Do subsidies keep the cost of food down?
First, it's an inefficient way of delivering the subsidy to those most in need. In so far as the cost is reduced, the benefit reaches the rich as well as the poor. It would be better to ensure that the least well off members of the population had enough money for a balanced and nutritious diet.
Second, the protectionist aspects of the CAP mean that consumers in the EU do not have unhindered access to agricultural goods at world market prices. The EU price is generally above the world price.
What about the single farm payment? Much of the money spent on the CAP does not reach the farmer, but is absorbed by administrative costs (pushed up by the complexity of the system), traders and food processors (just look at how much Tate & Lyle has received) and by criminal organisations.
As a rational decision-maker, the farmer is not going to use the subsidy to lower prices. He may invest in the farm, thereby benefiting suppliers of agricultural equipment which is why they favour the CAP so much. Or he may decide to use the money for personal consumption - or more likely some mixture of both.
Some of my colleagues at Warwick HRI think that food is too cheap. By that they mean that it costs so little that people do not value it enough and the emphasis is on low cost production, regardless of some of the negative externalities. There is something in this, but one has to consider the impact of higher food prices on the constrained budget of the typical family.