A Moon Space Elevator Is Actually Feasible and Inexpensive: Study | Observer
Dangling the space elevator at this height would eliminate the need to place a large counterweight near Earth’s orbit to balance out the planet’s massive gravitational pull if the elevator were to be built from ground up. This method would also prevent any relative motion between Earth’s surface and space below the geostationary orbit area from bending or twisting the elevator.
These won’t be problems for the moon because the lunar gravitational pull is significantly smaller and the moon’s orbit is tidally locked, meaning that the moon keeps the same face turned toward Earth during its orbit, therefore no relative motion of the anchor point.
After doing the math, the researchers estimated that the simplest version of the lunar elevator would be a cable thinner than a pencil and weigh about 88,000 pounds, which is within the payload capacity of the next-generation NASA or SpaceX rocket.
Except it solves a problem that you DON'T really have with Moon... A space elevator would be really useful on Earth (90-95% of spaceload if used to get in orbit) but it is impossible (partially for the same reasons that it would be so useful : atmosphere, breaking hazard...) On the other hand, on the Moon, landing and lifting is easy, no atmosphere, a 1/6 of Earth gravitational pull... So is a elevator really needed ?