(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.0'

                    Mathematica-Compatible Notebook

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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).

NOTE: If you modify the data for this notebook not in a Mathematica-
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*******************************************************************)

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Notebook[{
Cell[BoxData[{
    StyleBox[\(Mathematica\ Tutorial\ for\ APEC\ 720\),
      "Subtitle"], "\[IndentingNewLine]", 
    StyleBox[\(Klaus\ Moeltner\ /\ Spring\ 2004\),
      "SmallText"]}], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    StyleBox[\(Formatting\ text\),
      "Subsubtitle"]], "Input"],

Cell[TextData[{
  StyleBox["Format titles and text:",
    FontVariations->{"Underline"->True}],
  "\nTo quickly change the font size of text in your notebook, choose \n>",
  StyleBox["Format /Style /",
    FontSlant->"Italic"],
  "and select a style type. \nAlternatively, select ",
  StyleBox[">Format /ShowToolbar",
    FontSlant->"Italic"],
  ", and choose from the toolbar menu of text styles.\n\nAlso, for input \
that's just plain text or comments (like this one), it's best to select \
\"text\" from the Format / Style - menu. For actual input to be evaluated, \
select \"input\"."
}], "Text"],

Cell[TextData[{
  StyleBox["To start a new cell",
    FontVariations->{"Underline"->True}],
  " WITHOUT \"evaluating\" any input, just place your browser beneath the \
current cell,and click (the browser should change to a horizontal line with a \
reversed arrow on each end).\nTo evaluate input, press \"Shift+Enter\"."
}], "Text",
  PageWidth->WindowWidth]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    StyleBox[\(Grouping\ Cells\),
      "Subsubtitle"]], "Input"],

Cell[TextData[{
  StyleBox["To group cells: ",
    FontVariations->{"Underline"->True}],
  "\nStart with  ",
  StyleBox[">Cell /Cell Grouping / Manual Grouping.",
    FontSlant->"Italic"],
  " This will give you full control over cell grouping.\nSelect adjacent \
cells you want to group by clicking on the cell outline (vertical bar to the \
very right in your window) while holding the \"Shift\" key.\n\nA longer cell \
outline will appear to the right of the original ones. To \"compress\" the \
grouped cells, just double-click on the long outline.  To \"expand\", just \
double-click on the \"arrow\" head of the compressed outline. All of the \
following topics are grouped."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    StyleBox[\(Creating\ Palettes\ and\ Symbolizing\ subscripts\),
      "Subsubtitle"]], "Input"],

Cell[BoxData[
    RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(and\\)\\\" is similar to existing symbol \
\\\"\\!\\(And\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"],

Cell[BoxData[
    \(and\ Creating\ Palettes\ subscripts\ Symbolizing\)], "Output"],

