qu.1.topic=First Order Conditions@

qu.1.1.mode=Inline@
qu.1.1.name=Optimization - Two Variables with S.O.C. no steps@
qu.1.1.comment=<p>The critical values (stationary points) can be found by:<br />
(1) Finding the derivative by X.<br />
(2) Finding the derivative by Y.<br />
(3) Setting both equations equal to zero and solving for X and Y.</p>
<p>The nature of the critical values can be found by:<br />
(1) Finding the Hessian matrix.<br />
(2) Finding the eigenvalues of the Hessian matrix.<br />
(3) If they are both positive, it is a minimum.&nbsp; If they are both negative, it is a maximum.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=range(-100,100);
condition:not(eq($a,0));
$b=range(-100,100);
condition:not(eq($b,0));
$c=range(-100,100);
condition:not(eq($c,0));
$d=range(-100,100);
condition:not(eq($d,0));
$e=range(-100,100);
condition:not(eq($e,0));
$g=range(-100,100);
condition:not(eq($g,0));
$v=maple("
if $d>0 and $g>0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g<0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d>0 and $g<0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g>0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
end if:
fx:=diff(F,X):
fy:=diff(F,Y):
fxx:=diff(fx,X):
fxy:=diff(fx,Y):
fyy:=diff(fy,Y):
det:=fxx*fyy-fxy^2:
if fxx>0 and det>0 then k:=0
elif fxx<0 and det>0 then k:=1
else k:=2
end if:
ratio:=solve(fx=fy,X):
Ycrit:=solve(eval(fy,X=ratio)=0,Y):
Xcrit:=eval(ratio,Y=Ycrit):
Fpretty:=MathML[ExportPresentation](F):
convert(F,string),convert(fx,string),convert(fy,string),convert(fxx,string),convert(fxy,string),convert(fyy,string),det,k,ratio,Ycrit,Xcrit,Fpretty
");
$F=switch(11,$v);
$fx=switch(1,$v);
$fy=switch(2,$v);
$fxx=switch(3,$v);
$fxy=switch(4,$v);
$fyy=switch(5,$v);
$det=switch(6,$v);
$k=switch(7,$v);
$Ycrit=switch(9,$v);
$Xcrit=switch(10,$v);
$ans2=switch($k,'min','max','indeterminate');
$wrong1=switch($k,'max','indeterminate','min');
$wrong2=switch($k,'indeterminate','min','max');@
qu.1.1.uid=7b27ffe3-d72d-4106-981a-6aa0ceb6696d@
qu.1.1.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Two Variables;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.1.weighting=1,1,1@
qu.1.1.numbering=alpha@
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qu.1.1.part.1.editing=useHTML@
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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=$Xcrit@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.answer.units=@
qu.1.1.part.2.numStyle=   arithmetic@
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.showUnits=false@
qu.1.1.part.2.err=0.05@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.mode=Numeric@
qu.1.1.part.2.grading=toler_abs@
qu.1.1.part.2.negStyle=minus@
qu.1.1.part.2.answer.num=$Ycrit@
qu.1.1.part.3.grader=exact@
qu.1.1.part.3.name=sro_id_3@
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.display.permute=true@
qu.1.1.part.3.answer.3=$wrong2@
qu.1.1.part.3.question=(Unset)@
qu.1.1.part.3.answer.2=$wrong1@
qu.1.1.part.3.answer.1=$ans2@
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qu.1.1.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">What are the stationary points (critical values)? (Enter your answers to at least 2 decimal places.)</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p align="left"><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>Ycrit</mi></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Is the critical point a maximum, minimum or is it indeterminate?</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>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.1.2.mode=Inline@
qu.1.2.name=Optimization - Two Variables no steps@
qu.1.2.comment=<p>The critical values (stationary points) can be found by:<br />
(1) Finding the derivative by X.<br />
(2) Finding the derivative by Y.<br />
(3) Setting both equations equal to zero and solving for X and Y.</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=range(-100,100);
condition:not(eq($a,0));
$b=range(-100,100);
condition:not(eq($b,0));
$c=range(-100,100);
condition:not(eq($c,0));
$d=range(-100,100);
condition:not(eq($d,0));
$e=range(-100,100);
condition:not(eq($e,0));
$g=range(-100,100);
condition:not(eq($g,0));
$v=maple("
if $d>0 and $g>0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g<0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d>0 and $g<0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g>0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
end if:
fx:=diff(F,X):
fy:=diff(F,Y):
fxx:=diff(fx,X):
fxy:=diff(fx,Y):
fyy:=diff(fy,Y):
det:=fxx*fyy-fxy^2:
if det>0 then k:=0
elif det<0 then k:=1
elif det=0 then k:=2
end if:
ratio:=solve(fx=fy,X):
Ycrit:=solve(eval(fy,X=ratio)=0,Y):
Xcrit:=eval(ratio,Y=Ycrit):
Fpretty:=MathML[ExportPresentation](F):
convert(F,string),convert(fx,string),convert(fy,string),convert(fxx,string),convert(fxy,string),convert(fyy,string),det,k,ratio,Ycrit,Xcrit,Fpretty
");
$F=switch(11,$v);
$fx=switch(1,$v);
$fy=switch(2,$v);
$fxx=switch(3,$v);
$fxy=switch(4,$v);
$fyy=switch(5,$v);
$det=switch(6,$v);
$k=switch(7,$v);
$Ycrit=switch(9,$v);
$Xcrit=switch(10,$v);
$ans2=switch($k,'min','max','unknown');
$wrong1=switch($k,'max','unknown','min');
$wrong2=switch($k,'unknown','min','max');@
qu.1.2.uid=25733ac5-a6d6-4fb3-8986-6a4ecafa47be@
qu.1.2.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Two Variables;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.1.2.weighting=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.err=0.05@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.mode=Numeric@
qu.1.2.part.1.grading=toler_abs@
qu.1.2.part.1.negStyle=both@
qu.1.2.part.1.answer.num=$Xcrit@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.answer.units=@
qu.1.2.part.2.numStyle=   arithmetic@
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.showUnits=false@
qu.1.2.part.2.err=0.05@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.mode=Numeric@
qu.1.2.part.2.grading=toler_abs@
qu.1.2.part.2.negStyle=minus@
qu.1.2.part.2.answer.num=$Ycrit@
