I think the topic of wage growth and inflation is one of those topics that is muddled by what I call the "Wizard of Oz" theory of the Fed. Or, maybe a better name would be the "Pet Rock" Fed. This is the notion that interest rates are primarily set by the Fed, so that if we move from .25% to, say, 3% short term rates over the next few years, that reflects intentions of the Fed regarding inflation control. I would say that most of that movement is determined, essentially, by the market, and that, if the Fed is lucky, they keep their target near the market rate as we move along.
The Fed is like a gas station. In the literal sense, they can go out to the curb each day to set the price. But, in another sense, they are guessing at what the right price is, and tweaking it one direction or another. If they get outside a given range for too long, bad stuff is going to happen.
Here is a graph showing wage growth, short term rates, and inflation.
Next is a graph estimating wage growth and short term interest rates, net of inflation. Generally there is a relationship between wages and interest rates, both nominally and in real terms.
But, I think, instead of thinking of wage growth as a cause of inflation, which, in turn, causes Fed tightening through rising interest rates, I think it makes more sense to simply think of wages and interest rates as two symptoms of a thriving investment market. Firms with a high demand for investment are bidding up both the price of capital and of labor.
Inflation may reflect Fed policy, but real interest rates and wages tend to rise and fall together in both high and low inflationary periods. Wage growth might signal interest rate increases, but not necessarily because the Fed plans it that way.
On the other hand, during the 1980's, wages did tend to move pretty tightly with inflation, while debt retained a high real premium. Perhaps the secular downward pressures on real interest rates are so strong that we will see the reverse over the next decade or two, with growing wages while interest rates remain pinned near inflation.
Could it be that when the baby boomers were young, healthy, and poor, capital was more scarce than labor, but now that baby boomers are old, retiring, and flush with capital, labor is becoming more scarce than capital?