It's more complex, and I really wish people wouldn't try to look at this purely from an economic perspective.
If you see excessive workers in any government run agency, whether it's Willie Brown appointing 100 "mayoral advisors" in San Francisco, or 30 people standing around pointing towards an exit sign at a train station, then you are watching patronage networks in action. It's not like the station can't afford signage, or that those running the station are too stupid to allocate labor efficiently. We should not look down on these countries and their capabilities, they are just stuck in a different equilibrium.
Poor nations, if they could avoid it, would always choose to spend less money per government service delivered, as it would allow them to deliver more service. It's not true that labor is cheap in poor nations. It's cheap for a rich nation, but not for the poor nation. But they are often governed in such a way that in order to obtain and keep power, you have adopt a clientist governance model, which is so common throughout the world, and this is a big reason why the nation in question is poor in the first place.
Then you can ask "Why are nations governed on clientist principles?" And the answer is often related to the nation not having any other "glue" holding people together or incentivizing them to respect rule of law. To support a leader, they want some money in their pocket given to them by that leader. If he says no, another leader will arise that will offer to put that money in your pocket, and then that will be the one you support. The easiest way to give someone money is to give them a job where they get paid for doing very little. This continues in the private sector -- you are opening a tourist hotel and need good relations with a local notable. Well, he will expect that you hire many of his relatives or supporters. Then those supporters have a reason to support that leader -- because again he puts money in their pocket. The result is a different view of labor -- where jobs are given to people based on who they know and which patron is looking out for them. Such a system provides great stability -- Rome was built on these types of patronage networks. They may even be the oldest form of labor organization around -- the idea that random people should be hired based solely on whether they were needed or had qualifications -- that's a relative new, and in some sense dehumanizing way of looking at labor. In a clientist model, a jobs are assigned based on relationships.
To counter this requires a strong cultural opposition to seeking this type of self-interest, and very few nations or regions have this. Even the U.S. was plagued by clientism after the large migration waves at the end of the 19th Century, as many cities became run by various machines that catered primarily to those ethnic groups that followed clientism in their native lands. It was what they expected from a leader in the old country, and so it was carried over here. Those old machines turned out to be incredibly hard to dismantle, requiring almost a century long struggle, and in many cities like San Francisco or Philadelphia, these machines are still in operation. So this is not at all related to the GDP, except for the fact that clientist nations tend to have lower GDP, on average, than the more rule of law nations.