How do I calculate how much the relitive stiffness or bend resistance
of a wire, of the same material (say copper), goes up as the wire
gauge increases?  From say 22AWG to 20AWG, as an example.

Stiffnes of the circular beam  increases is 4th power of diameter,

  so, if you want, as square of crossection area.

  For general crossesction it depends (by coincidence)
 on the moment of inertia of the  crosssection, as shown here:

 http://pergatory.mit.edu/2.007/Resources/calculations/bending/bending.html
  
 The complete expression for deflection :
-----------------------------------------
We will consider only the simplest, a rectangular beam of width b
, height 2c and length between supports L, with a single concentrated
load W at the centre


 The deflection at the centre of the beam is ? = WL3/48IY. We have
made a lot of assumptions,
  but the result agrees well with practice. For beams of other shapes,
   the results are the same if the proper moment of inertia of area is used.
 
  where I is the moment of inertia of area about the neutral axis, 
 
 where Y = (3? + 2?)?/(? + ?) is called Young's Modulus. A more common
symbol for it is E.
 
 Details and picture:
 
 http://www.du.edu/~jcalvert/tech/elastic.htm

Simple, concise answer.  The link with a spreadsheet was a plus.

Simple answer:

Yes, an object's resistance to bending is proportional to the moment
of inertia along the neuteral axis.  With a bit of simplification, the
moment of inertia for an object with a circular cross section (wire
for example) is proportional to the radius^4, or the area-squared.