qu.1.2.question=<p>Given the following 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>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>$F</p><p>&nbsp;</p><p align="left">What are the stationary points (critical values)? (Enter your answers to at least 2 decimal places.)</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p align="left"><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>Ycrit</mi></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p align="left">&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p>@

qu.1.3.mode=Inline@
qu.1.3.name=Optimization - Two Variables with S.O.C. and steps@
qu.1.3.comment=<p>The critical values (stationary points) can be found by setting both derivates&nbsp;equal to zero and solving for X and Y.</p>
<p>The nature of the critical values can be found by finding the eigenvalues of the Hessian matrix.&nbsp; If they are both positive, it is a minimum. If they are both negative, it is a maximum.</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=range(-100,100);
condition:not(eq($a,0));
$b=range(-100,100);
condition:not(eq($b,0));
$c=range(-100,100);
condition:not(eq($c,0));
$d=range(-100,100);
condition:not(eq($d,0));
$e=range(-100,100);
condition:not(eq($e,0));
$g=range(-100,100);
condition:not(eq($g,0));
$v=maple("
if $d>0 and $g>0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g<0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d>0 and $g<0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g>0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
end if:
fx:=diff(F,X):
fy:=diff(F,Y):
fxx:=diff(fx,X):
fxy:=diff(fx,Y):
fyy:=diff(fy,Y):
det:=fxx*fyy-fxy^2:
if fxx>0 and det>0 then k:=0
elif fxx<0 and det>0 then k:=1
else k:=2
end if:
ratio:=solve(fx=fy,X):
Ycrit:=solve(eval(fy,X=ratio)=0,Y):
Xcrit:=eval(ratio,Y=Ycrit):
Fpretty:=MathML[ExportPresentation](F):
convert(F,string),convert(fx,string),convert(fy,string),convert(fxx,string),convert(fxy,string),convert(fyy,string),det,k,ratio,Ycrit,Xcrit,Fpretty
");
$F=switch(11,$v);
$fx=switch(1,$v);
$fy=switch(2,$v);
$fxx=switch(3,$v);
$fxy=switch(4,$v);
$fyy=switch(5,$v);
$det=switch(6,$v);
$k=switch(7,$v);
$Ycrit=switch(9,$v);
$Xcrit=switch(10,$v);
$ans2=switch($k,'min','max','indeterminate');
$wrong1=switch($k,'max','indeterminate','min');
$wrong2=switch($k,'indeterminate','min','max');@
qu.1.3.uid=246bd788-0eed-4a70-9273-96ced2b8b39b@
qu.1.3.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Two Variables;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Walks Students Through Steps;
@
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evalb(($ANSWER)=(resp));@
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qu.1.3.part.5.question=(Unset)@
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qu.1.3.part.5.grading=toler_abs@
qu.1.3.part.5.negStyle=both@
qu.1.3.part.5.answer.num=$Xcrit@
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qu.1.3.part.6.numStyle=   arithmetic@
qu.1.3.part.6.editing=useHTML@
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qu.1.3.part.6.err=0.05@
qu.1.3.part.6.question=(Unset)@
qu.1.3.part.6.mode=Numeric@
qu.1.3.part.6.grading=toler_abs@
qu.1.3.part.6.negStyle=minus@
qu.1.3.part.6.answer.num=$Ycrit@
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qu.1.3.part.7.maple_answer=$fxx@
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qu.1.3.part.7.question=(Unset)@
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evalb(($ANSWER)=(resp));@
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qu.1.3.part.11.credit.3=0.0@
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qu.1.3.part.11.credit.1=1.0@
qu.1.3.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">Find the first order conditions for optimizing this function:</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span> </span><1><span> </span>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi mathvariant='normal'>Y</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><3><span> </span>=<span>&nbsp;</span><4><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above first order conditions, what are the stationary points (critical values)? (Enter your answers to at least 2 decimal places.)</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><5><span>&nbsp;</span></p><p align="left"><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>Ycrit</mi></mrow></mstyle></math>=<span>&nbsp;</span><6><span>&nbsp;</span></span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">What are the second derivatives?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span> </span><7><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>Y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>XY</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><9><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>YX</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><10><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Given the above second derivatives, is the critical point a maximum, minimum or is it indeterminate?</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>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><11><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.2.topic=Second Order Conditions@

qu.2.1.mode=Inline@
qu.2.1.name=Optimization - Two Variables with S.O.C. no steps@
qu.2.1.comment=<p>The critical values (stationary points) can be found by:<br />
(1) Finding the derivative by X.<br />
(2) Finding the derivative by Y.<br />
(3) Setting both equations equal to zero and solving for X and Y.</p>
<p>The nature of the critical values can be found by:<br />
(1) Finding the Hessian matrix.<br />
(2) Finding the eigenvalues of the Hessian matrix.<br />
(3) If they are both positive, it is a minimum.&nbsp; If they are both negative, it is a maximum.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=range(-100,100);
condition:not(eq($a,0));
$b=range(-100,100);
condition:not(eq($b,0));
$c=range(-100,100);
condition:not(eq($c,0));
$d=range(-100,100);
condition:not(eq($d,0));
$e=range(-100,100);
condition:not(eq($e,0));
$g=range(-100,100);
condition:not(eq($g,0));
$v=maple("
if $d>0 and $g>0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g<0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d>0 and $g<0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g>0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
end if:
fx:=diff(F,X):
fy:=diff(F,Y):
fxx:=diff(fx,X):
fxy:=diff(fx,Y):
fyy:=diff(fy,Y):
det:=fxx*fyy-fxy^2:
if fxx>0 and det>0 then k:=0