Cell[TextData[{
  StyleBox["Creating customized Palettes:\n",
    FontVariations->{"Underline"->True}],
  "Build-in palettes can be made visible by selecting >File / Palettes /. A \
good one to have open at all times is \"BasicInput\".",
  StyleBox["\n",
    FontVariations->{"Underline"->True}],
  "It's convenient to create your own palettes with characters and symbols \
you expect to use frequently, especially those with superscripts or \
subscripts that are tedious to input manually. Palette buttons can also be \
programmed to execute commands, much like a \"macro\" button in MS Office \
packages.\n\nTo start, select >Input / Create Table,Matrix,Palette and select \
\"Palette\", and choose the number of rows and columns (you can always add \
more buttons later).\nWe'll create a palette customized to a \
Utility-maximization problem with 2 agents and goods and a Cobb-Douglas \
U-function in a pure-exchange economy."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[GridBox[{
        {
          ButtonBox[\(p\_1\)], 
          ButtonBox[\(p\_2\)]},
        {
          ButtonBox[\(x\_11\)], 
          ButtonBox[\(x\_21\)]},
        {
          ButtonBox[\(x\_12\)], 
          ButtonBox[\(x\_22\)]},
        {
          ButtonBox[\(\[Omega]\_11\)], 
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          ButtonBox[\(\[Omega]\_12\)], 
          ButtonBox[\(\[Omega]\_22\)]},
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          ButtonBox[\(U\_1\)], 
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      RowSpacings->0,
      ColumnSpacings->0,
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      RowLines->True,
      ColumnLines->True,
      GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]]], "Input"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{
        RowBox[{"{", 
          RowBox[{
            ButtonBox[\(p\_1\)], ",", 
            ButtonBox[\(p\_2\)]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            ButtonBox[\(x\_11\)], ",", 
            ButtonBox[\(x\_21\)]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            ButtonBox[\(x\_12\)], ",", 
            ButtonBox[\(x\_22\)]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            ButtonBox[\(\[Omega]\_11\)], ",", 
            ButtonBox[\(\[Omega]\_21\)]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            ButtonBox[\(\[Omega]\_12\)], ",", 
            ButtonBox[\(\[Omega]\_22\)]}], "}"}], ",", 
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          RowBox[{
            ButtonBox[\(U\_1\)], ",", 
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        RowBox[{"{", 
          RowBox[{
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            ButtonBox[\(\[ScriptCapitalL]\_1\)], ",", 
            ButtonBox[\(\[ScriptCapitalL]\_2\)]}], "}"}], ",", 
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          RowBox[{
            ButtonBox[\(\[Beta]\_1\)], ",", 
            ButtonBox[\(\[Beta]\_2\)]}], "}"}]}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
  "Note: ",
  StyleBox["Crtl+Enter ",
    FontSlant->"Italic"],
  "adds a row, ",
  StyleBox["Cntrl+,",
    FontSlant->"Italic"],
  " (comma) adds a column (at any point in the palette you choose - just put \
your browser there).\nAfter defining each button, your palette is \"active\". \
To show your palette in a separate window, select your palette, then :",
  StyleBox[">File/Generate Palette from selection",
    FontSlant->"Italic"],
  ". To save your palette, just close the palette window (by clicking on the \
\"x\" in the upper right hand corner), and follow the prompts. The palette \
will be saved with a  \"notebook\" .nb extension, just like a regular \
notebook.  To add stuff or edit your palette, open it and save it as an \
actual notebook using >File / Generate notebook from palette."
}], "Text"],

Cell[TextData[{
  StyleBox["Symbolizing subscripts:",
    FontVariations->{"Underline"->True}],
  "\nSubscripted code is typical in economics, and generally much neater than \
just adding numbers to a variable (as in \"x11\"). However, for a variety of \
reasons, some ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " commands don't recognize subscripted elements as \"symbols\", i.e  \
function inputs etc.  A way around this is to symbolize all subscripted \
elements you'd like to use as follows:"
}], "Text"],

Cell[BoxData[
    \(<< Utilities`Notation`\)], "Input"],

Cell["\<\
Note the \"Notation Palette\" that just opened, probably in the upper right \
hand corner. Use the \"Symbolize[]\" for each subscripted element.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(p\_1\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(p\_2\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(x\_11\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(x\_21\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(x\_22\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(x\_12\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \
\\\"\\!\\(x\[UnderBracket]Subscript\[UnderBracket]12\\)\\\" is similar to \
existing symbol \\\"\\!\\(x\\_21\\)\\\". \\!\\(\\*ButtonBox[\\\"More\
\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"]
}, Open  ]],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(\[Omega]\_11\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(\[Omega]\_21\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(\[Omega]\_22\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(\[Lambda]\_1\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Symbolize", "[", 
        TagBox[\(\[Lambda]\_2\),
          NotationBoxTag,
          TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input"]
}, Open  ]],

Cell["\<\
To use symbolized elements in definitions of \"pure\" functions, you need to \
add a \":\" after each subscripted term as shown:\
\>", "Text",
  Background->RGBColor[0.972549, 0.992157, 0.733333]],

Cell[CellGroupData[{

Cell[BoxData[
    \(g[\(x\_11\) : _, \(x\_21\) : _] = x\_11\^2 + x\_21\^2\)], "Input"],

