qu.1.topic=Applications@

qu.1.1.mode=Inline@
qu.1.1.name=2 Variables Constrained Optimization - CD cost minimization no steps@
qu.1.1.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $q-$Pk*K-$Pl*L + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$Cpretty].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>K</mi></mrow></mfrac></mrow></mstyle></math>: $derKpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>L</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $dermpretty=0</p>
<p>Optimal K and L are:</p>
<p>X=$K</p>
<p>Y=$L</p>@
qu.1.1.editing=useHTML@
qu.1.1.hint.1=The objective function goes in the first part of the Lagrangean. In this case, the objective function is the cost.@
qu.1.1.hint.2=The constraint here is that the firm wants to make a certain amount of output.@
qu.1.1.solution=@
qu.1.1.algorithm=$a=range(2,4);
$b=frac($a-1,$a);
$e=1-1/$a;
$Q=range(5,10);
$Pk=range(2,4);
$Pl=range(2,4);
$q=range(10,20);
$v=maple("
randomize():
C:=K^(1/$a)*L^($b):
Lag:=$Pk*K+$Pl*L+m*($q-C):
v1:=diff(Lag,K):
v2:=diff(Lag,L):
v3:=diff(Lag,m):
v4:=round(((($b*$a)*($Pk/$Pl))^(-$b))*$q*100)/100:
v5:=round(((($b*$a)*($Pk/$Pl))^(1/$a))*$q*100)/100:
convert(v1,string),convert(v2,string),convert(v3,string),C,convert(C,string),convert(v1,string),convert(v2,string),convert(v3,string),v4,v5
");
$derK=switch(0,$v);
$derL=switch(1,$v);
$derm=switch(2,$v);
$Cmath=switch(3,$v);
$Cstring=switch(4,$v);
$derKstring=switch(5,$v);
$derLstring=switch(6,$v);
$dermstring=switch(7,$v);
$Cpretty=mathml("$Cstring");
$derKpretty=mathml("$derKstring");
$derLpretty=mathml("$derLstring");
$dermpretty=mathml("$dermstring");
$K=switch(8,$v);
$L=switch(9,$v);@
qu.1.1.uid=c1d98430-8f46-43d5-96c9-cbe408a370b0@
qu.1.1.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Minimization - Cd;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.1.weighting=5,5@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=0.05@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$K@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.answer.units=@
qu.1.1.part.2.numStyle=   @
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.showUnits=false@
qu.1.1.part.2.err=0.05@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.mode=Numeric@
qu.1.1.part.2.grading=toler_abs@
qu.1.1.part.2.negStyle=both@
qu.1.1.part.2.answer.num=$L@
qu.1.1.question=<p>A firm has the following production function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>K</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>L</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Cpretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>K</mi></mrow></msub></mrow></mstyle></math>=$Pk, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>L</mi></mrow></msub></mrow></mstyle></math>=$Pl. The firm wishes to produce $q units for the least cost possible.</p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi></mrow></mstyle></math> should the firm use?</p><p>(Round answers to the nearest two decimal places. For example, 1.6666 becomes 1.67.)</p><p>K=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>L=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.2.mode=Inline@
qu.1.2.name=Corner Solution - No Steps@
qu.1.2.comment=<p>This question has a corner solution. If the price of capital is low  enough, it is best to put all the firm's labour into producing good Y.  Otherwise, it is best to put all the firm's labour into producing good  X.</p>
<p>In this case, the price of capital is $P. Therefore, it is best to choose:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Lx</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mi>Y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Ly</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$K</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$X</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Y</p>
<p>Which gives a profit of:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Profit</p>@
qu.1.2.editing=useHTML@
qu.1.2.hint.1=Check for a corner solution.@
qu.1.2.hint.2=Check whether profit is higher when only X is produced or when only Y is produced.@
qu.1.2.solution=@
qu.1.2.algorithm=$P=range(1,5);
$v=maple("
if $P < 3 then k := 0 
elif $P > 2 then k := 1
end if:
k
");
$X = switch($v, 0, 10);
$Y = switch($v, 50/$P, 0);
$Lx = switch($v, 0, 10);
$Ly = switch($v, 10, 0);
$K = switch($v, ((50/$P)^2)/100, 0);
$Profit = switch($v, 25/$P, 10);@
qu.1.2.uid=235d830b-d10d-41b5-8456-e7de3ee271ee@
qu.1.2.info=  Course=Introductory Mathematical Economics;
  Topic=Constrained Optimization;
  Sub-Topic=Corner Solutions;
  Author=Katherine Dare;
  Difficulty=Very Hard;
@
qu.1.2.weighting=1,1,1,1,1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.answer.units=@
qu.1.2.part.1.numStyle=   @
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.showUnits=false@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.mode=Numeric@
qu.1.2.part.1.grading=exact_value@
qu.1.2.part.1.negStyle=both@
qu.1.2.part.1.answer.num=$Lx@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.answer.units=@
qu.1.2.part.2.numStyle=   @
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.showUnits=false@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.mode=Numeric@
qu.1.2.part.2.grading=exact_value@
qu.1.2.part.2.negStyle=both@
qu.1.2.part.2.answer.num=$Ly@
qu.1.2.part.3.name=sro_id_3@
qu.1.2.part.3.answer.units=@
qu.1.2.part.3.numStyle=   @
qu.1.2.part.3.editing=useHTML@
qu.1.2.part.3.showUnits=false@
qu.1.2.part.3.question=(Unset)@
qu.1.2.part.3.mode=Numeric@
qu.1.2.part.3.grading=exact_value@
qu.1.2.part.3.negStyle=both@
qu.1.2.part.3.answer.num=$K@
qu.1.2.part.4.name=sro_id_4@
qu.1.2.part.4.answer.units=@
qu.1.2.part.4.numStyle=   @
qu.1.2.part.4.editing=useHTML@
qu.1.2.part.4.showUnits=false@
qu.1.2.part.4.question=(Unset)@
qu.1.2.part.4.mode=Numeric@
qu.1.2.part.4.grading=exact_value@
qu.1.2.part.4.negStyle=both@
qu.1.2.part.4.answer.num=$X@
qu.1.2.part.5.name=sro_id_5@
qu.1.2.part.5.answer.units=@
qu.1.2.part.5.numStyle=   @
qu.1.2.part.5.editing=useHTML@
qu.1.2.part.5.showUnits=false@
qu.1.2.part.5.question=(Unset)@
qu.1.2.part.5.mode=Numeric@
qu.1.2.part.5.grading=exact_value@
qu.1.2.part.5.negStyle=both@
qu.1.2.part.5.answer.num=$Y@
qu.1.2.part.6.name=sro_id_6@
qu.1.2.part.6.answer.units=@
qu.1.2.part.6.numStyle=  dollars @
qu.1.2.part.6.editing=useHTML@
qu.1.2.part.6.showUnits=false@
qu.1.2.part.6.question=(Unset)@
qu.1.2.part.6.mode=Numeric@
qu.1.2.part.6.grading=exact_value@
qu.1.2.part.6.negStyle=both@
qu.1.2.part.6.answer.num=$Profit@
qu.1.2.question=<div align="left"><p>A firm produces and sells two goods:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Both goods sell for a price of $1 each.</p><p>The production function for good&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>L</mi><mrow><mi>X</mi></mrow></msub></mrow></mstyle></math></p><p>The production function for good&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msubsup><mi>L</mi><mrow><mi>Y</mi></mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msubsup><mi>K</mi></mrow></mstyle></math></p><p>The firm divides its labour between the two products:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>L</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>L</mi><mrow><mi>Y</mi></mrow></msub></mrow></mstyle></math> and gets its labour for free, but can only use 10 units of it or less <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><msub><mi>L</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>L</mi><mrow><mi>Y</mi></mrow></msub></mrow></mstyle></math>.</p><p>The price of capital is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>K</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$P</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>How much labour should the firm use to produce good <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><span><span>How much labour should the firm use to produce good <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>?</span></span></span></p><p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mi>Y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p><span>How much capital should the firm use?</span></p><p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p><p><span>How much of good&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> should the firm produce?</span></p><p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p><p>&nbsp;</p><p><span>How much of good&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should the firm produce?</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><5><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span>How much profit does the firm make?</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><6><span>&nbsp;</span></span></p><p>&nbsp;</p></div>@

qu.1.3.mode=Inline@
qu.1.3.name=2 Variables Constrained Optimization@
qu.1.3.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Upretty + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$m-$Px*X-$Py*Y].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p>Optimal X and Y are:</p>
<p>X=$X</p>
<p>Y=$Y</p>@
qu.1.3.editing=useHTML@
qu.1.3.hint.1=To solve for X and Y, use the first two First Order Conditions to find X as a function of Y. Then plug this into the budget constraint.@
qu.1.3.solution=@
qu.1.3.algorithm=$a=range(2,4);
$b=range(2,4);
$X=range(5,10);
$Y=range(5,10);
$m=range(20,40);
$Px=decimal(2,((1/$a)*$m)/((1/$a+1/$b)*$X));
$Py=decimal(2,($m-$Px*$X)/($Y));
$v=maple("
randomize():
U:=RandomTools[Generate](choose({X^(1/$a)*Y^(1/$b), (1/$a)*ln(X)+(1/$b)*ln(Y)})):
Lag:=U+L*($m-$Px * X-$Py * Y):
v1:=diff(Lag,X):
v2:=diff(Lag,Y):
v3:=diff(Lag,L):
convert(v1,string),convert(v2,string),convert(v3,string),convert(U,string),convert(U,string),convert(v1,string),convert(v2,string),convert(v3,string)
");
$derX=switch(0,$v);
$derY=switch(1,$v);
$derL=switch(2,$v);
$Umath=switch(3,$v);
$Ustring=switch(4,$v);
$derXstring=switch(5,$v);
$derYstring=switch(6,$v);
$derLstring=switch(7,$v);
$Upretty=mathml("$Ustring");
$derXpretty=mathml("$derXstring");
$derYpretty=mathml("$derYstring");
$derLpretty=mathml("$derLstring");@
qu.1.3.uid=00436a06-5c58-4aa3-a2ca-460352dcc1c7@
qu.1.3.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Maximization;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Switches Between Cobb-Douglas And Ln Utility;
@
qu.1.3.weighting=1,1,1,1,1,1,1,1,5,5@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.maple_answer=$Umath@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.libname=@
qu.1.3.part.1.mode=Maple@
qu.1.3.part.1.allow2d=1@
qu.1.3.part.1.plot=@
qu.1.3.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.1.type=formula@
qu.1.3.part.2.name=sro_id_2@
qu.1.3.part.2.maple_answer=$m-$Px*X-$Py*Y@
qu.1.3.part.2.editing=useHTML@
