I think there is a tendency to see the Amarr as a bit backward (because they operate with something roughly like a feudal structure, and have slaves). Technologically though, they lead the cluster in some areas (implants and cybernetics) and are at least on-par in a lot of others. Some of the ship types they use might be decades or centuries old, but they work, they are expensive to replace, and they probably get periodic upgrades (you can see something like this in most modern navies. Hell, the US still uses B-52s).
I think that slavery isn't economically efficient (for the reasons others have covered), but I think the Empire is also stuck. I expect there are probably quite a few people in the Empire who realize it isn't efficient. Things like tradition and religion help keep it in place - and of course simple inertia. Even if an Empire magically without slaves would be more efficient, the slave are there, and they are inseparably tangled up with the Imperial economy.