Why is the agricultural sector still protected in the US?

While most industrialized economies are strong advocates for free trade, they somehow manage to make an exception for the agricultural sector. I cannot think of one country that would not subsidize its farms in some major way and this despite the facts that we are far from witnessing a food shortage and that the agricultural sector by now constitutes a voting block that can safely be called small. So why do elected official keep pandering to farmers?

Marc Bellemare and Nicholas Carnes look at this question in the case of the United States. Using roll call votes from the US Congress and congress member ratings by the Farm Bureau, they find that electoral incentives in fact still do matter, while personal preferences and lobbying are less important. Could it be that the median voter is a farmer? I do not think this is what the paper is saying. Indeed, the variable "electoral incentive" is based on the proportion of the electorate that works in the agricultural sector. But it is above half, or approaching this, in few districts. In fact, Bellemare and Carnes restrict the variable to farm owners and managers, who are the ones really benefiting from the subsidies, and they comprise a tiny portion of the electorate in every district. (Of course, this measure is correlated with the total farm population.) For this influence of such a minority to still carry the vote, it must be that there is still lobbying going on, and of the sort that is not captured by the agricultural political actions committees that the authors use to control for lobbying. Maybe individual donations? Somehow, it always boils down to lobbying in the United States.