qu.1.3.part.2.question=(Unset)@
qu.1.3.part.2.libname=@
qu.1.3.part.2.mode=Maple@
qu.1.3.part.2.allow2d=1@
qu.1.3.part.2.plot=@
qu.1.3.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.2.type=formula@
qu.1.3.part.3.name=sro_id_3@
qu.1.3.part.3.maple_answer=$derX@
qu.1.3.part.3.editing=useHTML@
qu.1.3.part.3.question=(Unset)@
qu.1.3.part.3.libname=@
qu.1.3.part.3.mode=Maple@
qu.1.3.part.3.allow2d=1@
qu.1.3.part.3.plot=@
qu.1.3.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.3.type=formula@
qu.1.3.part.4.name=sro_id_4@
qu.1.3.part.4.answer.units=@
qu.1.3.part.4.numStyle=   @
qu.1.3.part.4.editing=useHTML@
qu.1.3.part.4.showUnits=false@
qu.1.3.part.4.question=(Unset)@
qu.1.3.part.4.mode=Numeric@
qu.1.3.part.4.grading=exact_value@
qu.1.3.part.4.negStyle=both@
qu.1.3.part.4.answer.num=0@
qu.1.3.part.5.name=sro_id_5@
qu.1.3.part.5.maple_answer=$derY@
qu.1.3.part.5.editing=useHTML@
qu.1.3.part.5.question=(Unset)@
qu.1.3.part.5.libname=@
qu.1.3.part.5.mode=Maple@
qu.1.3.part.5.allow2d=1@
qu.1.3.part.5.plot=@
qu.1.3.part.5.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.5.type=formula@
qu.1.3.part.6.name=sro_id_6@
qu.1.3.part.6.answer.units=@
qu.1.3.part.6.numStyle=   @
qu.1.3.part.6.editing=useHTML@
qu.1.3.part.6.showUnits=false@
qu.1.3.part.6.question=(Unset)@
qu.1.3.part.6.mode=Numeric@
qu.1.3.part.6.grading=exact_value@
qu.1.3.part.6.negStyle=both@
qu.1.3.part.6.answer.num=0@
qu.1.3.part.7.name=sro_id_7@
qu.1.3.part.7.maple_answer=$derL@
qu.1.3.part.7.editing=useHTML@
qu.1.3.part.7.question=(Unset)@
qu.1.3.part.7.libname=@
qu.1.3.part.7.mode=Maple@
qu.1.3.part.7.allow2d=1@
qu.1.3.part.7.plot=@
qu.1.3.part.7.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.3.part.7.type=formula@
qu.1.3.part.8.name=sro_id_8@
qu.1.3.part.8.answer.units=@
qu.1.3.part.8.numStyle=   @
qu.1.3.part.8.editing=useHTML@
qu.1.3.part.8.showUnits=false@
qu.1.3.part.8.question=(Unset)@
qu.1.3.part.8.mode=Numeric@
qu.1.3.part.8.grading=exact_value@
qu.1.3.part.8.negStyle=both@
qu.1.3.part.8.answer.num=0@
qu.1.3.part.9.name=sro_id_9@
qu.1.3.part.9.answer.units=@
qu.1.3.part.9.numStyle=   @
qu.1.3.part.9.editing=useHTML@
qu.1.3.part.9.showUnits=false@
qu.1.3.part.9.err=0.05@
qu.1.3.part.9.question=(Unset)@
qu.1.3.part.9.mode=Numeric@
qu.1.3.part.9.grading=toler_abs@
qu.1.3.part.9.negStyle=both@
qu.1.3.part.9.answer.num=$X@
qu.1.3.part.10.name=sro_id_10@
qu.1.3.part.10.answer.units=@
qu.1.3.part.10.numStyle=   @
qu.1.3.part.10.editing=useHTML@
qu.1.3.part.10.showUnits=false@
qu.1.3.part.10.err=0.05@
qu.1.3.part.10.question=(Unset)@
qu.1.3.part.10.mode=Numeric@
qu.1.3.part.10.grading=toler_abs@
qu.1.3.part.10.negStyle=both@
qu.1.3.part.10.answer.num=$Y@
qu.1.3.question=<p>An individual values two goods,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$Px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$Py. The individual has $m dollars.</p><p>&nbsp;</p><p>What is the Lagrangean for this problem?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>=<span>&nbsp;</span><1><span> </span>+<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[<span>&nbsp;</span><2><span>&nbsp;</span>]</p><p>&nbsp;</p><p>What are the first order conditions? Use L for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>.</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><3><span> </span>=<span>&nbsp;</span><4><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><5><span> </span>=<span>&nbsp;</span><6><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mi>&lambda;</mi></mrow></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><7><span> </span>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p>&nbsp;</p><p>How much X and Y should this individual buy?</p><p>(Round answers to the nearest whole number.)</p><p>X=<span>&nbsp;</span><9><span>&nbsp;</span></p><p>Y=<span>&nbsp;</span><10><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.4.mode=Inline@
qu.1.4.name=2 Variables Constrained Optimization - CD without steps@
qu.1.4.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Upretty + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$m-$Px*X-$Py*Y].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p>Optimal X and Y are:</p>
<p>X=$X</p>
<p>Y=$Y</p>@
qu.1.4.editing=useHTML@
qu.1.4.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.1.4.solution=@
qu.1.4.algorithm=$a=range(2,4);
$b=range(2,4);
$X=range(5,10);
$Y=range(5,10);
$m=range(20,40);
$Px=decimal(2,((1/$a)*$m)/((1/$a+1/$b)*$X));
$Py=decimal(2,($m-$Px*$X)/($Y));
$v=maple("
randomize():
U:=X^(1/$a)*Y^(1/$b):
Lag:=U+L*($m-$Px * X-$Py * Y):
v1:=diff(Lag,X):
v2:=diff(Lag,Y):
v3:=diff(Lag,L):
convert(v1,string),convert(v2,string),convert(v3,string),convert(U,string),convert(U,string),convert(v1,string),convert(v2,string),convert(v3,string)
");
$derX=switch(0,$v);
$derY=switch(1,$v);
$derL=switch(2,$v);
$Umath=switch(3,$v);
$Ustring=switch(4,$v);
$derXstring=switch(5,$v);
$derYstring=switch(6,$v);
$derLstring=switch(7,$v);
$Upretty=mathml("$Ustring");
$derXpretty=mathml("$derXstring");
$derYpretty=mathml("$derYstring");
$derLpretty=mathml("$derLstring");@
qu.1.4.uid=6287b7b5-2c92-4f2b-b399-e7bb6cb3c5ad@
qu.1.4.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Maximization - Cd;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.4.weighting=5,5@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.answer.units=@
qu.1.4.part.1.numStyle=   @
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.showUnits=false@
qu.1.4.part.1.err=0.05@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.mode=Numeric@
qu.1.4.part.1.grading=toler_abs@
qu.1.4.part.1.negStyle=both@
qu.1.4.part.1.answer.num=$X@
qu.1.4.part.2.name=sro_id_2@
qu.1.4.part.2.answer.units=@
qu.1.4.part.2.numStyle=   @
qu.1.4.part.2.editing=useHTML@
qu.1.4.part.2.showUnits=false@
qu.1.4.part.2.err=0.05@
qu.1.4.part.2.question=(Unset)@
qu.1.4.part.2.mode=Numeric@
qu.1.4.part.2.grading=toler_abs@
qu.1.4.part.2.negStyle=both@
qu.1.4.part.2.answer.num=$Y@
qu.1.4.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$Px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$Py. The individual has $m dollars.</p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should this individual buy?</p><p>(Round answers to the nearest whole number.)</p><p>X=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>Y=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.5.mode=Inline@
qu.1.5.name=2 Variables Constrained Optimization - ln with steps@
qu.1.5.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Upretty + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$m-$Px*X-$Py*Y].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p>Optimal X and Y are:</p>
<p>X=$X</p>
<p>Y=$Y</p>@
qu.1.5.editing=useHTML@
qu.1.5.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.1.5.solution=@
qu.1.5.algorithm=$a=range(2,4);
$b=range(2,4);
$X=range(5,10);
$Y=range(5,10);
$m=range(20,40);
$Px=decimal(2,((1/$a)*$m)/((1/$a+1/$b)*$X));
$Py=decimal(2,($m-$Px*$X)/($Y));
$v=maple("
randomize():
U:=(1/$a)*ln(X)+(1/$b)*ln(Y):
Lag:=U+L*($m-$Px * X-$Py * Y):
v1:=diff(Lag,X):
v2:=diff(Lag,Y):
v3:=diff(Lag,L):
convert(v1,string),convert(v2,string),convert(v3,string),U,convert(U,string),convert(v1,string),convert(v2,string),convert(v3,string)
");
$derX=switch(0,$v);
$derY=switch(1,$v);
$derL=switch(2,$v);
$Umath=switch(3,$v);
$Ustring=switch(4,$v);
$derXstring=switch(5,$v);
$derYstring=switch(6,$v);
$derLstring=switch(7,$v);
$Upretty=mathml("$Ustring");
$derXpretty=mathml("$derXstring");
$derYpretty=mathml("$derYstring");
$derLpretty=mathml("$derLstring");@
qu.1.5.uid=a351d5f9-bca2-4edc-ba4d-04b854b29b2f@
qu.1.5.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Maximization - Ln;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Walks Students Through Steps;
@
qu.1.5.weighting=1,1,1,1,1,1,1,1,5,5@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.maple_answer=$Umath@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.libname=@
qu.1.5.part.1.mode=Maple@
qu.1.5.part.1.allow2d=1@
qu.1.5.part.1.plot=@
qu.1.5.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.5.part.1.type=formula@
qu.1.5.part.2.name=sro_id_2@
qu.1.5.part.2.maple_answer=$m-$Px*X-$Py*Y@
qu.1.5.part.2.editing=useHTML@
qu.1.5.part.2.question=(Unset)@
qu.1.5.part.2.libname=@
qu.1.5.part.2.mode=Maple@
qu.1.5.part.2.allow2d=1@
qu.1.5.part.2.plot=@
qu.1.5.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.5.part.2.type=formula@
qu.1.5.part.3.name=sro_id_3@
qu.1.5.part.3.maple_answer=$derX@
qu.1.5.part.3.editing=useHTML@
qu.1.5.part.3.question=(Unset)@
qu.1.5.part.3.libname=@
qu.1.5.part.3.mode=Maple@
qu.1.5.part.3.allow2d=1@
qu.1.5.part.3.plot=@
qu.1.5.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.5.part.3.type=formula@
qu.1.5.part.4.name=sro_id_4@
qu.1.5.part.4.answer.units=@
qu.1.5.part.4.numStyle=   @
qu.1.5.part.4.editing=useHTML@
qu.1.5.part.4.showUnits=false@
qu.1.5.part.4.question=(Unset)@
qu.1.5.part.4.mode=Numeric@
qu.1.5.part.4.grading=exact_value@
qu.1.5.part.4.negStyle=both@
qu.1.5.part.4.answer.num=0@
qu.1.5.part.5.name=sro_id_5@
qu.1.5.part.5.maple_answer=$derY@
qu.1.5.part.5.editing=useHTML@
qu.1.5.part.5.question=(Unset)@
qu.1.5.part.5.libname=@
qu.1.5.part.5.mode=Maple@
qu.1.5.part.5.allow2d=1@
qu.1.5.part.5.plot=@
qu.1.5.part.5.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.5.part.5.type=formula@
qu.1.5.part.6.name=sro_id_6@
qu.1.5.part.6.answer.units=@
qu.1.5.part.6.numStyle=   @
qu.1.5.part.6.editing=useHTML@
qu.1.5.part.6.showUnits=false@
qu.1.5.part.6.question=(Unset)@
qu.1.5.part.6.mode=Numeric@
qu.1.5.part.6.grading=exact_value@
qu.1.5.part.6.negStyle=both@
qu.1.5.part.6.answer.num=0@
qu.1.5.part.7.name=sro_id_7@
qu.1.5.part.7.maple_answer=$derL@
qu.1.5.part.7.editing=useHTML@
qu.1.5.part.7.question=(Unset)@
qu.1.5.part.7.libname=@
qu.1.5.part.7.mode=Maple@
qu.1.5.part.7.allow2d=1@
qu.1.5.part.7.plot=@
qu.1.5.part.7.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.5.part.7.type=formula@
qu.1.5.part.8.name=sro_id_8@
qu.1.5.part.8.answer.units=@
qu.1.5.part.8.numStyle=   @
qu.1.5.part.8.editing=useHTML@
qu.1.5.part.8.showUnits=false@
qu.1.5.part.8.question=(Unset)@
qu.1.5.part.8.mode=Numeric@
qu.1.5.part.8.grading=exact_value@
qu.1.5.part.8.negStyle=both@
qu.1.5.part.8.answer.num=0@
qu.1.5.part.9.name=sro_id_9@
qu.1.5.part.9.answer.units=@
qu.1.5.part.9.numStyle=   @
qu.1.5.part.9.editing=useHTML@
qu.1.5.part.9.showUnits=false@
qu.1.5.part.9.err=0.05@
qu.1.5.part.9.question=(Unset)@
qu.1.5.part.9.mode=Numeric@
qu.1.5.part.9.grading=toler_abs@
qu.1.5.part.9.negStyle=both@
qu.1.5.part.9.answer.num=$X@
qu.1.5.part.10.name=sro_id_10@
qu.1.5.part.10.answer.units=@
qu.1.5.part.10.numStyle=   @
qu.1.5.part.10.editing=useHTML@
qu.1.5.part.10.showUnits=false@
qu.1.5.part.10.err=0.05@
qu.1.5.part.10.question=(Unset)@
