A bog-standard free-market analysis of this problem holds that these firms (grocery stores, and now financial institutions) observe marginal revenue below average variable cost. In other words, the firm identifies locations at which the day-to-day costs of operating the location are greater than the day-to-day revenue generated by the location. The firm then decides that it is profitable to shutter the location. Causal explanations for this phenomenon usually invoke higher shrinkage rates (theft, spoilage), or basic demand deficiencies (the poor would rather eat fast food) for the services involved. Further, market fundamentalists say that residents of the affected areas who desire the abandoned services are incentivized (and therefore likely) to relocate to areas where such services are more readily available.
This analysis fails to account for the often high costs involved with relocation. Let's try a thought experiment. If we disaggregate the population of a low-income food desert into "employed" and "unemployed," we can quickly see the positive correlation between "employed" and "available financial resources to relocate." We may even be able to infer a positive correlation between "employed" and "demand for fresh food," though this controversial and problematic inference is not material in this thought experiment.
Regardless, let's assume both of these correlations hold for now. If those who most want the fresh food are most able to relocate, then perhaps these closures are the efficient outcome. Stop for a moment, however, and consider our relocation criteria. We observed that the employed have the most resources at their disposal in order to move. On the other hand, the employed have the most to lose by moving, namely, their jobs. I think it's safe to assume that the employed fraction of a low-income population have the most tenuous hold on their jobs of any employed person (who isn't the drummer for Spinal Tap). Often these workers must commute far to their jobs, and often by public transport. Unless the representative low-income employed person who seeks to relocate to escape a food desert has the good fortune of being willing and able to move closer to their job, it's unlikely that any fraction of the population of "food deserts" is made better off by these closures.
Is the best solution to this problem the "government takeover of grocery stores" that Mr Johnson so fears? Probably not. Thankfully, no one is suggesting it is. If, on the other hand, you believe that we elect a government to fix problems, then a small federal program to incentivize grocers seems like a staggeringly pragmatic solution.
See also:
Schuetz, Jenny & Kolko, Jed & Meltzer, Rachel, 2012.
Elsevier, vol. 42(1-2), pages 269-285.

