Look a little harder.
Of course. How is this hard to understand? Some inflation comes from money growth, which raises nominal incomes. Some inflation comes from supply shocks (like the oil shocks of the 70s), and raises prices without raising income.
Here's why it should be super obvious to you: if inflation through excessive money growth could raise prices without a corresponding raise in wages, then monetary policy can permanently change the real wage. There's your first hint, since it violates the classical dichotomy. Your second hint is that this results in a permanent Phillips curve trade-off. Through setting the rate of inflation at an appropriate level the central bank can permanently affect the real wage such that no unemployment happens. They can permanently keep the economy at full employment by accepting more inflation. This was the theory that 70s stagflation disproved.
To summarize, this is what you need to address: 1) Monetary inflation must necessarily raise incomes in the same proportion as prices otherwise the Phillips curve holds. 2) Wages will not keep up with "inflation" when that inflation comes from supply shocks rather than demand shocks.
1) That is a non sequitur. Inflation affects different wages and different prices differently. There is nothng uniform about it.
Well no, because again that violates the classical dichotomy. Even so, it's still not non-sequitur. If prices can increase from monetary inflation without wages increases, then monetary policy can permanently change the real wage. It follows that the Phillips curve must hold.
2) The source of the "shock" is irrelevant as the source of inflation is monetary. If the money supply is constant then a rise in the price of one input will be off set by a fall in demand. That is supposed to be what happens.
You think following a price rise, demand falls and brings the price back to normal? Is that what you're saying? Because that's ridiculous. The quantity demanded, distinct from the demand schedule, will fall endogenously, but the price will still be higher.
Say the supply of money is constant. If there is a supply shock, a shock that reduces the quantity of output we can produce, prices must rise as the natural reaction to increased scarcity. The quantity of money is the same, output has fallen, so output is now more scarce relative to money than before.
Seriously, just think about the quantity theory of money for half a second. MV = PY, V = constant. If we set M constant, MV is constant. PY isn't constant though. Y, real output, isn't constant. If Y grows, P must fall. If Y falls, P must rise.
Yeah, if you raise the price of something you sell less of it. It's kinda basic.
Put away the textbook until you actually understand what you're writing.