Cell[BoxData[
    \(x\_11\%2 + x\_21\%2\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(g[a, b]\)], "Input"],

Cell[BoxData[
    \(a\^2 + b\^2\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(g[2, 4]\)], "Input"],

Cell[BoxData[
    \(20\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    StyleBox[\(Saving\ Notebooks\ and\ notebook\ elements\),
      "Subsubtitle"]], "Input"],

Cell[TextData[{
  StyleBox["Saving notebooks and free-standing palettes:",
    FontVariations->{"Underline"->True}],
  "\nEasy -just select >File / Save or Save As, and select a file name and \
destination folder just like with any other software.  Note that the library \
server reverse-maps your computer as v:\\ drive. "
}], "Text"],

Cell[TextData[{
  StyleBox["Saving other elements:",
    FontVariations->{"Underline"->True}],
  "\nOften times, you may want to save programs, commands, or functions you \
create to quickly re-load it later for other projects.  For example, let's \
create a gradient function that takes first derivatives of any possible input \
function with respect to any possible number of variables (as shown in \
Stinespring, p. 54):"
}], "Text"],

Cell[BoxData[
    \(G[s_, vars_List] := Map[Function[{v}, D[s, v]], vars]\)], "Input"],

Cell[TextData[{
  "The generic command to save programs is ",
  StyleBox[">Save[\"filename\",elements to be saved]", "Input"],
  ". However, this would save your stuff to a pre-mapped destination drive \
(D:\\, according to Greg) on the library server. You may not find it again, \
and other users could easily alter, replace, or delete it. Thus, it is a good \
idea to create a folder on your computer or CD-rom for all such saved \
elements. Example:"
}], "Text"],

Cell[BoxData[
    \(Save["\<V:\Klaus\APEC720\Mtica\code\G\>", G]\)], "Input"],

Cell[TextData[{
  "This saves G under the filename \"G\" to the folder \"code\" following the \
specified directory (in this case the c:\\ drive on my computer). Note: Don't \
label any folder \"functions\" - For some reason ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " has problems with that. To load G later on, type:"
}], "Text"],

Cell[BoxData[
    \(<< V:\\Klaus\\APEC720\\Mtica\\code\\G\)], "Input"],

Cell[TextData[{
  "You can also save multiple elements under the same filename: ",
  StyleBox["(general note: \";\" at the end of a command line suppresses \
output)",
    FontWeight->"Bold"]
}], "Text"],

Cell[BoxData[
    \(\({a = 3, b = 1};\)\)], "Input"],

Cell[BoxData[
    \(Save["\<V:\Klaus\APEC720\Mtica\code\parkinglot\>", a, b]\)], "Input"],

Cell[BoxData[
    \(Clear[a, b]\)], "Input"],

Cell[BoxData[
    \(\(<< V:\\Klaus\\APEC720\\Mtica\\code\\parkinglot;\)\)], "Input"],

Cell["To see what's in a specific file, type:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(!! V:\\Klaus\\APEC720\\Mtica\\code\\parkinglot\)\)], "Input"],

Cell["\<\
a = 3
 
b = 1\
\>", "Print"]
}, Open  ]],

Cell[BoxData[
    \(Clear[a, b]\)], "Input"],

Cell["To delete a file, type:", "Text"],

Cell[BoxData[
    \(\(DeleteFile["\<V:\Klaus\APEC720\Mtica\code\parkinglot\>"];\)\)], \
"Input"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    StyleBox[\(Clearing\ and\ Removing\ elements\),
      "Subsubtitle"]], "Input"],

Cell[TextData[{
  StyleBox["Clearing elements:\n",
    FontVariations->{"Underline"->True}],
  "There are 2 ways to dis-associate (\"clear\") an element from a previously \
assigned value. The first is using \"Clear[], the second is simply assigning \
\".\" to the element. Example:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(x = 3\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(x\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(Clear[x]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(x\)], "Input"],