qu.1.5.part.10.mode=Numeric@
qu.1.5.part.10.grading=toler_abs@
qu.1.5.part.10.negStyle=both@
qu.1.5.part.10.answer.num=$Y@
qu.1.5.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$Px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$Py. The individual has $m dollars.</p><p>&nbsp;</p><p>What is the Lagrangean for this problem?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>=<span>&nbsp;</span><1><span> </span>+<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[<span>&nbsp;</span><2><span>&nbsp;</span>]</p><p>&nbsp;</p><p>What are the first order conditions? Type L for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>.</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><3><span> </span>=<span>&nbsp;</span><4><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><5><span> </span>=<span>&nbsp;</span><6><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mi>&lambda;</mi></mrow></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><7><span> </span>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should this individual buy?</p><p>(Round answers to the nearest whole number.)</p><p>X=<span>&nbsp;</span><9><span>&nbsp;</span></p><p>Y=<span>&nbsp;</span><10><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.6.mode=Inline@
qu.1.6.name=2 Variables Constrained Optimization - CES no steps@
qu.1.6.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Upretty + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$m-$Px*X-$Py*Y].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p>Optimal X and Y are:</p>
<p>X=$X</p>
<p>Y=$Y</p>@
qu.1.6.editing=useHTML@
qu.1.6.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.1.6.solution=@
qu.1.6.algorithm=$A=range(2,10);
$s=(range(2,8));
$r=frac($s-1,$s);
$rd=decimal(2,$r);
$fracr=frac($s,$s-1);
$fracrd=decimal(2,$fracr);
$delta=frac(1,range(2,4));
$deltad=decimal(2,$delta);
$py=range(2,5);
$px=range(2,5);
$m=range(10,20);
$U=$A*((X^(-($rd))*($deltad)+decimal(2,(1-$deltad))*Y^(-($rd)))^(-$fracrd));
$Upretty=mathml($U);
$v=maple("
U:=$A*((X^(-($r))*($delta)+(1-$delta)*Y^(-($r)))^(-$fracr)):
ans:=Optimization[Maximize](U, {$m-$px*X-$py*Y>=0,X>=0,Y>=0}):
v1:=ans[2]:
v2:=eval(X,v1[1]):
v3:=eval(Y,v1[2]):
derX:=diff(U,X):
derXpretty:=MathML[ExportPresentation](derX):
derY:=diff(U,Y):
derYpretty:=MathML[ExportPresentation](derY):
derL:=diff(U+L*($m-$px*X-$py*Y),L):
derLpretty:=MathML[ExportPresentation](derL):
v2,v3,convert(derX,string),derXpretty,convert(derY,string),derYpretty,convert(derL,string),derLpretty
");
$Xtemp=switch(0,$v);
$Ytemp=switch(1,$v);
$X=decimal(2,$Xtemp);
$Y=decimal(2,$Ytemp);
$derX=switch(2,$v);
$derXpretty=switch(3,$v);
$derY=switch(4,$v);
$derYpretty=switch(5,$v);
$derL=switch(6,$v);
$derLpretty=switch(7,$v);@
qu.1.6.uid=b10ce753-0e5d-4508-85dc-cbc774cb5ae2@
qu.1.6.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Maximization - Ces;
  Author=Katherine Dare;
  Difficulty=Hard;
@
qu.1.6.weighting=5,5@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_9@
qu.1.6.part.1.answer.units=@
qu.1.6.part.1.numStyle=   @
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.showUnits=false@
qu.1.6.part.1.err=0.1@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.mode=Numeric@
qu.1.6.part.1.grading=toler_abs@
qu.1.6.part.1.negStyle=both@
qu.1.6.part.1.answer.num=$X@
qu.1.6.part.2.name=sro_id_10@
qu.1.6.part.2.answer.units=@
qu.1.6.part.2.numStyle=   @
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.showUnits=false@
qu.1.6.part.2.err=0.1@
qu.1.6.part.2.question=(Unset)@
qu.1.6.part.2.mode=Numeric@
qu.1.6.part.2.grading=toler_abs@
qu.1.6.part.2.negStyle=both@
qu.1.6.part.2.answer.num=$Y@
qu.1.6.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$py. The individual has $m dollars.</p><p>&nbsp;</p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should this individual buy?</p><p>(Round answers to the nearest two decimals. For example, 1.666 rounds to 1.67.)</p><p>X=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>Y=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.7.mode=Inline@
qu.1.7.name=2 Variables Constrained Optimization - CD with steps@
qu.1.7.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Upretty + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$m-$Px*X-$Py*Y].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p>Optimal X and Y are:</p>
<p>X=$X</p>
<p>Y=$Y</p>@
qu.1.7.editing=useHTML@
qu.1.7.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.1.7.solution=@
qu.1.7.algorithm=$a=range(2,4);
$b=range(2,4);
$X=range(5,10);
$Y=range(5,10);
$m=range(20,40);
$Px=decimal(2,((1/$a)*$m)/((1/$a+1/$b)*$X));
$Py=decimal(2,($m-$Px*$X)/($Y));
$v=maple("
randomize():
U:=X^(1/$a)*Y^(1/$b):
Lag:=U+L*($m-$Px * X-$Py * Y):
v1:=diff(Lag,X):
v2:=diff(Lag,Y):
v3:=diff(Lag,L):
convert(v1,string),convert(v2,string),convert(v3,string),convert(U,string),convert(U,string),convert(v1,string),convert(v2,string),convert(v3,string)
");
$derX=switch(0,$v);
$derY=switch(1,$v);
$derL=switch(2,$v);
$Umath=switch(3,$v);
$Ustring=switch(4,$v);
$derXstring=switch(5,$v);
$derYstring=switch(6,$v);
$derLstring=switch(7,$v);
$Upretty=mathml("$Ustring");
$derXpretty=mathml("$derXstring");
$derYpretty=mathml("$derYstring");
$derLpretty=mathml("$derLstring");@
qu.1.7.uid=4085ac32-8b0c-440b-92e9-63da9920c57f@
qu.1.7.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Maximization - Cd;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Walks Students Through Steps;
@
qu.1.7.weighting=1,1,1,1,1,1,1,1,5,5@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.maple_answer=$Ustring@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.libname=@
qu.1.7.part.1.mode=Maple@
qu.1.7.part.1.allow2d=1@
qu.1.7.part.1.plot=@
qu.1.7.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.7.part.1.type=formula@
qu.1.7.part.2.name=sro_id_2@
qu.1.7.part.2.maple_answer=$m-$Px*X-$Py*Y@
qu.1.7.part.2.editing=useHTML@
qu.1.7.part.2.question=(Unset)@
qu.1.7.part.2.libname=@
qu.1.7.part.2.mode=Maple@
qu.1.7.part.2.allow2d=1@
qu.1.7.part.2.plot=@
qu.1.7.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.7.part.2.type=formula@
qu.1.7.part.3.name=sro_id_3@
qu.1.7.part.3.maple_answer=$derX@
qu.1.7.part.3.editing=useHTML@
qu.1.7.part.3.question=(Unset)@
qu.1.7.part.3.libname=@
qu.1.7.part.3.mode=Maple@
qu.1.7.part.3.allow2d=1@
qu.1.7.part.3.plot=@
qu.1.7.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.7.part.3.type=formula@
qu.1.7.part.4.name=sro_id_4@
qu.1.7.part.4.answer.units=@
qu.1.7.part.4.numStyle=   @
qu.1.7.part.4.editing=useHTML@
qu.1.7.part.4.showUnits=false@
qu.1.7.part.4.question=(Unset)@
qu.1.7.part.4.mode=Numeric@
qu.1.7.part.4.grading=exact_value@
qu.1.7.part.4.negStyle=both@
qu.1.7.part.4.answer.num=0@
qu.1.7.part.5.name=sro_id_5@
qu.1.7.part.5.maple_answer=$derY@
qu.1.7.part.5.editing=useHTML@
qu.1.7.part.5.question=(Unset)@
qu.1.7.part.5.libname=@
qu.1.7.part.5.mode=Maple@
qu.1.7.part.5.allow2d=1@
qu.1.7.part.5.plot=@
qu.1.7.part.5.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.7.part.5.type=formula@
qu.1.7.part.6.name=sro_id_6@
qu.1.7.part.6.answer.units=@
qu.1.7.part.6.numStyle=   @
qu.1.7.part.6.editing=useHTML@
qu.1.7.part.6.showUnits=false@
qu.1.7.part.6.question=(Unset)@
qu.1.7.part.6.mode=Numeric@
qu.1.7.part.6.grading=exact_value@
qu.1.7.part.6.negStyle=both@
qu.1.7.part.6.answer.num=0@
qu.1.7.part.7.name=sro_id_7@
qu.1.7.part.7.maple_answer=$derL@
qu.1.7.part.7.editing=useHTML@
qu.1.7.part.7.question=(Unset)@
qu.1.7.part.7.libname=@
qu.1.7.part.7.mode=Maple@
qu.1.7.part.7.allow2d=1@
qu.1.7.part.7.plot=@
qu.1.7.part.7.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.7.part.7.type=formula@
qu.1.7.part.8.name=sro_id_8@
qu.1.7.part.8.answer.units=@
qu.1.7.part.8.numStyle=   @
qu.1.7.part.8.editing=useHTML@
qu.1.7.part.8.showUnits=false@
qu.1.7.part.8.question=(Unset)@
qu.1.7.part.8.mode=Numeric@
qu.1.7.part.8.grading=exact_value@
qu.1.7.part.8.negStyle=both@
qu.1.7.part.8.answer.num=0@
qu.1.7.part.9.name=sro_id_9@
qu.1.7.part.9.answer.units=@
qu.1.7.part.9.numStyle=   @
qu.1.7.part.9.editing=useHTML@
qu.1.7.part.9.showUnits=false@
qu.1.7.part.9.err=0.05@
qu.1.7.part.9.question=(Unset)@
qu.1.7.part.9.mode=Numeric@
qu.1.7.part.9.grading=toler_abs@
qu.1.7.part.9.negStyle=both@
qu.1.7.part.9.answer.num=$X@
qu.1.7.part.10.name=sro_id_10@
qu.1.7.part.10.answer.units=@
qu.1.7.part.10.numStyle=   @
qu.1.7.part.10.editing=useHTML@
qu.1.7.part.10.showUnits=false@
qu.1.7.part.10.err=0.05@
qu.1.7.part.10.question=(Unset)@
qu.1.7.part.10.mode=Numeric@
qu.1.7.part.10.grading=toler_abs@
qu.1.7.part.10.negStyle=both@
qu.1.7.part.10.answer.num=$Y@
qu.1.7.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$Px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$Py. The individual has $m dollars.</p><p>&nbsp;</p><p>What is the Lagrangean for this problem?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>=<span>&nbsp;</span><1><span> </span>+<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[<span>&nbsp;</span><2><span>&nbsp;</span>]</p><p>&nbsp;</p><p>What are the first order conditions? Type L for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>.</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><3><span> </span>=<span>&nbsp;</span><4><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><5><span> </span>=<span>&nbsp;</span><6><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mi>&lambda;</mi></mrow></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><7><span> </span>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should this individual buy?</p><p>(Round answers to the nearest whole number.)</p><p>X=<span>&nbsp;</span><9><span>&nbsp;</span></p><p>Y=<span>&nbsp;</span><10><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.8.mode=Inline@
qu.1.8.name=2 Variables Constrained Optimization - CD cost minimization with steps@
qu.1.8.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Pk*K+$Pl*L + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>$q-[$Cpretty].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>K</mi></mrow></mfrac></mrow></mstyle></math>: $derKpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>L</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $dermpretty=0</p>
<p>Optimal K and L are:</p>
<p>X=$K</p>
<p>Y=$L</p>@
qu.1.8.editing=useHTML@
qu.1.8.hint.1=The objective function goes in the first part of the Lagrangean. In this case, the objective function is the cost.@
qu.1.8.hint.2=The constraint here is that the firm wants to make a certain amount of output.@
qu.1.8.solution=@
qu.1.8.algorithm=$a=range(2,4);
$b=frac($a-1,$a);
$e=1-1/$a;
$Q=range(5,10);
$Pk=range(2,4);
$Pl=range(2,4);
$q=range(10,20);
$v=maple("
randomize():
C:=K^(1/$a)*L^($b):
Lag:=$Pk*K+$Pl*L+m*($q-C):
v1:=diff(Lag,K):
v2:=diff(Lag,L):
v3:=diff(Lag,m):
