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"your solution might have a caveat your boss" this is very possible, but most of the time we discuss about something like that and he tells me those caveats, this time he is on full block.
Perfect answer. Just translate your needs to simple cost. Show them the break-even point and how much money they safe just by investing 3k in a very beefy rig. I've done this, now I'm rocking a 64-Core Machine and everything went better then expected for all of us.
What should not be forgotten is that being a programmer is more than an "Good programming ability in language.", he should most likely be familiar with different programming techniques/patterns/antipatterns/conventions, too.