58.
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
u i (?) defined on
R. Our objective is to divide
R into
n sub-regions
R i so as to “balance” the overall utilities on the regions, which are given by the integrals
\(\iint _{R_{i}}f(x)u_{i}(x)\, dA\). 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
u i (
x) ?
u j (
x) or the ratio
u i (
x)/
u j (
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
u i (?). 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.
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