A quick rant / ramble from my years in (IT) business.
"People don't want the cheapest - they want the best perceived value for money. The problem is is that a lot of people ASSUME that money is the most important thing, so when someone makes contact and asks for a quote - the seller assumes its all about money & gives them a price. At that point the ONLY piece of information the potential client has to base a decision on is the price. So it's not that they're a price-shopper when they called -- it's the seller/provider who's turned them INTO a price shopper.
Perceived value for money covers a lot of things; it covers the price of the transaction - it covers friendliness - it covers trust - it covers recommendations from friends. If they don't like you or don't feel they can trust you then they're not going to buy from you at any cost. Conversely, if they really like you and really feel they can trust you - and they like your work - then that will command a higher price.
So when they make contact, be friendly - be respectful - ask them lots of questions (set the stage with "before I can give you a price, I need to really understand what your needs are -- so would it be OK if I ask you a few questions"?) (which lets them be reassured by the fact that you're taking the time to understand their real needs).
And in the end, don't give them a "price" or a "quote" ...
... Give them a Recommendation & Proposal".
Don't just tell them you charge more because you're better - articulate to them (with examples) of WHY you're better - and what the benefit of that is to them.