elif fxx<0 and det>0 then k:=1
else k:=2
end if:
ratio:=solve(fx=fy,X):
Ycrit:=solve(eval(fy,X=ratio)=0,Y):
Xcrit:=eval(ratio,Y=Ycrit):
Fpretty:=MathML[ExportPresentation](F):
convert(F,string),convert(fx,string),convert(fy,string),convert(fxx,string),convert(fxy,string),convert(fyy,string),det,k,ratio,Ycrit,Xcrit,Fpretty
");
$F=switch(11,$v);
$fx=switch(1,$v);
$fy=switch(2,$v);
$fxx=switch(3,$v);
$fxy=switch(4,$v);
$fyy=switch(5,$v);
$det=switch(6,$v);
$k=switch(7,$v);
$Ycrit=switch(9,$v);
$Xcrit=switch(10,$v);
$ans2=switch($k,'min','max','indeterminate');
$wrong1=switch($k,'max','indeterminate','min');
$wrong2=switch($k,'indeterminate','min','max');@
qu.2.1.uid=7b27ffe3-d72d-4106-981a-6aa0ceb6696d@
qu.2.1.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Two Variables;
  Author=Katherine Dare;
  Difficulty=Medium;
@
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qu.2.1.part.1.grading=toler_abs@
qu.2.1.part.1.negStyle=both@
qu.2.1.part.1.answer.num=$Xcrit@
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qu.2.1.part.2.answer.units=@
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qu.2.1.part.2.editing=useHTML@
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qu.2.1.part.2.err=0.05@
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qu.2.1.part.2.grading=toler_abs@
qu.2.1.part.2.negStyle=minus@
qu.2.1.part.2.answer.num=$Ycrit@
qu.2.1.part.3.grader=exact@
qu.2.1.part.3.name=sro_id_3@
qu.2.1.part.3.editing=useHTML@
qu.2.1.part.3.display.permute=true@
qu.2.1.part.3.answer.3=$wrong2@
qu.2.1.part.3.question=(Unset)@
qu.2.1.part.3.answer.2=$wrong1@
qu.2.1.part.3.answer.1=$ans2@
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qu.2.1.part.3.credit.1=1.0@
qu.2.1.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">What are the stationary points (critical values)? (Enter your answers to at least 2 decimal places.)</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p align="left"><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>Ycrit</mi></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Is the critical point a maximum, minimum or is it indeterminate?</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>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.2.2.mode=Inline@
qu.2.2.name=Optimization - Two Variables with S.O.C. and steps@
qu.2.2.comment=<p>The critical values (stationary points) can be found by setting both derivates&nbsp;equal to zero and solving for X and Y.</p>
<p>The nature of the critical values can be found by finding the eigenvalues of the Hessian matrix.&nbsp; If they are both positive, it is a minimum. If they are both negative, it is a maximum.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=range(-100,100);
condition:not(eq($a,0));
$b=range(-100,100);
condition:not(eq($b,0));
$c=range(-100,100);
condition:not(eq($c,0));
$d=range(-100,100);
condition:not(eq($d,0));
$e=range(-100,100);
condition:not(eq($e,0));
$g=range(-100,100);
condition:not(eq($g,0));
$v=maple("
if $d>0 and $g>0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g<0 then
F:=($a)+($b)*X+($c)*Y+($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d>0 and $g<0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
elif $d<0 and $g>0 then
F:=($a)+($b)*X+($c)*Y-($d)*(X^2)+($e)*X*Y+($g)*(Y^2)
end if:
fx:=diff(F,X):
fy:=diff(F,Y):
fxx:=diff(fx,X):
fxy:=diff(fx,Y):
fyy:=diff(fy,Y):
det:=fxx*fyy-fxy^2:
if fxx>0 and det>0 then k:=0
elif fxx<0 and det>0 then k:=1
else k:=2
end if:
ratio:=solve(fx=fy,X):
Ycrit:=solve(eval(fy,X=ratio)=0,Y):
Xcrit:=eval(ratio,Y=Ycrit):
Fpretty:=MathML[ExportPresentation](F):
convert(F,string),convert(fx,string),convert(fy,string),convert(fxx,string),convert(fxy,string),convert(fyy,string),det,k,ratio,Ycrit,Xcrit,Fpretty
");
$F=switch(11,$v);
$fx=switch(1,$v);
$fy=switch(2,$v);
$fxx=switch(3,$v);
$fxy=switch(4,$v);
$fyy=switch(5,$v);
$det=switch(6,$v);
$k=switch(7,$v);
$Ycrit=switch(9,$v);
$Xcrit=switch(10,$v);
$ans2=switch($k,'min','max','indeterminate');
$wrong1=switch($k,'max','indeterminate','min');
$wrong2=switch($k,'indeterminate','min','max');@
qu.2.2.uid=246bd788-0eed-4a70-9273-96ced2b8b39b@
qu.2.2.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization;
  Sub-Topic=Two Variables;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Walks Students Through Steps;
@
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qu.2.2.part.3.maple_answer=$fy@
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evalb(($ANSWER)=(resp));@
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qu.2.2.part.5.question=(Unset)@
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qu.2.2.part.5.grading=toler_abs@
qu.2.2.part.5.negStyle=both@
qu.2.2.part.5.answer.num=$Xcrit@
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qu.2.2.part.6.answer.num=$Ycrit@
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qu.2.2.part.7.maple_answer=$fxx@
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evalb(($ANSWER)=(resp));@
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qu.2.2.part.8.maple_answer=$fyy@
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evalb(($ANSWER)=(resp));@
qu.2.2.part.8.type=formula@
qu.2.2.part.9.name=sro_id_9@
qu.2.2.part.9.maple_answer=$fxy@
qu.2.2.part.9.editing=useHTML@
qu.2.2.part.9.question=(Unset)@
qu.2.2.part.9.libname=@
qu.2.2.part.9.mode=Maple@
qu.2.2.part.9.allow2d=1@
qu.2.2.part.9.plot=@
qu.2.2.part.9.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.2.part.9.type=formula@
qu.2.2.part.10.name=sro_id_10@
qu.2.2.part.10.maple_answer=$fxy@
qu.2.2.part.10.editing=useHTML@
qu.2.2.part.10.question=(Unset)@
qu.2.2.part.10.libname=@
qu.2.2.part.10.mode=Maple@
qu.2.2.part.10.allow2d=1@
qu.2.2.part.10.plot=@
qu.2.2.part.10.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.2.2.part.10.type=formula@
qu.2.2.part.11.grader=exact@
qu.2.2.part.11.name=sro_id_11@
qu.2.2.part.11.editing=useHTML@
qu.2.2.part.11.display.permute=true@
qu.2.2.part.11.answer.3=$wrong2@
qu.2.2.part.11.question=(Unset)@
qu.2.2.part.11.answer.2=$wrong1@
qu.2.2.part.11.answer.1=$ans2@
qu.2.2.part.11.mode=List@
qu.2.2.part.11.display=menu@
qu.2.2.part.11.credit.3=0.0@
qu.2.2.part.11.credit.2=0.0@
qu.2.2.part.11.credit.1=1.0@
