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I'm (still) reading the microeconomics textbook of Mas-Colell et al. On p. 135, the profit maximization problem (PMP) for producers is introduced; characterizing the technology as $Y = \{ y \in \mathbb{R}^L : F(y) \le 0\}$ for a suitable transformation function $F$, the PMP is, for a given price vector $p$, to maximize $p'y$ s.t. $y \in Y$, i.e. $F(y) \le 0$.

On p. 136, the authors give the necessary FOCs for the case of a differentiable $F$ as $p = \lambda \nabla F(y^*)$ ((5.C.1); here, $y^* \in Y$ is a profit-maximizing production plan, and $\lambda$ is the KKT multiplier of the constraint).

What I don't understand is why this always has to hold with strict equality. Usually when you do constrained optimization under inequality constraints, you get inequalities for the FOCs as well (e.g. (5.C.2) on p. 137, where the authors study the important special case of a single output good). Why not here? Why can we say that the above FOC must always hold with equality (when $F$ is differentiable)?

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Having thought about this some more, I believe that the answer relates to the nature of the transformation function. $F$ does not have an intuitive interpretation (the way that, say, a production function does), but is rather a mathematical tool for characterizing the transformation frontier, i.e. the border $\partial Y$ of the production set $Y$ (which is assumed to be non-empty, closed, and satisfying the free disposal property for this section of the book).

In any case, if the constraint is slack, that is if $F(y) < 0$, then there is some $\varepsilon$-environment of $y$ that is wholly in $Y$, and thus there is a way of increasing profits by increasing the net outputs of at least good (while not decreasing the net outputs of any other); as such $y$ could not be profit-maximizing, and we can conclude $F(y) = 0$ necessarily holds at the maximum so that the FOC also holds with equality.

I'd still love to hear others' thoughts on this.

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