The Lagrange multiplier is not the shadow value of the limiting resource in the presence of strategically interacting agents

In the case of a single net-benefit maximizing agent facing a resource constraint, the economic interpretation of the Lagrange multiplier is that of the shadow value of the constraining resource. The formal justification for this economic interpretation is by way of the classical envelope theorem. Once an environment of strategically interacting agents is contemplated, however, the Lagrange multiplier no longer represents the shadow value of the resource to an agent. A concise proof of this claim and a revised economic interpretation of the Lagrange multiplier are given in this note.