[MalcolmDiggs]

I'd recommend breaking down the time lost due to the slower machine, and spelling out how that equates to dollars down the drain. Here's an example (with arbitrary numbers):

"With a 16GB machine, it takes my test suite 5 minutes to run. With 4GB, that suite runs in 20 minutes. I need to run it on average, 10 times a day. So by giving me the 4GB machine, you're having me sit idle for an unnecessary 15 minutes, 10 times a day. So that's 2.5 hours per day that you're paying me to spin around in my chair and surf HackerNews. That's X wasted dollars every day, which is significantly more than the cost of the upgrade from 4GB to 16GB. So you'll save money by giving me a faster machine".

If there's a way to speak dollars-and-cents to the bureaucrats in your org, then there's a better chance of them approving your request.