v4:=round(((($b*$a)*($Pk/$Pl))^(-$b))*$q*100)/100:
v5:=round(((($b*$a)*($Pk/$Pl))^(1/$a))*$q*100)/100:
convert(v1,string),convert(v2,string),convert(v3,string),convert(C,string),convert(C,string),convert(v1,string),convert(v2,string),convert(v3,string),v4,v5
");
$derK=switch(0,$v);
$derL=switch(1,$v);
$derm=switch(2,$v);
$Cmath=switch(3,$v);
$Cstring=switch(4,$v);
$derKstring=switch(5,$v);
$derLstring=switch(6,$v);
$dermstring=switch(7,$v);
$Cpretty=mathml("$Cstring");
$derKpretty=mathml("$derKstring");
$derLpretty=mathml("$derLstring");
$dermpretty=mathml("$dermstring");
$K=switch(8,$v);
$L=switch(9,$v);@
qu.1.8.uid=42ea0ae6-6a9a-4e55-a506-457c5279a8e3@
qu.1.8.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Minimization - Cd;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Walks Students Through Steps;
@
qu.1.8.weighting=1,1,1,1,1,1,1,1,1,5,5@
qu.1.8.numbering=alpha@
qu.1.8.part.1.grader=exact@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.display.permute=true@
qu.1.8.part.1.answer.3=solve@
qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.answer.2=max@
qu.1.8.part.1.answer.1=min@
qu.1.8.part.1.mode=List@
qu.1.8.part.1.display=menu@
qu.1.8.part.1.credit.3=0.0@
qu.1.8.part.1.credit.2=0.0@
qu.1.8.part.1.credit.1=1.0@
qu.1.8.part.2.name=sro_id_2@
qu.1.8.part.2.maple_answer=$Pk*K+$Pl*L@
qu.1.8.part.2.editing=useHTML@
qu.1.8.part.2.question=(Unset)@
qu.1.8.part.2.libname=@
qu.1.8.part.2.mode=Maple@
qu.1.8.part.2.allow2d=1@
qu.1.8.part.2.plot=@
qu.1.8.part.2.maple=resp:=subs({k=K,l=L,M=m},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.8.part.2.type=formula@
qu.1.8.part.3.name=sro_id_3@
qu.1.8.part.3.maple_answer=$q-$Cstring@
qu.1.8.part.3.editing=useHTML@
qu.1.8.part.3.question=(Unset)@
qu.1.8.part.3.libname=@
qu.1.8.part.3.mode=Maple@
qu.1.8.part.3.allow2d=1@
qu.1.8.part.3.plot=@
qu.1.8.part.3.maple=resp:=subs({k=K,l=L,M=m},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.8.part.3.type=formula@
qu.1.8.part.4.name=sro_id_4@
qu.1.8.part.4.maple_answer=$derK@
qu.1.8.part.4.editing=useHTML@
qu.1.8.part.4.question=(Unset)@
qu.1.8.part.4.libname=@
qu.1.8.part.4.mode=Maple@
qu.1.8.part.4.allow2d=1@
qu.1.8.part.4.plot=@
qu.1.8.part.4.maple=resp:=subs({k=K,l=L,M=m},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.8.part.4.type=formula@
qu.1.8.part.5.name=sro_id_5@
qu.1.8.part.5.answer.units=@
qu.1.8.part.5.numStyle=   @
qu.1.8.part.5.editing=useHTML@
qu.1.8.part.5.showUnits=false@
qu.1.8.part.5.question=(Unset)@
qu.1.8.part.5.mode=Numeric@
qu.1.8.part.5.grading=exact_value@
qu.1.8.part.5.negStyle=both@
qu.1.8.part.5.answer.num=0@
qu.1.8.part.6.name=sro_id_6@
qu.1.8.part.6.maple_answer=$derL@
qu.1.8.part.6.editing=useHTML@
qu.1.8.part.6.question=(Unset)@
qu.1.8.part.6.libname=@
qu.1.8.part.6.mode=Maple@
qu.1.8.part.6.allow2d=1@
qu.1.8.part.6.plot=@
qu.1.8.part.6.maple=resp:=subs({k=K,l=L,M=m},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.8.part.6.type=formula@
qu.1.8.part.7.name=sro_id_7@
qu.1.8.part.7.answer.units=@
qu.1.8.part.7.numStyle=   @
qu.1.8.part.7.editing=useHTML@
qu.1.8.part.7.showUnits=false@
qu.1.8.part.7.question=(Unset)@
qu.1.8.part.7.mode=Numeric@
qu.1.8.part.7.grading=exact_value@
qu.1.8.part.7.negStyle=both@
qu.1.8.part.7.answer.num=0@
qu.1.8.part.8.name=sro_id_8@
qu.1.8.part.8.maple_answer=$derm@
qu.1.8.part.8.editing=useHTML@
qu.1.8.part.8.question=(Unset)@
qu.1.8.part.8.libname=@
qu.1.8.part.8.mode=Maple@
qu.1.8.part.8.allow2d=1@
qu.1.8.part.8.plot=@
qu.1.8.part.8.maple=resp:=subs({k=K,l=L,M=m},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.8.part.8.type=formula@
qu.1.8.part.9.name=sro_id_9@
qu.1.8.part.9.answer.units=@
qu.1.8.part.9.numStyle=   @
qu.1.8.part.9.editing=useHTML@
qu.1.8.part.9.showUnits=false@
qu.1.8.part.9.question=(Unset)@
qu.1.8.part.9.mode=Numeric@
qu.1.8.part.9.grading=exact_value@
qu.1.8.part.9.negStyle=both@
qu.1.8.part.9.answer.num=0@
qu.1.8.part.10.name=sro_id_10@
qu.1.8.part.10.answer.units=@
qu.1.8.part.10.numStyle=   @
qu.1.8.part.10.editing=useHTML@
qu.1.8.part.10.showUnits=false@
qu.1.8.part.10.err=0.05@
qu.1.8.part.10.question=(Unset)@
qu.1.8.part.10.mode=Numeric@
qu.1.8.part.10.grading=toler_abs@
qu.1.8.part.10.negStyle=both@
qu.1.8.part.10.answer.num=$K@
qu.1.8.part.11.name=sro_id_11@
qu.1.8.part.11.answer.units=@
qu.1.8.part.11.numStyle=   @
qu.1.8.part.11.editing=useHTML@
qu.1.8.part.11.showUnits=false@
qu.1.8.part.11.err=0.05@
qu.1.8.part.11.question=(Unset)@
qu.1.8.part.11.mode=Numeric@
qu.1.8.part.11.grading=toler_abs@
qu.1.8.part.11.negStyle=both@
qu.1.8.part.11.answer.num=$L@
qu.1.8.question=<p>A firm has the following production function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mfenced open='(' close=')' separators=','><mrow><mi>K</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>L</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Cpretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>K</mi></mrow></msub></mrow></mstyle></math>=$Pk, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>L</mi></mrow></msub></mrow></mstyle></math>=$Pl. The firm wishes to produce $q units for the least cost possible.</p><p>&nbsp;</p><p>What is the Lagrangean for this problem?</p><p><span>&nbsp;</span><1><span>&nbsp;</span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>=<span>&nbsp;</span><2><span> </span>+<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[<span>&nbsp;</span><3><span>&nbsp;</span>]</p><p>&nbsp;</p><p>What are the first order conditions? <strong>Note: use m for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math> (because in this question L means labour).</strong></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>K</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><4><span> </span>=<span>&nbsp;</span><5><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>L</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><6><span> </span>=<span>&nbsp;</span><7><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mi>&lambda;</mi></mrow></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><8><span> </span>=<span>&nbsp;</span><9><span>&nbsp;</span></p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi></mrow></mstyle></math> should the firm use?</p><p>(Round answers to the nearest two decimal places. For example, 1.6666 becomes 1.67.)</p><p>K=<span>&nbsp;</span><10><span>&nbsp;</span></p><p>L=<span>&nbsp;</span><11><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.9.mode=Inline@
qu.1.9.name=2 Variables Constrained Optimization - ln without steps@
qu.1.9.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Upretty + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$m-$Px*X-$Py*Y].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p>Optimal X and Y are:</p>
<p>X=$X</p>
<p>Y=$Y</p>@
qu.1.9.editing=useHTML@
qu.1.9.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.1.9.solution=@
qu.1.9.algorithm=$a=range(2,4);
$b=range(2,4);
$X=range(5,10);
$Y=range(5,10);
$m=range(20,40);
$Px=decimal(2,((1/$a)*$m)/((1/$a+1/$b)*$X));
$Py=decimal(2,($m-$Px*$X)/($Y));
$v=maple("
randomize():
U:=(1/$a)*ln(X)+(1/$b)*ln(Y):
Lag:=U+L*($m-$Px * X-$Py * Y):
v1:=diff(Lag,X):
v2:=diff(Lag,Y):
v3:=diff(Lag,L):
convert(v1,string),convert(v2,string),convert(v3,string),U,convert(U,string),convert(v1,string),convert(v2,string),convert(v3,string)
");
$derX=switch(0,$v);
$derY=switch(1,$v);
$derL=switch(2,$v);
$Umath=switch(3,$v);
$Ustring=switch(4,$v);
$derXstring=switch(5,$v);
$derYstring=switch(6,$v);
$derLstring=switch(7,$v);
$Upretty=mathml("$Ustring");
$derXpretty=mathml("$derXstring");
$derYpretty=mathml("$derYstring");
$derLpretty=mathml("$derLstring");@
qu.1.9.uid=032907f4-5c7d-4aa2-961b-3abd15fabbec@
qu.1.9.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Maximization - Ln;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.9.weighting=5,5@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_9@
qu.1.9.part.1.answer.units=@
qu.1.9.part.1.numStyle=   @
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.showUnits=false@
qu.1.9.part.1.err=0.05@
qu.1.9.part.1.question=(Unset)@
qu.1.9.part.1.mode=Numeric@
qu.1.9.part.1.grading=toler_abs@
qu.1.9.part.1.negStyle=both@
qu.1.9.part.1.answer.num=$X@
qu.1.9.part.2.name=sro_id_10@
qu.1.9.part.2.answer.units=@
qu.1.9.part.2.numStyle=   @
qu.1.9.part.2.editing=useHTML@
qu.1.9.part.2.showUnits=false@
qu.1.9.part.2.err=0.05@
qu.1.9.part.2.question=(Unset)@
qu.1.9.part.2.mode=Numeric@
qu.1.9.part.2.grading=toler_abs@
qu.1.9.part.2.negStyle=both@
qu.1.9.part.2.answer.num=$Y@
qu.1.9.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$Px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$Py. The individual has $m dollars.</p><p>&nbsp;</p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should this individual buy?</p><p>(Round answers to the nearest whole number.)</p><p>X=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>Y=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.1.10.mode=Inline@
qu.1.10.name=Corner Solution - Steps@
qu.1.10.comment=<p>This question has a corner solution. If the price of capital is low enough, it is best to put all the firm's labour into producing good Y. Otherwise, it is best to put all the firm's labour into producing good X.</p>
<p>In this case, the price of capital is $P. Therefore, it is best to choose:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Lx</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mi>Y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Ly</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$K</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$X</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Y</p>
<p>Which gives a profit of:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Profit</p>@
qu.1.10.editing=useHTML@
qu.1.10.hint.1=Check which (all X or all Y) gives a higher profit.@
qu.1.10.solution=@
qu.1.10.algorithm=$P=range(1,5);
$v=maple("
if $P < 3 then k := 0 
elif $P > 2 then k := 1
end if:
k
");
$X = switch($v, 0, 10);
$Y = switch($v, 50/$P, 0);
$Lx = switch($v, 0, 10);
$Ly = switch($v, 10, 0);
$K = switch($v, ((50/$P)^2)/100, 0);
$Profit = switch($v, 25/$P, 10);@
qu.1.10.uid=f2216428-290d-4071-9131-82d6d8c67c56@
qu.1.10.info=  Course=Introductory Mathematical Economics;
  Topic=Constrained Optimization;
  Sub-Topic=Corner Solutions;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Walks Students Through Steps;
@
qu.1.10.weighting=1,1,1,1,1,1,1,1,1,1,1,1@
qu.1.10.numbering=alpha@
qu.1.10.part.1.name=sro_id_1@
qu.1.10.part.1.answer.units=@
qu.1.10.part.1.numStyle=   @
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.showUnits=false@
qu.1.10.part.1.question=(Unset)@
qu.1.10.part.1.mode=Numeric@
qu.1.10.part.1.grading=exact_value@
qu.1.10.part.1.negStyle=both@
qu.1.10.part.1.answer.num=0@
qu.1.10.part.2.name=sro_id_2@