qu.2.2.question=<p>Given the following function:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p><p>&nbsp;</p><p align="left">Find the first order conditions for optimizing this function:</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>X</mi></mrow></mfrac></mrow></mstyle></math>=<span> </span><1><span> </span>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi mathvariant='normal'>Y</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><3><span> </span>=<span>&nbsp;</span><4><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">Given the above first order conditions, what are the stationary points (critical values)? (Enter your answers to at least 2 decimal places.)</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>Xcrit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><5><span>&nbsp;</span></p><p align="left"><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>Ycrit</mi></mrow></mstyle></math>=<span>&nbsp;</span><6><span>&nbsp;</span></span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">What are the second derivatives?</p><p align="left">&nbsp;</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span> </span><7><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><msup><mi>Y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><8><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>XY</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><9><span>&nbsp;</span></p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mrow><mn>2</mn></mrow></msup><mi>F</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&PartialD;</mo><mi>YX</mi></mrow></mfrac></mrow></mstyle></math>=<span>&nbsp;</span><10><span>&nbsp;</span></p><p align="left">&nbsp;</p><p align="left">&nbsp;</p><p align="left">Given the above second derivatives, is the critical point a maximum, minimum or is it indeterminate?</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>Xcrit</mi></mrow></mstyle></math> is a <span>&nbsp;</span><11><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.topic=Word Problems - Set Up Only@

qu.3.1.mode=Inline@
qu.3.1.name=utility max word problem -- give model@
qu.3.1.comment=<p>The consumer wants to maximize utility subject to the budget constraint. The utility is given by $u, and the budget constraint is $b. Therefore, we form the Lagrangean by taking the utility and adding to&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>J</mi></mrow></mstyle></math>(the Lagrange multiplier)&nbsp; times the budget constraint (rearranged so that it is all on the right hand side of the equation), to get $Ans.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$A=maple("
U:=(x^2)*(y^2):
B:=p1*x +p2*y:
u:=MathML[ExportPresentation](U):
b:=MathML[ExportPresentation](B=M):
answer:=((x^2)*(y^2))+J*(M-'P1'*x-'P2'*y):
a1:=MathML[ExportPresentation](answer):
convert(answer,string),a1,u,b
");
$ans=switch(0,$A);
$Ans=switch(1,$A);
$u=switch(2,$A);
$b=switch(3,$A);@
qu.3.1.uid=30646cc1-1978-49d9-bda1-a52a57abfdd6@
qu.3.1.weighting=1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.answer=$ans@
qu.3.1.part.1.mode=Formula@
qu.3.1.question=<p>A consumer wants to maximize her utility given by&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>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><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> by choosing amounts 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>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>to purchase. The price of good x 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>P1</mi></mrow></mstyle></math> , the price 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>P2</mi></mrow></mstyle></math> is , and the consumer has a total 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>M</mi></mrow></mstyle></math> to spend.</p><p>&nbsp;</p><p>Form the Lagrangean that best depicts this situation.<span> Use&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>J</mi></mrow></mstyle></math> to denote the Lagrange multiplier, since&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>&lambda;</mi></mrow></mstyle></math> is not available in the answer field.&nbsp; </span><1><span>&nbsp;</span></p>@

qu.3.2.mode=Inline@
qu.3.2.name=Word problem - Cost minimization - with SOC@
qu.3.2.comment=<p>To find the Lagrangean&nbsp; for the firm's cost minimization, we take the firm's cost $F1 and add to it <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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> times the constraint of producing <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>units. But in this case we use "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>" for the Lagrange multiplier. We find the Lagrangean to be $A2.</p>
<p>We find the first order conditions by differentiating the Lagrangean with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>J</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>respectively, to find:</p>
<p>$fk</p>
<p>$fl</p>
<p>$fj</p>
<p>These equations in many instances can be solved by taking the first two equations and moving the Lagrange multiplier term to the right hand side, and then dividing one equation by the other. The main purpose of this step is the cancel the Lagrange multiplier on the right hand side of the equation, as follows:</p>
<p>&nbsp;$f1k / $f1l =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$f2k <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&sdot;</mo><mi>J</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></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><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$f2l <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&sdot;</mo><mi>J</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math></p>
<p>or $foc. We take this equation along with the last equation to find our solutions <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$AnsK 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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$AnsL.</p>
<p>&nbsp;</p>
<p>To check that this is a minimum, we need to find the bordered Hessian, evaluate it at the solution, and check that its determinant is negative. We find the bordered Hessian by differentiating the Lagrangean again with all the variables, and arranging these second derivatives in a matrix. The bordered Hessian is $G,</p>
