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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[ 117569, 2249]*) (*NotebookOutlinePosition[ 122890, 2401]*) (* CellTagsIndexPosition[ 122769, 2393]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["3", "ChapterLine", CellTags->"1.1"], Cell["\<\ Properties of Information Market Equilibrium under Partly \ Informative Price\ \>", "Title", CellTags->"1.1"], Cell[CellGroupData[{ Cell["\<\ 3.1 Evaluation of Terms in Table 2 (Utility Responses to Signal \ Acquisition, Appendix H)\ \>", "Section", CellTags->"1.1"], Cell[TextData[{ StyleBox["To evaluate the terms in Table 2, revisit results from the \ preceding notebook ", "Text"], StyleBox["infgauss2.nb", "Text", FontWeight->"Bold"], StyleBox[". The results were", "Text"] }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell["for News Subscribers", "BulletListItem", CellTags->"1.2"], Cell[BoxData[{ \(\(expNewsSubscribers = \(xbar\ \[Gamma]\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\)\ \((iInv\^2\ nSig\ \[Lambda]\^2 + \[Gamma]\^2\ \[Sigma]\^2\ \ \[Omega]\^2)\)\)\/\(iInv\^3\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + iInv\ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2\);\)\), "\[IndentingNewLine]", \(\(varNewsSubscribers = \(\[Gamma]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\ \)\ \[Omega]\^2\ \((iInv\^2\ nSig\ \[Gamma]\^2\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Sigma]\^4\ \[Omega]\^2 + \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((iInv\^2\ \ nSig\ \[Lambda] + \[Gamma]\^2\ \[Sigma]\^2\ \ \[Omega]\^2)\)\^2)\)\)\/\((iInv\^3\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + iInv\ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2;\)\), "\[IndentingNewLine]", \(\(tausqNewsSubscribers = 1\/\(nSig\/\[Sigma]\^2 + 1\/\[Tau]\^2\);\)\)}], "Input"], Cell[BoxData[{ \(\(elastexpNewsSubscribers = \(nSig\ \[Gamma]\^2\ \((\(-1\) + \[Lambda])\ \)\ \[Sigma]\^2\ \[Tau]\^2\ \[Omega]\^2\ \((iInv\^2\ nSig\^2\ \[Lambda]\^2\ \ \[Tau]\^2 - \[Gamma]\^2\ \[Sigma]\^4\ \[Omega]\^2)\)\)\/\(\((\[Sigma]\^2 + \ nSig\ \[Tau]\^2)\)\ \((iInv\^2\ nSig\ \[Lambda]\^2 + \[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)\)\);\)\), "\[IndentingNewLine]", \(\(elastvarNewsSubscribers = \(-\(\(nSig\ \[Gamma]\^2\ \((\(-1\) + \ \[Lambda])\)\ \[Sigma]\^4\ \[Omega]\^2\ \((iInv\^4\ nSig\ \[Lambda]\^2\ \((1 \ + \[Lambda])\)\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((\[Sigma]\^2 + 2\ nSig\ \[Tau]\^2)\) + iInv\^2\ \[Gamma]\^2\ \[Sigma]\^2\ \((\((1 + \[Lambda])\)\ \ \[Sigma]\^4 + nSig\ \((2 + \[Lambda]\ \((5 + \[Lambda])\))\)\ \ \[Sigma]\^2\ \[Tau]\^2 + 6\ nSig\^2\ \[Lambda]\ \[Tau]\^4)\)\ \[Omega]\^2 + 2\ \[Gamma]\^4\ \[Sigma]\^4\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \[Omega]\^4)\)\)\/\(\((\[Sigma]\ \^2 + nSig\ \[Tau]\^2)\)\ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \ \((iInv\^4\ nSig\^2\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Sigma]\^2\ \((\((1 + \ \[Lambda]\^2)\)\ \[Sigma]\^2 + 2\ nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2 + \ \[Gamma]\^4\ \[Sigma]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \[Omega]\^4)\)\)\)\);\)\), "\ \[IndentingNewLine]", \(\(elasttauNewsSubscribers = \(-\(\(nSig\ \[Tau]\^2\)\/\(\[Sigma]\^2 + nSig\ \[Tau]\^2\)\)\);\)\), "\[IndentingNewLine]", \(\(focEtermNewsSubscribers = \(-\((\((1 + R)\)\ xbar\^2\ \[Gamma]\^2\ \[Sigma]\^2\ \[Tau]\^4\ \ \((iInv\^2\ nSig\ \[Lambda]\^2 + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2)\)\ \ \((iInv\^4\ nSig\^2\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \ \((2\ \[Sigma]\^2 - nSig\ \((\(-3\) + \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^4)\))\)\)/\ \((2\ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \ \((iInv\^6\ nSig\^2\ \((1 + R)\)\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\ \)\^2 + iInv\^4\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 \ + nSig\ \[Tau]\^2)\)\ \((2\ \((1 + R)\)\ \[Sigma]\^2 + nSig\ \((1 + 2\ \((1 + R)\)\ \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + iInv\^2\ \[Gamma]\^4\ \[Sigma]\^4\ \((\((1 + R)\)\ \[Sigma]\^4 + nSig\ \((1 + \[Lambda]\ \((2 + 2\ R + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^2 + nSig\^2\ \[Lambda]\ \((2 + \[Lambda] + R\ \[Lambda])\)\ \[Tau]\^4)\)\ \[Omega]\^4 + \ \[Gamma]\^6\ \[Sigma]\^6\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \[Omega]\^6)\))\);\)\), "\ \[IndentingNewLine]", \(\(focVartermNewsSubscribers = \(-\((iInv\^2\ \((1 + R)\)\ \[Gamma]\^2\ \[Sigma]\^2\ \[Tau]\^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)\)\ \ \((iInv\^6\ nSig\^3\ \[Lambda]\^4\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^4\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + 3\ nSig\ \[Lambda]\^2\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\ \[Tau]\^4)\)\ \[Omega]\^2 + iInv\^2\ \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + \ \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \((5 + \ \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\ \[Omega]\^4 \ + \[Gamma]\^6\ \[Sigma]\^6\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^6)\))\)\)/\ \((2\ \((iInv\^6\ nSig\^2\ \((1 + R)\)\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\)\^2 + iInv\^4\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((2\ \((1 + R)\)\ \[Sigma]\^2 + nSig\ \((1 \ + 2\ \((1 + R)\)\ \[Lambda])\)\ \[Tau]\^2)\)\ \[Omega]\^2 + iInv\^2\ \[Gamma]\ \^4\ \[Sigma]\^4\ \((\((1 + R)\)\ \[Sigma]\^4 + nSig\ \((1 + \[Lambda]\ \((2 \ + 2\ R + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + nSig\^2\ \[Lambda]\ \((2 + \ \[Lambda] + R\ \[Lambda])\)\ \[Tau]\^4)\)\ \[Omega]\^4 + \[Gamma]\^6\ \ \[Sigma]\^6\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \ \[Omega]\^6)\)\^2)\);\)\), "\[IndentingNewLine]", \(\(focDelfactorNewsSubscribers = R + \(\((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((iInv\^2\ nSig\ \ \[Lambda]\^2 + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2)\)\ \((iInv\^4\ nSig\ \ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^4\ \[Sigma]\^4\ \ \[Tau]\^2\ \[Omega]\^2\ \((\(-xbar\^2\) + \[Omega]\^2)\) + iInv\^2\ \ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2\ \[Omega]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2\ \((\(-xbar\^2\)\ \[Lambda] + 2\ \[Omega]\^2)\))\))\)\)\/\((iInv\^3\ \ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\ \[Gamma]\^2\ \ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \^2;\)\)}], "Input"], Cell["and for Price Watchers", "BulletListItem", CellTags->"1.2"], Cell[BoxData[{ \(\(expPriceWatchers = \(iInv\^2\ nSig\ xbar\ \[Gamma]\ \[Lambda]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\) + xbar\ \[Gamma]\^3\ \[Sigma]\^4\ \ \[Omega]\^2\)\/\(iInv\^3\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + iInv\ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2\);\)\), "\[IndentingNewLine]", \(\(varPriceWatchers = \(iInv\^2\ nSig\ \[Gamma]\^4\ \[Lambda]\^2\ \ \[Sigma]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \[Omega]\^4 + \[Gamma]\^6\ \ \[Sigma]\^8\ \[Omega]\^6\)\/\((iInv\^3\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + \ nSig\ \[Tau]\^2)\) + iInv\ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\^2;\)\), "\[IndentingNewLine]", \(\(tausqPriceWatchers = \(\[Tau]\^2\ \((iInv\^2\ nSig\ \[Lambda]\^2\ \ \[Sigma]\^2 + \[Gamma]\^2\ \[Sigma]\^4\ \[Omega]\^2)\)\)\/\(iInv\^2\ nSig\ \ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^4\ \ \[Omega]\^2\);\)\)}], "Input"], Cell[BoxData[{ \(\(elastexpPriceWatchers = \(iInv\^2\ nSig\^3\ \[Gamma]\^2\ \[Lambda]\^3\ \ \[Sigma]\^2\ \[Tau]\^4\ \[Omega]\^2 - nSig\ \[Gamma]\^4\ \[Lambda]\ \ \[Sigma]\^6\ \[Tau]\^2\ \[Omega]\^4\)\/\(\((iInv\^2\ nSig\ \[Lambda]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[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)\)\);\)\), "\[IndentingNewLine]", \(\(elastvarPriceWatchers = \(nSig\ \[Lambda]\ \((\(-iInv\^4\)\ nSig\ \ \[Lambda]\^3\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((\[Sigma]\^2 + 2\ nSig\ \ \[Tau]\^2)\) - iInv\^2\ \[Gamma]\^2\ \[Lambda]\ \[Sigma]\^4\ \((\[Sigma]\^2 + \ nSig\ \((2 + \[Lambda])\)\ \[Tau]\^2)\)\ \[Omega]\^2 - 2\ \[Gamma]\^4\ \ \[Sigma]\^6\ \[Tau]\^2\ \[Omega]\^4)\)\)\/\(\((iInv\^2\ nSig\ \[Lambda]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[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)\)\);\)\), "\[IndentingNewLine]", \(\(elasttauPriceWatchers = \(-\(\(nSig\^2\ \[Tau]\^2\ \((iInv\^4\ nSig\ \ \[Lambda]\^4 + 2\ iInv\^2\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \ \[Omega]\^2)\)\)\/\(\((iInv\^2\ nSig\ \[Lambda]\^2 + \[Gamma]\^2\ \[Sigma]\^2\ \ \[Omega]\^2)\)\ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^4\ \ \[Omega]\^2)\)\)\)\);\)\), "\[IndentingNewLine]", \(\(focEtermPriceWatchers = \(-\((\((1 + R)\)\ xbar\^2\ \[Gamma]\^2\ \[Lambda]\ \[Sigma]\^2\ \ \[Tau]\^4\ \((iInv\^6\ nSig\^3\ \[Lambda]\^5\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^4\ nSig\^2\ \[Gamma]\^2\ \[Lambda]\^3\ \[Sigma]\^2\ \ \((3\ \[Sigma]\^2 - nSig\ \((\(-2\) + \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + 2\ iInv\^2\ nSig\ \[Gamma]\^4\ \[Lambda]\ \((1 + \ \[Lambda])\)\ \[Sigma]\^6\ \[Omega]\^4 + 2\ \[Gamma]\^6\ \[Sigma]\^8\ \[Omega]\^6)\))\)\)/\((2\ \ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \ \((iInv\^6\ nSig\^2\ \((1 + R)\)\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\ \)\^2 + 2\ iInv\^4\ nSig\ \((1 + R)\)\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2 + iInv\^2\ \[Gamma]\^4\ \[Sigma]\^4\ \((\((1 + R)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((2 + 2\ R + \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\^2 + nSig\^2\ \((1 + R)\)\ \[Lambda]\^2\ \[Tau]\^4)\)\ \[Omega]\^4 + \ \[Gamma]\^6\ \[Sigma]\^8\ \[Tau]\^2\ \[Omega]\^6)\))\);\)\), "\ \[IndentingNewLine]", \(\(focVartermPriceWatchers = \(-\(\(iInv\^2\ \((1 + R)\)\^2\ \[Gamma]\^4\ \ \[Lambda]\ \[Sigma]\^6\ \[Tau]\^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)\)\ \ \((iInv\^4\ nSig\ \[Lambda]\^3\ \((\[Sigma]\^2 + 3\ nSig\ \[Tau]\^2)\) + iInv\^2\ \[Gamma]\^2\ \[Lambda]\ \[Sigma]\^2\ \((\[Sigma]\ \^2 + nSig\ \((4 + \[Lambda])\)\ \[Tau]\^2)\)\ \[Omega]\^2 + 2\ \[Gamma]\^4\ \[Sigma]\^4\ \[Tau]\^2\ \ \[Omega]\^4)\)\)\/\(2\ \((iInv\^6\ nSig\^2\ \((1 + R)\)\ \[Lambda]\^4\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\)\^2 + 2\ iInv\^4\ nSig\ \((1 + R)\)\ \[Gamma]\ \^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2 + iInv\^2\ \ \[Gamma]\^4\ \[Sigma]\^4\ \((\((1 + R)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((2 \ + 2\ R + \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\^2 + nSig\^2\ \((1 + R)\)\ \ \[Lambda]\^2\ \[Tau]\^4)\)\ \[Omega]\^4 + \[Gamma]\^6\ \[Sigma]\^8\ \[Tau]\^2\ \ \[Omega]\^6)\)\^2\)\)\);\)\), "\[IndentingNewLine]", \(\(focDelfactorPriceWatchers = R + \(\((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^4\ \[Omega]\^2)\)\ \((iInv\^4\ nSig\ \ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^4\ \[Sigma]\^4\ \ \[Tau]\^2\ \[Omega]\^2\ \((\(-xbar\^2\) + \[Omega]\^2)\) + iInv\^2\ \ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2\ \[Omega]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2\ \((\(-xbar\^2\)\ \[Lambda] + 2\ \[Omega]\^2)\))\))\)\)\/\((iInv\^3\ \ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\ \[Gamma]\^2\ \ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \^2;\)\)}], "Input"], Cell[TextData[{ StyleBox["The first derivative of ", "Text"], StyleBox["focEtermNewsSubscribers", FontWeight->"Bold"], StyleBox[" with respect to is ", "Text"], Cell[BoxData[ \(\(\(xbar\^2\)\(\ \)\)\)]], StyleBox["(", "Text"], StyleBox["xbarsq", FontWeight->"Bold"], ") is" }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ D[\(-\((\((1 + R)\)\ xbarsq\ \[Gamma]\^2\ \[Sigma]\^2\ \[Tau]\^4\ \ \((iInv\^2\ nSig\ \[Lambda]\^2 + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2)\)\ \ \((iInv\^4\ nSig\^2\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \ \((2\ \[Sigma]\^2 - nSig\ \((\(-3\) + \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^4)\))\)\)/\ \((2\ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \ \((iInv\^6\ nSig\^2\ \((1 + R)\)\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\ \)\^2 + iInv\^4\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 \ + nSig\ \[Tau]\^2)\)\ \((2\ \((1 + R)\)\ \[Sigma]\^2 + nSig\ \((1 + 2\ \((1 + R)\)\ \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + iInv\^2\ \[Gamma]\^4\ \[Sigma]\^4\ \((\((1 + R)\)\ \[Sigma]\^4 + nSig\ \((1 + \[Lambda]\ \((2 + 2\ R + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^2 + nSig\^2\ \[Lambda]\ \((2 + \[Lambda] + R\ \[Lambda])\)\ \[Tau]\^4)\)\ \[Omega]\^4 + \ \[Gamma]\^6\ \[Sigma]\^6\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \[Omega]\^6)\))\), xbarsq]]\)], "Input"], Cell[BoxData[ \(\(-\(\((\((1 + R)\)\ \[Gamma]\^2\ \[Sigma]\^2\ \[Tau]\^4\ \((iInv\^2\ nSig\ \ \[Lambda]\^2 + \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2)\)\ \((iInv\^4\ nSig\^2\ \ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((2\ \ \[Sigma]\^2 - nSig\ \((\(-3\) + \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^4)\))\)/\((2\ \ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \ \((iInv\^6\ nSig\^2\ \((1 + R)\)\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \^2 + iInv\^4\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((2\ \((1 + R)\)\ \[Sigma]\^2 + nSig\ \((1 + 2\ \((1 + R)\)\ \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + iInv\^2\ \[Gamma]\^4\ \[Sigma]\^4\ \((\((1 + R)\)\ \[Sigma]\^4 + nSig\ \((1 + \[Lambda]\ \((2 + 2\ R + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\ \^2 + nSig\^2\ \[Lambda]\ \((2 + \[Lambda] + R\ \[Lambda])\)\ \[Tau]\^4)\)\ \[Omega]\^4 + \ \[Gamma]\^6\ \[Sigma]\^6\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \[Omega]\^6)\))\)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The restriction on ", "Text"], Cell[BoxData[ \(\(\(\ \)\(\[Omega]\^2\)\)\)]], StyleBox[" (", "Text"], StyleBox["omegasq", FontWeight->"Bold"], ")", StyleBox[" for a positive derivative ", "Text"], "is" }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(FullSimplify[ Solve[\((iInv\^4\ nSig\^2\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((2\ \ \[Sigma]\^2 - 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4\ \((3\ nSig\ \ \[Lambda]\^2\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + 3\ \ nSig\ \[Lambda]\^2\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\ \ \[Tau]\^4)\) + \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ \ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\^2)\)\^3)\))\))\)\^\(1/3\) + 2\^\(2/3\)\ \((iInv\^6\ \[Gamma]\^12\ \[Sigma]\^12\ \((2\ \ \((\(-1\) + \[Lambda]\^2)\)\^3\ \[Sigma]\^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) \ + \[Lambda]\^2)\)\^2\ \((\(-6\) + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\ \^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \((\(-720\) + \[Lambda]\ \((52 + \ \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \[Lambda])\))\))\))\))\))\)\ \ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \[Lambda]\^4\ \((21 + \[Lambda]\ \ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \((62 + 5\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\ \^6\ \[Tau]\^12)\) + \[Sqrt]\((iInv\^12\ \[Gamma]\^24\ \[Sigma]\^24\ \((\((2\ \ \((\(-1\) + \[Lambda]\^2)\)\^3\ \[Sigma]\^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) \ + \[Lambda]\^2)\)\^2\ \((\(-6\) + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\ \^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \((\(-720\) + \[Lambda]\ \((52 + \ \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \[Lambda])\))\))\))\))\))\)\ \ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \[Lambda]\^4\ \((21 + \[Lambda]\ \ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \((62 + 5\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\ \^6\ \[Tau]\^12)\)\^2 - 4\ \((3\ nSig\ \[Lambda]\^2\ \[Tau]\^2\ \((\((\(-1\) \ + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \((\((\(-1\) \ + \[Lambda]\^2)\)\ \[Sigma]\^4 + 3\ nSig\ \[Lambda]\^2\ \[Sigma]\^2\ \ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\ \[Tau]\^4)\) + \((\((\(-1\) + \ \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \((5 + \ \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\^2\ \ \[Tau]\^4)\)\^2)\)\^3)\))\))\)\^\(1/3\))\)\)}, {omegasq \[Rule] \(\(1\/\(12\ \ \[Gamma]\^6\ \[Sigma]\^6\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\)\)\((4\ iInv\^2\ \ \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\ \))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\) - \((2\ \ \[ImaginaryI]\ 2\^\(1/3\)\ \((\(-\[ImaginaryI]\) + \@3)\)\ iInv\^4\ \ \[Gamma]\^8\ \[Sigma]\^8\ \((\((\(-1\) + \[Lambda]\^2)\)\^2\ \[Sigma]\^8 + nSig\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\ \((1 + \ \[Lambda])\)\ \((\(-6\) + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^6\ \ \[Tau]\^2 + nSig\^2\ \[Lambda]\^2\ \((3 + \[Lambda]\ \((\(-33\) + \ \[Lambda]\ \((16 + \[Lambda]\ \((31 + \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \ \[Tau]\^4 + 3\ nSig\^3\ \[Lambda]\^3\ \((\(-9\) + \[Lambda]\ \((16 + 5\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^6 + 18\ nSig\^4\ \[Lambda]\^4\ \[Tau]\^8)\))\)/\((iInv\^6\ \ \[Gamma]\^12\ \[Sigma]\^12\ \((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) \ + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\) + \[Sqrt]\((iInv\^12\ \ \[Gamma]\^24\ \[Sigma]\^24\ \((\((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \[Sigma]\ \^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) + \ \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\)\^2 - 4\ \((3\ nSig\ \ \[Lambda]\^2\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + 3\ \ nSig\ \[Lambda]\^2\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\ \ \[Tau]\^4)\) + \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ \ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\^2)\)\^3)\))\))\)\^\(1/3\) + \ \[ImaginaryI]\ 2\^\(2/3\)\ \((\[ImaginaryI] + \@3)\)\ \((iInv\^6\ \ \[Gamma]\^12\ \[Sigma]\^12\ \((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) \ + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\) + \[Sqrt]\((iInv\^12\ \ \[Gamma]\^24\ \[Sigma]\^24\ \((\((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \[Sigma]\ \^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) + \ \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\)\^2 - 4\ \((3\ nSig\ \ \[Lambda]\^2\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + 3\ \ nSig\ \[Lambda]\^2\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\ \ \[Tau]\^4)\) + \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ \ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\^2)\)\^3)\))\))\)\^\(1/3\))\)\)}, \ {omegasq \[Rule] \(\(1\/\(12\ \[Gamma]\^6\ \[Sigma]\^6\ \[Tau]\^2\ \((\((\(-1\ \) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\)\)\((4\ iInv\^2\ \ \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\ \))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\) + \((2\ \ \[ImaginaryI]\ 2\^\(1/3\)\ \((\[ImaginaryI] + \@3)\)\ iInv\^4\ \[Gamma]\^8\ \ \[Sigma]\^8\ \((\((\(-1\) + \[Lambda]\^2)\)\^2\ \[Sigma]\^8 + nSig\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\ \((1 + \ \[Lambda])\)\ \((\(-6\) + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^6\ \ \[Tau]\^2 + nSig\^2\ \[Lambda]\^2\ \((3 + \[Lambda]\ \((\(-33\) + \ \[Lambda]\ \((16 + \[Lambda]\ \((31 + \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \ \[Tau]\^4 + 3\ nSig\^3\ \[Lambda]\^3\ \((\(-9\) + \[Lambda]\ \((16 + 5\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^6 + 18\ nSig\^4\ \[Lambda]\^4\ \[Tau]\^8)\))\)/\((iInv\^6\ \ \[Gamma]\^12\ \[Sigma]\^12\ \((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) \ + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\) + \[Sqrt]\((iInv\^12\ \ \[Gamma]\^24\ \[Sigma]\^24\ \((\((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \[Sigma]\ \^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) + \ \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\)\^2 - 4\ \((3\ nSig\ \ \[Lambda]\^2\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + 3\ \ nSig\ \[Lambda]\^2\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\ \ \[Tau]\^4)\) + \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ \ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\^2)\)\^3)\))\))\)\^\(1/3\) - 2\^\(2/3\)\ \((1 + \[ImaginaryI]\ \@3)\)\ \((iInv\^6\ \ \[Gamma]\^12\ \[Sigma]\^12\ \((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) \ + \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\) + \[Sqrt]\((iInv\^12\ \ \[Gamma]\^24\ \[Sigma]\^24\ \((\((2\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \[Sigma]\ \^12 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((\(-6\) + \ \[Lambda]\ \((7 + 8\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 + 6\ nSig\^2\ \ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + \[Lambda] + 34\ \[Lambda]\^2 + 4\ \[Lambda]\^3)\))\)\ \[Sigma]\ \^8\ \[Tau]\^4 + nSig\^3\ \[Lambda]\^3\ \((81 + \[Lambda]\ \((117 + \[Lambda]\ \ \((\(-720\) + \[Lambda]\ \((52 + \[Lambda]\ \((537 + \[Lambda]\ \((93 + 2\ \ \[Lambda])\))\))\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + 9\ nSig\^4\ \ \[Lambda]\^4\ \((21 + \[Lambda]\ \((\(-114\) + \[Lambda]\ \((80 + \[Lambda]\ \ \((62 + 5\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 54\ nSig\^5\ \ \[Lambda]\^5\ \((\(-7\) + \[Lambda]\ \((13 + 3\ \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 162\ nSig\^6\ \[Lambda]\^6\ \[Tau]\^12)\)\^2 - 4\ \((3\ nSig\ \ \[Lambda]\^2\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \ \[Lambda]\ \[Tau]\^2)\)\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + 3\ \ nSig\ \[Lambda]\^2\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\ \ \[Tau]\^4)\) + \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ \ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\)\^2)\)\^3)\))\))\)\^\(1/3\))\)\)}}\)], \ "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.6 Proof of Theorem 1 (Appendix I)", "Section", CellTags->"1.1"], Cell[TextData[{ StyleBox["To assess the utility effect of signals on Price Watchers, it \ suffices to evaluate the term ", "Text"], Cell[BoxData[ \(d + b \((d - g)\)\)]], ", where ", StyleBox[" ", "Text"], Cell[BoxData[ \(b = \(tausqPriceWatchers\/\(1 + R\)\) varPriceWatchers\)]], ", ", StyleBox[" ", "Text"], Cell[BoxData[ \(d = \(1\/nSig\) \((\(1\/2\) elasttauPriceWatchers + elastexpPriceWatchers)\)\)]], ", ", StyleBox[" and ", "Text"], Cell[BoxData[ \(g = \(1\/nSig\) \((\(1\/2\) elasttauPriceWatchers + \(1\/2\) elastvarPriceWatchers)\)\)]], "." }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[\(1\/nSig\) \((\(1\/2\) elasttauPriceWatchers + elastexpPriceWatchers)\) + \(1\/nSig\) \ \(tausqPriceWatchers\/\(1 + R\)\) varPriceWatchers \((\((\(1\/2\) elasttauPriceWatchers + elastexpPriceWatchers)\) - \((\(1\/2\) elasttauPriceWatchers + \(1\/2\) elastvarPriceWatchers)\))\)]\)], "Input"], Cell[BoxData[ \(\(-\(\((\[Lambda]\ \[Tau]\^2\ \((iInv\^8\ nSig\^4\ \((1 + R)\)\ \[Lambda]\^7\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \^2 + 2\ iInv\^6\ nSig\^3\ \((1 + R)\)\ \[Gamma]\^2\ \[Lambda]\^5\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((2\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \[Omega]\^2 + iInv\^4\ nSig\^2\ \[Gamma]\^4\ \[Lambda]\^3\ \[Sigma]\^4\ \((\ \((5 - \((\(-2\) + \[Lambda])\)\ \[Lambda] + R\ \((5 + 2\ \[Lambda])\))\)\ \[Sigma]\^4 + 2\ nSig\ \((2 - \((\(-2\) + \[Lambda])\)\ \[Lambda] + 2\ R\ \((1 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\ \^2 - nSig\^2\ \((1 + R)\)\ \((\(-2\) + \[Lambda])\)\ \[Lambda]\ \[Tau]\ \^4)\)\ \[Omega]\^4 + 2\ iInv\^2\ nSig\ \[Gamma]\^6\ \[Lambda]\ \[Sigma]\^8\ \ \((\((1 + R + 2\ \((1 + R)\)\ \[Lambda] - \[Lambda]\^2)\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \((2 - \[Lambda] + R\ \((2 + \[Lambda])\))\)\ \[Tau]\^2)\)\ \[Omega]\ \^6 + \[Gamma]\^8\ \[Sigma]\^10\ \((\((2 + 2\ R - \[Lambda])\)\ \[Sigma]\^2 + 2\ nSig\ R\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^8)\))\)/\ \((2\ \((1 + R)\)\ \((iInv\^2\ nSig\ \[Lambda]\^2 + \[Gamma]\^2\ \ \[Sigma]\^2\ \[Omega]\^2)\)\ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[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)\)\^2)\)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["So the term ", "Text"], Cell[BoxData[ \(d + b \((d - g)\)\)]], " is strictly negative, so