"We are not the cheapest, but we are the best"
That is the motto of my local pizza shop that has been in business for 30+ years. Firstly, he actually does sell a good product. Secondly, when a competitor introduced $5 pizzas he increased the price of his and improved the product, so while the other guy was selling more pizzas at slimmer margins, he was selling fewer but with a larger profit. The other guy eventually went out of business.
So the moral of this story is when caught in a race to the bottom don't take the lead.