qu.1.10.part.2.answer.units=@
qu.1.10.part.2.numStyle=   @
qu.1.10.part.2.editing=useHTML@
qu.1.10.part.2.showUnits=false@
qu.1.10.part.2.question=(Unset)@
qu.1.10.part.2.mode=Numeric@
qu.1.10.part.2.grading=exact_value@
qu.1.10.part.2.negStyle=both@
qu.1.10.part.2.answer.num=10@
qu.1.10.part.3.name=sro_id_3@
qu.1.10.part.3.maple_answer=($P*(Y^2))/100@
qu.1.10.part.3.editing=useHTML@
qu.1.10.part.3.question=(Unset)@
qu.1.10.part.3.libname=@
qu.1.10.part.3.mode=Maple@
qu.1.10.part.3.allow2d=1@
qu.1.10.part.3.plot=@
qu.1.10.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.10.part.3.type=formula@
qu.1.10.part.4.name=sro_id_4@
qu.1.10.part.4.maple_answer=Y-($P*(Y^2))/100@
qu.1.10.part.4.editing=useHTML@
qu.1.10.part.4.question=(Unset)@
qu.1.10.part.4.libname=@
qu.1.10.part.4.mode=Maple@
qu.1.10.part.4.allow2d=1@
qu.1.10.part.4.plot=@
qu.1.10.part.4.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.10.part.4.type=formula@
qu.1.10.part.5.name=sro_id_5@
qu.1.10.part.5.answer.units=@
qu.1.10.part.5.numStyle=   arithmetic@
qu.1.10.part.5.editing=useHTML@
qu.1.10.part.5.showUnits=false@
qu.1.10.part.5.err=1.0@
qu.1.10.part.5.question=(Unset)@
qu.1.10.part.5.mode=Numeric@
qu.1.10.part.5.grading=toler_abs@
qu.1.10.part.5.negStyle=minus@
qu.1.10.part.5.answer.num=50/$P@
qu.1.10.part.6.name=sro_id_6@
qu.1.10.part.6.answer.units=@
qu.1.10.part.6.numStyle=   arithmetic@
qu.1.10.part.6.editing=useHTML@
qu.1.10.part.6.showUnits=false@
qu.1.10.part.6.err=1.0@
qu.1.10.part.6.question=(Unset)@
qu.1.10.part.6.mode=Numeric@
qu.1.10.part.6.grading=toler_abs@
qu.1.10.part.6.negStyle=minus@
qu.1.10.part.6.answer.num=25/$P@
qu.1.10.part.7.name=sro_id_7@
qu.1.10.part.7.answer.units=@
qu.1.10.part.7.numStyle=   @
qu.1.10.part.7.editing=useHTML@
qu.1.10.part.7.showUnits=false@
qu.1.10.part.7.question=(Unset)@
qu.1.10.part.7.mode=Numeric@
qu.1.10.part.7.grading=exact_value@
qu.1.10.part.7.negStyle=both@
qu.1.10.part.7.answer.num=$Lx@
qu.1.10.part.8.name=sro_id_8@
qu.1.10.part.8.answer.units=@
qu.1.10.part.8.numStyle=   @
qu.1.10.part.8.editing=useHTML@
qu.1.10.part.8.showUnits=false@
qu.1.10.part.8.question=(Unset)@
qu.1.10.part.8.mode=Numeric@
qu.1.10.part.8.grading=exact_value@
qu.1.10.part.8.negStyle=both@
qu.1.10.part.8.answer.num=$Ly@
qu.1.10.part.9.name=sro_id_9@
qu.1.10.part.9.answer.units=@
qu.1.10.part.9.numStyle=   @
qu.1.10.part.9.editing=useHTML@
qu.1.10.part.9.showUnits=false@
qu.1.10.part.9.question=(Unset)@
qu.1.10.part.9.mode=Numeric@
qu.1.10.part.9.grading=exact_value@
qu.1.10.part.9.negStyle=both@
qu.1.10.part.9.answer.num=$K@
qu.1.10.part.10.name=sro_id_10@
qu.1.10.part.10.answer.units=@
qu.1.10.part.10.numStyle=thousands scientific  arithmetic@
qu.1.10.part.10.editing=useHTML@
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qu.1.10.part.10.question=(Unset)@
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qu.1.10.part.10.grading=exact_value@
qu.1.10.part.10.negStyle=minus@
qu.1.10.part.10.answer.num=$X@
qu.1.10.part.11.name=sro_id_11@
qu.1.10.part.11.answer.units=@
qu.1.10.part.11.numStyle=   @
qu.1.10.part.11.editing=useHTML@
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qu.1.10.part.11.question=(Unset)@
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qu.1.10.part.11.grading=exact_value@
qu.1.10.part.11.negStyle=both@
qu.1.10.part.11.answer.num=$Y@
qu.1.10.part.12.name=sro_id_12@
qu.1.10.part.12.answer.units=@
qu.1.10.part.12.numStyle=   @
qu.1.10.part.12.editing=useHTML@
qu.1.10.part.12.showUnits=false@
qu.1.10.part.12.question=(Unset)@
qu.1.10.part.12.mode=Numeric@
qu.1.10.part.12.grading=exact_value@
qu.1.10.part.12.negStyle=both@
qu.1.10.part.12.answer.num=$Profit@
qu.1.10.question=<div align="left"><p>A firm produces and sells two goods:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Both goods sell for a price of $1 each.</p><p>The production function for good X is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi mathvariant='normal'>L</mi><mrow><mn>X</mn></mrow></msub></mrow></mstyle></math>.</p><p>The production function for good Y is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>K</mi><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msub><mi>L</mi><mrow><mn>Y</mn></mrow></msub></mrow></mstyle></math>.</p><p>The firm divides its labour between the two products: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>L</mi><mrow><mi mathvariant='normal'>X</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>L</mi><mrow><mi mathvariant='normal'>Y</mi></mrow></msub></mrow></mstyle></math> and gets its labour for free, but can only use 10 units of it or less. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>10</mn></mrow></mstyle></math>.</p><p>The price of capital is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>p</mi><mrow><mi>K</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>What is the firm's cost function if it puts all its labour into <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>? (Assuming it does not try to use more than 10 units of labour.)</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the firm's profit if it puts all its labour into <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi mathvariant='normal'>L</mi><mrow><mn>X</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>10</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the firm's cost function if it puts all its labour into <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>?</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mfenced open='(' close=')' separators=','><mrow><mn>Y</mn></mrow></mfenced></mrow></mstyle></math>= </span><3><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the firm's profit function if it puts all its labour into <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>L</mi><mrow><mn>Y</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>10</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p><p>&nbsp;</p><p>How much&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should the firm produce if it puts all its labour into <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><5><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the firm's profit if it puts all its labour into <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>L</mi><mrow><mn>Y</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>10</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><6><span>&nbsp;</span></p><p>&nbsp;</p><p><span>Given the above results, how much&nbsp;</span>of each input should the firm use, and how much of each output should it produce?</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mn>X</mn></mrow></msub></mrow></mstyle></math>=<span><span>&nbsp;</span><7><span>&nbsp;</span></span></span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>L</mi><mrow><mn>Y</mn></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><9><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>=<span>&nbsp;</span><10><span>&nbsp;</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><11><span>&nbsp;</span></p><p>&nbsp;</p><p><span>How much profit does the firm make?</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><12><span>&nbsp;</span><br /></span></p><p>&nbsp;</p></div>@

qu.1.11.mode=Inline@
qu.1.11.name=2 Variables Constrained Optimization - CES with steps@
qu.1.11.comment=<p>The Lagrangean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow></mrow></mstyle></math>= $Upretty + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[$m-$Px*X-$Py*Y].</p>
<p>The first order conditions are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty=0</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>&lambda;</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty=0</p>
<p>Optimal X and Y are:</p>
<p>X=$X</p>
<p>Y=$Y</p>@
qu.1.11.editing=useHTML@
qu.1.11.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.1.11.solution=@
qu.1.11.algorithm=$A=range(2,10);
$s=(range(2,8));
$r=frac($s-1,$s);
$rd=decimal(2,$r);
$fracr=frac($s,$s-1);
$fracrd=decimal(2,$fracr);
$delta=frac(1,range(2,4));
$deltad=decimal(2,$delta);
$py=range(2,5);
$px=range(2,5);
$m=range(10,20);
$tmp=decimal(2,(1-$deltad));
$U="$A*((X^(-($rd))*($deltad)+$tmp*Y^(-($rd)))^(-$fracrd))";
$Upretty=mathml($U);
$v=maple("
U:=$A*((X^(-($r))*($delta)+(1-$delta)*Y^(-($r)))^(-$fracr)):
ans:=Optimization[Maximize](U, {$m-$px*X-$py*Y>=0,X>=0,Y>=0}):
v1:=ans[2]:
v2:=eval(X,v1[1]):
v3:=eval(Y,v1[2]):
derX:=diff(U,X):
derXpretty:=MathML[ExportPresentation](derX):
derY:=diff(U,Y):
derYpretty:=MathML[ExportPresentation](derY):
derL:=diff(U+L*($m-$px*X-$py*Y),L):
derLpretty:=MathML[ExportPresentation](derL):
v2,v3,convert(derX,string),derXpretty,convert(derY,string),derYpretty,convert(derL,string),derLpretty
");
$Xtemp=switch(0,$v);
$Ytemp=switch(1,$v);
$X=decimal(2,$Xtemp);
$Y=decimal(2,$Ytemp);
$derX=switch(2,$v);
$derXpretty=switch(3,$v);
$derY=switch(4,$v);
$derYpretty=switch(5,$v);
$derL=switch(6,$v);
$derLpretty=switch(7,$v);@
qu.1.11.uid=6dda0ef4-ba93-460b-ad48-88ba13df0a87@
qu.1.11.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Constrained Optimization - Utility Maximization - Ces;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Walks Students Through Steps;
@
qu.1.11.weighting=1,1,1,1,1,1,1,1,5,5@
qu.1.11.numbering=alpha@
qu.1.11.part.1.name=sro_id_1@
qu.1.11.part.1.maple_answer=$U@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.question=(Unset)@
qu.1.11.part.1.libname=@
qu.1.11.part.1.mode=Maple@
qu.1.11.part.1.allow2d=1@
qu.1.11.part.1.plot=@
qu.1.11.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.11.part.1.type=formula@
qu.1.11.part.2.name=sro_id_2@
qu.1.11.part.2.maple_answer=$m-$px*X-$py*Y@
qu.1.11.part.2.editing=useHTML@
qu.1.11.part.2.question=(Unset)@
qu.1.11.part.2.libname=@
qu.1.11.part.2.mode=Maple@
qu.1.11.part.2.allow2d=1@
qu.1.11.part.2.plot=@
qu.1.11.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.11.part.2.type=formula@
qu.1.11.part.3.name=sro_id_3@
qu.1.11.part.3.maple_answer=$derX@
qu.1.11.part.3.editing=useHTML@
qu.1.11.part.3.question=(Unset)@
qu.1.11.part.3.libname=@
qu.1.11.part.3.mode=Maple@
qu.1.11.part.3.allow2d=1@
qu.1.11.part.3.plot=@
qu.1.11.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.11.part.3.type=formula@
qu.1.11.part.4.name=sro_id_4@
qu.1.11.part.4.answer.units=@
qu.1.11.part.4.numStyle=   @
qu.1.11.part.4.editing=useHTML@
qu.1.11.part.4.showUnits=false@
qu.1.11.part.4.question=(Unset)@
qu.1.11.part.4.mode=Numeric@
qu.1.11.part.4.grading=exact_value@
qu.1.11.part.4.negStyle=both@
qu.1.11.part.4.answer.num=0@
qu.1.11.part.5.name=sro_id_5@
qu.1.11.part.5.maple_answer=$derY@
qu.1.11.part.5.editing=useHTML@
qu.1.11.part.5.question=(Unset)@
qu.1.11.part.5.libname=@
qu.1.11.part.5.mode=Maple@
qu.1.11.part.5.allow2d=1@
qu.1.11.part.5.plot=@