Cell[BoxData[
    \(x\)], "Output"]
}, Open  ]],

Cell["\<\
Note that only the second route works for elements with subscripts. To use \
\"Clear[]\" with subscripts, you have to symbolize those lements first (see \
above), otherwise you'll get an eror message.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(x\_11 = 3\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(Clear[x\_11]\)], "Input"],

Cell[BoxData[
    \(x\_11 =. \)], "Input"],

Cell["\<\
Once you have successfully cleared an association, and you try to do it again \
later (without having re-assigned any value to the variable), you may get an \
error message:\
\>", "Text"],

Cell[BoxData[
    \(x\_11 =. \)], "Input"],

Cell["\<\
...but that's OK. No harm done. It's better to clear often, so you don't risk \
carrying unwanted associations throughout your program. To clear multiple \
elements:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \({x = 3, y = 4, z = 6}\)], "Input"],

Cell[BoxData[
    \({3, 4, 6}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \({x, y, z}\)], "Input"],

Cell[BoxData[
    \({3, 4, 6}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(Clear[x, y, z]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \({x, y, z}\)], "Input"],

Cell[BoxData[
    \({x, y, z}\)], "Output"]
}, Open  ]],

Cell["or, alternatively,", "Text"],

Cell[BoxData[
    \(\({x = 3, y = 4, z = 6};\)\)], "Input"],

Cell[BoxData[
    \(x =. ; \ y =. ; \ z =. ;\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \({x, y, z}\)], "Input"],

Cell[BoxData[
    \({x, y, z}\)], "Output"]
}, Open  ]],

Cell["\<\
Important: \"Clear\" only deletes numerical values associated with a \
variable, but not the variable itself.  To completely erase a variable or \
functyion, use \"Remove\". This is especially important if you want to \
re-define a function (perhaps because you originally coded it wrong). As an \
example, let's use function \"g\" from above:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(g[a, b]\)], "Input"],

Cell[BoxData[
    \(a\^2 + b\^2\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(Remove[g]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(g[a, b]\)], "Input"],

Cell[BoxData[
    \(g[a, b]\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "While for this function (and other simple examples), \"Clear\" would have \
done the job as well, ",
  StyleBox["it's always a good idea to \"Remove\" elements that you want to \
redefine.",
    FontWeight->"Bold"]
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    StyleBox[\(An\ \(Example : \ MWG\), \ Ex . \ 15.  B  .1, \ p . \ 519\),
      "Subsubtitle"]], "Input"],

Cell["\<\
Define C/D utility functions in Log form (note: I will define these as \"pure\
\" funtions, with generic inputs. This allows the function to be used for any \
inputs, symbolic or numeric).\
\>", "Text"],

Cell[BoxData[
    \(\[Alpha] =. ; p\_2 =. ;\)], "Input"],

Cell[BoxData[
    \(\(U\_1[\(x\_11\) : _, \(x\_21\) : _] := 
        Log[x\_11\^\[Alpha]*x\_21\^\(1 - \[Alpha]\)];\)\)], "Input"],

Cell[BoxData[
    \(\(U\_2[\(x\_12\) : _, \(x\_22\) : _] := 
        Log[x\_12\^\[Alpha]*x\_22\^\(1 - \[Alpha]\)];\)\)], "Input"],

Cell[TextData[{
  "Note: ",
  StyleBox["For superscripts, press Cntrl+6, for subscripts, press Cntrl+-.",
    FontWeight->"Bold"],
  " To get back to \"normal font\", press the right hand arrow. You have to \
do this before evaluating any input. Now specify the remaining model inputs:\n\
Note: Instead of writing each definition as a separate cell, we can use",
  StyleBox[" lists:",
    FontWeight->"Bold"]
}], "Text"],

Cell[BoxData[
    \(\({{\[Omega]\_11, \[Omega]\_21}, {\[Omega]\_12, \[Omega]\_22}} = {{1, 
            2}, {2, 1}};\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\[Omega]\_1\)\&_ = \[Omega]\_11 + \[Omega]\_12\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\[Omega]\_2\)\&_ = \[Omega]\_21 + \[Omega]\_22\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[ScriptCapitalL]\_1 = 
      U\_1[x\_11, 
          x\_21] + \[Lambda]\_1*\((p\_1*\[Omega]\_11 + p\_2*\[Omega]\_21 - 
              p\_1*x\_11 - p\_2*x\_21)\)\)], "Input"],

