Note---An Objective Function Perturbation with Economic Interpretations

For an ordinary linear program it is well known that, if the resources are evaluated at marginal prices determined by an optimal dual solution, then this imputed value is identical with the value of the primal objective function. For a convex program with a nonlinear objective function and linear constraints this identity in general does not hold. The resulting difference is due to a returns to scale associated with the objective function, as earlier pointed out by Balinski and Baumol (Balinski, M. L., W. J. Baumol. 1968. The dual in nonlinear programming and its economic interpretation. Rev. Econom. Studies 35 237--256.). In this paper we consider a certain perturbation of the objective function that characterizes the difference between the objective function value and imputed marginal cost This perturbation, when applied to a certain class of profit maximizing monopolies, explains the difference between the monopoly price and the marginal production cost.