The current subsidies have the direct effect of reducing supply (since they are not subsidies on the growth of food, but on letting farmland remain fallow...the way to get the money is to have farmland and not grow food on it).
This is only sort of right. IWADFUSDA (I was a developer for the USDA).
My main application issued 2 types of subsidies, to the tune of $4b a year. The first was a direct subsidy. The government says "we will pay you X dollars per ${unit_of_measure} of Y commodity. You grew Z ${units_of_measure}, therefore you get X*Z dollars." There were complex eligibility and attribution rules, but that was the basic idea. This subsidy program was not renewed in the latest Farm Bill.
The other subsidy happened after harvest and market. The government would say "We wanted the market price of Y commodity to be A dollars per ${unit_of_measure}. It was, in fact, A - B dollars. Therefore, in addition to your direct subsidy, we will pay you a "counter cyclical" subsidy of B dollars." If the market price of the commodity was higher than the targetted price, no payment was issued. This subsidy program expired in 2011 or 2012 (I don't remember exactly), and, like the above, was not renewed in the Farm Bill.
I did also do some work on a conservation program. A farmer goes to the government and says "I think these acres on my farm are wetlands/${some_other_environmental_gem}." The government says "We will pay you X dollars per year to not grow crops on this land for Y years." I didn't spend much time on this one, so I don't know the finer points.
I'm not saying these were the only programs around, but the types of programs varied widely. The direct payment program was a big player in the USDA though.
My old coworkers tell me that the latest Farm Bill has shifted emphasis dramatically from subsidy programs to crop insurance programs.