What good would a space elevator do anyway? Our civilization wouldn't use one yet to the capacity that would justify 3 trillion dollars. At least the war gets back the gain. The money goes to build weapons, which improves the economy of the areas where the weapons manufactures are at. Then there is the gain of the plundered resources...
A single space elevator offers you enough cheap, clean solar power to fulfill every energy need of the human race many times over. Solar panels would be so much more efficient in space that it would easily outperform fossil fuels and if they were in space they would last. Power could be transmitted anywhere on the earth's surface via microwaves, greatly reducing the need for infastructure and redundency.
Plus, once you're in space you can start cheaply harvesting asteroids to build space satilite habitats along the thoughts of the great O'Niel. Cruising around the solar system is easy once you're out of the atmosphere and gravity well of earth. Built in zero-g, your space stantions, would be much cheaper places to perform any industrial process, including satilite building, then earth. Centripital force easily allows for earthlike gravity and their engineered and controled design could be made to have a perfect climate and ecosystem.
All this could be done with existing technology. No sci-fi warp engines or fancy new materials (save the space elevator itself), just rockets and steel. There's enough materials in the asteroid belt alone to make 3,000 earth's worth of space station habitat and enough sunlight to power more stations then that. Exponetial growth tells us that if stations are devoting a substantial fraction of their efforts to building new stations, we could have enough stations to house the entire human race within a hundred years of a 10k person station being built.
So, trading from one Earth to 3,000 Edens. Enough resources to end every social ill. An end to wars over resource scarcity. That's what a space elevator will give you.
War might offer economic stimulus but spending such money on social projects or space elevators would offer the same economic stimulus, and have actually directly improved your populations welfare. Plus the economic consequences of deficit spending can outwiegh any stimulus. The motivations for war are more psycological and political then economic.
But... pillars and cave ins, I'd say put them back in eventually but don't make it a priority. There's plenty of cooler stuff to do first.