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Mathematical explanation of the intuition behind the Lagrange multiplier

From the f.o.c. for utility maximization under a budget constraint, we have $$\frac{\partial U}{\partial x} = \lambda p_x,\;\;\;\frac{\partial U}{\partial y} = \lambda p_y$$ that holds at the optimal ...
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Stationarity in KKT condition

In general, you don't know before you find them, which, if any, of the minimizers $\ \hat{x}\ $ will satisfy the condition $\ G(\hat{x})<0\ $, and if $\ \hat{x}\ $ is a local minimizer for which $\ ...
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