss more. Incidentally, please point out typos if you find any.

Best wishes.

ddg

Dear Sir,

In exercise 2 for proving U to be concave do I also have to first proof the differentiability of U (for we are using differentiability to construct the Hessian) ?

With regards,

Bikramaditya Datta

Dear Bikramaditya:

Since U is really a redefinition of u and the latter is differentiable by assumption, one need not prove that U is differentiable.

ddg

Dear Sir,

In the proof of proposition 1, in the 3rd line, when you say that the series is absolutely convergent , the result you have stated – is it the same as what is known as Abel’s Convergence test – and are you applying that test to the absolute value of U in the proof ?

With regards,

Bikramaditya Datta

Dear Sir,

Regarding the last comment – I have found the relevant theorem – Walter Rudin, theorem 3.24 (Page 60).

With regards,

Bikramaditya Datta

Dear Sir,

In the last line of page 9 where you have written there exists a number k(subscript 0 and superscript t) with – I think that the orders of t and 0 should be interchanged that is, it should read there exists a number k(subscript t and superscript 0).

With regards,

Bikramaditya Datta

Done.

Dear Sir,

I am attaching the partial solution of question 1 and the complete solution of question 2 – can you please take a look at them and comment –

Proof of Exercise 1

Proof of Exercise 2

With regards,

Bikramaditya Datta

Dear Bikramaditya:

I will check your solution and reply.

ddg

I went through both solutions. They are correct, though, in my opinion, you should search for shorter proofs. However, as I said, there is nothing wrong with the solutions.

Hint for the natural choice for a bound. Suppose bar{k} is the bound you established in your proof. Since k_0 is exogenously specified, is there any reason to suppose that k_0 < bar{k}?

ddg

Regarding the choice for k(bar) in the exercise 1, I think the answer should be k(bar) = max { k(0) , k* } where k(0) is the initial value of per capita capital stock and k* is the solution to the equation f(k*) = (n + delta)k*.

The reasoning is as follows – Consider the path of pure accumulation defined by c(t) = 0 for all t. Then from equation (12) we get, an equation whereby k(t+1) is expressed in terms of k(t) and other parameters of the model (This equation is obtained by setting the LHS of (12) as 0 and then bringing k(t+1) to the LHS). Then we can say that k(t+1) > ( ( < or =) (n + delta)k(t). So suppose k(0) k*. In that case k will keep on falling till it reaches k*. So maximum value of k = k(0). In the last case when k(0) = k*, the value of k will remain unchanged at this value and so maximum value = k(0) = k*.

Is this answer & reasoning correct ?

With regards,

Bikramaditya Datta