A lunar space elevator will never happen. First of all the gravity is unstable. And even if you do so, the main advantage of an elevator can also be had with a

A lunar space elevator will never happen. First of all the gravity is unstable. And even if you do so, the main advantage of an elevator can also be had with an accelerator, there is no need to build an elevator. Because there is no atmosphere you can simply fling stuff into orbit for pretty much exactly the same price but far, far, far, far less investment. The one place elevator actually makes sense is Mars potentially. But that is a very long way off.