Minimaliseer de kosten bij een gegeven outputvolume
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Stap 1: Formuleer het minimalisatieprobleem.
\\begin{eqnarray}
\\min_{A,K} C(A,K) &=& w_A \\cdot A + w_K \\cdot K\\\\
&=& \\var{wA} \\cdot A + \\var{wK} \\cdot K\\\\
\\mbox{als} && \\phi(A,K) = q \\\\
\\mbox{d.w.z. als} && \\var{c} \\cdot A^{\\var{alpha}}\\cdot K^{\\var{beta}} = \\var{geg}
\\end{eqnarray}
Stap 2: Bepaal de Lagrangefunctie.
\\begin{eqnarray}
L(A, K, \\lambda) &=& C(A, K)+\\lambda (q - \\phi(A,K)) \\\\
&=& \\var{wA} \\cdot A + \\var{wK} \\cdot K + \\lambda (\\var{geg} - \\var{c} \\cdot A^{\\var{alpha}}\\cdot K^{\\var{beta}})
\\end{eqnarray}
Stap 3: Schrijf de eerste orde voorwaarden neer om de kritische punten van deze Lagrangefunctie te berekenen.
\\begin{eqnarray}
\\mbox{(1) }\\quad \\frac{\\partial {{L}}}{\\partial A} & = & \\var{wA} - \\lambda \\cdot \\var{c} \\cdot \\var{alpha} \\cdot A^{\\var{alpha-1}}\\cdot K^{\\var{beta}} = 0\\\\
\\mbox{(2) }\\quad \\frac{\\partial {{L}}}{\\partial K} & = & \\var{wK} - \\lambda \\cdot \\var{c} \\cdot \\var{beta} \\cdot A^{\\var{alpha}}\\cdot K^{\\var{beta-1}}= 0 \\\\
\\mbox{(3) }\\quad \\frac{\\partial {{L}}}{\\partial \\lambda} & = & \\var{geg} - \\var{c} \\cdot A^{\\var{alpha}}\\cdot K^{\\var{beta}} = 0
\\end{eqnarray}
Stap 4: Los dit stelsel vergelijkingen op naar $A$, $K$, en $\\lambda$.
Het oplossen van $\\lambda$ uit de vergelijkingen (1) en (2) levert na gelijkstelling:
\\begin{eqnarray}
\\frac{\\var{wA}}{\\var{c} \\cdot \\var{alpha} \\cdot A^{\\var{alpha-1}}\\cdot K^{\\var{beta}}} & = & \\frac{\\var{wK}}{\\var{c} \\cdot \\var{beta} \\cdot A^{\\var{alpha}}\\cdot K^{\\var{beta-1}}}\\\\
K & = & \\frac{\\var{wA*beta/gcd(wA*beta,wK*alpha)}}{\\var{wK*alpha/gcd(wA*beta,wK*alpha)}} \\cdot A
\\end{eqnarray}
Substitutie hiervan in (3) geeft de optimale waarde van $A$:
\\[ A^* = \\var{Aopl} ,\\]
waardoor
\\[ K^* = \\var{Kopl} .\\]
De minimale kosten bedragen dan \\[ \\var{wA} \\cdot \\var{Aopl} + \\var{wK} \\cdot \\var{Kopl} = \\var{wA*Aopl+wK*Kopl} \\]
\n{toonfiguur()}
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