<p>Having thought about this some more, I believe that the answer relates to the nature of the transformation function. <span class="math-container">$F$</span> 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 <span class="math-container">$\partial Y$</span> of the production set <span class="math-container">$Y$</span> (which is assumed to be non-empty, closed, and satisfying the free disposal property for this section of the book).</p>
<p>In any case, if the constraint is slack, that is if <span class="math-container">$F(y) &lt; 0$</span>, then there is some <span class="math-container">$\varepsilon$</span>-environment of <span class="math-container">$y$</span> that is wholly in <span class="math-container">$Y$</span>, 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 <span class="math-container">$y$</span> could not be profit-maximizing, and we can conclude <span class="math-container">$F(y) = 0$</span> necessarily holds at the maximum so that the FOC also holds with equality.</p>
<p>I'd still love to hear others' thoughts on this.</p>

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2023-11-24T13:55:46.360Z
2023-11-24T13:55:46.360Z
chsk
<p>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 <span class="math-container">$Y = \{ y \in \mathbb{R}^L : F(y) \le 0\}$</span> for a suitable transformation function <span class="math-container">$F$</span>, the PMP is, for a given price vector <span class="math-container">$p$</span>, to maximize <span class="math-container">$p'y$</span> s.t. <span class="math-container">$y \in Y$</span>, i.e. <span class="math-container">$F(y) \le 0$</span>.</p>
<p>On p. 136, the authors give the necessary FOCs for the case of a differentiable <span class="math-container">$F$</span> as <span class="math-container">$p = \lambda \nabla F(y^*)$</span> ((5.C.1); here, <span class="math-container">$y^* \in Y$</span> is a profit-maximizing production plan, and <span class="math-container">$\lambda$</span> is the KKT multiplier of the constraint).</p>
<p>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 <span class="math-container">$F$</span> is differentiable)?</p>

<microeconomics><optimization><profit-maximization>
FOCs for profit maximization using a transformation function
chsk
# FOCs for profit maximization using a transformation function

(Asked by: username_0 on 2023-11-24)

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)?

**Tags:** microeconomics, optimization, profit-maximization


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## 1 answers

### Answer by username_0 on 2023-11-24. Score: 0

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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