<p>which is evaluated at the solution <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>K</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>L</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$AnsK, $AnsL);&nbsp; and finally, taking the determinant, we find $E1</p>
<p>which is indeed negative, giving us a minimum.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$A=maple("
F:=w*L+v*K+J*(Q-(K^a)*(L^b)):
F1:=w*L+v*K:
f1:=MathML[ExportPresentation](F1):
F2:=(Q-(K^a)*(L^b)):
F0:=MathML[ExportPresentation](F):
FK:=diff(F,K):
FL:=diff(F,L):
FJ:=diff(F,J):
fk:=MathML[ExportPresentation](FK=0):
fl:=MathML[ExportPresentation](FL=0):
fj:=MathML[ExportPresentation](FJ=0):
F1K:=diff(F1,K):
F1L:=diff(F1,L):
F2K:=diff(F2,K):
F2L:=diff(F2,L):
f1k:=MathML[ExportPresentation](F1K):
f1l:=MathML[ExportPresentation](F1L):
f2k:=MathML[ExportPresentation](F2K):
f2l:=MathML[ExportPresentation](F2L):
FOC1:=(F1K)/(F1L):
FOC11:=simplify(FOC1):
FOC2:=(F2K)/(F2L):
FOC22:=simplify(FOC2):
foc:=MathML[ExportPresentation](FOC11=FOC22):
ansL:=((b*v)/(a*w))^(b/(a+b))*Q^(1/(a+b)):
ansK:=((a*w)/(b*v))^(a/(a+b))*Q^(1/(a+b)):
AnsK:=MathML[ExportPresentation](ansK):
AnsL:=MathML[ExportPresentation](ansL):
F,F0,FK,FL,FJ,fk,fl,fj,convert(ansK,string),convert(ansL,string),AnsK,AnsL,f1,f1k,f1l,f2k,f2l,foc,x
");
$A1=switch(0,$A);
$A2=switch(1,$A);
$FK=switch(2,$A);
$FL=switch(3,$A);
$FJ=switch(4,$A);
$fk=switch(5,$A);
$fl=switch(6,$A);
$fj=switch(7,$A);
$ansK=switch(8,$A);
$ansL=switch(9,$A);
$AnsK=switch(10,$A);
$AnsL=switch(11,$A);
$f1=switch(12,$A);
$f1k=switch(13,$A);
$f1l=switch(14,$A);
$f2k=switch(15,$A);
$f2l=switch(16,$A);
$foc=switch(17,$A);
$B=maple("
F:=$A1:
H:=Matrix([[diff(diff(F,K),K),diff(diff(F,K),L),diff(diff(F,K),J)],[diff(diff(F,L),K),diff(diff(F,L),L),diff(diff(F,L),J)],[diff(diff(F,J),K),diff(diff(F,J),L),diff(diff(F,J),J)]]):
G:=MathML[ExportPresentation](H):
R:=eval(H,[K=$ansK, L=$ansL]):
R1:=simplify(R):
E0:=LinearAlgebra[Determinant](R):
E:=simplify(E0):
E1:=MathML[ExportPresentation]($E):
convert(H,string), G, convert(R,string),convert(E,string),E1
");
$H=switch(0,$B);
$G=switch(1,$B);
$E=switch(3,$B);
$E1=switch(4,$B);@
qu.3.2.uid=01636975-a2aa-4fcf-9ce8-79d0bd2eb3ec@
qu.3.2.info=  Course=Introductory Mathematical Economics;
  Topic=Word Problem - Cost Minimization;
  Sub-Topic=Cobb Douglas Production;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.3.2.weighting=1,1,1,1,1,1,1,1,1,1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.answer=$A1@
qu.3.2.part.1.mode=Formula@
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.answer=$FK@
qu.3.2.part.2.mode=Formula@
qu.3.2.part.3.editing=useHTML@
qu.3.2.part.3.question=(Unset)@
qu.3.2.part.3.name=sro_id_3@
qu.3.2.part.3.answer=$FL@
qu.3.2.part.3.mode=Formula@
qu.3.2.part.4.editing=useHTML@
qu.3.2.part.4.question=(Unset)@
qu.3.2.part.4.name=sro_id_4@
qu.3.2.part.4.answer=$FJ@
qu.3.2.part.4.mode=Formula@
qu.3.2.part.5.editing=useHTML@
qu.3.2.part.5.question=(Unset)@
qu.3.2.part.5.name=sro_id_5@
qu.3.2.part.5.answer=$ansK@
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qu.3.2.part.6.question=(Unset)@
qu.3.2.part.6.name=sro_id_6@
qu.3.2.part.6.answer=$ansL@
qu.3.2.part.6.mode=Formula@
qu.3.2.part.7.grader=exact@
qu.3.2.part.7.name=sro_id_7@
qu.3.2.part.7.editing=useHTML@
qu.3.2.part.7.display.permute=true@
qu.3.2.part.7.answer.3=zero@
qu.3.2.part.7.question=(Unset)@
qu.3.2.part.7.answer.2=positive@
qu.3.2.part.7.answer.1=negative@
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qu.3.2.part.7.credit.1=1.0@
qu.3.2.part.8.name=sro_id_8@
qu.3.2.part.8.maple_answer=printf("$G")@
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qu.3.2.part.8.libname=@
qu.3.2.part.8.mode=Maple@
qu.3.2.part.8.allow2d=2@
qu.3.2.part.8.plot=@
qu.3.2.part.8.maple=ans:=$H:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] - $RESPONSE[i,j]=0
then grade:=grade+0.111111111111111112:
end if;
end;
end;
grade;@
qu.3.2.part.8.type=maple@
qu.3.2.part.9.editing=useHTML@
qu.3.2.part.9.question=(Unset)@
qu.3.2.part.9.name=sro_id_9@
qu.3.2.part.9.answer=$E@
qu.3.2.part.9.mode=Formula@
qu.3.2.part.10.grader=exact@
qu.3.2.part.10.name=sro_id_10@
qu.3.2.part.10.editing=useHTML@
qu.3.2.part.10.display.permute=true@
qu.3.2.part.10.answer.3=is inconclusive@
qu.3.2.part.10.question=(Unset)@
qu.3.2.part.10.answer.2=indicates a maximum@
qu.3.2.part.10.answer.1=indicates a minimum@
qu.3.2.part.10.mode=List@
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qu.3.2.part.10.credit.3=0.0@
qu.3.2.part.10.credit.2=0.0@
qu.3.2.part.10.credit.1=1.0@
qu.3.2.question=<p>A firm has a technology <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>K</mi><mrow><mi>a</mi></mrow></msup><msup><mi>L</mi><mrow><mi>b</mi></mrow></msup></mrow></mstyle></math>. If the price of labour is denoted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>w</mi></mrow></mstyle></math>and 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><mi>v</mi></mrow></mstyle></math>, and the firm wants to produce <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>units of output, give the Lagrangean used in solving the firm's cost minimization problem. Since&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>&lambda;</mi></mrow></mstyle></math> is not available in the solution field, please use "J" for the Lagrange multiplier.</p><p><span>Lagrangean&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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math> </span><1><span> </span>.</p><p>The first order conditions are (starting with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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 Lagrange multiplier <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>)</p><p><span>&nbsp;</span><2><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='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p><span>&nbsp;</span><3><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='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p><span>&nbsp;</span><4><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='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p>Solving these we find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><5><span>&nbsp;</span>, 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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><6><span>&nbsp;</span>.</p><p>To confirm that this is indeed a minimum, the determinant of the bordered Hessian must be&nbsp;<span> </span><7><span> </span>.</p><p>The bordered Hessian is <span>&nbsp;</span><8><span>. </span></p><p>Evaluating at the solution, and taking the determinant gives&nbsp; <9><span>. We can conclude that the bordered Hessian <span>&nbsp;</span><10><span>. <br /></span></span></p><p>&nbsp;</p>@