qu.1.11.part.5.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.11.part.5.type=formula@
qu.1.11.part.6.name=sro_id_6@
qu.1.11.part.6.answer.units=@
qu.1.11.part.6.numStyle=   @
qu.1.11.part.6.editing=useHTML@
qu.1.11.part.6.showUnits=false@
qu.1.11.part.6.question=(Unset)@
qu.1.11.part.6.mode=Numeric@
qu.1.11.part.6.grading=exact_value@
qu.1.11.part.6.negStyle=both@
qu.1.11.part.6.answer.num=0@
qu.1.11.part.7.name=sro_id_7@
qu.1.11.part.7.maple_answer=$derL@
qu.1.11.part.7.editing=useHTML@
qu.1.11.part.7.question=(Unset)@
qu.1.11.part.7.libname=@
qu.1.11.part.7.mode=Maple@
qu.1.11.part.7.allow2d=1@
qu.1.11.part.7.plot=@
qu.1.11.part.7.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.11.part.7.type=formula@
qu.1.11.part.8.name=sro_id_8@
qu.1.11.part.8.answer.units=@
qu.1.11.part.8.numStyle=   @
qu.1.11.part.8.editing=useHTML@
qu.1.11.part.8.showUnits=false@
qu.1.11.part.8.question=(Unset)@
qu.1.11.part.8.mode=Numeric@
qu.1.11.part.8.grading=exact_value@
qu.1.11.part.8.negStyle=both@
qu.1.11.part.8.answer.num=0@
qu.1.11.part.9.name=sro_id_9@
qu.1.11.part.9.answer.units=@
qu.1.11.part.9.numStyle=   @
qu.1.11.part.9.editing=useHTML@
qu.1.11.part.9.showUnits=false@
qu.1.11.part.9.err=0.1@
qu.1.11.part.9.question=(Unset)@
qu.1.11.part.9.mode=Numeric@
qu.1.11.part.9.grading=toler_abs@
qu.1.11.part.9.negStyle=both@
qu.1.11.part.9.answer.num=$X@
qu.1.11.part.10.name=sro_id_10@
qu.1.11.part.10.answer.units=@
qu.1.11.part.10.numStyle=   @
qu.1.11.part.10.editing=useHTML@
qu.1.11.part.10.showUnits=false@
qu.1.11.part.10.err=0.1@
qu.1.11.part.10.question=(Unset)@
qu.1.11.part.10.mode=Numeric@
qu.1.11.part.10.grading=toler_abs@
qu.1.11.part.10.negStyle=both@
qu.1.11.part.10.answer.num=$Y@
qu.1.11.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$py. The individual has $m dollars.</p><p>&nbsp;</p><p>What is the Lagrangean for this problem?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>=<span>&nbsp;</span><1><span> </span>+<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>[<span>&nbsp;</span><2><span>&nbsp;</span>]</p><p>&nbsp;</p><p>What are the first order conditions? Use L for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi></mrow></mstyle></math>.</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><3><span> </span>=<span>&nbsp;</span><4><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><5><span> </span>=<span>&nbsp;</span><6><span>&nbsp;</span></p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mi>&lambda;</mi></mrow></mrow></mfrac></mrow></mstyle></math>:<span>&nbsp;</span><7><span> </span>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p>&nbsp;</p><p>How much <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> should this individual buy?</p><p>(Round answers to the nearest two decimals. For example, 1.666 rounds to 1.67.)</p><p>X=<span>&nbsp;</span><9><span>&nbsp;</span></p><p>Y=<span>&nbsp;</span><10><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.2.topic=Bordered Hessians@

qu.2.1.mode=Inline@
qu.2.1.name=Cost minimization with SOC@
qu.2.1.comment=<p>The firm wants to minimize cost subject to a production constraint. So, the Lagrangean is $w*L+$r*K+<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>q - $A1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>. To find the First order conditions we differentiate the Lagrangean with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math> Letting <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>J</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>denote <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>, we find</p>
<p>$f1</p>
<p>$f2</p>
<p>$f3</p>
<p>The Hessian for this problem is $H</p>
<p>&nbsp;</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=decimal(1,range(.1,.5,.1));
$b=decimal(1,range(.1,.5,.1));
$F=(K^$a)*(L^$b);
$Fq=q-$F;
$r=range(1,5,1);
$w=range(1,5,1);
$c=($r*K)+($w*L);
$O=$c+J*($Fq);
$A=maple("
a1:=MathML[ExportPresentation]($F):
a2:=MathML[ExportPresentation](($r*K)+($w*L)):
fk:=diff($O,K):
fl:=diff($O,L):
fj:=diff($O,J):
FK:=MathML[ExportPresentation](fk=0):
FL:=MathML[ExportPresentation](fl=0):
FJ:=MathML[ExportPresentation](fj=0):
fkk:=diff($fk,K):
fkl:=diff($fk,L):
fkj:=diff($fk,J):
fll:=diff($fl,L):
flj:=diff($fl,J):
s:=Matrix([[($fkk),($fkl),($fkj)], [($fkl),($fll),($flj)],[($fkj),($flj),(0)]]):
S:=MathML[ExportPresentation]($s):
a1,a2,fk,fl,fj, FK,FL,FJ,convert(s,string),S
");
$A1=switch(0,$A);
$A2=switch(1,$A);
$F1=switch(2,$A);
$F2=switch(3,$A);
$F3=switch(4,$A);
$f1=switch(5,$A);
$f2=switch(6,$A);
$f3=switch(7,$A);
$Ans=switch(8,$A);
$H=switch(9,$A);@
qu.2.1.uid=f9f093a3-85b7-4fae-b2e6-3ca21b264723@
qu.2.1.info=  Course=Introductory Mathematical Economics;
  Topic=Constrained Optimization;
  Sub-Topic=Cost Minimization;
  Difficulty=Medium;
  Author=Asha Sadanand;
@
qu.2.1.weighting=1,1,1,1,1,1,1,1,1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.name=sro_id_1@
qu.2.1.part.1.maple_answer=$c@
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.libname=@
qu.2.1.part.1.mode=Maple@
qu.2.1.part.1.allow2d=1@
qu.2.1.part.1.plot=@
qu.2.1.part.1.maple=resp:=subs({k=K,l=L,j=J},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.1.type=formula@
qu.2.1.part.2.name=sro_id_2@
qu.2.1.part.2.maple_answer=$Fq@
qu.2.1.part.2.editing=useHTML@
qu.2.1.part.2.question=(Unset)@
qu.2.1.part.2.libname=@
qu.2.1.part.2.mode=Maple@
qu.2.1.part.2.allow2d=1@
qu.2.1.part.2.plot=@
qu.2.1.part.2.maple=resp:=subs({k=K,l=L,j=J},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.2.type=formula@
qu.2.1.part.3.name=sro_id_3@
qu.2.1.part.3.maple_answer=$F1@
qu.2.1.part.3.editing=useHTML@
qu.2.1.part.3.question=(Unset)@
qu.2.1.part.3.libname=@
qu.2.1.part.3.mode=Maple@
qu.2.1.part.3.allow2d=1@
qu.2.1.part.3.plot=@
qu.2.1.part.3.maple=resp:=subs({k=K,l=L,j=J},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.3.type=formula@
qu.2.1.part.4.editing=useHTML@
qu.2.1.part.4.question=(Unset)@
qu.2.1.part.4.name=sro_id_4@
qu.2.1.part.4.answer=0@
qu.2.1.part.4.mode=Formula@
qu.2.1.part.5.name=sro_id_5@
qu.2.1.part.5.maple_answer=$F2@
qu.2.1.part.5.editing=useHTML@
qu.2.1.part.5.question=(Unset)@
qu.2.1.part.5.libname=@
qu.2.1.part.5.mode=Maple@
qu.2.1.part.5.allow2d=1@
qu.2.1.part.5.plot=@
qu.2.1.part.5.maple=resp:=subs({k=K,l=L,j=J},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.5.type=formula@
qu.2.1.part.6.editing=useHTML@
qu.2.1.part.6.question=(Unset)@
qu.2.1.part.6.name=sro_id_6@
qu.2.1.part.6.answer=0@
qu.2.1.part.6.mode=Formula@
qu.2.1.part.7.name=sro_id_7@
qu.2.1.part.7.maple_answer=$F3@
qu.2.1.part.7.editing=useHTML@
qu.2.1.part.7.question=(Unset)@
qu.2.1.part.7.libname=@
qu.2.1.part.7.mode=Maple@
qu.2.1.part.7.allow2d=1@
qu.2.1.part.7.plot=@
qu.2.1.part.7.maple=resp:=subs({k=K,l=L,j=J},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.1.part.7.type=formula@
qu.2.1.part.8.editing=useHTML@
qu.2.1.part.8.question=(Unset)@
qu.2.1.part.8.name=sro_id_8@
qu.2.1.part.8.answer=0@
qu.2.1.part.8.mode=Formula@
qu.2.1.part.9.name=sro_id_9@
qu.2.1.part.9.maple_answer=printf("$H")@
qu.2.1.part.9.editing=useHTML@
qu.2.1.part.9.question=(Unset)@
qu.2.1.part.9.libname=@
qu.2.1.part.9.mode=Maple@
qu.2.1.part.9.allow2d=2@
qu.2.1.part.9.plot=@
qu.2.1.part.9.maple=resp:=subs({k=K,l=L,j=J},$RESPONSE);
ans:=$Ans:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if evalb(simplify(ans[i,j] - resp[i,j])=0)
then grade:=grade+0.111112:
end if;
end;
end;
grade;@
qu.2.1.part.9.type=maple@
qu.2.1.question=<p>A firm produces using the technology $A1. It wants to minimize its cost of producing a quantity, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>. The price of capital is $r and the wage is $w. Give the Lagrangean associated with this problem:<span>&nbsp;</span><1><span>&nbsp;</span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></mstyle></math><span>&nbsp;</span><2><span> .</span></p><p><span>Letting <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><ms>J</ms></mrow></mstyle></math>denote the Lagrange multiplier, give the first order conditions starting with the one for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi></mrow></mstyle></math> and ending with the one for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>J</mi></mrow></mstyle></math></span>.</p><p><span>&nbsp;</span><3><span> </span>=<span>&nbsp;</span><4><span> </span>,</p><p><span>&nbsp;</span><5><span> </span>=<span>&nbsp;</span><6><span> </span>, and</p><p><span>&nbsp;</span><7><span> </span>=<span>&nbsp;</span><8><span> </span>.</p><p><span>The Hessian is <span>&nbsp;</span><9><span> </span>. (Enter it in the order K, L, J.)<br /></span></p><p>&nbsp;</p>@

qu.2.2.mode=Inline@
qu.2.2.name=evaluate Hessian and determine SOC - 2x2@
qu.2.2.comment=<p>The Hessian is found by twice differentiating the function $F, and arranging the second derivatives in a matrix. $G</p>
<p>Then we evaluate the matrix at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T to find $R.</p>
<p>We know that if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T satisfied the first order conditions, then for this to be a minimum a sufficient condition is that the Hessian is positive semi-definite. This requires that the principal minors are all positive. Recall that the principal minors are determinants of matrices formed from the Hessian by starting with the 1x1 matrix containing the upper left most entry. Then we look at the 2x2 by adding one row from below and one column from the right. We keep going until we finally have the entire Hessian matrix.</p>
<p>For a maximum the principal minors are alternating in sign starting with negative.</p>
<p>For any other situation the Hessian is inconclusive and the point may be a maximum a minimum or neither.</p>
<p>In this instance since the Hessian is 2x2 we look at the upper left hand entry and then the determinant of the whole matrix. Looking at the upper left entry&nbsp; $R1&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi></mrow></mstyle></math>$R = $D, we find that the Hessian is $Y, which $Z.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=range(0,1,1);
$b=range(0,1,1);
$A=maple("
randomize():
F:=randpoly([X,Y], degree = 3):
f:=MathML[ExportPresentation](F):
JX:=diff($F,X):
Jx:=MathML[ExportPresentation](JX):
JY:=diff($F,X):
Jy:=MathML[ExportPresentation](JY):
H:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y)],[diff(diff($F,Y),X),diff(diff($F,Y),Y)]]):
H1:=diff(diff($F,X),X):
G:=MathML[ExportPresentation](H):
G1:=MathML[ExportPresentation](H1):
Q:=eval(H, [X=($a),Y=($b)]):
Q1:=eval(H1, [X=($a),Y=($b)]):
R:=MathML[ExportPresentation](Q):
R1:=MathML[ExportPresentation](Q1):
E:=LinearAlgebra[Determinant](Q):
S:=Matrix([$a,$b]):
T:=MathML[ExportPresentation](S):
with(LinearAlgebra):
if IsDefinite(Q, query = 'indefinite') = true then i := 2 
