There's never really been any competition for cheap rockets in the past. The US expendable rockets could be expensive because the bulk of their launches went to the military who'd pay almost anything. The only serious attempt to cut costs was the original plan to turn the Shuttle into a space truck, but the Shuttle proved to be about the most expensive way imaginable of putting stuff into orbit and the commercial launches were all suspended anyway after the Challenger explosion.
The Russians did produce cheap rockets because they needed to launch several hundred rockets EVERY year in the most appalling conditions. So their Soyuz family of rockets was built on an assembly line out of cheap components with fairly loose tolerances. However, Soyuz has had trouble getting into the commercial business because of concerns over technology transfer to the Russians and the lack of an equatorial launch site which limits the amount of payload that can be fired into the most valuable geostationary orbits.
Soyuz is by far the best cheap launcher at the moment, but it's worth remembering it is based on half century old technology - all those small thrust chambers and strap-on boosters make it inherently less effective than a rocket designed today with more advanced materials and engines. So there is a market for a competitor, but the Soyuz IS going to be a tough act to follow.