San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

The Lagrangian Multiplier Method

An Example of the Use of
the Lagrangian Multiplier Method
to Solve a Constrained Maximization Problem

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.

Problem: Find the combination of L and K that maximizes output subject to the constraint that the cost of resources used is C; i.e., maximize Q with respect to L and K subject to the constraint that vL+rK=C.

Note that maximizing a monotonically increasing function of a variable is equivalent to maximizing the variable itself. Therefore ln(Q)=(2/3)ln(L)+(1/3)ln(K), a more convenient expression, is the same as maximizing Q. Therefore the objective function for the optimization problem is ln(Q)=(2/3)ln(L)+(1/3)ln(K).

For second order conditions for
a maximum or minimum visit

2nd Order Conditions


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