I recently finished the game on Medium difficulty without changing the price in Drug Casebook (F2), because whenever I increased the price even 1%, a lot of patients would go home after visiting the GP office. I also tried lowering prices when I had enough to handle more patients, but it didn't seem to do much. I kept 25% of the customers according to the Progress Report and Status (F8) and didn't notice a big increase.

It seems like a good idea to increase price if you're having problem dealing with the flow of patients, but how much? What's your experience?

It does seem finicky. I completed the game and just didn't bother with changing prices. There was a level where the objective even mentioned raising prices because the hospital was in a rich area. But people still kept leaving if I upped it a few %. Perhaps you're just supposed to expect a few rebel protestors so long as the majority still pay (but I just ended up lowering my price to the default again as I wanted to cure everyone).

I never tried lowering the price but according to one tips guide someone posted online:

I finished game on easy and I never changed prices. Now, I'm playing on medium and I'm not touching them as well :). In my experience, if you have happy patients, more of them will come. I always run my hospital with 2 GP's office and I leave some space in hospital to build second Pharmacy/Psychiatrist in case of emergency or greater demand :). That way I have steady cash flow.

Tried to increase prices in the mission where we were instructed to, but it just resulted in a LOT fewer patients. Even at 100% across the board there was the odd patient that refused to pay throughout the levels.

At one point I did increase it a bit though, think to 105%, because there were queues and epidemics everywhere, and I had to contain it somehow.

Have never seen the need to decrease prices to get patients. It's always slow in the start, but they come in droves soon enough, and money will fly in. You get a snowballing effect too, because with a well-run hospital with the right equipment you will treat most patients, which increases your reputation, which increases the patient flow. After a couple of years money will roll in, and reputation will likely be touching on 1000 (max).