"Almost" subsidy-free spatial pricing in a multi-dimensional setting

Consider a population of citizens uniformly spread over the entire plane. The population faces a problem of locating public facilities financed by its users, who face an idiosyncratic private access cost to the facility. We show that, under mild assumptions, an external intervention that covers a tiny portion of the facility cost is sufficient to guarantee secession-proofness or no cross-subsidization, where no group of individuals is charged more than the cost incurred if it had acted on its own. Moreover, we demonstrate that in this case the Rawlsian access pricing is the only mechanism that rules out secession threats.