(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 116397, 3542]*) (*NotebookOutlinePosition[ 190857, 6046]*) (* CellTagsIndexPosition[ 190813, 6042]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["A 2 by 2 Economy", "Subtitle", FontSize->16, FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Symbolizing", FontSize->14]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(<< Utilities`Notation`\)], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(z\_11\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", 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RowBox[{"Symbolize", "[", TagBox[\(w\_2\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(\[Alpha]\_1\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(\[Alpha]\_2\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(\[Beta]\_1\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(\[Beta]\_2\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(\[Lambda]\_1\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[\(\[Lambda]\_2\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Set-up", FontSize->14]], "Text"], Cell["Consider a 2x2 economy with the folowing features:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f\_1[\(z\_11\) : _, \(z\_21\) : _] = z\_11\^\[Alpha]\_1*z\_21\^\[Alpha]\_2\)], "Input"], Cell[BoxData[ \(z\_11\%\(\[Alpha]\_1\)\ z\_21\%\(\[Alpha]\_2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f\_2[\(z\_12\) : _, \(z\_22\) : _] = z\_12\^\[Beta]\_1*z\_22\^\[Beta]\_2\)], "Input"], Cell[BoxData[ \(z\_12\%\(\[Beta]\_1\)\ z\_22\%\(\[Beta]\_2\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\({\[Alpha]\_1 = 1/2, \[Alpha]\_2 = 1/2, \[Beta]\_1 = 2/3, \[Beta]\_2 = 1/3};\)\)], "Input"], Cell[BoxData[ \(\({\(z\_1\)\&_ = 10, \(z\_2\)\&_ = 2};\)\)], "Input"], Cell[BoxData[ \(\({p\_1 = 4, p\_2 = 3};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Numerical Solutions", FontSize->14]], "Text"], Cell["\<\ 1) Find the technical coefficients as a function of factor prices. \ \>", "Text"], Cell["Let's set the problem up as unit-cost-minimization:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c\_1 = w\_1*a\_11 + w\_2*a\_21\)], "Input"], Cell[BoxData[ \(a\_11\ w\_1 + a\_21\ w\_2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalL]\_1 = c\_1 + \[Lambda]\_1*\((1 - a\_11\^\[Alpha]\_1*a\_21\^\[Alpha]\_2)\)\)], "Input"], Cell[BoxData[ \(a\_11\ w\_1 + a\_21\ w\_2 + \((1 - \@a\_11\ \@a\_21)\)\ \[Lambda]\_1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[{FOC\_1 = D[\[ScriptCapitalL]\_1, a\_11], FOC\_2 = D[\[ScriptCapitalL]\_1, a\_21], FOC\_3 = D[\[ScriptCapitalL]\_1, \[Lambda]\_1]}]\)], "Input"], Cell[BoxData[ \({w\_1 - \(\@a\_21\ \[Lambda]\_1\)\/\(2\ \@a\_11\), w\_2 - \(\@a\_11\ \[Lambda]\_1\)\/\(2\ \@a\_21\), 1 - \@a\_11\ \@a\_21}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = Solve[{FOC\_1 \[Equal] 0, FOC\_2 \[Equal] 0, FOC\_3 \[Equal] 0}, {a\_11, a\_21, \[Lambda]\_1}]\)], "Input"], Cell[BoxData[ \({{a\_11 \[Rule] \(-\(\@w\_2\/\@w\_1\)\), a\_21 \[Rule] \(-\(\@w\_1\/\@w\_2\)\), \[Lambda]\_1 \[Rule] \(-2\)\ \ \@w\_1\ \@w\_2}, {a\_11 \[Rule] \@w\_2\/\@w\_1, a\_21 \[Rule] \@w\_1\/\@w\_2, \[Lambda]\_1 \[Rule] 2\ \@w\_1\ \@w\_2}}\)], "Output"] }, Open ]], Cell["\<\ We want positive quantities, so the second set of solutions applies.