Does the state make seigniorage profits when it prints money?

That’s a nice simple exam question. My answer for what it’s worth is as follows.

If the state prints money and does helipcopter drops, the state gains nothing. All that happens (at least initially) is that the private sector’s paper assets rise in value.

If the state prints money and and that money just boosts CURRENT public spending (e.g. hiring more state employed bureaucrats), then again the state gains nothing. There are no seigniorage profits there.

A third possibility is that the state prints money and purchases items of a CAPITAL nature, e.g. roads, bridges, etc. In that case the state’s assets do rise in value, so there are seigniorage profits there.