What is wrong with Marty G's analysis? In short, S1 is not, as he claims, a supply curve.
In the real world, it’s different. A new supply curve (S1) can be plotted that is the amount of the subsidy below the original supply line. Where the new supply line intersects with D gives the new level of price and quantity – note, it’s not as low as the full subsidy and quantity is less than what would be demanded if the full subsidy was passed on. (emphasis added)
I have redrawn Marty G's diagram below with an extra point on it, P2. Let us assume that we are at the new equilibrium P1,Q1. At this point the quantity traded is Q1 and the price paid by the consumer is P1. Question: If Q1 is to be supplied, what price must the producer receive? We know that the consumer price and the producer price must differ by the amount of the subsidy and that the consumer pays P1. Thus the producer receives P1 + the subsidy, which equals P2 on my diagram. But note that the P2,Q1 combination corresponds to the curve S, NOT the curve S1. That is, to induce the firm to supply an amount Q1 it must receive the price P2. Therefore S is the supply curve after the subsidy is introduced, as well as before it is. But if S is the supply curve both pre and post subsidy, then S1 is NOT the supply curve in either case. That is, the supply does not move with the introduction of the subsidy.
If we don't move the supply curve from S to S1, how do we find the new quantity traded, Q1, when the subsidy is put on? Note that the consumer and producer prices differ by the amount of the subsidy, so we increase the quantity traded starting from Q until the vertical distance between the supply and demand curves is equal to the amount of the subsidy. This is will give Q1 without having to move the supply curve.
Update: Matt Nolan over at TVHE has some additional issue with Subsidises and complaints.