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(Without using differentiation) When $w \leq w'$ it follows that $pf(x) − w · x \geq pf(x) − w' · x$ and so $\pi(p,w) \geq \pi(p,w')$. EDIT: If the output price $p$ is endogenous, as it seems to be the case in your question, the issue has be treated by: Heiner, R. A. (1982): “Theory of the Firm in “Short-Run” Industry Equilibrium,” American Economic Review, ...


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From FOC, we know that: \begin{align} \nabla_x\pi(\mathbf{x},\mathbf{w})=p\nabla f(\mathbf{x})-\mathbf{w}=\mathbf{0} \tag{1} \end{align} This will be true at equilibrium, i.e. for any given $\mathbf{w}$, the input vector $\mathbf{x}$ will adjust so that the above holds. Now consider $d\pi(\mathbf{x},\mathbf{w})/d w_i$ (and using $(1)$): \begin{align} \frac{d\...


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