Cell[BoxData[
    \(\((p\_1 + 2\ p\_2 - p\_1\ x\_11 - p\_2\ x\_21)\)\ \[Lambda]\_1 + 
      Log[x\_11\%\[Alpha]\ x\_21\%\(1 - \[Alpha]\)]\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Note how ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "replaces the \[Omega]-symbols with numerical values. This is always the \
case once numbers are assigned to symbols."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[ScriptCapitalL]\_2 = 
      U\_2[x\_12, 
          x\_22] + \[Lambda]\_2*\((p\_1*\[Omega]\_12 + p\_2*\[Omega]\_22 - 
              p\_1*x\_12 - p\_2*x\_22)\)\)], "Input"],

Cell[BoxData[
    \(\((2\ p\_1 + p\_2 - p\_1\ x\_12 - p\_2\ x\_22)\)\ \[Lambda]\_2 + 
      Log[x\_12\%\[Alpha]\ x\_22\%\(1 - \[Alpha]\)]\)], "Output"]
}, Open  ]],

Cell["\<\
Now re-define or load in your G-function,as defined& saved above.Then you can \
take derivatives& set FOC's to zero all in 1 step.Note the double-equal sign \
required by the \"Solve\" command.Note the[[1]] at the end avoids a double \
{{}} for output& makes life easier later on.\
\>", "Text"],

Cell[BoxData[
    \(G[s_, vars_List] := Map[Function[{v}, D[s, v]], vars]\)], "Input"],

Cell["or, if you have saved it before:", "Text"],

Cell[BoxData[
    \(<< V:\\Klaus\\APEC720\\Mtica\\code\\G\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(sol1 = \(Solve[
          G[\[ScriptCapitalL]\_1, {x\_11, x\_21, \[Lambda]\_1}] \[Equal] 
            0, {x\_11, x\_21, \[Lambda]\_1}]\)[\([1]\)]\)], "Input"],

Cell[BoxData[
    \({\[Lambda]\_1 \[Rule] 1\/\(p\_1 + 2\ p\_2\), 
      x\_11 \[Rule] \(\((p\_1 + 2\ p\_2)\)\ \[Alpha]\)\/p\_1, 
      x\_21 \[Rule] \(p\_1 + 2\ p\_2 - p\_1\ \[Alpha] - 2\ p\_2\ \
\[Alpha]\)\/p\_2}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The ",
  StyleBox["generic approach to solve FOC's",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" ",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 1]],
  "would involve the following 2 steps:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \({FOC\_1 = D[\[ScriptCapitalL]\_1, x\_11], 
      FOC\_2 = D[\[ScriptCapitalL]\_1, x\_21], 
      FOC\_3 = D[\[ScriptCapitalL]\_1, \[Lambda]\_1]}\)], "Input"],

Cell[BoxData[
    \({\[Alpha]\/x\_11 - p\_1\ \[Lambda]\_1, \(1 - \[Alpha]\)\/x\_21 - 
        p\_2\ \[Lambda]\_1, 
      p\_1 + 2\ p\_2 - p\_1\ x\_11 - p\_2\ x\_21}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(sol1alt = \(Solve[{FOC\_1 \[Equal] 0, FOC\_2 \[Equal] 0, 
            FOC\_3 \[Equal] 0}, {x\_11, 
            x\_21, \[Lambda]\_1}]\)[\([1]\)]\)], "Input"],

