Advances in the theoretical literature have extended the Hotelling model of spatial competition from a uniform distribution of consumers to the family of log-concave distributions. While a closed form has been found for the equilibrium locations for symmetric log-concave distributions, the literature contains no closed form solution for the socially optimal locations. We provide a closed form solution for the socially optimal locations: one mean-deviation away from the median. We also derive a formula for the excess differentiation ratio which complements the bounds previously derived in the literature, and establish the invariance of this ratio to a form of mean preserving spread. The equilibrium duopoly locations of several types of commonly used distributions were discussed in [1]. This paper provides the closed form solutions for the socially optimal locations to the same set of distributions. We calculate welfare improvements arising from regulation of firm location and show how these vary with the distribution of consumers. While regulating firm locations is sufficient to optimize welfare for symmetric distributions, additional price regulation is required to ensure social optimality for asymmetric distributions. These results are significant for urban policy over firm/store locations.
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