I've been in a few science fiction NRPs; every so often, you get one or two roleplayers who seem adamant on believing that a synthetic fighting force is dramatically more expensive than an organic force. 

This is wholly incorrect.

People seem to assume that humans are free and cost nothing. This is wrong. Let's use a real-life example. On average, it costs about 277,000 USD to raise a child from birth to adulthood in the United States; this cost does not factor in college or other expenses that are not required for a healthy upbringing. If the child opts to enroll within college, the cost for them to actually reach adulthood and obtain an enhanced education dramatically increases. You could easily be looking at a 330,000+ price tag for a single kid. 

So let's say that our child wants to serve their country by shooting brown people in Afghanistan. Someone has to preserve the freedom of US citizens! Our child has opted to enroll within the US military, which means that their training, food, and equipment is not free. We shouldn't forget the medical bills either, because our child is bound to get hurt at some point doing PT, firing range exercises or hand-to-hand sparring drills. When they are finally done being trained in the art of warfare and killing, and after having racked up a training bill of several thousand dollars, they're finally deployed to Afghanistan, and will now proceed to be a logistical and financial burden on the US budget. 815,000 dollars or so per year, actually.

But that's real life. This is a roleplay. Realistically, and with advancements in robotics and manufacturing techniques, there's no reason why a synthetic soldier should be more expensive than a human soldier--especially if you're equipping your human soldier with powered armor which is really nothing more than a wearable robot to begin with.

EDIT: And if we look at this from a cost-effective standpoint, a robotic soldier beats out an organic simply because it isn't bound by the much stricter limitations imposed on the organic body. And no, powered armor does not mitigate these limitations. PA just gives you a strength boost and, to some extent, a protection boost as well, but run too fast in it, and your legs will still pull muscles and break bones.