// Numbas version: finer_feedback_settings
{"name": "SG_Elasticiteiten", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "SG_Elasticiteiten", "tags": [], "metadata": {"description": "<p>Bereken elasticiteiten horend bij een gegeven Stone-Geary nutsfunctie.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<h3>Een individu heeft als nutsfunctie \\[ U(q_1,q_2)={\\var{c1}} {\\ } (q_1-{\\var{a1}})^{\\frac{1}{\\var{n1}}} {\\ }&nbsp;&nbsp; (q_2-{\\var{a2}})^&nbsp;&nbsp; {\\frac{\\var{n1min1}}{\\var{n1}}}\\]waarbij $q_1$ en $q_2$ de hoeveelheden voorstellen van respectievelijk de producten 1 en 2. <br/>Beschouw verder het punt $P^*$ \\[ (p_1 , p_2 , y) = ( \\var{gegp1} , \\var{gegp2} , \\var{gegy} ) .\\]</h3>", "advice": "<p><strong>Stap 1: Formuleer het maximalisatieprobleem.</strong><br/>Noteer voor de&nbsp; 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). <br/>\\begin{eqnarray}<br/>\\max_{q_1,q_2} U(q_1,q_2) &amp;=&amp; {\\var{c1}} \\cdot (q_1-{\\var{a1}})^{1 / {\\var{n1}}}&nbsp; \\cdot (q_2-{\\var{a2}})^{{\\var{n1min1}} / {\\var{n1}}}<br/>\\mbox{als} &amp;&amp;&nbsp; p_1 \\cdot q_1 + p_2 \\cdot q_2 = y<br/>\\end{eqnarray}<br/>wat equivalent is met<br/>\\begin{eqnarray}<br/>\\max_{q_1,q_2} ln U(q_1,q_2) &amp;=&amp; ln({\\var{c1}}) + {\\frac{1}{\\var{n1}}}&nbsp;\\cdot ln(q_1-{\\var{a1}}) + {\\frac{\\var{n1min1}}{\\var{n1}}}&nbsp;\\cdot ln(q_2-{\\var{a2}}) \\\\<br/>\\mbox{als} &amp;&amp;&nbsp;&nbsp; p_1&nbsp;\\cdot q_1 + p_2\\cdot q_2 = y<br/>\\end{eqnarray}</p>\n<p><br/><strong>Stap 2: Bepaal de Lagrangefunctie.</strong><br/>\\begin{eqnarray}<br/>L(q_1, q_2, \\lambda) &amp;=&amp; ln U(q_1,q_2) + \\lambda \\cdot (y - p_1 \\cdot q_1 - p_2 \\cdot&nbsp; q_2) <br/>&amp;=&amp; ln({\\var{c1}}) + {\\frac{1}{\\var{n1}}} \\cdot&nbsp; ln(q_1-{\\var{a1}}) + {\\frac{\\var{n1min1}}{\\var{n1}}} \\cdot&nbsp;&nbsp; ln(q_2-{\\var{a2}}) + \\lambda \\cdot ( y - p_1 \\cdot&nbsp; q_1 - p_2\\cdot&nbsp; q_2 )<br/>\\end{eqnarray}</p>\n<p><strong>Stap 3: Schrijf de eerste orde voorwaarden neer om de kritische punten van deze Lagrangefunctie te berekenen.</strong><br/>\\begin{eqnarray}<br/>\\mbox{(1) } \\quad \\frac{\\partial {L}}{\\partial q_1} &amp; = &amp; {\\frac{1}{\\var{n1}}} \\cdot&nbsp; {\\frac {1} {q_1-{\\var{a1}}} } - \\lambda \\cdot p_1 = 0 \\\\<br/>\\mbox{(2) } \\quad \\frac{\\partial {L}}{\\partial q_2} &amp; = &amp; {\\frac{\\var{n1min1}}{\\var{n1}}} \\cdot {\\frac {1} {q_2-{\\var{a2}}}}&nbsp; -&nbsp; \\lambda \\cdot p_2 = 0 \\\\<br/>\\mbox{(3) } \\quad \\frac{\\partial {L}}{\\partial \\lambda} &amp; = &amp; y - p_1 \\cdot q_1 - p_2 \\cdot q_2 = 0<br/>\\end{eqnarray}</p>\n<p><strong>Stap 4: Los dit stelsel vergelijkingen op naar $q_1$, $q_2$ en $\\lambda$.