Comments on Chapters 20 and 21 ( Calculus of Variations and Optimal Control Theory ) of ” Introduction to Mathematical Economics ” ( third edition

The textbook ”Introduction to Mathematical Economics” (third edition, 2008) by E.T. Dowling, ISBN 0-07-135896 (henceforth IME) is intended for economics and business students who seek to enrich their knowledge of mathematics in sofar as it is immediately applicable to economic and business problems in academics. In this connection IME can be used both for self-study or as a supplement to other texts. It is part of the Schaum Outlines Series, published by McGraw-Hill and, like all other publications in this series, it is characterized by the Outlines’ learning-by-doing philosophy. Indeed, about each of its subjects IME presents just a little theory, which is then supported by a number of explicitly solved problems (of these the present book contains no less than 1600). In 2011 I used IME’s chapters 20 and 21 (in particular the former) as an initial stepping stone towards optimal control and dynamic programming in my Utrecht course on Optimization. I did this because, for the first time in that course, the students had received very diverse training in mathematics: my audience consisted of a mix of third year Bachelor students of Mathematics and Research Master students of Economics. For this reason my comments below are strictly limited to chapters 20 and 21 of IME. Perhaps, as time progresses, I will have the opportunity to take a similarly sharp look at the other chapters. Because I found IME to be very useful for my course, I hope that my remarks, which I give in the critical but certainly constructive spirit that is traditional among mathematicians, may help to make its chapters 20-21 even more helpful and clear for its users. While I realize quite well that the average reader of IME is not looking for mathematical precision per se, one main thrust of my remarks below is that certainly when confusion threatens, a modest amount of precision should be exercised. Another such thrust is that at several junctures the material in chapters 20-21 can easily be enriched for the student at a minimum