qu.3.3.mode=Inline@
qu.3.3.name=word problem - rate of return regulation@
qu.3.3.comment=<p>The revenue is found by first inverting the demand to find&nbsp; $Pr1. Then we find revenue by multiplying price times quantity, $R1. Profit is found by subtracting from revenue the cost of the monopoly of the labour and capital, $P1.</p>
<p>For rate of return regulation we first find the total return on capital, that is the revenue less the total cost of labour. Then we divide that amount by the amount of capital used. This gives us the return per unit of capital, $c1.</p>
<p>We combine everything to find the Lagrangean. $P1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></mstyle></math>($C1)</p>
<p>&nbsp;</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$R="(A-(1/m)*(K^a)*(L^b))*((K^a)*(L^b))";
$P="$R-(r*K)-(w*L)";
$c="($R-(w*L))/K";
$C="s-$c";
$Z=maple("
z1:=MathML[ExportPresentation](P=A-q/m):
z2:=MathML[ExportPresentation]($R):
z3:=MathML[ExportPresentation]($P):
z4:=MathML[ExportPresentation]($c):
z5:=MathML[ExportPresentation]($C):
z1,z2,z3,z4,z5
");
$Pr1=switch(0,$Z);
$R1=switch(1,$Z);
$P1=switch(2,$Z);
$c1=switch(3,$Z);
$C1=switch(4,$Z);@
qu.3.3.uid=d00baaae-d5a5-41a7-8cd0-9ffda3001cf0@
qu.3.3.info=  Course=Introductory Mathematical Economics;
  Topic=Word Problem;
  Sub-Topic=Rate Of Return Regulation;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.3.3.weighting=1,1,1,1,1@
qu.3.3.numbering=alpha@
qu.3.3.part.1.editing=useHTML@
qu.3.3.part.1.question=(Unset)@
qu.3.3.part.1.name=sro_id_1@
qu.3.3.part.1.answer=$R@
qu.3.3.part.1.mode=Formula@
qu.3.3.part.2.editing=useHTML@
qu.3.3.part.2.question=(Unset)@
qu.3.3.part.2.name=sro_id_2@
qu.3.3.part.2.answer=$P@
qu.3.3.part.2.mode=Formula@
qu.3.3.part.3.editing=useHTML@
qu.3.3.part.3.question=(Unset)@
qu.3.3.part.3.name=sro_id_3@
qu.3.3.part.3.answer=$c@
qu.3.3.part.3.mode=Formula@
qu.3.3.part.4.editing=useHTML@
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qu.3.3.question=<p>&nbsp;</p><p>A regulated monopoly faces as demand <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>D</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>mP</mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>P</mi></mrow></mstyle></math>is the market price and produces according to the technology<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>K</mi><mrow><mi>a</mi></mrow></msup><msup><mi>L</mi><mrow><mi>b</mi></mrow></msup></mrow></mstyle></math>. Give an expression for the firm's revenue as a function of capital, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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 labour, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p><p>Revenue =<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>If we use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>w</mi></mrow></mstyle></math> to denote the wage rate 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>r</mi></mrow></mstyle></math> to denote the price of capital, the firm's profit without regulation is&nbsp; <span>&nbsp;</span><2><span> </span>.</p><p>The regulatory agency allows the regulated firm to earn a return of&nbsp;&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>s</mi></mrow></mstyle></math> per unit of capital. To find the total return on capital we take the revenue minus the total amount paid to labour. This is the total return on capital. The regulation requires that the per unit return on capital can be at most <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>s</mi></mrow></mstyle></math>. This is a constraint that the monopoly faces. Write down this constraint as an equality. <span>&nbsp;</span><3><span> =s.</span></p><p>The monopoly's constrained problem can be written using a Lagrange multiplier as</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>Max</mi><mrow><mi>K</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>L</mi></mrow></msub></mrow></mstyle></math><span>&nbsp;</span><4><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><5><span> </span>.</p>@

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qu.3.4.name=Word problem -- profit@
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qu.3.4.uid=39ad882b-774d-4310-abe0-e01928c52c2e@
qu.3.4.info=  Course=Introductory Mathematical Economics;
  Topic=Word Question - Profit Maximization;
  Sub-Topic=Ces Technology;
  Author=Asha Sadanand;
  Difficulty=Easy;
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qu.3.4.question=<p>A competitive firm wants to maximize its profit. It receives a market price 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>p</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>and the prices of labour (<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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 capital (<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>) are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>w</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>v</mi></mrow></mstyle></math>, respectively. The firm's production technology 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>Q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>AK</mi><mrow><mi>d</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>A</mi></mrow></mfenced><msup><mi>L</mi><mrow><mi>d</mi></mrow></msup></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>d</mi></mrow></mfrac></mrow></mfenced></mrow></msup></mrow></mstyle></math> and it has a fixed cost <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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></mrow></mstyle></math>. Give the expression for the firm's profit, that would be need to be maximized to solve this problem. Profit =<span>&nbsp;</span><1><span>&nbsp;</span></p>@

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qu.3.5.name=Word problem --profit and tax@
