Well I cant seem to work it out without a simulation. I have looked at at various parts of the problem and I think it would be a very worth while endeavor if there is profit on the moon.
The cable would provide a very energy efficient means of moving material from the moon to earth. For most of its length it will have Earth pulling gravity on which move elevators down using the energy in the moons orbit and earths gravity. They may well be under brakes thus providing more energy. Gases and liquids could even be siphoned from the moon.
Going up to the moon via cable is a little energy expensive. But if the net mass movement is to the Earth, it should need no energy.
One other problem is that the orbital plane of the earth end would also vary and an Earth space cable would be on another plane. Optimal transfer between the two ends would be cyclic. Masses would be accumulated at the ends of both cable and released every few months to keep energy cost down. All transfers will require orbital change.
So a moon to earth orbit (near geo stationary) cable can provide free transport to and from the moon. Energy extracted from the moons potential energy.
Building one is somewhat difficult. I like the boot strap method. Instead of creating one massive cable with counter weight to establish a moon cable which can then be used to extend toward earth.
I would start with a micro cable, may be a few millimeters or less in diameter. Then when that is connected to the moon and suspended toward earth (not all the way), the free energy available can the be used to transport material up the cable to increase its mass and strength. But we would need an established moon base where all the materials and processing can be supplied. Not cheap.