</strong><br/>Het oplossen van $\\lambda$ uit de vergelijkingen (1) en (2) levert na gelijkstelling:<br/>\\begin{eqnarray}<br/>{\\frac{1}{\\var{n1}}}\\cdot{\\frac {1} {p_1 \\cdot(q_1-{\\var{a1}})} }&amp; = &amp; {\\frac{\\var{n1min1}}{\\var{n1}}}\\cdot{\\frac {1} {p_2 \\cdot(q_2-{\\var{a2}})}}\\\\ \\\\<br/>{p_2} \\cdot {(q_2-{\\var{a2}})} &amp; = &amp; {\\var{n1min1}} \\cdot {p_1} \\cdot {(q_1-{\\var{a1}})} \\\\ \\\\<br/>\\mbox{(4)} \\quad \\quad \\quad {p_2} \\cdot {q_2} &amp; = &amp;&nbsp;&nbsp; {\\var{a2}} \\cdot&nbsp; {p_2} +&nbsp; {\\var{n1min1}} \\cdot {p_1} \\cdot {(q_1-{\\var{a1}})}&nbsp; &nbsp;<br/>\\end{eqnarray}<br/>d.w.z.<br/>\\begin{eqnarray}<br/>{q_2} &amp; = &amp; {\\var{a2}} + {\\frac {1} {p_2}} \\cdot {\\var{n1min1}} \\cdot {p_1} \\cdot {(q_1-{\\var{a1}})}<br/>\\end{eqnarray}<br/>Substitueer (4) in de budgetrestrictie:<br/>\\begin{eqnarray}<br/>{p_1} \\cdot {q_1} + {\\var{a2}} \\cdot&nbsp; {p_2} +&nbsp; {\\var{n1min1}} \\cdot {p_1}\\cdot {(q_1-{\\var{a1}})} &amp; = &amp; y\\\\<br/>{p_1} \\cdot {q_1} + {\\var{a2}} \\cdot&nbsp; {p_2} +&nbsp; {\\var{n1min1}} \\cdot {p_1} \\cdot{q_1} -&nbsp; {\\var{n1min1}} \\cdot {\\var{a1}} \\cdot {p_1} &amp; = &amp; y\\\\<br/>{\\var{n1}} \\cdot {p_1}\\cdot {q_1}&nbsp; &amp; = &amp; y&nbsp; + {\\var{n1min1maala1}} \\cdot {p_1} - {\\var{a2}} \\cdot&nbsp; {p_2} \\\\<br/>{q_1}&nbsp; &amp; = &amp; {\\frac {1}{\\var{n1} \\cdot {p_1}}} \\cdot { ( y&nbsp; + {\\var{n1min1maala1}} \\cdot {p_1} - {\\var{a2}} \\cdot&nbsp; {p_2} ) } \\\\<br/>{q_1}&nbsp; &amp; = &amp; {\\frac {y&nbsp; + {\\var{n1min1maala1}} \\cdot {p_1} - {\\var{a2}} \\cdot&nbsp; {p_2}}{\\var{n1} \\cdot {p_1}}}<br/>\\end{eqnarray}<br/>Substitutie hiervan in bovenstaande uitdrukking voor $q_2$ laat toe om deze&nbsp; $q_2$ te bepalen:<br/>\\begin{eqnarray}<br/>{q_2} &amp; = &amp; \\frac{\\var{t2}\\cdot y-\\var{a1maalt2} \\cdot {p_1} + \\var{a2}\\cdot{p_2}}{\\var{n1} \\cdot {p_2}}<br/>\\end{eqnarray}<br/>De vraagfuncties zijn dus<br/>\\begin{eqnarray}<br/>{q_1}&nbsp; &amp; = &amp; {\\frac {y&nbsp; + {\\var{n1min1maala1}} &nbsp;\\cdot {p_1} - {\\var{a2}} \\cdot {p_2}}{\\var{n1} \\cdot {p_1}}}\\\\<br/>{q_2} &amp; = &amp; \\frac{\\var{t2}\\, y-\\var{a1maalt2}\\cdot{p_1} + \\var{a2}\\cdot {p_2}}{\\var{n1} \\cdot {p_2}}<br/>\\end{eqnarray}</p>\n<p><strong>Stap 5: Bereken de elasticiteiten.