Background
As I'm sure everyone knows by now, Barack Obama has ordered the abandonment of the American Constellation rocket program, whose principal aim was the American return to the Moon (
http://www.space.com/news/obama-nasa-space-plan-reactions-100128.html). Indeed, NASA will soon lack a manned orbital space launch capability: Obama is counting on commercial spaceflight to provide this service, as it eventually will -- though note that this probably won't be until the mid-2010's or later, leaving years during which we'll be dependent upon the Russians to put men in orbit.
These cuts will do very little to balance the budget: they are trivial by comparison with the size of Obama's spending increases. But they do mean that America is, at least until we get a real President again in 2013, out of the Second Moon Race. Unless we recover quickly in the mid- to late-2010's, America will not lead the first human return to Luna or plant the first permanent outposts there, which will put us behind in the more serious race to control the Moon.
Why Luna is Important
One very common reaction is that control of Luna will never have any meaningful economic, military or social effects on Earth, so why should we care who controls the Moon? This is incredibly short-sighted.
Astronomical Position
To begin with, Luna is the closest other world to our own, by two orders of magnitude. (Since people who don't think that Luna is relevant to Man usually aren't very science- or math-savvy, I should explain that an "order of magnitude" means a "factor of 10," so what I'm pointing out is that the next-closest worlds, Venus and Mars, are hundreds of times as far away as is our natural satellite). This means that, barring some "magic" spacedrive that lets us teleport to any world regardless of distance, if we engage in any significant expansion off the Earth then we will assuredly colonize Luna, BECAUSE IT IS SO NEAR TO US.
Can you seriously envision a future, one with continued technological progress, in which the vast majority of the human race remain bound to Just One Planet for thousands of years to come? We can already (just) reach Luna: almost any technological advances would make it easier than it is now. Given these facts, it's fairly inevitable that a population will trickle from Earth to establish Lunar settlements: given any major advances in propulsion or life support technology, that trickle will become a flood. In a world in which technology advanced slowly compared to the last century of progress, for a long time Luna might be the only significant extraterrestrial colony world, but it would certainly be colonized, because the propulsion and life support problems are trivial compared to what would be required to reach (say) Mars, or the Galilean moons of Jupiter.
Mineral Resources
Secondly, Luna is a whole world, and contains a whole world's worth of untapped mineral resources. It's true that a different and less deeply-active geology has led to less segregation and concentration of ores than is the case on the Earth, but this also means that almost any random piece of regolith is likely to contain at least some quantity of any element which is reasonably common on Luna. Such elements include iron, titanium (useful for high-quality steels) and thorium (a fissionable which can be used to make nuclear batteries and even fuel some types of reactors). There are also tiny but commercially-harvestable quantities of tri-helium (aka "Helium 3"), an element much rarer on Earth than on Luna, which will someday with deuterium fuel advanced, radiation-free nuclear fusion reactors (we are currently working on deuterium-tritium reactions, which have the defect of producing significant radiation, though they are cleaner than nuclear fission reactors).
Now, note two things. Firstly, there is some segregation and concentration of ores -- the idea of Luna as utterly homogenous because of her lack of tectonic history is outmoded, see "Composition of the Moon's Crust," for example (
http://www.psrd.hawaii.edu/Dec04/LunarCrust.html) and especially the accompanying maps of iron, titanium and thorium abundance curves over the Lunar surface. Secondly, this is simply what we know now -- there is every reason to believe that a really comprehensive geological survey of the Moon, conducted by human geologists on the Lunar surface, will reveal far vaster riches. This is, after all, a whole virgin world.
Will the cost of Lunar travel ever drop to a degree that renders Lunar metals competitive with those available on Earth? Trivially, yes, because our per capita energy wealth increases year by year, and eventually we'll reach a point where the energy cost of Lunar travel becomes low compared to the demand for resources, and furthermore as our astronautics technology advances, the non-energy component of the cost of Lunar travel will also drop relative to the (irreducible) energy component of the cost. But more importantly, this question makes the huge assumption that Lunar raw materials have to be used on Earth.
Interplanetary Commerce
But why do they? We intend to operate elsewhere in the Solar System. Why can't Lunar materials be used to build spaceships, habs, and other items which will be used in space and on other worlds? Luna offers the huge advantages to spaceflight of no atmosphere (so no need for streamlining or heat-shielding) and 1/6 Earth gravity (thus less fuel needed to launch into orbit or achieve escape velocity). IMO, in a mature interplanetary economy, most bulk exports from the Terrestrial system to other lunar systems will be from Luna; with only small specialized items being exported from Earth, largely because of the Lunar advantage of a shallower gravity well.
Consider interplanetary commerce in general. An interplanetary freighter need not be streamlined; it need not even be strong enough to endure a 1G field, as a nuclear-ion or nuclear-plasma drive can produce a small steady thrust for a long period of time, rather than the swift but violent acceleration of a chemical rocket. But it would probably be able to endure a one-sixth gravity field, and even more so the tidal stresses of a close orbit around a world as small as Luna. The Moon would be the obvious entrepot to the Terrestrial System for interplanetary trade: goods would be offloaded there and then reloaded onto Lunar shuttles for transport to Earth-orbital stations. Offloading at Luna has the very significant astrogational advantage that one is not venturing deeply into Earth's gravity well: Luna's gentler field is much cheaper to overcome, in terms of energy aand reaction mass.
Nations Yet Unborn
Finally, Luna is a world, and because it has enough resources (including, apparently, water ice, see
http://en.wikipedia.org/wiki/Lunar_water) for large-scale and long-term settlement, its population will grow (both by immigration and natural increase) past its initial colonization. The day will come, centuries or millennia from now, when hundreds of millions of humans dwell there. This will be a sizable nation or nations, and those who control Luna over the next century or two will control just from which culture or cultures these future natures are descended. This is very long-term thinking, but it is also a very long-term prize. Consider how important the British victory in the Seven Years' War turned out to be to the future of North America -- and eventually, of the whole world. This might be equivalent.