Cell[BoxData[
    \({\[Lambda]\_1 \[Rule] 1\/\(p\_1 + 2\ p\_2\), 
      x\_11 \[Rule] \(\((p\_1 + 2\ p\_2)\)\ \[Alpha]\)\/p\_1, 
      x\_21 \[Rule] \(p\_1 + 2\ p\_2 - p\_1\ \[Alpha] - 2\ p\_2\ \
\[Alpha]\)\/p\_2}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The solution for ",
  Cell[BoxData[
      \(TraditionalForm\`x\_11\)]],
  "is identical to the first element in the offer curve in MWG, p. 519, but \
the expression for ",
  Cell[BoxData[
      \(TraditionalForm\`x\_21\)]],
  "appears to be different.  Often times, this is just an issue of \
representation. Rarely will ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " produce results exactly in the form you would like them. Sometimes, \
\"Simplify\" can help. Note [%] refers to \"last evaluation\"."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[%]\)], "Input"],

Cell[BoxData[
    \({\[Lambda]\_1 \[Rule] 1\/\(p\_1 + 2\ p\_2\), 
      x\_11 \[Rule] \[Alpha] + \(2\ p\_2\ \[Alpha]\)\/p\_1, 
      x\_21 \[Rule] \(-\(\(\((p\_1 + 
                    2\ p\_2)\)\ \((\(-1\) + \[Alpha])\)\)\/p\_2\)\)}\)], \
"Output"]
}, Open  ]],

Cell["\<\
Close enough - you will realize that these results are equivalent to the form \
given in the textbook. Let's optimize agent 2's problem. Note that we can do \
command and simplification all in one step.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[
      sol2 = \(Solve[
            G[\[ScriptCapitalL]\_2, {x\_12, x\_22, \[Lambda]\_2}] \[Equal] 
              0, {x\_12, x\_22, \[Lambda]\_2}]\)[\([1]\)]]\)], "Input"],

Cell[BoxData[
    \({\[Lambda]\_2 \[Rule] 1\/\(2\ p\_1 + p\_2\), 
      x\_12 \[Rule] \(\((2\ p\_1 + p\_2)\)\ \[Alpha]\)\/p\_1, 
      x\_22 \[Rule] \(-\(\(\((2\ p\_1 + 
                    p\_2)\)\ \((\(-1\) + \[Alpha])\)\)\/p\_2\)\)}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    StyleBox[\(Generic\ \(\(approach\)\(:\)\)\),
      "Text",
      FontColor->RGBColor[1, 0, 0]]], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \({FOC\_1 = D[\[ScriptCapitalL]\_2, x\_12], 
      FOC\_2 = D[\[ScriptCapitalL]\_2, x\_22], 
      FOC\_3 = D[\[ScriptCapitalL]\_2, \[Lambda]\_2]}\)], "Input"],

Cell[BoxData[
    \({\[Alpha]\/x\_12 - p\_1\ \[Lambda]\_2, \(1 - \[Alpha]\)\/x\_22 - 
        p\_2\ \[Lambda]\_2, 
      2\ p\_1 + p\_2 - p\_1\ x\_12 - p\_2\ x\_22}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(sol2alt = 
      Simplify[\(Solve[{FOC\_1 \[Equal] 0, FOC\_2 \[Equal] 0, 
              FOC\_3 \[Equal] 0}, {x\_12, 
              x\_22, \[Lambda]\_2}]\)[\([1]\)]]\)], "Input"],

Cell[BoxData[
    \({\[Lambda]\_2 \[Rule] 1\/\(2\ p\_1 + p\_2\), 
      x\_12 \[Rule] \(\((2\ p\_1 + p\_2)\)\ \[Alpha]\)\/p\_1, 
      x\_22 \[Rule] \(-\(\(\((2\ p\_1 + 
                    p\_2)\)\ \((\(-1\) + \[Alpha])\)\)\/p\_2\)\)}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Now define offer curves for both agents. Note that ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " will NOT automatically interpret estimation results as actual \
definitions. So for example, at this point, the program \"doesn't \
understand\" that ",
  Cell[BoxData[
      \(x\_11 = \[Alpha] + \(2\ \[Alpha]\ p\_2\)\/p\_1\)]],
  ". We need to define these optimal solutions from scratch, using cut & \
paste or - more conveniently, the replacement operator \"/.\": We'll denote \
equilibrium levels for x's with \"*\". That way we don't need to \"undo\" the \
optimality definition for the x's in later steps, where we just need the \
generic specifications."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \({\(x\_11\^*\), \(x\_21\^*\)} = {x\_11, x\_21} /. sol1alt\)], "Input"],

