A model of spatial flows and growth of capital and labor stocks

This article deals with optimal spatio-temporal development of capital and labour stocks in a production economy with spatial extension. Current stocks of capital and labour are used to produce a commodity, partly invested to replace worn capital, partly consumed. These stocks can be relocated in space, but relocation uses up some of the inputs themselves. Under these constraints the objective is to maximize a utility measure derived from per capita consumption and aggregated over individuals, space and time.The necessary conditions for optimum are derived as Euler equations of a continuous variational problem. They concern choice of production scale and technology, rate of reinvestment, and optimal flows of labour and produced commodities through space. The Lagrangian multipliers of the constraints are interpreted as imputed wages and commodity prices.The whole structure of optimum depends on these imputed wages and prices, and their solution can be derived from a pair of dependent non-linear partial differential equations. The spatial flow portrait at each moment depends on the time parameter and on the parameters of the model (net reproduction and capital depreciation rates). It can undergo sudden changes described by the elliptic and hyperbolic umbilic catastrophes.