Deflation would be a bigger problem. It would mean lots of people with no jobs, still trying to afford high prices.
The problem right now is that wages are only catching up slowly, because labor bargaining power is squat due to low union participation and legal protection.
I would argue the heart of the problem- or at least a major component, is that so few companies control the market. I can go to a different grocery store if the prices are insane at only the nearest one. However, pretty much all grocery stores carry mostly the same products. Those products are owned by just a handful of companies.
Want a different cell service? There's just 3 companies left. Internet? You probably don't even have a choice.
It's pretty easy to jack up prices if there are no competitors left to swoop up your customers.