qu.3.5.comment=<p>The firm's profit is $A1. If we impose and excise tax, the profit now becomes $A2. Instead if there is a profit tax the firm's after tax profit is $A3.</p>@
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qu.3.5.info=  Course=Introductory Mathematical Economics;
  Topic=Optimization One Variable;
  Sub-Topic=Profit Max;
  Difficulty=Medium;
  Author=Asha Sadanand;
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qu.3.5.question=<p>A firm receives a total revenue of&nbsp;&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><mn>50</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>q</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>q</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>, and its total costs are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>q</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>10</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>q</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>50</mn></mrow></mstyle></math>.&nbsp; Give an expression for its profit. Profit=<span>&nbsp;</span><1><span> </span>.</p><p><span>If the government levied an excise tax 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>t</mi></mrow></mstyle></math> dollars per unit sold, now what would its profit be? </span></p><p><span>Profit =</span><span>&nbsp;</span><2><span> </span>.</p><p>Instead if the government imposed a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi></mrow></mstyle></math>% profit tax, what would the firm's profits be?</p><p>Proift =<span>&nbsp;</span><3><span>&nbsp;</span></p>@

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qu.3.6.name=word problem - excise tax@
qu.3.6.comment=<p>Excess demand is demand minus supply. Since&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>p</mi></mrow></mstyle></math> denotes the price producers receive, the price paid by the consumers 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>p</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>t</mi></mrow></mstyle></math>, which is what is used for the consumers' price in the demand. So, Supply is $ST and Demand is $DT. To find excess demand, subtract supply from demand.&nbsp; $Ans1 Now solve for the price where excess demand is zero. $Ans2</p>
<p>To find the equilibrium quantity, remember the above price is the producers' price. Since the question asks to use the demand to find equilibrium quantity,&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>t</mi></mrow></mstyle></math>must be added to the price, and then it is substituted into demand. $Ans3</p>
<p>(If instead the question had asked to use the supply curve, then the above price could be directly substituted, since it is the producers' price. $AltAns3. Be sure to verify for yourself&nbsp; that this quantity is the same as the one found using the demand curve.)</p>
<p>Tax revenue is found by multiplying the equilibrium quantity by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi></mrow></mstyle></math>. $Ans4</p>@
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qu.3.6.uid=0b77984a-8eef-4a52-a887-e89c6ac4cebb@
qu.3.6.info=  Course=Introductory Mathematical Economics;
  Topic=Word Problem;
  Sub-Topic=Tax Revenue;
  Author=Asha Sadanand;
  Difficulty=Hard;
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qu.3.6.part.4.answer=$ans4@
qu.3.6.part.4.mode=Formula@
qu.3.6.question=<p>A market for 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> has demand <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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 mathvariant='normal'>D</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>r</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi></mrow></mstyle></math> and supply <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>S</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>s</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>p</mi></mrow></mstyle></math>is the price. The government wants to generate some revenue by introducing an excise tax <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi></mrow></mstyle></math>per unit sold. Give an expression for market clearing under the tax, by setting excess demand equal to zero. Naturally, with an excise tax, the price paid by the consumers will be different from the price received by the producers. Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>p</mi></mrow></mstyle></math> denote the price received by producers.</p><p>Excess Demand =<span>&nbsp;</span><1><span> </span>=0.</p><p>Solve this expression for the equilibrium price producers receive under the excise tax <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi></mrow></mstyle></math>. Equilibrium Producer Price=<span>&nbsp;</span><2><span> </span>.</p><p>Substitute this into the market demand to find the quantity under the excise tax <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi></mrow></mstyle></math>. Equilibrium Quantity=<span>&nbsp;</span><3><span>&nbsp;</span></p><p>Find an expression for the government revenue from this tax. Revenue =<span>&nbsp;</span><4><span> </span>.</p><p>&nbsp;</p>@

qu.3.7.mode=Inline@
qu.3.7.name=Word problem - Cost minimization@
qu.3.7.comment=<p>To find the Lagrangean&nbsp; for the firm's cost minimization, we take the firm's cost $F1 and add to it <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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> times the constraint of producing <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>units. But in this case we use "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>" for the Lagrange multiplier. We find the Lagrangean to be $A2.</p>
<p>We find the first order conditions by differentiating the Lagrangean with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>J</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>respectively, to find:</p>
<p>$fk</p>
<p>$fl</p>
<p>$fj</p>
<p>These equations in many instances can be solved by taking the first two equations and moving the Lagrange multiplier term to the right hand side, and then dividing one equation by the other. The main purpose of this step is the cancel the Lagrange multiplier on the right hand side of the equation, as follows:</p>
