You will never be able to please everyone. The cost of living varies drastically depending on the city you live in. I think it’s best to find a price that works for you and find the students willing to pay that price, and of course make sure that your quality matches your price.
Yes, many students who pay lower prices will not be happy if you raise your prices. The reason this happens is that students who pay prices that low tend to be more interested in low prices than in building a relationship with a teacher. Maybe they cannot afford to build a relationship at the cost of higher prices, or maybe they just don’t care. It’s effectively a different market when you increase your prices. This is why I don’t believe in starting very low and raising your prices. I think your initial prices need to be at the low-average end of what you’re willing to accept, so that when you raise your prices you’re still in the same range and targeting the same students. And the lower your prices are, the smaller the range is. A person who can only afford $10 per hour might not be able to afford $15, but a person who can afford $50 can probably afford $55 or even $60. What matters is the percentage increase. A person who can only pay $10 per hour probably values each dollar more than a person who can pay $50.