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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[ 35165, 875]*) (*NotebookOutlinePosition[ 38448, 975]*) (* CellTagsIndexPosition[ 38310, 966]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["1", "ChapterLine", CellTags->"1.1"], Cell["Financial Market Equilibrium", "Title", CellTags->"1.1"], Cell[CellGroupData[{ Cell["1.1 Derivation of Lemma 1", "Section", CellTags->"1.1"], Cell[TextData[{ StyleBox["To derive the financial-market equilibrium under partly \ informative prices, match the coefficients in equation (10) with the \ coefficients in equation (5) and solve the non-linear equation system in the \ three unknowns ", "Text"], Cell[BoxData[ \(TraditionalForm\`\[Pi]\_0\)]], ",", Cell[BoxData[ \(TraditionalForm\`\[Pi]\_S\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Pi]\_X\)\(.\)\)\)]], " To make the math more intuitive, I introduce the variable ", StyleBox["u", FontSlant->"Italic"], " in addition, as in Appendix D. In the following equations, I write ", StyleBox["u", FontWeight->"Bold"], ", ", StyleBox["p0", FontWeight->"Bold"], ", ", StyleBox["ps", FontWeight->"Bold"], ", and ", StyleBox["px", FontWeight->"Bold"], " instead of ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\_0\)]], ",", Cell[BoxData[ \(TraditionalForm\`\[Pi]\_S\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\_X\)]], " to keep ", StyleBox["Mathematica", FontSlant->"Italic"], " from confusing the parameters with the number \[Pi]. For similar reasons, \ the number of signals is defined as ", StyleBox["nSig", FontWeight->"Bold"], ", the number of investors as ", StyleBox["iInv", FontWeight->"Bold"], ", and the sum of all signal realizations as ", StyleBox["sumSig", FontWeight->"Bold"], "." }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Solve[{u \[Equal] 1\/\[Tau]\^2 + \((1 - \[Lambda])\) \(\(\(\[Lambda]\ nSig\)\/\(u\ \ \[Sigma]\^2\)\) \((\(\[Lambda]\ nSig\)\/\(u\ \[Sigma]\^2\) - 1)\)\)\/\(\(\((\ \[Lambda]\/\(u\ \[Sigma]\^2\))\)\^2\) nSig\ \[Sigma]\^2 + \(\((\(\[Gamma]\ \ \[Omega]\)\/iInv)\)\^2\) 1\/u\^2\) + \[Lambda]\ nSig\/\[Sigma]\^2}, \ {u}];\)\), "\[IndentingNewLine]", \(u = FullSimplify[u /. 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xbar\ \ \[Gamma]\ \((\(-1\) + \[Lambda])\)\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Mu]\ \ \[Sigma]\^4\ \[Omega]\^2\)\/\(iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + \ nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The covariance between ", "Text"], StyleBox["RP", "Text", FontWeight->"Bold"], StyleBox[" and the fundamental, and the covariance response to ", "Text"], StyleBox["nSig", "Text", FontWeight->"Bold"], StyleBox[" and ", "Text"], StyleBox["\[Lambda]", FontWeight->"Bold"], ",", StyleBox[" is", "Text"] }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[{ \(FullSimplify[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \((\ \(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig\ \[Tau]\^2]\), "\[IndentingNewLine]", \(FullSimplify[ D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig\ \[Tau]\^2, nSig]]\), "\[IndentingNewLine]", \(FullSimplify[ D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig\ \[Tau]\^2, \[Lambda]\ ]]\), "\[IndentingNewLine]", \(FullSimplify[ D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig\ \[Tau]\^2, nSig] - D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig\ \[Tau]\^2, \[Lambda]\ ]]\)}], "Input"], Cell[BoxData[ \(\(nSig\ \[Lambda]\ \[Tau]\^4\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ \ nSig\ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\)], "Output"], Cell[BoxData[ \(\(\[Lambda]\ \[Sigma]\^2\ \[Tau]\^4\ \((iInv\^4\ nSig\^2\ \[Lambda]\^3 \ + 2\ iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\ \[Sigma]\^2\ \[Omega]\^2 + \ \[Gamma]\^4\ \[Sigma]\^4\ \[Omega]\^4)\)\)\/\((iInv\^2\ nSig\ \[Lambda]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2\)], "Output"], Cell[BoxData[ \(\(nSig\ \[Gamma]\^2\ \[Sigma]\^4\ \[Tau]\^4\ \[Omega]\^2\ \((\(-iInv\^2\ \)\ nSig\ \((\(-2\) + \[Lambda])\)\ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \ \[Omega]\^2)\)\)\/\((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2\)], "Output"], Cell[BoxData[ \(\(\[Sigma]\^2\ \[Tau]\^4\ \((iInv\^4\ nSig\^2\ \[Lambda]\^4 + iInv\^2\ \ nSig\ \[Gamma]\^2\ \[Lambda]\ \((nSig\ \((\(-2\) + \[Lambda])\) + 2\ \ \[Lambda])\)\ \[Sigma]\^2\ \[Omega]\^2 + \[Gamma]\^4\ \((\(-nSig\) + \ \[Lambda])\)\ \[Sigma]\^4\ \[Omega]\^4)\)\)\/\((iInv\^2\ nSig\ \[Lambda]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The covariance between ", "Text"], StyleBox["RP", "Text", FontWeight->"Bold"], StyleBox[" and the sum of the signals, and its response to ", "Text"], StyleBox["nSig", "Text", FontWeight->"Bold"], StyleBox[" and ", "Text"], StyleBox["\[Lambda]", FontWeight->"Bold"], ",", StyleBox[" is", "Text"] }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[{ \(FullSimplify[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \((\ \(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)]\), "\[IndentingNewLine]", \(FullSimplify[ D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\), nSig]]\), "\[IndentingNewLine]", \(FullSimplify[ D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\), \[Lambda]\ ]]\), "\ \[IndentingNewLine]", \(FullSimplify[ D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\), nSig] - D[\(\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\), \[Lambda]\ ]]\)}], "Input"], Cell[BoxData[ \(\(nSig\ \[Lambda]\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\ \^2 + \(iInv\^2\ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \ \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\)], "Output"], Cell[BoxData[ \(\(\[Lambda]\ \[Tau]\^2\ \((iInv\^4\ nSig\^2\ \[Lambda]\^3\ \ \((\[Sigma]\^4 + 2\ nSig\ \[Lambda]\ \[Sigma]\^2\ \[Tau]\^2 + nSig\^2\ \ \[Lambda]\ \[Tau]\^4)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\ \[Sigma]\^2\ \ \((2\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((3 + \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\ \^2 + 2\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\ \[Omega]\^2 + \[Gamma]\^4\ \ \[Sigma]\^4\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\^2\ \ \[Omega]\^4)\)\)\/\((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2\)], "Output"], Cell[BoxData[ \(\(nSig\ \[Tau]\^2\ \((iInv\^4\ nSig\^3\ \[Lambda]\^4\ \[Tau]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\ \ \[Sigma]\^2\ \((\(-\((\(-2\) + \[Lambda])\)\)\ \[Sigma]\^4 + 3\ nSig\ \ \[Lambda]\ \[Sigma]\^2\ \[Tau]\^2 + 2\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\ \ \[Omega]\^2 + \[Gamma]\^4\ \[Sigma]\^4\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\^2\ \[Omega]\^4)\)\)\/\((iInv\^2\ nSig\ \[Lambda]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2\)], "Output"], Cell[BoxData[ \(\((\[Tau]\^2\ \((iInv\^4\ nSig\^2\ \[Lambda]\^4\ \((\[Sigma]\^4 - nSig\ \((nSig - 2\ \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\^2 + nSig\^2\ \((\(-nSig\) + \[Lambda])\)\ \[Tau]\^4)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\ \[Sigma]\^2\ \((\((nSig\ \ \((\(-2\) + \[Lambda])\) + 2\ \[Lambda])\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\)\ nSig + \[Lambda]\ \((3 + \ \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 2\ nSig\^2\ \[Lambda]\^2\ \((\(-nSig\) + \[Lambda])\)\ \ \[Tau]\^4)\)\ \[Omega]\^2 - \[Gamma]\^4\ \((nSig - \[Lambda])\)\ \[Sigma]\^4\ \ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\^2\ \ \[Omega]\^4)\))\)/\((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The variance of ", "Text"], StyleBox["RP", "Text", FontWeight->"Bold"], StyleBox[", and its response to ", "Text"], StyleBox["nSig", "Text", FontWeight->"Bold"], StyleBox[" and ", "Text"], StyleBox["\[Lambda]", FontWeight->"Bold"], ",", StyleBox[" is ", "Text"] }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[{ \(FullSimplify[\(\((\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \((\(-1\) + \ \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \ \[Omega]\^2\))\))\)\))\)\^2\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\) + \(\(\[Gamma]\ \ \[Tau]\^2\)\/\(iInv\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) \[Omega]\^2]\), "\[IndentingNewLine]", \(FullSimplify[ D[\(\((\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ \ nSig\ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2\))\))\)\))\)\^2\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\) + \(\(\[Gamma]\ \ \[Tau]\^2\)\/\(iInv\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) \[Omega]\^2, nSig]]\), "\[IndentingNewLine]", \(FullSimplify[ D[\(\((\(\[Lambda]\ \[Tau]\^2\)\/\(\[Sigma]\^2\ \((1 + nSig\ \[Lambda]\ \ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ \ nSig\ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2\))\))\)\))\)\^2\) nSig \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\) + \(\(\[Gamma]\ \ \[Tau]\^2\)\/\(iInv\ \((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2\))\))\)\)\) \[Omega]\^2, \[Lambda]\ ]]\)}], "Input"], Cell[BoxData[ \(\(\[Tau]\^2\ \((\(nSig\ \[Lambda]\^2\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\)\/\[Sigma]\^4 + \(\[Gamma]\ \[Omega]\^2\ \((1 + nSig\ \ \[Lambda]\ \((\[Tau]\^2\/\[Sigma]\^2 + \(iInv\^2\ \((\(-1\) + \ \[Lambda])\)\)\/\(iInv\^2\ nSig\ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \ \[Omega]\^2\))\))\)\)\/iInv)\)\)\/\((1 + nSig\ \[Lambda]\ \((\[Tau]\^2\/\ \[Sigma]\^2 + \(iInv\^2\ \((\(-1\) + \[Lambda])\)\)\/\(iInv\^2\ nSig\ \ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2\))\))\)\^2\)], "Output"], Cell[BoxData[ \(\((\[Lambda]\ \[Sigma]\^2\ \[Tau]\^2\ \((iInv\^7\ nSig\^3\ \[Lambda]\^5\ \ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \((\(-1\) + 2\ \[Lambda])\)\ \[Tau]\^2)\) - iInv\^5\ nSig\^2\ \[Gamma]\ \[Lambda]\^3\ \[Tau]\^2\ \((iInv\ \ nSig\ \[Lambda]\ \[Sigma]\^2 - 3\ \[Gamma]\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((iInv\ nSig - \[Gamma]\ \((1 + 2\ \[Lambda])\)\ \[Sigma]\^2)\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 - iInv\^3\ nSig\ \[Gamma]\^3\ \[Lambda]\^2\ \[Sigma]\^2\ \((iInv\ \ \((\(-1\) + \[Lambda])\)\ \[Sigma]\^4 + \[Sigma]\^2\ \((3\ iInv\ nSig\ \ \[Lambda] + \[Gamma]\ \((\(-4\) + \[Lambda])\)\ \[Sigma]\^2)\)\ \[Tau]\^2 + 3\ nSig\ \[Lambda]\ \((iInv\ nSig - \[Gamma]\ \ \[Sigma]\^2)\)\ \[Tau]\^4)\)\ \[Omega]\^4 - iInv\ \[Gamma]\^5\ \[Sigma]\^4\ \((iInv\ \((\(-1\) + \[Lambda])\ \)\ \[Sigma]\^4 + \[Lambda]\ \[Sigma]\^2\ \((iInv\ \((nSig + 2\ nSig\ \[Lambda])\) - \[Gamma]\ \ \[Sigma]\^2)\)\ \[Tau]\^2 + nSig\ \[Lambda]\^2\ \((3\ iInv\ nSig - \[Gamma]\ \[Sigma]\ \^2)\)\ \[Tau]\^4)\)\ \[Omega]\^6 - \[Gamma]\^7\ \[Sigma]\^6\ \[Tau]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^8)\))\)/\((iInv\ \ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \ \[Omega]\^2)\)\^3)\)\)], "Output"], Cell[BoxData[ \(\((nSig\ \[Tau]\^2\ \((iInv\^7\ nSig\^4\ \[Lambda]\^6\ \[Tau]\^4\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\) - 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