</strong><br/>\\begin{eqnarray}<br/>{\\varepsilon}^{q_1}_{p_1} &amp; = &amp; {\\frac {{\\partial}{q_1}} {{\\partial} {p_1}}} \\cdot {\\frac {p_1} {q_1}} = {\\frac {1} {{\\var{n1}} }} \\cdot&nbsp;&nbsp;&nbsp; {\\frac {{\\partial}} {{\\partial} {p_1}}}&nbsp; { ( {\\var{n1min1maala1}}&nbsp; + {\\frac&nbsp; {y - {\\var{a2}}\\cdot {p_2}} {p_1}}&nbsp;&nbsp; ) } \\cdot {\\frac {p_1} {q_1}} =&nbsp; - {\\frac {y - {\\var{a2}} \\cdot {p_2}} {{\\var{n1}} \\cdot {p_1} \\cdot {q_1} } } \\quad, {\\mbox{na evaluatie in }}P^*: \\var{epsq1p1} \\\\<br/><br/>{\\varepsilon}^{q_1}_{p_2} &amp; = &amp; {\\frac {{\\partial}{q_1}} {{\\partial} {p_2}}} \\cdot {\\frac {p_2} {q_1}} = - {\\frac {\\var{a2}} {{\\var{n1}} \\cdot {p_1}} } \\cdot {\\frac {p_2} {q_1}}&nbsp; \\quad, {\\ } \\mbox{na evaluatie in }&nbsp;P^*: \\var{epsq1p2} \\\\<br/><br/>{\\varepsilon}^{q_2}_{p_1} &amp; = &amp; {\\frac {{\\partial}{q_2}} {{\\partial} {p_1}}} \\cdot {\\frac {p_1} {q_2}} = - {\\frac {\\var{n1min1maala1}} {{\\var{n1}} \\cdot {p_2}} } \\cdot {\\frac {p_1} {q_2}}&nbsp;\\quad , \\mbox{na evaluatie in }P^*: \\var{epsq2p1} \\\\<br/><br/>{\\varepsilon}^{q_2}_{p_2} &amp; = &amp; {\\frac {{\\partial}{q_2}} {{\\partial} {p_2}}} \\cdot {\\frac {p_2} {q_2}} = {\\frac {1} {{\\var{n1}} }} \\cdot&nbsp;&nbsp;&nbsp; {\\frac {{\\partial}} {{\\partial} {p_2}}}&nbsp; { (&nbsp; {\\var{a2}} + {\\frac&nbsp; {{\\var{n1min1}} \\cdot y - {\\var{n1min1maala1}} \\cdot {p_1}} {p_2}}&nbsp;&nbsp; ) } \\cdot {\\frac {p_2} {q_2}} =&nbsp; - {\\frac {{\\var{n1min1}} \\cdot y - {\\var{n1min1maala1}} \\cdot {p_1}} {{\\var{n1}} \\cdot {p_2} \\cdot {q_2} } }&nbsp;\\quad , \\mbox{na evaluatie in } P^*: \\var{epsq2p2} \\\\<br/><br/>{\\varepsilon}^{q_1}_{y} &amp; = &amp; {\\frac {{\\partial}{q_1}} {{\\partial} {y}}} \\cdot {\\frac {y} {q_1}} =&nbsp; {\\frac {1} {{\\var{n1}} \\cdot {p_1}} } \\cdot {\\frac {y} {q_1}}&nbsp; , \\mbox{na evaluatie in }P^*: \\var{epsq1y}\\\\<br/><br/>{\\varepsilon}^{q_2}_{y} &amp; = &amp; {\\frac {{\\partial}{q_2}} {{\\partial} {y}}} \\cdot {\\frac {y} {q_2}} =&nbsp; {\\frac {1} {{\\var{n1}}}} \\cdot {\\frac {1} {{\\var{n1min1}} \\cdot {p_2}} } \\cdot {\\frac {y} {q_2}} \\quad , \\mbox{na evaluatie in } P^*: \\var{epsq2y}\\\\<br/><br/>\\end{eqnarray}</p>\n<p><strong>Antwoorden:</strong></p>\n<p>\\[ {\\varepsilon}^{q_1}_{p_1}&nbsp;{( \\var{gegp1} , \\var{gegp2} , \\var{gegy} )} \\approx \\var{benepsq1p1} \\]<br/>\\[ {\\varepsilon}^{q_2}_{p_2}&nbsp;{( \\var{gegp1} , \\var{gegp2} , \\var{gegy} )} \\approx \\var{benepsq2p2} \\]<br/>\\[ {\\varepsilon}^{q_1}_{y}&nbsp;{( \\var{gegp1} , \\var{gegp2} , \\var{gegy} )} \\approx \\var{benepsq1y} 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