A nonlinear variational problem of profit maximization with convex costs

In this article some optimal production plans are obtained by a deterministic profit maximization, arriving at a inflation scenery description. The industrial (production/storage) costing rate is assumed to be a positive, increasing, convex, fourth degree function of the production itself. By maximizing the integral of profit, in lack of inflation, the variational calculus leads to a Euler-Lagrange nonlinear boundary value problem which we turn to an auxiliary Cauchy nonlinear problem whose solution is proved to be always strictly increasing. Under inflation, introducing further constraints between costs and inflation itself, one is driven to a Duffing equation. In both contexts anyway, optimized production plans within a finite time horizon are amenable to exact treatments and formulated by means of Jacobi elliptic functions.