<p>&nbsp;$f1k / $f1l =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$f2k <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&sdot;</mo><mi>J</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></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><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$f2l <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&sdot;</mo><mi>J</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math></p>
<p>or $foc. We take this equation along with the last equation to find our solutions <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>K</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$AnsK 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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$AnsL.</p>
<p>&nbsp;</p>
<p>To check that this is a minimum, we need to find the bordered Hessian, evaluate it at the solution, and check that its determinant is negative. We find the bordered Hessian by differentiating the Lagrangean again with all the variables, and arranging these second derivatives in a matrix. The bordered Hessian is $G,</p>
<p>which is evaluated at the solution <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>K</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>L</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$AnsK, $AnsL);&nbsp; and finally, taking the determinant, we find $E1</p>
<p>which is indeed negative, giving us a minimum.</p>@
qu.3.7.editing=useHTML@
qu.3.7.solution=@
qu.3.7.algorithm=$A=maple("
F:=w*L+v*K+J*(Q-(K^a)*(L^b)):
F1:=w*L+v*K:
f1:=MathML[ExportPresentation](F1):
F2:=(Q-(K^a)*(L^b)):
F0:=MathML[ExportPresentation](F):
FK:=diff(F,K):
FL:=diff(F,L):
FJ:=diff(F,J):
fk:=MathML[ExportPresentation](FK=0):
fl:=MathML[ExportPresentation](FL=0):
fj:=MathML[ExportPresentation](FJ=0):
F1K:=diff(F1,K):
F1L:=diff(F1,L):
F2K:=diff(F2,K):
F2L:=diff(F2,L):
f1k:=MathML[ExportPresentation](F1K):
f1l:=MathML[ExportPresentation](F1L):
f2k:=MathML[ExportPresentation](F2K):
f2l:=MathML[ExportPresentation](F2L):
FOC1:=(F1K)/(F1L):
FOC11:=simplify(FOC1):
FOC2:=(F2K)/(F2L):
FOC22:=simplify(FOC2):
foc:=MathML[ExportPresentation](FOC11=FOC22):
ansL:=((b*v)/(a*w))^(b/(a+b))*Q^(1/(a+b)):
ansK:=((a*w)/(b*v))^(a/(a+b))*Q^(1/(a+b)):
AnsK:=MathML[ExportPresentation](ansK):
AnsL:=MathML[ExportPresentation](ansL):
F,F0,FK,FL,FJ,fk,fl,fj,convert(ansK,string),convert(ansL,string),AnsK,AnsL,f1,f1k,f1l,f2k,f2l,foc,x
");
$A1=switch(0,$A);
$A2=switch(1,$A);
$FK=switch(2,$A);
$FL=switch(3,$A);
$FJ=switch(4,$A);
$fk=switch(5,$A);
$fl=switch(6,$A);
$fj=switch(7,$A);
$ansK=switch(8,$A);
$ansL=switch(9,$A);
$AnsK=switch(10,$A);
$AnsL=switch(11,$A);
$f1=switch(12,$A);
$f1k=switch(13,$A);
$f1l=switch(14,$A);
$f2k=switch(15,$A);
$f2l=switch(16,$A);
$foc=switch(17,$A);
$B=maple("
F:=$A1:
H:=Matrix([[diff(diff(F,K),K),diff(diff(F,K),L),diff(diff(F,K),J)],[diff(diff(F,L),K),diff(diff(F,L),L),diff(diff(F,L),J)],[diff(diff(F,J),K),diff(diff(F,J),L),diff(diff(F,J),J)]]):
G:=MathML[ExportPresentation](H):
R:=eval(H,[K=$ansK, L=$ansL]):
R1:=simplify(R):
E0:=LinearAlgebra[Determinant](R):
E:=simplify(E0):
E1:=MathML[ExportPresentation]($E):
convert(H,string), G, convert(R,string),convert(E,string),E1
");
$H=switch(0,$B);
$G=switch(1,$B);
$E=switch(3,$B);
$E1=switch(4,$B);@
qu.3.7.uid=ff9ff849-48a3-445e-9adf-a14b93fea4c5@
qu.3.7.info=  Course=Introductory Mathematical Economics;
  Topic=Word Problem - Cost Minimization;
  Sub-Topic=Cobb Douglas Production;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.3.7.weighting=1,1,1,1,1,1@
qu.3.7.numbering=alpha@
qu.3.7.part.1.editing=useHTML@
qu.3.7.part.1.question=(Unset)@
qu.3.7.part.1.name=sro_id_1@
qu.3.7.part.1.answer=$A1@
qu.3.7.part.1.mode=Formula@
qu.3.7.part.2.editing=useHTML@
qu.3.7.part.2.question=(Unset)@
qu.3.7.part.2.name=sro_id_2@
qu.3.7.part.2.answer=$FK@
qu.3.7.part.2.mode=Formula@
qu.3.7.part.3.editing=useHTML@
qu.3.7.part.3.question=(Unset)@
qu.3.7.part.3.name=sro_id_3@
qu.3.7.part.3.answer=$FL@
qu.3.7.part.3.mode=Formula@
qu.3.7.part.4.editing=useHTML@
qu.3.7.part.4.question=(Unset)@
qu.3.7.part.4.name=sro_id_4@
qu.3.7.part.4.answer=$FJ@
qu.3.7.part.4.mode=Formula@
qu.3.7.part.5.editing=useHTML@
qu.3.7.part.5.question=(Unset)@
qu.3.7.part.5.name=sro_id_5@
qu.3.7.part.5.answer=$ansK@
qu.3.7.part.5.mode=Formula@
qu.3.7.part.6.editing=useHTML@
qu.3.7.part.6.question=(Unset)@
qu.3.7.part.6.name=sro_id_6@
qu.3.7.part.6.answer=$ansL@
qu.3.7.part.6.mode=Formula@
qu.3.7.question=<p>A firm has a technology <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>K</mi><mrow><mi>a</mi></mrow></msup><msup><mi>L</mi><mrow><mi>b</mi></mrow></msup></mrow></mstyle></math>. If the price of labour is denoted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>w</mi></mrow></mstyle></math>and 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><mi>v</mi></mrow></mstyle></math>, and the firm wants to produce <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>units of output, give the Lagrangean used in solving the firm's cost minimization problem. Since&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>&lambda;</mi></mrow></mstyle></math> is not available in the solution field, please use "J" for the Lagrange multiplier.</p><p><span>Lagrangean&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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math> </span><1><span> </span>.</p><p>The first order conditions are (starting with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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 Lagrange multiplier <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>)</p><p><span>&nbsp;</span><2><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='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p><span>&nbsp;</span><3><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='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p><span>&nbsp;</span><4><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='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p><p>Solving these we find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><5><span>&nbsp;</span>, 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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><6><span>&nbsp;</span>.</p><p>&nbsp;<span> </span></p><p>&nbsp;</p>@

