Stronger wages → stronger consumption → higher demand-pull inflation.
But higher inflation implying that “rates should go up” is central bank doctrine. It’s not a general law of how economies function.
Central banks intervening with interest rate adjustments is what distorts the prices of equities downward, when inflation rises.
Without central bank intervention, inflation should theoretically push equities higher (a highly-inflated economy driven by rising demand is by definition a well-performing economy!).
Central banks intervene because runaway inflation can be harmful to wage-earners (they save in dollars, not assets).
But I’m not sure if a 2–3% inflation target is ideal. It seems to me that this arbitrarily low inflation target restricts the growth of the economy in ways that might affect wage-earners, defeating the stated purpose of monetary policy, since higher rates also have the effect of curbing job growth as well as raising the cost of servicing mortgages.
Let's put it this way then: the central bank can raise rates or it can crash the economy into a brick wall. In that sense, rates should be raised. We have the least competent regime in history right now though, so they might choose the latter option.
Uh, no. If you have no central bank, more consumption and more employment means more demand for money. Ceteris paribus, that will raise rates. (Our own history with free banking is more complicated since the only inflationary period was driven by specie introduction from California's gold rush. The predominant problem in antebellum America was deflation and bank collapses.)
You're correct inasmuch as central banks quicken this reaction, and–when done properly–dampen it. But the fundamental engine is emergent, at least for nominal rates.