# Dector-An Example of the Use of the Lagrangian Multiplier Method

May 13th, 2015
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Let Q=output, L=labor input and K=capital input where Q = L2/3K1/3. The cost of resources used is C=wL+rK, where w .is the wage rate and r is the rental rate for capital

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An Example of the Use ofthe Lagrangian MultiplierMethodto Solve a ConstrainedMaximization ProblemLet Q=output, L=labor input and K=capital input where Q= L2/3K1/3. The cost of resources used is C=wL+rK, where w.is the wage rate and r is the rental rate for capitalProblem: Find the combination of L and K that maximizesoutput subject to the constraint that the cost of resourcesused is C; i.e., maximize Q with respect to L and K subject to.the constraint that vL+rK=CNote that maximizing a monotonically increasing function ofa variable is equivalent to maximizing the variable itself.Therefore ln(Q)=(2/3)ln(L)+(1/3)ln(K), a more convenientexpression, is the same as maximizing Q. Therefore theobjective function for the optimization problem is.(ln(Q)=(2/3)ln(L)+(1/3)ln(KStep 1: Form the Langrangian function by subtracting from the objective function a multiple ofthe difference between the cost of the resources and,.the budget allowed for resources; i.e(G= ln(Q) - (wL+rK-C(G= (2/3)ln(L) + (1/3)ln(K) - (wL+rK-Cwhere is called the Lagrangian multiplier. In effect,this method imposes a penalty upon any proposedsolution that is proportional to the e

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