Cell[BoxData[
    \({\(\((p\_1 + 2\ p\_2)\)\ \[Alpha]\)\/p\_1, \(p\_1 + 2\ p\_2 - p\_1\ \
\[Alpha] - 2\ p\_2\ \[Alpha]\)\/p\_2}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \({\(x\_12\^*\), \(x\_22\^*\)} = {x\_12, x\_22} /. sol2alt\)], "Input"],

Cell[BoxData[
    \({\(\((2\ p\_1 + p\_2)\)\ \[Alpha]\)\/p\_1, \(-\(\(\((2\ p\_1 + 
                  p\_2)\)\ \((\(-1\) + \[Alpha])\)\)\/p\_2\)\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \({OC\_1, 
        OC\_2} = {{\(x\_11\^*\), \(x\_21\^*\)}, {\(x\_12\^*\), \(x\_22\^*\)}}\
\)], "Input"],

Cell[BoxData[
    \({{\(\((p\_1 + 2\ p\_2)\)\ \[Alpha]\)\/p\_1, \(p\_1 + 2\ p\_2 - p\_1\ \
\[Alpha] - 2\ p\_2\ \[Alpha]\)\/p\_2}, {\(\((2\ p\_1 + p\_2)\)\ \[Alpha]\)\/p\
\_1, \(-\(\(\((2\ p\_1 + 
                    p\_2)\)\ \((\(-1\) + \[Alpha])\)\)\/p\_2\)\)}}\)], \
"Output"]
}, Open  ]],

Cell["\<\
To solve for the Walrasian equilibrium price ratio, we use the \"market \
clearing\" condition:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[\(x\_11\^*\) + \(x\_12\^*\) == \(\[Omega]\_1\)\&_, 
      p\_1]\)], "Input"],

Cell[BoxData[
    \({{p\_1 \[Rule] \(-\(\(p\_2\ \[Alpha]\)\/\(\(-1\) + \[Alpha]\)\)\)}}\)], \
"Output"]
}, Open  ]],

Cell["\<\
With a bit of massaging we get the relationship 15.B.2 on p. 520. Let's check \
that both markets clear at these prices. Note the \"substitution\" operator, \
which is essentially the replacement operator \"/.\" in combination with \"->\
\"\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(check1 = 
      Simplify[\(x\_11\^*\) + \ \(x\_12\^*\) - \(\[Omega]\_1\)\&_ /. 
          p\_1 -> \(-\(\(\[Alpha]\ p\_2\)\/\(\(-1\) + \[Alpha]\)\)\)]\)], \
"Input"],

Cell[BoxData[
    \(0\)], "Output"],

Cell[BoxData[
    \(check2\  = \ 
      Simplify[\(x\_21\^*\) + \ \(x\_22\^*\) - \(\[Omega]\_2\)\&_ /. 
          p\_1 -> \(-\(\(\[Alpha]\ p\_2\)\/\(\(-1\) + \[Alpha]\)\)\)]\)], \
"Input"],

Cell[BoxData[
    \(0\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Note: Always make sure there is a space to the right and left of a \"+\" \
or \"-\" operator. This might be a v.5 glitch, but for some reason ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " did not create a space after the \"+\" in the check1=equation, so the \
operator was interpreted as multiplication. Just something to check if you \
get weird results."
}], "Text",
  Background->RGBColor[1, 0.866667, 0.866667]],

Cell["\<\
Now let's verify that the marginal rates of substitution are equal for both \
agents at equilibrium. We'll use the derivative function \"D\" for this step. \
 Had we not used then \"*\" notation for the x's, we would now have to undo \
the optimality definition by simply clearing the x's:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(MRS\_1 = \((D[U\_1[x\_11, x\_21], x\_11])\)/\((D[U\_1[x\_11, x\_21], 
            x\_21])\)\)], "Input"],

