Kenneth Arrow and the golden age of economic theory
Gee, it&#x27;s fun to read about Arrow&#x27;s work!<p>From the OP, I see that I&#x27;ve been closer
to Arrow&#x27;s work than I knew!<p>In grad school, I had a relatively severe
introduction to optimization including
both linear and non-linear programming.<p>The non-linear programming was mostly
about the Kuhn-Tucker conditions, and
there the work was mostly about the
Kuhn-Tucker necessary conditions.  Kuhn
and Tucker were long at Princeton.  The
guy who was the Chair of my Ph.D. oral
exam had been a Tucker student.<p>Before I got to that grad program, I had
carefully studied W. Rudin, &#x27;Principles of
Mathematical Analysis&#x27; (a.k.a. Baby Rudin)
and W. Fleming, &#x27;Functions of Several
Variables&#x27;. In my first year of grad
school, I also had a severe course in H.
Royden, &#x27;Real Analysis&#x27;, the real part of
W. Rudin, &#x27;Real and Complex Analysis&#x27;,
Neveu, Breiman, Chung, etc.<p>So, that background gave tools that helped
attack the Kuhn-Tucker conditions.<p>Intuitive view of the Kuhn-Tucker
conditions (KTC):  You are in a cave with
an uneven floor and vertical walls and you
want to find the lowest point.  If you put
down a marble and it starts to roll, then
you are not at the lowest point.  So, to
be at the lowest point, it is necessary
that the marble not roll.<p>But K-T wanted more:  They wanted to say
that necessarily the slope of the floor
(calculus gradient) and the slopes of the
constraints that define the walls are such
that the slopes from the walls block
moving along the slope of the floor.
Right, the slopes from the walls form a
cone that contains the slope from the
floor (or its negative depending on
maximizing or minimizing and the direction
of the constraints, etc.)  -- it&#x27;s all
about a cone.<p>Well, this stronger statement is true in
<i>nice</i> cases, and for a <i>nice</i> case have
to have some assumptions that the
constraints are <i>nice</i>. For that there are
various KI <i>constraint qualifications</i>,
KTCQ, that are enough to make the KT
statements about slopes true.<p>There are lots of KT CQs, and one question
was, which imply the others?<p>For two famous KT CQs, one due to KT and
one due to Zangwill, it was not known if
they were independent.<p>So, as a grad student, I settled that --
they are independent.  The proof was by
counterexample -- I found some bizarre
constraints.<p>To know that such bizarre (goofy,
pathological, etc.) constraints could
exist, I needed essentially a theorem<p>For a positive integer n, the real numbers
R, Euclidean n-space R^n with the usual
topology, and a subset C of R^n closed in
that topology, there exists a function<p>f: R^n --&gt; R<p>such that f is zero on closed set C,
strictly positive otherwise, and
infinitely differentiable.  So, I proved
that.  For the KT CQ I didn&#x27;t need all of
infinitely differentiable, but I got that
also.<p>This result is curious in part because
some examples of a close set C can be
surprisingly intricate, e.g., the
Mandelbrot set, a sample path of Brownian
motion, Cantor sets of positive measure,
etc.<p>As I went to publish, I discovered that my
work also answered a question asked but
not answered in a paper by Arrow, Hurwicz,
and Uzawa.<p>Of course Arrow got his Nobel Prize.  A
few years ago, so did Hurwicz.  Last I
heard, Uzawa was still waiting! Cute:  As
a grad student I answered a question asked
but not answered by Arrow, Hurwicz, and
Uzawa.  Reading Rudin and Fleming helped!<p>Gee, in the OP, I see that Arrow was also
interested in decision making under
uncertainty.  Well, my dissertation
research was in best decision making over
time under uncertainty -- stochastic
optimal control.<p>I never took a course in economics.  My
Ph.D. advisor thought that I would need
such a course if only later in my career
to fend off nonsense objections from
economists -- I&#x27;ve never needed that!<p>So, I signed up for an econ course, went
the first day, sat in the front row, said
nothing, and took careful notes.  After
the class when just the professor and I
were there, I asked him what he was
assuming for his supply and demand curves
-- continuous, uniformly continuous,
differentiable, continuously
differentiable, infinitely differentiable,
convex, pseudo-convex, quasi-convex, etc.?
He said nothing.<p>Soon I got a call from my department
secretary to call my Ph.D. advisor -- I
was out of the econ course!<p>Still, the OP shows that I was closer to
some mathematical economics than I knew!<p>Maybe someday some people in data science
or artificial intelligence will exploit
the KTC!