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In the proof of Shepherd's Lemma (production and supply), we have the following step:

$$\frac{\partial \mathcal{L}}{\partial w} = \frac{\partial \ [wl + rk + \lambda(q - f(l,k))]}{\partial w} = l^c(w,r,q)$$

The other variables $l, k$ are also dependent on $w$, why do we treat them as constants here instead of calculating it as $\frac{\partial L}{\partial w} = w \frac{\partial l^c}{\partial w} + l^c + r \frac{\partial k^c}{\partial w}$?

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In the Lagrangian function $\mathcal{L}(w,r,q,l,k,\lambda)$, the arguments $l$ and $k$ are independent variables, so they do not depend on $w$.

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