In my own studies I have noticed that many of the constrained optimization type questions that are often covered in intermediate/advanced microeconomics courses are seldom seen in more empirical types of work.

under what circumstances would I apply the use of the Legrangian multiplier to an real world problem?

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    $\begingroup$ Well there is the Lagrange multiplier test in econometrics. $\endgroup$ – Herr K. Apr 4 '18 at 16:27
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    $\begingroup$ I had put this as an answer, but came back and noticed that you had put the microeconomics tag. One place where multipliers are used in empirical work is Macroeconomics. Modern macroeconomics papers usually contain a fairly sophisticated theoretical model along with empirical analysis. Think of modern DSGE and New Keynesian models which have numerous multipliers (shadow variables). The parameters are then ``calibrated" and the model estimated. See e.g. eml.berkeley.edu/~ygorodni/CGH_store_switching.pdf $\endgroup$ – user11305 Apr 4 '18 at 21:15

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