elif IsDefinite(Q, query = 'negative_definite') = true then i := 0 
elif IsDefinite(Q, query = 'negative_semidefinite') = true then i := 1 
elif IsDefinite(Q, query = 'positive_definite') = true then i := 4 
elif IsDefinite(Q, query = 'positive_semidefinite') = true then i := 3 
end if:
f,F,convert(H,string),H1,G, G1,convert(Q,string),Q1,R,R1,T,i,E
");
$F=switch(0,$A);
$H=switch(2,$A);
$G=switch(4,$A);
$R=switch(8,$A);
$R1=switch(9,$A);
$T=switch(10,$A);
$i=switch(11,$A);
$D=switch(12,$A);
$Y=switch(($i),"negative definite","negative semi-definite","indefinite","positive semi-definite","positive definite");
$N1=switch(($i),"negative semi-definite","indefinite","positive semi-definite","positive definite","negative definite");
$N2=switch(($i),"indefinite","positive semi-definite","positive definite","negative definite","negative semi-definite");
$N3=switch(($i),"positive semi-definite","positive definite","negative definite","negative semi-definite","indefinite");
$N4=switch(($i),"positive definite","negative definite","negative semi-definite","indefinite","positive semi-definite");
$X=switch(($i),"be a maximum","be inconclusive","be inconclusive","be inconclusive","be a minimum");
$M1=switch(($i),"be a maximum","be a minimum","be a minimum","be a minimum","be a minimum");
$M2=switch(($i),"be inconclusive","be a maximum","be a maximum","be a maximum","be inconclusive");
$Z=switch(($i),"indicates a maximum","is inconclusive","is inconclusive","is inconclusive","indicates a minimum");@
qu.2.2.uid=670277f9-a72c-43ab-8e4c-39e299d55424@
qu.2.2.info=  Course=Introductory Mathematical Economics;
  Topic=Constrained Optimization;
  Sub-Topic=Second Order Conditions;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.2.2.weighting=1,1,1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.maple_answer=printf("$G");@
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.libname=@
qu.2.2.part.1.mode=Maple@
qu.2.2.part.1.allow2d=2@
qu.2.2.part.1.plot=@
qu.2.2.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
ans:=$H:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if evalb(simplify(ans[i,j] - resp[i,j])=0)
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.2.2.part.1.type=maple@
qu.2.2.part.2.grader=exact@
qu.2.2.part.2.name=sro_id_2@
qu.2.2.part.2.editing=useHTML@
qu.2.2.part.2.answer.5=$N4@
qu.2.2.part.2.display.permute=true@
qu.2.2.part.2.answer.4=$N3@
qu.2.2.part.2.answer.3=$N2@
qu.2.2.part.2.question=(Unset)@
qu.2.2.part.2.answer.2=$N1@
qu.2.2.part.2.answer.1=$Y@
qu.2.2.part.2.mode=List@
qu.2.2.part.2.display=menu@
qu.2.2.part.2.credit.5=0.0@
qu.2.2.part.2.credit.4=0.0@
qu.2.2.part.2.credit.3=0.0@
qu.2.2.part.2.credit.2=0.0@
qu.2.2.part.2.credit.1=1.0@
qu.2.2.part.3.grader=exact@
qu.2.2.part.3.name=sro_id_3@
qu.2.2.part.3.editing=useHTML@
qu.2.2.part.3.display.permute=true@
qu.2.2.part.3.answer.3=$M2@
qu.2.2.part.3.question=(Unset)@
qu.2.2.part.3.answer.2=$M1@
qu.2.2.part.3.answer.1=$X@
qu.2.2.part.3.mode=List@
qu.2.2.part.3.display=menu@
qu.2.2.part.3.credit.3=0.0@
qu.2.2.part.3.credit.2=0.0@
qu.2.2.part.3.credit.1=1.0@
qu.2.2.question=<p>Consider the function $F. Find the Hessian matrix associated with this function. <span>&nbsp;</span><1><span>.</span></p><p><span>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T the Hessian is&nbsp;<span>&nbsp;</span><2><span>; if the first order conditions held at this point it would&nbsp;</span></span>&nbsp;<3><span>.</span></p>@

qu.2.3.mode=Inline@
qu.2.3.name=2 Variables Constrained Optimization - bordered Hessian@
qu.2.3.comment=@
qu.2.3.editing=useHTML@
qu.2.3.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.2.3.solution=@
qu.2.3.algorithm=$a=range(2,4);
$b=range(2,4);
$X=range(5,10);
$Y=range(5,10);
$m=range(20,40);
$Px=decimal(2,((1/$a)*$m)/((1/$a+1/$b)*$X));
$Py=decimal(2,($m-$Px*$X)/($Y));
$v=maple("
randomize():
U:=X^(1/$a)*Y^(1/$b):
Lag:=U+L*($m-$Px * X-$Py * Y):
v1:=diff(Lag,X):
v2:=diff(Lag,Y):
v3:=diff(Lag,L):
v4:=diff(v1,X):
v5:=diff(v2,Y):
v6:=diff(v1,Y):
v7:=diff(v1,L):
v8:=diff(v2,L):
v9:=diff(v3,L):
H:=Matrix([[v4,v6,v7],[v6,v5,v8],[v7,v8,v9]]):
J:=MathML[ExportPresentation](H):
convert(v1,string),convert(v2,string),convert(v3,string),U,convert(U,string),convert(H,string),J
");
$derX=switch(0,$v);
$derY=switch(1,$v);
$derL=switch(2,$v);
$Umath=switch(3,$v);
$Ustring=switch(4,$v);
$Upretty=mathml("$Ustring");
$derXpretty=mathml("$derX");
$derYpretty=mathml("$derY");
$derLpretty=mathml("$derL");
$H=switch(5,$v);
$Hpretty=switch(6,$v);@
qu.2.3.uid=94214ad2-a3f2-4095-b120-81b32009f3ad@
qu.2.3.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Second Order Conditions;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.2.3.weighting=1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.maple_answer=printf("$Hpretty");@
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.question=(Unset)@
qu.2.3.part.1.libname=@
qu.2.3.part.1.mode=Maple@
qu.2.3.part.1.allow2d=2@
qu.2.3.part.1.plot=@
qu.2.3.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if evalb(simplify(ans[i,j] - resp[i,j])=0)
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.2.3.part.1.type=maple@
qu.2.3.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$Px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$Py. The individual has $m dollars.</p><p>&nbsp;</p><p>If the first order derivatives are the following, (where L is lambda)</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>L</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty</p><p>&nbsp;</p><p>Find the bordered Hessian:</p><p>&nbsp;</p><p>&nbsp; (To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <span style="vertical-align: -4px"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mi>m</mi></mrow></mstyle></math></span>and set the dimensions yourself.</p><p>Also, please make sure to use capital X and Y and be careful about order of operations when entering answers.)</p><p>&nbsp;H=<span>&nbsp;</span><1><span> (Enter in the order X, Y, lambda.)<br /></span></p>@

qu.2.4.mode=Inline@
qu.2.4.name=2 Variables Constrained Optimization - evaluate bordered Hessian@
qu.2.4.comment=@
qu.2.4.editing=useHTML@
qu.2.4.hint.1=To solve for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, use the first two First Order Conditions to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> as a function of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>. Then plug this into the budget constraint.@
qu.2.4.solution=@
qu.2.4.algorithm=$a=range(2,4);
$b=range(2,4);
$X=range(5,10);
$Y=range(5,10);
$X1=range(5,10);
$Y1=range(5,10);
$m=range(20,40);
$Px=decimal(2,((1/$a)*$m)/((1/$a+1/$b)*$X));
$Py=decimal(2,($m-$Px*$X)/($Y));
$v=maple("
randomize():
U:=X^(1/$a)*Y^(1/$b):
Lag:=U+L*($m-$Px * X-$Py * Y):
v1:=diff(Lag,X):
v2:=diff(Lag,Y):
v3:=diff(Lag,L):
v4:=diff(v1,X):
v5:=diff(v2,Y):
v6:=diff(v1,Y):
v7:=diff(v1,L):
v8:=diff(v2,L):
v9:=diff(v3,L):
H:=Matrix([[v4,v6,v7],[v6,v5,v8],[v7,v8,v9]]):
v41:=evalf(eval(v4, [X = $X, Y = $Y])):
v51:=evalf(eval(v5, [X = $X, Y = $Y])):
v61:=evalf(eval(v6, [X = $X, Y = $Y])):
v71:=evalf(eval(v7, [X = $X, Y = $Y])):
v81:=evalf(eval(v8, [X = $X, Y = $Y])):
v91:=evalf(eval(v9, [X = $X, Y = $Y])):
H1:=Matrix([[v41,v61,v71],[v61,v51,v81],[v71,v81,v91]]):
det:=LinearAlgebra[Determinant](H1):
Hpretty:=MathML[ExportPresentation](H):
convert(v1,string),convert(v2,string),convert(v3,string),U,convert(U,string),convert(H,string),det,convert(Hpretty,string)
");
$derX=switch(0,$v);
$derY=switch(1,$v);
$derL=switch(2,$v);
$Umath=switch(3,$v);
$Ustring=switch(4,$v);
$Upretty=mathml("$Ustring");
$derXpretty=mathml("$derX");
$derYpretty=mathml("$derY");
$derLpretty=mathml("$derL");
$H=switch(5,$v);
$Det=switch(6,$v);
$Hpretty=switch(7,$v);@
qu.2.4.uid=34bd92a3-82d1-4aa5-ab11-827873b26d26@
qu.2.4.info=  Course=Introductory Mathematical Economics;
  Topic=Derivatives - Two Variables;
  Sub-Topic=Second Order Conditions;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.2.4.weighting=1,1,1@
qu.2.4.numbering=alpha@
qu.2.4.part.1.name=sro_id_1@
qu.2.4.part.1.maple_answer=printf("$Hpretty");@
qu.2.4.part.1.editing=useHTML@
qu.2.4.part.1.question=(Unset)@
qu.2.4.part.1.libname=@
qu.2.4.part.1.mode=Maple@
qu.2.4.part.1.allow2d=2@
qu.2.4.part.1.plot=@
qu.2.4.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if evalb(simplify(ans[i,j] - resp[i,j])=0)
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.2.4.part.1.type=maple@
qu.2.4.part.2.name=sro_id_2@
qu.2.4.part.2.answer.units=@
qu.2.4.part.2.numStyle=   @
qu.2.4.part.2.editing=useHTML@
qu.2.4.part.2.showUnits=false@
qu.2.4.part.2.err=0.0050@
qu.2.4.part.2.question=(Unset)@
qu.2.4.part.2.mode=Numeric@
qu.2.4.part.2.grading=toler_abs@
qu.2.4.part.2.negStyle=both@
qu.2.4.part.2.answer.num=$Det@
qu.2.4.part.3.grader=exact@
qu.2.4.part.3.name=sro_id_3@
qu.2.4.part.3.editing=useHTML@
qu.2.4.part.3.display.permute=true@
qu.2.4.part.3.answer.3=indeterminate@
qu.2.4.part.3.question=(Unset)@
qu.2.4.part.3.answer.2=minimum@
qu.2.4.part.3.answer.1=maximum@
qu.2.4.part.3.mode=List@
qu.2.4.part.3.display=menu@
qu.2.4.part.3.credit.3=0.0@
qu.2.4.part.3.credit.2=0.0@
qu.2.4.part.3.credit.1=1.0@
qu.2.4.question=<p>An individual values two goods, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math>, according to the following utility function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$Upretty</p><p>The price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> is Px=$Px, and the price of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> is Py=$Py. The individual has $m dollars.</p><p>&nbsp;</p><p>If the first order derivatives are the following, (where L is lambda)</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>: $derXpretty</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>Y</mi></mrow></mfrac></mrow></mstyle></math>: $derYpretty</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mo lspace='0.0em' rspace='0.0em'>&Laplacetrf;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>L</mi></mrow></mfrac></mrow></mstyle></math>: $derLpretty</p><p>&nbsp;</p><p>Find the bordered Hessian:</p><p>&nbsp;</p><p>&nbsp; (To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <span style="vertical-align: -4px"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mi>m</mi></mrow></mstyle></math></span>and set the dimensions yourself.</p><p>Also, please make sure to use capital X and Y and be careful about order of operations when entering answers.)</p><p>&nbsp;H=<span>&nbsp;</span><1><span> (Enter in the order X, Y, lambda.)<br /></span></p><p>&nbsp;</p><p>&nbsp;Now evaluate the bordered Hessian if the critical values (stationary values) are X=$X and Y=$Y. What is the determinant of the bordered Hessian?</p><p>&nbsp;</p><p>(Enter your answer to at least two decimal places.)</p><p>det(H)=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>Given this determinant, do the values </span>X=$X and Y=$Y represent a maximum, minimum, or is it indeterminate?</p><p><span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.2.5.mode=Inline@
