This is a door which swings two ways. As the world technical expertise is rising, emerging markets are also rising. This means that while we have more competition, there is also rising demand to eat up that talent.
There may be a pain point in the transition, but I believe that domestic markets for developers will keep demand high and availability tight. Look at Chinese manufacturing as an example. China started out as the factory for the world but now the Chinese are looking more towards serving their domestic markets.
U.S. companies are seeing China as an opportunity for growth in sales rather than as a source for cheap labor. Today, developers are bringing in high paying gigs from the West and worrying about competition from developing nations. Tomorrow, developers are going to be competing for work originating from those same developing nations.
"and there are likely people in the world that are cheaper to hire than you that can do the same thing"
I don't think it's that simple. I'm a U.S. citizen living in the Philippines and I have found that live here isn't necessarily cheaper than anywhere else. In some cases things are more expensive. The reason people are living cheaper here isn't because of some national discount on everything for being a member of the "developing nation club," but rather it's because far more people are living in poverty. They have far lower expectations of "having stuff" than people in the U.S. have.
I have also found lately that as a Western developer, I'm not always the most expensive option. I have found that dev shops from India are often charging more than I do. In some cases it's because they have simply progressed to a point where they have been able to raise their rates by being better at selling, and in other cases it's because they don't have the same knowledge as I do in certain areas and aren't as accurate in their estimates.
A typical developer in the Philippines may start out cheap, but as he makes more money his expectations will rise. He will raise his rates as he is able to just like any other developer would. Eventually he is charging global market rates which is as much or more as you are charging. Again, the pain point is the period where that developer is competing with the same work you are competing for in your home country. Eventually he will be able to get those rates in the Philippines and you may be competing with him for work from his own country. The tables have turned!
Don't worry about the competition. Keep doing your thing and growing. As the world economy grows and technology progresses, there will be more opportunities. When this web thing got started, one guy would develop an entire site and he would call himeself a web master. Today it's getting more difficult to be the guy who can cover all bases. We can still do that for a simple site, but many of us are sticking to our strong points and letting other developers take care of the rest. These days I rarely work alone. This evolution creates new niche areas every day and in those areas you only have to run a little faster than the masses.