// Numbas version: finer_feedback_settings {"name": "SG_Vraagfuncties", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "SG_Vraagfuncties", "tags": [], "metadata": {"description": "

Bepaal vraagfuncties vanaf een Stone-Geary nutsfunctie

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Een individu heeft als nutsfunctie \\[ U(q_1,q_2)={\\var{c1}} \\cdot (q_1-{\\var{a1}})^{{1}/{\\var{n1}}} \\cdot (q_2-{\\var{a2}})^{{\\var{n1min1}}/{\\var{n1}}}  \\] waarbij $q_1$ en $q_2$ de hoeveelheden voorstellen van respectievelijk de producten 1 en 2.
Bepaal de bijhorende vraagfuncties.

\n

Let op: als je een index gebruikt zoals bij $p_1$ dien je dit te noteren via een underscore als p_1   !

", "advice": "

Stap 1: Formuleer het maximalisatieprobleem.
Noteer voor de  prijs per eenheid van deze producten resp. $p_1 $ en $p_2 $, met als budget $ y $ (dit budget wordt volledig gespendeerd aan deze beide producten).
\\begin{eqnarray}
\\max_{q_1,q_2} U(q_1,q_2) &=& {\\var{c1}} \\cdot (q_1-{\\var{a1}})^{1 / {\\var{n1}}}  \\cdot (q_2-{\\var{a2}})^{{\\var{n1min1}} / {\\var{n1}}}
\\mbox{als} &&  p_1 \\cdot q_1 + p_2 \\cdot q_2 = y
\\end{eqnarray}
wat equivalent is met
\\begin{eqnarray}
\\max_{q_1,q_2} \\ln U(q_1,q_2) &=& \\ln({\\var{c1}}) + {\\frac{1}{\\var{n1}}} \\cdot \\ln(q_1-{\\var{a1}}) + {\\frac{\\var{n1min1}}{\\var{n1}}} \\cdot \\ln(q_2-{\\var{a2}}) \\\\
\\mbox{als} &&   p_1 \\cdot q_1 + p_2\\cdot q_2 = y
\\end{eqnarray}

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Stap 2: Bepaal de Lagrangefunctie.
\\begin{eqnarray}
L(q_1, q_2, \\lambda) &=& \\ln U(q_1,q_2) + \\lambda \\cdot (y - p_1 \\cdot q_1 - p_2 \\cdot  q_2)
&=& \\ln({\\var{c1}}) + {\\frac{1}{\\var{n1}}} \\cdot  \\ln(q_1-{\\var{a1}}) + {\\frac{\\var{n1min1}}{\\var{n1}}} \\cdot   \\ln(q_2-{\\var{a2}}) + \\lambda \\cdot ( y - p_1 \\cdot  q_1 - p_2\\cdot  q_2 )
\\end{eqnarray}

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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 q_1} & = & {\\frac{1}{\\var{n1}}} \\cdot  {\\frac {1} {q_1-{\\var{a1}}} } - \\lambda \\cdot p_1 = 0 \\\\
\\mbox{(2) } \\quad \\frac{\\partial {L}}{\\partial q_2} & = & {\\frac{\\var{n1min1}}{\\var{n1}}} \\cdot {\\frac {1} {q_2-{\\var{a2}}}}  -  \\lambda \\cdot p_2 = 0 \\\\
\\mbox{(3) } \\quad \\frac{\\partial {L}}{\\partial \\lambda} & = & y - p_1 \\cdot q_1 - p_2 \\cdot q_2 = 0
\\end{eqnarray}

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Stap 4: Los dit stelsel vergelijkingen op naar $q_1$, $q_2$ en $\\lambda$.
Het oplossen van $\\lambda$ uit de vergelijkingen (1) en (2) levert na gelijkstelling:
\\begin{eqnarray}
{\\frac{1}{\\var{n1}}}\\cdot{\\frac {1} {p_1 \\cdot(q_1-{\\var{a1}})} }& = & {\\frac{\\var{n1min1}}{\\var{n1}}}\\cdot{\\frac {1} {p_2 \\cdot(q_2-{\\var{a2}})}}\\\\ \\\\
{p_2} \\cdot {(q_2-{\\var{a2}})} & = & {\\var{n1min1}} \\cdot {p_1} \\cdot {(q_1-{\\var{a1}})} \\\\ \\\\
\\mbox{(4)} \\quad \\quad \\quad {p_2} \\cdot {q_2} & = &   {\\var{a2}} \\cdot  {p_2} +  {\\var{n1min1}} \\cdot {p_1} \\cdot {(q_1-{\\var{a1}})}   
\\end{eqnarray}
d.w.z.
\\begin{eqnarray}
{q_2} & = & {\\var{a2}} + {\\frac {1} {p_2}} \\cdot {\\var{n1min1}} \\cdot {p_1} \\cdot {(q_1-{\\var{a1}})}
\\end{eqnarray}
Substitueer (4) in de budgetrestrictie:
\\begin{eqnarray}
{p_1} \\cdot {q_1} + {\\var{a2}} \\cdot  {p_2} +  {\\var{n1min1}} \\cdot {p_1}\\cdot {(q_1-{\\var{a1}})} & = & y\\\\
{p_1} \\cdot {q_1} + {\\var{a2}} \\cdot  {p_2} +  {\\var{n1min1}} \\cdot {p_1} \\cdot{q_1} -  {\\var{n1min1}} \\cdot {\\var{a1}} \\cdot {p_1} & = & y\\\\
{\\var{n1}} \\cdot {p_1}\\cdot {q_1}  & = & y  + {\\var{n1min1maala1}} \\cdot {p_1} - {\\var{a2}} \\cdot  {p_2} \\\\
{q_1}  & = & {\\frac {1}{\\var{n1} \\cdot {p_1}}} \\cdot { ( y  + {\\var{n1min1maala1}} \\cdot {p_1} - {\\var{a2}} \\cdot  {p_2} ) } \\\\
{q_1}  & = & {\\frac {y  + {\\var{n1min1maala1}} \\cdot {p_1} - {\\var{a2}} \\cdot  {p_2}}{\\var{n1} \\cdot {p_1}}}
\\end{eqnarray}
Substitutie hiervan in bovenstaande uitdrukking voor $q_2$ laat toe om deze  $q_2$ te bepalen:
\\begin{eqnarray}
{q_2} & = & \\frac{\\var{t2}\\cdot y-\\var{a1maalt2} \\cdot {p_1} + \\var{a2}\\cdot{p_2}}{\\var{n1} \\cdot {p_2}}
\\end{eqnarray}

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Antwoorden
De vraagfuncties zijn dus
\\begin{eqnarray}
{q_1}  & = & {\\frac {y  + {\\var{n1min1maala1}}  \\cdot {p_1} - {\\var{a2}} \\cdot {p_2}}{\\var{n1} \\cdot {p_1}}}\\\\
{q_2} & = & \\frac{\\var{t2}\\cdot y-\\var{a1maalt2}\\cdot{p_1} + \\var{a2}\\cdot {p_2}}{\\var{n1} \\cdot {p_2}}
\\end{eqnarray}

\n

 

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De vraagfunctie voor goed 1 wordt gegeven door $q_1=$   [[0]]

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De vraagfunctie voor goed 2 wordt gegeven door $q_2=$   [[0]]

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