What would the world be like with 2 billion people in it? Maybe it would be a lot like the world the last time it had 2 billion people in it.
According to charts and articles on Wikipedia, the Encyclopedia of Death and Dying, and Docstoc.com it seems the last time the global population was around 2 billion people was in the early part of the 20th century, between the two world wars. Setting economic, political and technological considerations aside for the moment, we can safely draw some parallels between that real era and this theoretical one.
First off, every human being back then had to be more self-sufficient. Almost everyone in 1920 knew how to plant a garden, build a fire, and find their way home using only the sun or the stars. If something broke, you fixed it yourself. You mended your own clothes, fixed your own car (if you were lucky enough to have one at all) and pulled your own teeth. Yes, there were people to do all these things just as there are now, but they were much fewer and farther between, and it was not practical for most people to access their services. In a world with 2 billion people it's likely that this would become the case again.
Break a bone? You'd better know how to put it in a splint, because there may be only one doctor in the nearest hundred miles.
In the 1920's families were large, but babysitters were unheard of. Today, one parent would most likely be forced to stay home to raise the children, and deal with the far more extensive household chores (no maid service, grass cutters, plumbers or handymen for all but the wealthiest people).
Two billion human beings simply means a shallower talent pool of people who are willing and able to provide many things we take for granted. Sure, anyone can grill the perfect steak, but how many people know how to properly butcher a cow? Or raise one? Will there be enough people who have the time and the inclination to learn to be doctors, astro-phyicists and engineers? Will there be enough people who know how to wire a light bulb, weld a piece of steel or build a house? Much mankind's long period of slow growth is owing to simply not having enough people having the skills to improve people's lives in meaningful ways. It's perfectly conceivable that progress would once again slow to a crawl, perhaps even reverse, if we did not actively manage and make optimal use of our collective knowledge.
A world of tomorrow with 2 billion people would have the advantage of technological advancements the people of the 1920's did not enjoy. Technology could (and currently often does) reduce the need for human involvement in many areas. Technology could sustain communications between people separated by great distances, and allow teachers, religious leaders and bureaucrats to reach their various constituencies. It could plow the roads, drive the cargo ships and mix a decent vodka martini, all with a minimal amount of human intervention.
But technology can't solve every challenge a population of 2 billion would face, nor is it likely that we would be able to completely optimise our use of technology - wealth would still drive the system, and the needs and wants of those with means would still come before the needs and wants of everyone else. What's worse, a less-populated world would have fewer checks and balances, so it would be easier to expropriate technology for nefarious ends (and stand a greater chance of getting away with it - there would be fewer people trying to catch you).
A world with 2 billion people in it would represent an awesome uphill challenge for the survival of humankind. True, we have done it before, but we're a much different species than we were back then. We're much more inter-dependent, specialized and isolated from our environment. While I'm confident that humankind could surive (and it would be the best thing that has ever happened to the Earth itself) I don't think we would enjoy it very much. In a deserted, crumbling world we'd feel lost, alone and insecure. We'd struggle to hold on to the old ways, only to find there aren't enough people to allow us the luxury to do the things we used to do. We'd be horrified to see the history of our ancestors disappearing with each passing year, and we'd probably have to cede our place as the dominant species in many areas of the planet, perhaps retreating to just one or two continents to ensure there were enough of us around just to keep each other company.