1. Econ 101 Richard A. Baum
Read over Spring Break Santa Barbara, CA 93101
There will be a Movie on it during Evaluations BaumRA@aol.com
You are Not responsible for this material March 8, 2016
A One Parameter Family of Catenoids
Let t    be time. Consider some interval of time ( , )  , 0  , with mean 0. An
open, non-compact, cover of time is ( , )t t t     with mean
2
2 2
t t t
t
   
  for each point t.
Let 0 s   denote some lifespan starting at time t, with strict equality implying a lifespan of zero.
Over the course of a lifetime with expected lifespan
0
t
s
s e dt

  comes an endowment at the
mean of the open subcover t , namely at some time t . Note it is possible one dies before receiving
one’s endowment. In a world of government grants and transfers, allow this grant to serve as one’s
endowment, so everyone has some endowment whose average value is amortized to age
2
s
, namely the
mean of the open cover
2
( , )
2 2
s
s s
     . Let this endowment amortized as if it were received at the
mean age
2
s
of one’s expected lifespan s be ( )
2
s
a s 
 
  
 
.
Consider an isothermal parameterization of the minimal surface conjugate to the helicoid, namely
the catenoid. An isothermal parameterization of a family of catenoids indexed by a(s) is:
( ) ( ( )cosh cos , ( )cosh sin , ( ) ) , .a s a s v u a s v u a s v u v        x
Let u be periodic effort expended on a 2 clock. Call u < 0 leisure and u > 0 labor .

2. Let v be an index of growth defined by ( ) ( )v v t r t t  where r(t) is a Gaussian density given
by
2
2
( )
2
t
e
r t


 for all t    and with cumulative density unity. Call capital ( )k s and recall
an endowment ( )a s was received at the midpoint
2
s
of one’s expected life interval (0, )s . The present
value at time t of half the endowment received at age
2
s
plus the accumulated value at time t of half the
endowment received at age
2
s
is simply:
   ( ) ( )
( ) ( ) ( ) ( )cosh ( )
2 2 2 2 2
v v v v
v va s a s e e e e
e e a s a s a s a s v k s
 
      
          
     
.
Perhaps we may agree to index time t for each lifetime so time t = 0 corresponds to the point at
which life begins. This one parameter family of catenoids is a production surface when capital, labor
and leisure jointly produce the product ( ) 1 2 3( , , )a s x x xx according to the above mapping. With the
restriction 0 , 0
2
u t

   , product is strictly positive.
Summary to take Home
A Catenoid is a three-dimensional surface we call a Minimal Surface. Another example of a
Minimal Surface is the Plane in three dimensions. A very nice property of Minimal Surfaces is they
Minimize Energy. By minimizing energy, we are saying the same thing as we Economize
with respect to a very precious commodity, name Energy, whether it be Nature’s or Our Own Energy
in producing the three products  1 2 3 1 2 3, , , , .x x and x which we write as x x xx
This may be an interesting area to explore if you go onto advanced study in Economics.
I hope you enjoy the movie (running time 41 minutes). Feel free to leave if you must.