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({a\_11, a\_21, \[Lambda]\_1} = {\@w\_2\/\@w\_1, \@w\_1\/\@w\_2, 2\ \@w\_1\ \@w\_2}\)], "Input"], Cell[BoxData[ \({\@w\_2\/\@w\_1, \@w\_1\/\@w\_2, 2\ \@w\_1\ \@w\_2}\)], "Output"] }, Open ]], Cell["Same procedure for second firm:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c\_2 = w\_1*a\_12 + w\_2*a\_22\)], "Input"], Cell[BoxData[ \(a\_12\ w\_1 + a\_22\ w\_2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalL]\_2 = c\_2 + \[Lambda]\_2*\((1 - a\_12\^\[Beta]\_1*a\_22\^\[Beta]\_2)\)\)], "Input"], Cell[BoxData[ \(a\_12\ w\_1 + a\_22\ w\_2 + \((1 - a\_12\%\(2/3\)\ a\_22\%\(1/3\))\)\ \[Lambda]\_2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[{FOC\_1 = D[\[ScriptCapitalL]\_2, a\_12], FOC\_2 = D[\[ScriptCapitalL]\_2, a\_22], FOC\_3 = D[\[ScriptCapitalL]\_2, \[Lambda]\_2]}]\)], "Input"], Cell[BoxData[ \({w\_1 - \(2\ a\_22\%\(1/3\)\ \[Lambda]\_2\)\/\(3\ a\_12\%\(1/3\)\), w\_2 - \(a\_12\%\(2/3\)\ \[Lambda]\_2\)\/\(3\ a\_22\%\(2/3\)\), 1 - a\_12\%\(2/3\)\ a\_22\%\(1/3\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sol2 = Solve[{FOC\_1 \[Equal] 0, FOC\_2 \[Equal] 0, FOC\_3 \[Equal] 0}, {a\_12, a\_22, \[Lambda]\_2}]\)], "Input"], Cell[BoxData[ \({{a\_12 \[Rule] \(-\(\(\((\(-2\))\)\^\(1/3\)\ \ w\_2\%\(1/3\)\)\/w\_1\%\(1/3\)\)\), a\_22 \[Rule] \(-\(\(\((\(-1\))\)\^\(1/3\)\ \ w\_1\%\(2/3\)\)\/\(2\^\(2/3\)\ w\_2\%\(2/3\)\)\)\), \[Lambda]\_2 \[Rule] \ \(-\(\(3\ \((\(-1\))\)\^\(1/3\)\ w\_1\%\(2/3\)\ w\_2\%\(1/3\)\)\/2\^\(2/3\)\)\ \)}, {a\_12 \[Rule] \(2\^\(1/3\)\ w\_2\%\(1/3\)\)\/w\_1\%\(1/3\), a\_22 \[Rule] w\_1\%\(2/3\)\/\(2\^\(2/3\)\ w\_2\%\(2/3\)\), \[Lambda]\_2 \[Rule] \ \(3\ w\_1\%\(2/3\)\ w\_2\%\(1/3\)\)\/2\^\(2/3\)}, {a\_12 \[Rule] \(\((\(-1\))\ \)\^\(2/3\)\ 2\^\(1/3\)\ w\_2\%\(1/3\)\)\/w\_1\%\(1/3\), a\_22 \[Rule] \(\((\(-\(1\/2\)\))\)\^\(2/3\)\ w\_1\%\(2/3\)\)\/w\_2\%\ \(2/3\), \[Lambda]\_2 \[Rule] \(3\ \((\(-1\))\)\^\(2/3\)\ w\_1\%\(2/3\)\ w\_2\ \%\(1/3\)\)\/2\^\(2/3\)}}\)], "Output"] }, Open ]], Cell["\<\ We want positive quantities, so the second set of solutions applies.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({a\_12, a\_22, \[Lambda]\_2} = {\(2\^\(1/3\)\ w\_2\%\(1/3\)\)\/w\_1\%\(1/3\), w\_1\%\(2/3\)\/\(2\^\(2/3\)\ w\_2\%\(2/3\)\), \(3\ w\_1\%\(2/3\)\ \ w\_2\%\(1/3\)\)\/2\^\(2/3\)}\)], "Input"], Cell[BoxData[ \({\(2\^\(1/3\)\ w\_2\%\(1/3\)\)\/w\_1\%\(1/3\), w\_1\%\(2/3\)\/\(2\^\(2/3\)\ w\_2\%\(2/3\)\), \(3\ w\_1\%\(2/3\)\ \ w\_2\%\(1/3\)\)\/2\^\(2/3\)}\)], "Output"] }, Open ]], Cell["\<\ 2) Plug into \"no-profit\" constraint (unit cost = unit price)and solve for \ equilibrium wages\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Solve[{a\_11*w\_1 + a\_21*w\_2 == p\_1, a\_12*w\_1 + a\_22*w\_2 == p\_2}, {w\_1, w\_2}]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({w\_2 \[Rule] 4, w\_1 \[Rule] 1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(w\_1\^*\), \(w\_2\^*\)} = {w\_1, w\_2} /. %\)], "Input"], Cell[BoxData[ \({1, 4}\)], "Output"] }, Open ]], Cell["3) Use this to solve for the technical coefficients", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\(a\_11\^*\), \(a\_21\^*\), \(a\_12\^*\), \(a\_22\^*\)} = {a\_11, a\_21, a\_12, a\_22} /. {w\_1 -> \(w\_1\^*\), w\_2 -> \(w\_2\^*\)}\)], "Input"], Cell[BoxData[ \({2, 1\/2, 2, 1\/4}\)], "Output"] }, Open ]], Cell["\<\ Which factor is more intensively used in the production of which good?\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({r\_1, r\_2} = {\(a\_11\^*\)/\(a\_21\^*\), \(a\_12\^*\)/\(a\_22\^*\)}\)], \ "Input"], Cell[BoxData[ \({4, 8}\)], "Output"] }, Open ]], Cell["\<\ This implies that factor 1 is used relatively more intensively in the \ production of the second good, and factor 2 is used relatively more \ intensively in the production of the first good.\ \>", "Text"], Cell["\<\ 4) Now use the market clearing condition to solve for equilibrium total \ output\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Solve[{q\_1*\(a\_11\^*\) + q\_2*\(a\_12\^*\) == \(z\_1\)\&_, q\_1*\(a\_21\^*\) + q\_2*\(a\_22\^*\) == \(z\_2\)\&_}, {q\_1, q\_2}]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({q\_1 \[Rule] 3, q\_2 \[Rule] 2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(q\_1\^*\), \(q\_2\^*\)} = {q\_1, q\_2} /. %\)], "Input"], Cell[BoxData[ \({3, 2}\)], "Output"] }, Open ]], Cell["\<\ 5) Next, combine technical coefficients and total output to derive total \ factor employment by the two firms:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\(z\_11\^*\), \(z\_21\^*\), \(z\_12\^*\), \(z\_22\^*\)} = \ {\(q\_1\^*\)*\(a\_11\^*\), \(q\_1\^*\)*\(a\_21\^*\), \ \(q\_2\^*\)*\(a\_12\^*\), \(q\_2\^*\)*\(a\_22\^*\)}\)], "Input"], Cell[BoxData[ \({6, 3\/2, 4, 1\/2}\)], "Output"] }, Open ]], Cell["\<\ Double-check on equilibrium output for each firm using the production \ function:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f\_1\^*\) = z\_11\^\[Alpha]\_1*z\_21\^\[Alpha]\_2 /. {z\_11 -> \(z\_11\^*\), z\_21 -> \(z\_21\^*\)}\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(f\_2\^*\) = z\_12\^\[Beta]\_1*z\_22\^\[Beta]\_2 /. {z\_12 -> \(z\_12\^*\), z\_22 -> \(z\_22\^*\)}\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Graphs", "Text", FontSize->14], Cell["\<\ First, graph the isoquants at equilibrium production levels.\ \>", "Text"], Cell[BoxData[ \(Clear[x, y]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(IQ\_1 = ContourPlot[f\_1[x, y], {x, 0, \(z\_1\)\&_}, {y, 0, \(z\_2\)\&_}, ContourShading \[Rule] False, 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