Cell[BoxData[
    \(\(x\_21\ \[Alpha]\)\/\(x\_11\ \((1 - \[Alpha])\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(MRS\_2 = \((D[U\_2[x\_12, x\_22], x\_12])\)/\((D[U\_2[x\_12, x\_22], 
            x\_22])\)\)], "Input"],

Cell[BoxData[
    \(\(x\_22\ \[Alpha]\)\/\(x\_12\ \((1 - \[Alpha])\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(check3 = \((MRS\_1/MRS\_2)\)\  /. {x\_11 -> \(x\_11\^*\), 
          x\_21 -> \(x\_21\^*\), x\_12 -> \(x\_12\^*\), 
          x\_22 -> \(x\_22\^*\)}\)], "Input"],

Cell[BoxData[
    \(\(-\(\(p\_1 + 2\ p\_2 - p\_1\ \[Alpha] - 
            2\ p\_2\ \[Alpha]\)\/\(\((p\_1 + 
                2\ p\_2)\)\ \((\(-1\) + \[Alpha])\)\)\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[%]\)], "Input"],

Cell[BoxData[
    \(1\)], "Output"]
}, Open  ]],

Cell[BoxData[
    StyleBox["Graphs",
      "Subsubtitle"]], "Input"],

Cell[TextData[{
  "To show this \"story\" graphically, we need to assign numerical values to \
the remaining exogenous elements, in this case the Cobb-Douglas exponents.  \
Also, since only relative prices are identified in this system, we're \
following standard convention by setting ",
  Cell[BoxData[
      \(TraditionalForm\`p\_2 = 1\)]],
  ". Let's solve for equilibrium price ",
  Cell[BoxData[
      \(TraditionalForm\`p\_1\)]],
  " using these numerical assignments and above results. Remember, v.5.0 \
requires use of fractions for exponents to avoid problems later."
}], "Text"],

Cell[BoxData[
    \(\({\[Alpha] = 1/5, p\_2 = 1};\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(p\_1\^*\) = 
      N[\(-\(\(\[Alpha]\ p\_2\)\/\(\(-1\) + \[Alpha]\)\)\)]\)], "Input"],

Cell[BoxData[
    \(0.25`\)], "Output"]
}, Open  ]],

Cell["\<\
As in Stinespring (p.84),we'll only show indifference curves associated with \
the original (endowment) urilities,and utilities at equilibrium. Note that, \
as expected, utilities are higher in equilibrium compared to endowment. The \
\"N[]+ function returns actual numbers (as opposed to decimals , \
Log-expressions etc).\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(N[{V\_1 = U\_1[\[Omega]\_11, \[Omega]\_21], \(V\_1\^*\) = 
            U\_1[\(x\_11\^*\), \(x\_21\^*\)]} /. 
        p\_1 -> \(p\_1\^*\)]\)], "Input"],

Cell[BoxData[
    \({0.5545177444479562`, 0.5877866649021194`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(N[{V\_2 = U\_2[\[Omega]\_12, \[Omega]\_22], \(V\_2\^*\) = 
            U\_2[\(x\_12\^*\), \(x\_22\^*\)]} /. 
        p\_1 -> \(p\_1\^*\)]\)], "Input"],

Cell[BoxData[
    \({0.13862943611198905`, 0.1823215567939548`}\)], "Output"]
}, Open  ]],

Cell["\<\
Let's plot the indifference curves for both individuals at these two \
U-levels. Note the \";\" at the end of the command suppresses the somewhat \
superfluous \"graphics\" output line (to see what I mean, try running this \
command without \";\" ).\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(IC\_1 = 
        ContourPlot[
          U\_1[x, y], {x, 0.001, \(\[Omega]\_1\)\&_}, {y, 
            0.001, \(\[Omega]\_2\)\&_}, ContourShading \[Rule] False, 
          Contours -> {V\_1, \(V\_1\^*\) /. p\_1 -> \(p\_1\^*\)}, 
          PlotPoints \[Rule] 100];\)\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
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Edgeworth Box is at {",
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load in the 'Graphics'-package. First, we'll plot and label endowment and \
equilibrium points."
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