Why couldn't a space elvator's cable be tapered? Why couldn't a space elevator's cable be tapered to solve the weight issue for the cable? After all you don't need a massive amount of tensile strength at the bottom of the cable.
 A: A tensile design for a space elevator cable requires there to be a counterweight some distance beyond the point of geostationary orbit.  The counterweight would be hurled with centrifugal force and would keep the entire cable taut.  The cable must not only support a payload, but also must be a tether for the counterweight, and provide for centripetal force keeping the counterweight from flying off into space.  Thus, there would be stress on all parts of the cable.
However, the downward pull of gravity would be offset by the outward pull of the counterweight at the point of geostationary orbit.  As the cable gets closer to the Earth's surface, net force of gravity increases, and as the cable gets closer to the counterweight, net centrifugal force increases.  At geostationary orbit, they cancel each other out.
So the thickest part of the cable would be at geostationary orbit.  Here is a report on a study performed for NASA on the concept: http://www.niac.usra.edu/files/studies/final_report/472Edwards.pdf.  The Wikipedia article seems to be a good resource, also: https://en.wikipedia.org/wiki/Space_elevator.
