But shipping from the moon to the earth should be much cheaper? I don't have the math (so someone please do?) but based on the size of the Apollo 11, it seems to be trivial to send stuff to earth.
Looking at our handy solar system delta-v map[1] it takes 2.5 km/s of delta-v to get from the Moon's surface to an Earth intercept course (you can aerobrake to a landing) and 14.6 to get up from the Earth surface to the Moon's (no aerobraking, you have to use rockets to land). Also without an atmosphere you can use railguns and such to take off on the Moon, if you get more than a few km/s of boost on the Earth that way you'll burn up.
>> ... based on the size of the Apollo 11, it seems to be trivial to send stuff to earth.
You mean the worlds largest rocket? That machine that launched from the moon first had to be brought to the moon, a task requiring an enormous rocket and untold support structures on earth. Of course one might say "build the rocket on the moon from moon rocks" but that requires significant handwaving. Try smelting aerospace-quality metal in your garage before suggesting that anyone simply park an aluminum smelting facility in a lunar crater.
The machine that does this first has to leave Earth and land softly on the moon before it can start boosting stuff back. You can’t skip the first two steps.
The machine that does this needs to be built by a machine that is carried by the machine that is first built and launched from earth before you can start boosting stuff back.
It is feasible to build a space elevator on the moon with existing materials.
[1]https://i.imgur.com/WGOy3qT.png