a signal to News Subscribers inflicts a strict \ negative externality on Price Watchers. A direct evaluation of Price \ Watchers' marginal utility is possible." }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[{ \(\(focEtermPriceWatchersRewr = \(-\((\((1 + R)\)\ xbarsq\ \[Gamma]\^2\ \[Lambda]\ \[Sigma]\^2\ \[Tau]\ \^4\ \((iInv\^6\ nSig\^3\ \[Lambda]\^5\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^4\ nSig\^2\ \[Gamma]\^2\ \[Lambda]\^3\ \[Sigma]\^2\ \ \((3\ \[Sigma]\^2 - nSig\ \((\(-2\) + \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^2 + 2\ iInv\^2\ nSig\ \[Gamma]\^4\ \[Lambda]\ \((1 + \ \[Lambda])\)\ \[Sigma]\^6\ \[Omega]\^4 + 2\ \[Gamma]\^6\ \[Sigma]\^8\ \[Omega]\^6)\))\)\)/\((2\ \ \((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2)\)\ \ \((iInv\^6\ nSig\^2\ \((1 + R)\)\ \[Lambda]\^4\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\ \)\^2 + 2\ iInv\^4\ nSig\ \((1 + R)\)\ \[Gamma]\^2\ \[Lambda]\^2\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \[Omega]\^2 + iInv\^2\ \[Gamma]\^4\ \[Sigma]\^4\ \((\((1 + R)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((2 + 2\ R + \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\^2 + nSig\^2\ \((1 + R)\)\ \[Lambda]\^2\ \[Tau]\^4)\)\ \[Omega]\^4 + \ \[Gamma]\^6\ \[Sigma]\^8\ \[Tau]\^2\ \[Omega]\^6)\))\);\)\), "\ \[IndentingNewLine]", \(\(focDelfactorPriceWatchersRewr = R + \(\((iInv\^2\ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^4\ \[Omega]\^2)\)\ \((iInv\^4\ nSig\ \ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^4\ \[Sigma]\^4\ \ \[Tau]\^2\ \[Omega]\^2\ \((\(-xbarsq\) + \[Omega]\^2)\) + iInv\^2\ \ \[Gamma]\^2\ \[Sigma]\^2\ \((\[Sigma]\^2\ \[Omega]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2\ \((\(-xbarsq\)\ \[Lambda] + 2\ \[Omega]\^2)\))\))\)\)\/\((iInv\^3\ \ nSig\ \[Lambda]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\ \[Gamma]\^2\ \ \[Sigma]\^2\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \ \[Omega]\^2)\)\^2;\)\), "\[IndentingNewLine]", \(FullSimplify[ Solve[{focEtermPriceWatchersRewr + focDelfactorPriceWatchersRewr\ focVartermPriceWatchers \[Equal] 0}, {xbarsq}]]\), "\[IndentingNewLine]", \(FullSimplify[ Limit[xbarsq /. 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2\ \[Lambda]\^2 - 4\ R\ \((\(-2\) + \[Lambda] + \[Lambda]\^2)\ \))\)\ \[Sigma]\^2\ \[Tau]\^2 - nSig\^2\ \((2 + R)\)\ \((\(-2\) + \[Lambda])\)\ \ \[Lambda]\^2\ \[Tau]\^4)\)\ \[Omega]\^8 - iInv\^2\ \[Gamma]\^10\ \[Sigma]\^14\ \((\((1 + R)\)\ \((\(-2\) + \[Lambda])\)\ \[Sigma]\^2 \ + nSig\ \[Lambda]\ \((\(-2\) + \[Lambda] + 5\ R\ \[Lambda])\)\ \[Tau]\^2)\)\ \ \[Omega]\^10 - 2\ R\ \[Gamma]\^12\ \[Sigma]\^16\ \[Tau]\^2\ \ \[Omega]\^12)\))\)\)\)}}\)], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ \(\(\((1 + R)\)\ \[Gamma]\^2\ \[Sigma]\^4\ \ DirectedInfinity[Sign[\[Tau]]\^2\/\(Sign[R]\ Sign[\[Gamma]]\^2\ \ Sign[\[Sigma]]\^4\)]\)\/\[Tau]\^2\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.5 Proof of Theorem 2 (Appendix J)", "Section", CellTags->"1.1"], Cell[TextData[{ StyleBox["Evaluate the term ", "Text"], Cell[BoxData[ \(D + B \((D - G)\)\)]], ", where ", StyleBox[" ", "Text"], Cell[BoxData[ \(B = \(tausqNewsSubscribers\/\(1 + R\)\) varNewsSubscribers\)]], ", ", StyleBox[" ", "Text"], Cell[BoxData[ \(D = \(1\/nSig\) \((\(1\/2\) elasttauNewsSubscribers + elastexpNewsSubscribers)\)\)]], ", ", StyleBox[" and ", "Text"], Cell[BoxData[ \(G = \(1\/nSig\) \((\(1\/2\) elasttauNewsSubscribers + \(1\/2\) elastvarNewsSubscribers)\)\)]], "." }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[{ \(criterionPosUtil = FullSimplify[\(1\/nSig\) \((\(1\/2\) elasttauNewsSubscribers + elastexpNewsSubscribers)\) + \(tausqNewsSubscribers\/\(1 + R\)\) varNewsSubscribers \((\(1\/nSig\) \((\(1\/2\) elasttauNewsSubscribers + elastexpNewsSubscribers)\) - \(1\/nSig\) \((\(1\/2\) elasttauNewsSubscribers + \(1\/2\) elastvarNewsSubscribers)\))\)]\[IndentingNewLine]\), \ "\[IndentingNewLine]", \(FullSimplify[ Limit[criterionPosUtil, \[Omega] -> 0]]\), "\[IndentingNewLine]", \(FullSimplify[ Limit[criterionPosUtil, \[Omega] -> \[Infinity]]]\)}], "Input"], Cell[BoxData[ \(\(-\(\((\[Tau]\^2\ \((iInv\^6\ nSig\^3\ \((1 + R)\)\ \[Lambda]\^6\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \^2 + 3\ iInv\^4\ nSig\^2\ \((1 + R)\)\ \[Gamma]\^2\ \[Lambda]\^4\ \[Sigma]\^2\ \((\ \[Sigma]\^2 + nSig\ \[Tau]\^2)\)\^2\ \[Omega]\^2 + iInv\^2\ nSig\ \[Gamma]\^4\ \[Lambda]\^2\ \[Sigma]\^4\ \ \((\((2 + R + 2\ \((1 + R)\)\ \[Lambda] - \[Lambda]\^2)\)\ \[Sigma]\^4 + 2\ nSig\ \((2 + R + 2\ \((1 + R)\)\ \[Lambda] - \[Lambda]\^2)\)\ \[Sigma]\ \^2\ \[Tau]\^2 - nSig\^2\ \((\(-2\) + \[Lambda]\ \((\(-2\) + R\ \((\(-4\) + \[Lambda])\) + \[Lambda])\))\ \)\ \[Tau]\^4)\)\ \[Omega]\^4 + \[Gamma]\^6\ \[Sigma]\^6\ \((\(-\((\(-2\) + \ \[Lambda])\)\)\ \[Lambda]\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\^2 + R\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\))\)\ \[Omega]\^6)\))\ \)/\((2\ \((1 + R)\)\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\ \((iInv\^2\ nSig\ \[Lambda]\^2 + \[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[ \(\((2\ iInv\^2\ nSig\ \[Gamma]\^2\ \((\(-1\) + \[Lambda])\)\ \ \[Lambda]\^2\ \[Sigma]\^2\ \[Tau]\^2\ \[Omega]\ \((iInv\^6\ nSig\^4\ \((1 + R)\)\ \[Lambda]\^6\ \[Tau]\^2\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + iInv\^4\ nSig\^2\ \[Gamma]\^2\ \[Lambda]\^4\ \[Sigma]\^2\ \ \((\((\(-1\) - 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+ \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^8\ \[Tau]\^2 + 6\ \ nSig\^2\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 \ + \[Lambda]\ \((\(-27\) + 2\ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \ \[Sigma]\^6\ \[Tau]\^4 - nSig\^3\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \ \((27 + \[Lambda]\ \((171 + \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + \ 2\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \ \[Lambda])\)\^2\ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \ \[Sigma]\^2\ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^5\ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \ \[Gamma]\^24\ \((\(-1\) + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\ \^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \ \[Sigma]\^12 + 2\ nSig\ \((1 + \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\ \)\ \[Lambda]\ \((\(-1\) + 3\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\ \^2\ \((1 + \[Lambda])\)\^2\ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \ \((45 + \[Lambda]\ \((77 + \((\(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^8\ \[Tau]\^4 - 6\ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \ \((\(-6\) + \[Lambda]\ \((\(-5\) + \[Lambda]\ \((48 + \[Lambda]\ \((4 + \ \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + \ nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \((\(-225\) + \[Lambda]\ \ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \[Lambda])\)\ \ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \[Lambda]\^4\ \((\ \(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^10 + 108\ \ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \[Tau]\^12)\))\))\)\^\(1/3\ \) + 2\^\(2/3\)\ \((iInv\^6\ \[Gamma]\^12\ \[Sigma]\^14\ \((\(-2\)\ \((\(-1\) \ + \[Lambda]\^2)\)\^3\ \[Sigma]\^10 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \ \[Lambda]\^2)\)\^2\ \((6 + \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \ \[Sigma]\^8\ \[Tau]\^2 + 6\ nSig\^2\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \((\(-27\) + 2\ \ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \[Sigma]\^6\ \[Tau]\^4 - nSig\^3\ \ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \((27 + \[Lambda]\ \((171 + \ \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + 2\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^5\ \ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \[Gamma]\^24\ \((\(-1\) \ + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\^4\ \((\((\(-1\) + 2\ \ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) \ + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \[Sigma]\^12 + 2\ nSig\ \((1 + \ \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\)\ \[Lambda]\ \((\(-1\) + 3\ \ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\^2\ \((1 + \[Lambda])\)\^2\ \ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \((45 + \[Lambda]\ \((77 + \((\ \(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^8\ \[Tau]\^4 - 6\ \ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \((\(-6\) + \[Lambda]\ \((\(-5\) + \ \[Lambda]\ \((48 + \[Lambda]\ \((4 + \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\ \))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \ \((\(-225\) + \[Lambda]\ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \ \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \ \[Lambda]\^4\ \((\(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 108\ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \ \[Tau]\^12)\))\))\)\^\(1/3\))\)\)}, {omegasq \[Rule] \(\(1\/\(12\ \[Gamma]\^6\ \ \[Sigma]\^6\ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\)\)\((\(-4\)\ iInv\^2\ \ \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \((5 + \[Lambda])\ \))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\) - \((2\ \ \[ImaginaryI]\ 2\^\(1/3\)\ \((\(-\[ImaginaryI]\) + \@3)\)\ iInv\^4\ \ \[Gamma]\^8\ \((\(-1\) + \[Lambda])\)\ \[Sigma]\^10\ \((\((\(-1\) + \ \[Lambda])\)\ \((1 + \[Lambda])\)\^2\ \[Sigma]\^6 - nSig\ \[Lambda]\ \((1 + \[Lambda])\)\ \((6 + \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^4\ \[Tau]\^2 + nSig\^2\ \[Lambda]\^2\ \((\(-3\) + \((\(-6\) + \ \[Lambda])\)\ \((\(-4\) + \[Lambda])\)\ \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\^4 \ - 3\ nSig\^3\ \((\(-3\) + \[Lambda])\)\ \[Lambda]\^3\ \[Tau]\^6)\))\)/\((iInv\ \^6\ \[Gamma]\^12\ \[Sigma]\^14\ \((\(-2\)\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^10 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((6 + \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^8\ \[Tau]\^2 + 6\ \ nSig\^2\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 \ + \[Lambda]\ \((\(-27\) + 2\ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \ \[Sigma]\^6\ \[Tau]\^4 - nSig\^3\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \ \((27 + \[Lambda]\ \((171 + \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + \ 2\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \ \[Lambda])\)\^2\ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \ \[Sigma]\^2\ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^5\ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \ \[Gamma]\^24\ \((\(-1\) + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\ \^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \ \[Sigma]\^12 + 2\ nSig\ \((1 + \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\ \)\ \[Lambda]\ \((\(-1\) + 3\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\ \^2\ \((1 + \[Lambda])\)\^2\ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \ \((45 + \[Lambda]\ \((77 + \((\(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^8\ \[Tau]\^4 - 6\ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \ \((\(-6\) + \[Lambda]\ \((\(-5\) + \[Lambda]\ \((48 + \[Lambda]\ \((4 + \ \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + \ nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \((\(-225\) + \[Lambda]\ \ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \[Lambda])\)\ \ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \[Lambda]\^4\ \((\ \(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^10 + 108\ \ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \[Tau]\^12)\))\))\)\^\(1/3\ \) + \[ImaginaryI]\ 2\^\(2/3\)\ \((\[ImaginaryI] + \@3)\)\ \((iInv\^6\ \ \[Gamma]\^12\ \[Sigma]\^14\ \((\(-2\)\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^10 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((6 + \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^8\ \[Tau]\^2 + 6\ \ nSig\^2\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 \ + \[Lambda]\ \((\(-27\) + 2\ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \ \[Sigma]\^6\ \[Tau]\^4 - nSig\^3\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \ \((27 + \[Lambda]\ \((171 + \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + \ 2\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \ \[Lambda])\)\^2\ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \ \[Sigma]\^2\ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^5\ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \ \[Gamma]\^24\ \((\(-1\) + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\ \^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \ \[Sigma]\^12 + 2\ nSig\ \((1 + \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\ \)\ \[Lambda]\ \((\(-1\) + 3\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\ \^2\ \((1 + \[Lambda])\)\^2\ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \ \((45 + \[Lambda]\ \((77 + \((\(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^8\ \[Tau]\^4 - 6\ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \ \((\(-6\) + \[Lambda]\ \((\(-5\) + \[Lambda]\ \((48 + \[Lambda]\ \((4 + \ \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + \ nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \((\(-225\) + \[Lambda]\ \ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \[Lambda])\)\ \ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \[Lambda]\^4\ \((\ \(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^10 + 108\ \ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \[Tau]\^12)\))\))\)\^\(1/3\ \))\)\)}, {omegasq \[Rule] \(-\(\(1\/\(12\ \[Gamma]\^6\ \[Sigma]\^6\ \ \[Tau]\^2\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\)\)\((4\ iInv\^2\ \ \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \((5 + \ \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\) + \((2\ 2\^\(1/3\ \)\ \((1 - \[ImaginaryI]\ \@3)\)\ iInv\^4\ \[Gamma]\^8\ \((\(-1\) + \ \[Lambda])\)\ \[Sigma]\^10\ \((\((\(-1\) + \[Lambda])\)\ \((1 + \ \[Lambda])\)\^2\ \[Sigma]\^6 - nSig\ \[Lambda]\ \((1 + \[Lambda])\)\ \((6 + \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^4\ \[Tau]\^2 + nSig\^2\ \[Lambda]\^2\ \((\(-3\) + \((\(-6\) + \ \[Lambda])\)\ \((\(-4\) + \[Lambda])\)\ \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\^4 \ - 3\ nSig\^3\ \((\(-3\) + \[Lambda])\)\ \[Lambda]\^3\ \[Tau]\^6)\))\)/\((iInv\ \^6\ \[Gamma]\^12\ \[Sigma]\^14\ \((\(-2\)\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^10 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((6 + \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^8\ \[Tau]\^2 + 6\ \ nSig\^2\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 \ + \[Lambda]\ \((\(-27\) + 2\ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \ \[Sigma]\^6\ \[Tau]\^4 - nSig\^3\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \ \((27 + \[Lambda]\ \((171 + \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + \ 2\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \ \[Lambda])\)\^2\ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \ \[Sigma]\^2\ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^5\ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \ \[Gamma]\^24\ \((\(-1\) + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\ \^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \ \[Sigma]\^12 + 2\ nSig\ \((1 + \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\ \)\ \[Lambda]\ \((\(-1\) + 3\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\ \^2\ \((1 + \[Lambda])\)\^2\ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \ \((45 + \[Lambda]\ \((77 + \((\(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^8\ \[Tau]\^4 - 6\ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \ \((\(-6\) + \[Lambda]\ \((\(-5\) + \[Lambda]\ \((48 + \[Lambda]\ \((4 + \ \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + \ nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \((\(-225\) + \[Lambda]\ \ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \[Lambda])\)\ \ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \[Lambda]\^4\ \((\ \(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^10 + 108\ \ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \[Tau]\^12)\))\))\)\^\(1/3\ \) + 2\^\(2/3\)\ \((1 + \[ImaginaryI]\ \@3)\)\ \((iInv\^6\ \[Gamma]\^12\ \ \[Sigma]\^14\ \((\(-2\)\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \[Sigma]\^10 + 3\ \ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((6 + \[Lambda]\ \((\(-13\) \ + 4\ \[Lambda])\))\)\ \[Sigma]\^8\ \[Tau]\^2 + 6\ nSig\^2\ \((\(-1\) + \ \[Lambda])\)\^2\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \ \((\(-27\) + 2\ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \[Sigma]\^6\ \[Tau]\^4 \ - nSig\^3\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \((27 + \[Lambda]\ \ \((171 + \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + 2\ \[Lambda])\))\))\ \))\)\ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^5\ \ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \[Gamma]\^24\ \((\(-1\) \ + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\^4\ \((\((\(-1\) + 2\ \ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) \ + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \[Sigma]\^12 + 2\ nSig\ \((1 + \ \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\)\ \[Lambda]\ \((\(-1\) + 3\ \ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\^2\ \((1 + \[Lambda])\)\^2\ \ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \((45 + \[Lambda]\ \((77 + \((\ \(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^8\ \[Tau]\^4 - 6\ \ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \((\(-6\) + \[Lambda]\ \((\(-5\) + \ \[Lambda]\ \((48 + \[Lambda]\ \((4 + \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\ \))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \ \((\(-225\) + \[Lambda]\ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \ \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \ \[Lambda]\^4\ \((\(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 108\ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \ \[Tau]\^12)\))\))\)\^\(1/3\))\)\)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ " Evaluate the expression ", StyleBox[" ", "Text"], Cell[BoxData[ \(D + B \((D - G)\)\)]], "at the cutoff for ", StyleBox[" ", "Text"], Cell[BoxData[ \(G\ > \ 0\)]], ":" }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ StyleBox["(", FontWeight-> "Bold"], \(iInv\^6\ nSig\^3\ \((1 + R)\)\ \[Lambda]\^6\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\^2 \ + 3\ iInv\^4\ nSig\^2\ \((1 + R)\)\ \[Gamma]\^2\ \[Lambda]\^4\ \[Sigma]\^2\ \(\((\[Sigma]\ \^2 + nSig\ \[Tau]\^2)\)\^2\) omegasq + iInv\^2\ nSig\ \[Gamma]\^4\ \[Lambda]\^2\ \[Sigma]\^4\ \((\((2 + R + 2\ \((1 + R)\)\ \[Lambda] - \[Lambda]\^2)\)\ \[Sigma]\^4 \ + 2\ nSig\ \((2 + R + 2\ \((1 + R)\)\ \[Lambda] - \[Lambda]\^2)\)\ \[Sigma]\^2\ \ \[Tau]\^2 + nSig\^2\ \((2 + \[Lambda]\ \((2 + R\ \((4 - \[Lambda])\) - \[Lambda])\))\)\ \ \[Tau]\^4)\)\ omegasq\^2 + \[Gamma]\^6\ \[Sigma]\^6\ \((\((2 - \[Lambda])\)\ \ \[Lambda]\ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\)\^2 - R\ \((\[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\ \((\((1 - 2\ \[Lambda])\)\ \[Sigma]\^2 - nSig\ \[Lambda]\ \[Tau]\^2)\))\) omegasq\^3\), ")"}], "/.", \(omegasq \[Rule] \(1\/\(6\ \[Gamma]\^6\ \[Sigma]\^6\ \[Tau]\^2\ \ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\)\) \((\(-2\)\ iInv\^2\ \ \[Gamma]\^4\ \[Sigma]\^4\ \((\((\(-1\) + \[Lambda]\^2)\)\ \[Sigma]\^4 + nSig\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \((5 + \ \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^2 + 3\ nSig\^2\ \[Lambda]\^2\ \[Tau]\^4)\) + \((2\ 2\^\(1/3\ \)\ iInv\^4\ \[Gamma]\^8\ \((\(-1\) + \[Lambda])\)\ \[Sigma]\^10\ \ \((\((\(-1\) + \[Lambda])\)\ \((1 + \[Lambda])\)\^2\ \[Sigma]\^6 - nSig\ \[Lambda]\ \((1 + \[Lambda])\)\ \((6 + \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^4\ \[Tau]\^2 + nSig\^2\ \[Lambda]\^2\ \((\(-3\) + \((\(-6\) + \ \[Lambda])\)\ \((\(-4\) + \[Lambda])\)\ \[Lambda])\)\ \[Sigma]\^2\ \[Tau]\^4 \ - 3\ nSig\^3\ \((\(-3\) + \[Lambda])\)\ \[Lambda]\^3\ \[Tau]\^6)\))\)/\((iInv\ \^6\ \[Gamma]\^12\ \[Sigma]\^14\ \((\(-2\)\ \((\(-1\) + \[Lambda]\^2)\)\^3\ \ \[Sigma]\^10 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \[Lambda]\^2)\)\^2\ \((6 + \ \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \[Sigma]\^8\ \[Tau]\^2 + 6\ \ nSig\^2\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 \ + \[Lambda]\ \((\(-27\) + 2\ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \ \[Sigma]\^6\ \[Tau]\^4 - nSig\^3\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \ \((27 + \[Lambda]\ \((171 + \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + \ 2\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \ \[Lambda])\)\^2\ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \ \[Sigma]\^2\ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^5\ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \ \[Gamma]\^24\ \((\(-1\) + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\ \^4\ \((\((\(-1\) + 2\ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \ \[Sigma]\^12 + 2\ nSig\ \((1 + \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\ \)\ \[Lambda]\ \((\(-1\) + 3\ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\ \^2\ \((1 + \[Lambda])\)\^2\ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \ \((45 + \[Lambda]\ \((77 + \((\(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^8\ \[Tau]\^4 - 6\ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \ \((\(-6\) + \[Lambda]\ \((\(-5\) + \[Lambda]\ \((48 + \[Lambda]\ \((4 + \ \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + \ nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \((\(-225\) + \[Lambda]\ \ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \[Lambda])\)\ \ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \[Lambda]\^4\ \((\ \(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \[Tau]\^10 + 108\ \ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \[Tau]\^12)\))\))\)\^\(1/3\ \) + 2\^\(2/3\)\ \((iInv\^6\ \[Gamma]\^12\ \[Sigma]\^14\ \((\(-2\)\ \((\(-1\) \ + \[Lambda]\^2)\)\^3\ \[Sigma]\^10 + 3\ nSig\ \[Lambda]\ \((\(-1\) + \ \[Lambda]\^2)\)\^2\ \((6 + \[Lambda]\ \((\(-13\) + 4\ \[Lambda])\))\)\ \ \[Sigma]\^8\ \[Tau]\^2 + 6\ nSig\^2\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^2\ \((1 + \[Lambda])\)\ \((6 + \[Lambda]\ \((\(-27\) + 2\ \ \[Lambda]\ \((8 + \[Lambda])\))\))\)\ \[Sigma]\^6\ \[Tau]\^4 - nSig\^3\ \ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^3\ \((27 + \[Lambda]\ \((171 + \ \[Lambda]\ \((\(-261\) + \[Lambda]\ \((\(-29\) + 2\ \[Lambda])\))\))\))\)\ \ \[Sigma]\^4\ \[Tau]\^6 + 9\ nSig\^4\ \((\(-1\) + \[Lambda])\)\^2\ \ \[Lambda]\^4\ \((\(-9\) + \[Lambda]\ \((24 + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^8 + 54\ nSig\^5\ \((\(-1\) + \[Lambda])\)\^2\ \[Lambda]\^5\ \ \[Tau]\^10)\) + 3\ \@3\ \[Sqrt]\((iInv\^12\ nSig\^2\ \[Gamma]\^24\ \((\(-1\) \ + \[Lambda])\)\^3\ \[Lambda]\^4\ \[Sigma]\^28\ \[Tau]\^4\ \((\((\(-1\) + 2\ \ \[Lambda])\)\ \[Sigma]\^2 + nSig\ \[Lambda]\ \[Tau]\^2)\)\^2\ \((\(-\((\(-1\) \ + \[Lambda])\)\)\ \((1 + \[Lambda])\)\^4\ \[Sigma]\^12 + 2\ nSig\ \((1 + \ \[Lambda])\)\^3\ \((2 + \((\(-3\) + \[Lambda])\)\ \[Lambda]\ \((\(-1\) + 3\ \ \[Lambda])\))\)\ \[Sigma]\^10\ \[Tau]\^2 - nSig\^2\ \((1 + \[Lambda])\)\^2\ \ \((\(-4\) + \[Lambda]\ \((\(-34\) + \[Lambda]\ \((45 + \[Lambda]\ \((77 + \((\ \(-53\) + \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^8\ \[Tau]\^4 - 6\ \ nSig\^3\ \[Lambda]\ \((1 + \[Lambda])\)\ \((\(-6\) + \[Lambda]\ \((\(-5\) + \ \[Lambda]\ \((48 + \[Lambda]\ \((4 + \((\(-26\) + \[Lambda])\)\ \[Lambda])\))\ \))\))\)\ \[Sigma]\^6\ \[Tau]\^6 + nSig\^4\ \[Lambda]\^2\ \((81 + \[Lambda]\ \ \((\(-225\) + \[Lambda]\ \((\(-438\) + \[Lambda]\ \((294 + \((169 - 9\ \ \[Lambda])\)\ \[Lambda])\))\))\))\)\ \[Sigma]\^4\ \[Tau]\^8 + 8\ nSig\^5\ \ \[Lambda]\^4\ \((\(-54\) + 5\ \[Lambda]\ \((9 + \[Lambda])\))\)\ \[Sigma]\^2\ \ \[Tau]\^10 + 108\ nSig\^6\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\^4\ \ \[Tau]\^12)\))\))\)\^\(1/3\))\)\)}], "]"}]], "Input"], Cell[BoxData[ \($Aborted\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.7 Limits for Derivation of Information Market Equilibrium", "Section", CellTags->"1.1"], Cell[TextData[{ StyleBox["Take the limits of the News 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