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Why can the Lagrangian Multiplier be dropped in the inverse demand function?

The Lagrangian is given by: $$ L = x_0 + \frac{1}{\mu} \sum_{j = 1}^J X_j^\mu - \lambda(x_0 + \sum_j \int_i p_j(i) x_j(i) di - m). $$ The first order condition (for an interior solution) with respect ...
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Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization

Let $\mathbf x \in \mathbb{R^n} \quad \mathbf x= (x_1 , x_2 , \cdots , x_n ) $ $A$, $B$ $m$x$n$ matrix, $\mathbf b \in \mathbb{R^m}$, $\mathbf c \in \mathbb{R^m}$. Let indicate with $< . , . >$ ...
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Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization

The Lagrange multiplier provides the "shadow price" of the constraint. In the case where the Lagrangian takes the form: $$ L = f(x,y) - \lambda(g(x,y) - c), $$ then $\lambda$ measures the ...
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