qu.2.5.name=evaluate Hessian and determine SOC 3x3@
qu.2.5.comment=<p>The Hessian is found by twice differentiating the function $F , and arranging the second derivatives in a matrix $G.</p>
<p>Then we evaluate the matrix at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T&nbsp; to find $R.</p>
<p>We know that if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T satisfied the first order conditions, then for this to be a minimum a sufficient condition is that the Hessian is positive semi-definite. This requires that the principal minors are all positive. Recall that the principal minors are determinants of matrices formed from the Hessian by starting with the 1x1 matrix containing the upper left most entry. Then we look at the 2x2 by adding one row from below and one column from the right. We keep going until we finally have the entire Hessian matrix.</p>
<p>For a maximum the principal minors are alternating in sign starting with negative.<br />
For any other situation the Hessian is inconclusive and the point may be a maximum a minimum or neither.</p>
<p>In this instance since the Hessian is 3x3 we look at the upper left hand entry, then the determinant of the upper left 2x2 matrix, and finally the determinant of the whole matrix. Looking at the upper left entry $R1, the determinant of the upper left 2x2 matrix, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$R2<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D2, and the whole matrix, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$R<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D, we find that the Hessian $Y, which is $Z.</p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$a=range(-1,1,1);
$b=range(-1,1,1);
$c=range(-1,1,1);
$A=maple("
randomize():
F:=randpoly([X,Y,Z], degree = 3):
f:=MathML[ExportPresentation](F):
JX:=diff($F,X):
Jx:=MathML[ExportPresentation](JX):
JY:=diff($F,X):
Jy:=MathML[ExportPresentation](JY):
JZ:=diff($F,X):
Jz:=MathML[ExportPresentation](JZ):
H:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y),diff(diff($F,X),Z)],[diff(diff($F,Y),X),diff(diff($F,Y),Y),diff(diff($F,Y),Z)],[diff(diff($F,Z),X),diff(diff($F,Z),Y),diff(diff($F,Z),Z)]]):
H1:=diff(diff($F,X),X):
H2:=Matrix([[diff(diff($F,X),X),diff(diff($F,X),Y)],[diff(diff($F,Y),X),diff(diff($F,Y),Y)]]):
G:=MathML[ExportPresentation](H):
G1:=MathML[ExportPresentation](H1):
G2:=MathML[ExportPresentation](H2):
Q:=eval(H, [X=($a),Y=($b), Z=($c)]):
Q1:=eval(H1, [X=($a),Y=($b), Z=($c)]):
Q2:=eval(H2, [X=($a),Y=($b), Z=($c)]):
E:=LinearAlgebra[Determinant](Q):
E2:=LinearAlgebra[Determinant](Q2):
R:=MathML[ExportPresentation](Q):
R1:=MathML[ExportPresentation](Q1):
R2:=MathML[ExportPresentation](Q2):
S:=Matrix([$a,$b,$c]):
T:=MathML[ExportPresentation](S):
with(LinearAlgebra):
if IsDefinite(Q, query = 'indefinite') = true then i := 2 
elif IsDefinite(Q, query = 'negative_definite') = true then i := 0 
elif IsDefinite(Q, query = 'negative_semidefinite') = true then i := 1 
elif IsDefinite(Q, query = 'positive_definite') = true then i := 4 
elif IsDefinite(Q, query = 'positive_semidefinite') = true then i := 3 
end if:
f,F,convert(H,string),convert(H2,string),H1,G, G1, G2,convert(Q,string),convert(Q2,string),Q1,R,R1,R2,T,i,E,E2
");
$F=switch(0,$A);
$H=switch(2,$A);
$G=switch(5,$A);
$R=switch(11,$A);
$T=switch(14,$A);
$i=switch(15,$A);
$D=switch(16,$A);
$D2=switch(17,$A);
$Y=switch(($i),"negative definite","negative semi-definite","indefinite","positive semi-definite","positive definite");
$N1=switch(($i),"negative semi-definite","indefinite","positive semi-definite","positive definite","negative definite");
$N2=switch(($i),"indefinite","positive semi-definite","positive definite","negative definite","negative semi-definite");
$N3=switch(($i),"positive semi-definite","positive definite","negative definite","negative semi-definite","indefinite");
$N4=switch(($i),"positive definite","negative definite","negative semi-definite","indefinite","positive semi-definite");
$X=switch(($i),"be a maximum","be inconclusive","be inconclusive","be inconclusive","be a minimum");
$M1=switch(($i),"be a maximum","be a minimum","be a minimum","be a minimum","be a minimum");
$M2=switch(($i),"be inconclusive","be a maximum","be a maximum","be a maximum","be inconclusive");
$Z=switch(($i),"indicates a maximum","is inconclusive","is inconclusive","is inconclusive","indicates a minimum");@
qu.2.5.uid=a609da79-33a2-4933-aac3-44a179bc0c7a@
qu.2.5.info=  Course=Introductory Mathematical Economics;
  Topic=Second Order Conditions;
  Sub-Topic=Hessian;
  Difficulty=Hard;
  Author=Asha Sadanand;
@
qu.2.5.weighting=1,1,1@
qu.2.5.numbering=alpha@
qu.2.5.part.1.name=sro_id_1@
qu.2.5.part.1.maple_answer=printf("$G");@
qu.2.5.part.1.editing=useHTML@
qu.2.5.part.1.question=(Unset)@
qu.2.5.part.1.libname=@
qu.2.5.part.1.mode=Maple@
qu.2.5.part.1.allow2d=2@
qu.2.5.part.1.plot=@
qu.2.5.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if evalb(simplify(ans[i,j] - resp[i,j])=0)
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.2.5.part.1.type=maple@
qu.2.5.part.2.grader=exact@
qu.2.5.part.2.name=sro_id_2@
qu.2.5.part.2.editing=useHTML@
qu.2.5.part.2.answer.5=$N4@
qu.2.5.part.2.display.permute=true@
qu.2.5.part.2.answer.4=$N3@
qu.2.5.part.2.answer.3=$N2@
qu.2.5.part.2.question=(Unset)@
qu.2.5.part.2.answer.2=$N1@
qu.2.5.part.2.answer.1=$Y@
qu.2.5.part.2.mode=List@
qu.2.5.part.2.display=menu@
qu.2.5.part.2.credit.5=0.0@
qu.2.5.part.2.credit.4=0.0@
qu.2.5.part.2.credit.3=0.0@
qu.2.5.part.2.credit.2=0.0@
qu.2.5.part.2.credit.1=1.0@
qu.2.5.part.3.grader=exact@
qu.2.5.part.3.name=sro_id_3@
qu.2.5.part.3.editing=useHTML@
qu.2.5.part.3.display.permute=true@
qu.2.5.part.3.answer.3=$M2@
qu.2.5.part.3.question=(Unset)@
qu.2.5.part.3.answer.2=$M1@
qu.2.5.part.3.answer.1=$X@
qu.2.5.part.3.mode=List@
qu.2.5.part.3.display=menu@
qu.2.5.part.3.credit.3=0.0@
qu.2.5.part.3.credit.2=0.0@
qu.2.5.part.3.credit.1=1.0@
qu.2.5.question=<p>Consider the function $F. Find the Hessian matrix associated with this function.&nbsp; <1></p><p><span>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mtd><mtd><mrow><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mtd><mtd><mrow><mi>Z</mi></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math></span>$T, the Hessian is<span>&nbsp;</span><2><span>; if the first order conditions held at this point it would&nbsp;&nbsp; <3><span>.<br /></span></span></p>@

qu.2.6.mode=Inline@
qu.2.6.name=evaluate bordered Hessian and determine SOC@
qu.2.6.comment=<p>To find the bordered Hessian, we first construct the Lagrangean, $F1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>&lambda;</mi><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$C1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>The Hessian is found by twice differentiating the Lagrangean, and arranging the second derivatives in a matrix, $G.</p>
<p>Then at the point <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T, the evaluated Hessian is $R. (If the first order conditions held at this point), a sufficient condition for the  point <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T to be a maximum is that the determinant if the evaluated Hessian matrix is positive; for a minimum it is negative; and with a zero determinant it is inconclusive.</p>
<p>In this case, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi></mrow></mstyle></math>$R<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D, which means that the Hessian $X.</p>@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=$a=range(0,1,1);
$b=range(0,1,1);
$p=range(0,9,1);
$q=range(0,9,1);
$I=range(0,20,1);
$A=maple("
randomize():
F1:=randpoly([X,Y], degree = 3):
f1:=MathML[ExportPresentation](F1):
C:=$I-(($p)*X+($q)*Y):
c:=MathML[ExportPresentation](C):
F:=F1+L*(C):
H:=Matrix([[diff(diff(F,X),X),diff(diff(F,X),Y),diff(diff(F,X),L)],[diff(diff(F,Y),X),diff(diff(F,Y),Y),diff(diff(F,Y),L)],
[diff(diff(F,L),X),diff(diff(F,L),Y),diff(diff(F,L),L)]]):
G:=MathML[ExportPresentation](H):
Q:=eval(H, [X=($a),Y=($b)]):
R:=MathML[ExportPresentation](Q):
S:=Matrix([$a,$b]):
T:=MathML[ExportPresentation](S):
E:=LinearAlgebra[Determinant](Q):
if E>0 then i:=0
elif E=0  then i := 1 
elif E<0 then i := 2
end if:
f1,F1,c,convert(H,string),G,convert(Q,string),R,T,i,E
");
$F1=switch(0,$A);
$C1=switch(2,$A);
$H=switch(3,$A);
$G=switch(4,$A);
$R=switch(6,$A);
$T=switch(7,$A);
$i=switch(8,$A);
$D=switch(9,$A);
$Y=switch(($i),"indicate a maximum","be inconclusive", "indicate a minimum");
$N1=switch(($i),"be inconclusive", "indicate a minimum","indicate a maximum");
$N2=switch(($i), "indicate a minimum","indicate a maximum","be inconclusive");
$X=switch(($i),"indicates a maximum","is inconclusive", "indicates a minimum");
$Q=switch(5,$A);@
qu.2.6.uid=257f8096-4e1f-4e51-b7ed-012a86e0aece@
qu.2.6.weighting=1,1,1@
qu.2.6.numbering=alpha@
qu.2.6.part.1.name=sro_id_1@
qu.2.6.part.1.maple_answer=printf("$G")@
qu.2.6.part.1.editing=useHTML@
qu.2.6.part.1.question=(Unset)@
qu.2.6.part.1.libname=@
qu.2.6.part.1.mode=Maple@
qu.2.6.part.1.allow2d=2@
qu.2.6.part.1.plot=@
qu.2.6.part.1.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if evalb(simplify(ans[i,j] - resp[i,j])=0)
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.2.6.part.1.type=maple@
qu.2.6.part.2.name=sro_id_2@
qu.2.6.part.2.maple_answer=printf("$R")@
qu.2.6.part.2.editing=useHTML@
qu.2.6.part.2.question=(Unset)@
qu.2.6.part.2.libname=@
qu.2.6.part.2.mode=Maple@
qu.2.6.part.2.allow2d=2@
qu.2.6.part.2.plot=@
qu.2.6.part.2.maple=ans:=$Q:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.2.6.part.2.type=maple@
qu.2.6.part.3.grader=exact@
qu.2.6.part.3.name=sro_id_3@
qu.2.6.part.3.editing=useHTML@
qu.2.6.part.3.display.permute=true@
qu.2.6.part.3.answer.3=$N2@
qu.2.6.part.3.question=(Unset)@
qu.2.6.part.3.answer.2=$N1@
qu.2.6.part.3.answer.1=$Y@
qu.2.6.part.3.mode=List@
qu.2.6.part.3.display=menu@
qu.2.6.part.3.credit.3=0.0@
qu.2.6.part.3.credit.2=0.0@
qu.2.6.part.3.credit.1=1.0@
qu.2.6.question=<p>Consider the function $F1, which is being optimized with the constraint $C1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>. Find the Bordered Hessian matrix associated with this problem. (Enter it in the order X ,Y ,lambda.)<span> </span><1><span>.</span></p><p><span>At <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$T, we evaluate the matrix to find <span>&nbsp;</span><2><span>&nbsp;</span>.</span></p><p><span><span>If the first order conditions held at this point the Hessian would </span></span><3><span>.<br /></span></p>@

