The health impacts of people buying soda/soft drinks with food stamps?

The USDA recently rejected a proposal from New York City to ban the use of food stamps to purchase soda/soft drinks. Grist has much more on the decision. It made me wonder what would be the health impacts of such a proposal?

There seems to be an awful lot of paternalistic assumptions underpinning the proposal. Further I doubt it would have much of an impact at a population level. What are your thoughts?

Related: Bad Food? Tax It, and Subsidize Vegetables [New York Times]