Here is the simplest argument for non-neutrality of money I can think of. It is actually an argument for the non-neutrality of price inflation, but it’s hard to think of changing prices without causing changes in price inflation. One way to do it would be to relabel all prices from the beginning of time until the end of the economic system. That would be a price change that would leave inflation unaffected — but it would be the only one.
In period n, a household can purchase a single period zero coupon (nominal) bond, paying in the next period, or it can spend dollars to purchase a single capital good, rent the capital good to a firm, and receive as the rental rate next period. Then, it can sell the capital good for .
Via arbitrage, we have
where is the expected future price of the capital good in the next period.
If all present prices double, then all future prices would need to double as well, in which case inflation is zero and becomes a real interest rate, rather than a nominal rate. But if inflation is not zero, then it is impossible to double all current period prices — the capital rental rate will not double, as this would allow for arbitrage. You can double wages, consumer goods prices, and capital goods prices, but then the relative price of capital rental to these other prices will shift, and this will have real effects.
The above doesn’t mean that there is no long run neutrality, only that this needs to be proved by making additional assumptions. It could well be the case that these non-linearities cancel out in a moving average sense, so that over the long run money is completely neutral. But it doesn’t automatically follow by looking at degrees of polynomials, or sheer dint of thought. Simple reflection tells us that any expected change in future (nominal) prices is going to cause a change in (present) real output as well as present relative prices.