Can One Really "Control" the Moon?
The Outer Space Treaty
At this point, someone is bound to remember the Outer Space Treaty of 1967, which "... explicitly forbids any country from claiming a celestial resource such as the Moon ..." (
http://en.wikipedia.org/wiki/Outer_Space_Treaty). Surely this Treaty will render the concept of "control" of the Moon meaningless?
And if you believe that, I have this nice bridge across the East River to sell to you. The fact is that whoever founds a Lunar settlement, and continues to staff and provide the population for that settlement, will "control" that settlement for all meaningful purposes. They will enforce their own national laws in that settlement; they will control access into whatever boundaries they claim and can defend for that settlement; they will control trade with and production of devices within that settlement (and if you don't believe that will include armaments, regardless of what some 20th-century treaty says, then you really should buy a piece of my wonderful 19th-century suspension bridge!).
I'm sure that at some point after the establishment of a Lunar settlement the Outer Space Treaty will come into conflict with the new realities. At that point, one of three things will happen:
(1) The Outer Space Treaty will be quietly forgotten, as no one will want to embarass themselves by raising an objection that would be considered laughable by that point.
(2) Someone will protest the violation of the Outer Space Treaty, but the protest will not be enforced, and eventually the treaty will be seen as a dead letter for that reason, or
(3) Someone will enforce the Outer Space Treaty -- but by doing so, of course, they will have gained control over the Moon, and the Outer Space Treaty will be relevant only as an excuse to maintain that control.
How Does One "Control" a World?
Of course, controlling Luna would require more than simply establishing and controlling a Lunar settlement, even a large Lunar settlement. A world is big, and it's certainly possible that multiple Powers will colonize the Moon, resulting in several colonial zones, at first with wide uninhabited spaces in between. What does "control" mean in this case?
Obviously, given that the Moon is not homogenous, some areas will have more crucial resources than others. An obvious example of this is that most Lunar ice is probably at the poles, and thus colonies which occupy polar positions will be at a considerable advantage for growth and trade. As we discover more of the location and nature of Lunar resources, we will probably find other key zones of concentration, produced perhaps by primal vulcanism or asteroidal impact.
Secondly, there is the issue of relative numbers and strength. A colony that remains under colonial administration will have power both from its own numbers and strength, and from the military might of its controlling Power. A colony that achieves independence may become a Lunar Power in its own right. Whether by outright war, or by debate in a federation, some Lunar settlements will have greater influence than others.
Finally, there is the possiblity of direct space force, whether by bombardment or blockade. A Power with a strong military presence in Earth-orbital or Lunar space could bombard rivals, or simply cut them off from commerce. The strategy and tactics of both aerial and naval warfare would probably apply here, with new elements for vector and orbital maneuver, and the medium of the void.
Who Are Our Rivals?
Russia
Russia currently has the second greatest space capability of any Power, and will have be the first-ranking space Power as soon as we retire the Shuttle Orbiters from service. The Russians have conducted research into both unmanned and manned armed spacecraft, and are planning to launch a series of "Luna Glob" unmanned Lunar probes starting in 2012. They currently have no explicit plans for manned Lunar flights or colonization, though this is the obvious long-term goal for which Luna Glob would perform the reconnaissance. Russia, under Vladimir Putin, is somewhat hostile to America, though she still far from being an outright enemy.
Red China
Red China has the third greatest space capability of any Power, and has launched manned orbital flights (the only country other than America and Russia to have accomplished such a feat, so far). The Red Chinese currently plan to launch the Chang'e-3 robot lander and rover, first of a series, in 2013, make a robot landing and sample return in 2017, and conduct manned Lunar landings by the late 2020's. China has done some research into armed spacecraft, and (like Russia) is somewhat hostile to America.
Japan
Japan famously launched the recent Kaguya orbital explorer, in 2007. The Japanese recently abandoned an explicit Lunar landing program, but continue to work on plans for the robotic construction of Moonbases which could then receive human tenants. Japan is the only country in the world with technology at least as advanced as America's, but is (fortunately) a friendly rival.
Private Programs
Richard Branson, founder of Virgin Galactic, has made his company's explicit long-term goal the establishment and operation of a permanent commercial Moonbase. There is as yet no definite time frame for this achievement, but the logic of his business plan and likely success rate implies sometime in the 2020's. Virgin's success would be generally good news for America, as it is a peaceful corporation on which in part NASA is relying for transportation under Obama's new plan -- but it would give a commercial entity fairly significant influence over our future Lunar colonization efforts, which might bother some people (not me).
Conclusion
The colonization of Luna is of utmost importance to the human future, and consequently control of the Moon should logically be important to us. Just because America may choose not to colonize the Moon does not mean that it will not be colonized -- merely that it will not be colonized by Americans.
Admittedly the choice is not that stark, because we are not deciding for all time. We may choose not to colonize the Moon now, but a future generation of Americans may reverse that choice.
But an early start can be important. Consider why Spanish is one of the world's most important languages.
It's not because of the relatively small country of Spain, which is roughly equivalent in size and population to (say) Poland.
Do we want American culture to be one of the world's most important cultures of the future? Or do we want to be left by the wayside, a "Poland" whose ways are confined to Just One Planet, while other cultures expand throughout the Universe?
That decision is one that we will probably have to make within a century or so, or lose the opportunity to make in the future, when other cultures have already snapped up all the near-Earth strategic locations.