п»їAn Sort of the Use of
the Lagrangian Multiplier MethodВ
to resolve a Restricted Maximization ProblemВ
Let Q=output, L=labor suggestions and K=capital input exactly where Q sama dengan L2/3K1/3. The cost of resources utilized is C=wL+rK, where watts is the wage rate and r may be the rental charge for capital. Problem: Get the combination of L and K that maximizes output subject to the constraint the cost of resources used can be C; we. e., take full advantage of Q with respect to L and K be subject to the constraint that vL+rK=C. Note that making the most of a monotonically increasing function of a variable is equivalent to increasing the changing itself. As a result ln(Q)=(2/3)ln(L)+(1/3)ln(K), a much more convenient manifestation, is the same as making the most of Q. Therefore the objective function for the optimization issue is ln(Q)=(2/3)ln(L)+(1/3)ln(K).
Step 1 : Form the Langrangian function by subtracting through the objective function a multiple of the big difference between the expense of the resources as well as the budget allowed for resources; my spouse and i. e.,
G= ln(Q) - О»(wL+rK-C)
G= (2/3)ln(L) & (1/3)ln(K) -- О»(wL+rK-C)В
in which О» is called the Lagrangian multiplier. Essentially, this method imposes a penalty after any proposed solution that may be proportional for the extent to which the constraint is violated. By choosing the of proportionality large enough the perfect solution can be forced into compliance with the limitation.
Step 2: Locate the unconstrained maximum ofВ GВ with respect to L and K for a fixed value of О» by finding the values of L and K such that the incomplete derivatives ofВ GВ are equal to actually zero.
в€‚G/в€‚L sama dengan (2/3)(1/L) -- О»w = 0В
в€‚G/в€‚K = (1/3)(1/K) - О»r = 0В
Step 3: Solve for the optimal L and K while function of О»; i actually. e.,
(2/3)(1/L)= О»w thus L = (2/3)/(О»w)В
(1/3)(1/K)= О»r therefore K = (1/3)/(О»r)В
Step four: Find a value of О» such that the constraint is satisfied. This is achieved by substituting the expressions to get L and K with regards to О» into the constraint and solving pertaining to О».
wL + rK = (2/3)(1/О») + (1/3)(1/О») =
1/О» = C so О»...