Shadow prices in territory division

We consider a geographic optimization problem in which we are given a region R, a probability density function f(·) defined on R, and a collection of n utility density functions ui(·) defined on R. Our objective is to divide R into n sub-regions Ri so as to “balance” the overall utilities on the regions, which are given by the integrals ̃ Ri f(x)ui(x) dA. Using a simple complementary slackness argument, we show that (depending on what we mean precisely by “balancing” the utility functions) the boundary curves between optimal sub-regions are level curves of either the difference function ui(x)−uj(x) or the ratio ui(x)/uj(x). This allows us to solve the problem of optimally partitioning the region efficiently by reducing it to a low-dimensional convex optimization problem. This result generalizes, and gives very short and constructive proofs of, several existing results in the literature on equitable partitioning for particular forms of f(·) and ui(·). We next give two economic applications of our results in which we show how to compute a market-clearing price vector in an aggregate demand system or a variation of the classical Fisher exchange market. Finally, we consider a dynamic problem in which the density function f(·) varies over time (simulating population migration or transport of a resource, for example) and derive a set of partial differential equations that describe the evolution of the optimal sub-regions over time. Numerical simulations for both static and dynamic problems confirm that such partitioning problems become tractable when using our methods.