Abstract
This paper proposes an optimal allocation problem with ramified transport technology in a spatial economy. Ramified transportation is used to model the transport economy of scale in group transportation observed widely in both nature and efficiently designed transport systems of branching structures. The ramified allocation problem aims at finding an optimal allocation plan as well as an associated optimal allocation path to minimize overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transportation literature in that the distribution of production among factories is not fixed but endogenously determined as observed in many allocation practices. It's shown that due to the transport economy of scale in ramified transportation, each optimal allocation plan corresponds equivalently to an optimal assignment map from households to factories. This optimal assignment map provides a natural partition of both households and allocation paths. We develop methods of marginal transportation analysis and projectional analysis to study properties of optimal assignment maps. These properties are then related to the search for an optimal assignment map in the context of state matrix.
Highlights
One of the lasting interests in economics is to study optimal resource allocation in a spatial economy
This paper proposes an optimal allocation problem with ramified transport technology in a spatial economy
A planner needs to find an optimal allocation plan as well as an associated optimal allocation path to minimize overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transportation literature in that the distribution of production among factories is not fixed but endogenously determined as in many allocation practices
Summary
One of the lasting interests in economics is to study optimal resource allocation in a spatial economy. A planner needs to find an efficient allocation plan such that demands from households will be met in a least cost way In this problem, the distribution of production over factories is not pre-determined as in Monge-Kantorovich or ramified transport problems, but endogenously determined by the distribution of demands from households as well as their relative locations to factories. Our search method uses properties about optimal assignment maps to update some nonzero entries with zeros in a state matrix This method is motivated by the observation that via group transportation under ramified transport technology, assignment of each household has a global effect on the allocation path as well as the associated assignment map. In some non-trivial cases, it’s shown that this method can exactly find an optimal assignment map as desired
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