This rapidly falls apart when you try to actually calculate it. Whose income do you use? Is inflation lower for doctors than burger flippers? What do you use as the retirement age? Does inflation go down if the retirement age is raised? What counts as "survive"? Does that mean the price of smartphones don't count toward inflation because you can theoretically survive without them?
Sticky prices are only important, when expectations are invalidated.
So to fix your sentence:
> If wages are stickier than expenses, then the gap between wages and expenses will represent recent unexpected inflation to some degree.