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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[ 117881, 2498]*) (*NotebookOutlinePosition[ 123973, 2672]*) (* CellTagsIndexPosition[ 123827, 2663]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["2", "ChapterLine", CellTags->"1.1"], Cell["\<\ Utility Responses to Signal Acquisition under Partly Informative \ Price\ \>", "Title", CellTags->"1.1"], Cell[CellGroupData[{ Cell["2.1 Elasticities of Key Term Moments (Appendix F)", "Section", CellTags->"1.1"], Cell[TextData[{ StyleBox["This section derives moments of ", "Text"], " the Updated Variance ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\[Tau]\^i\), "TraditionalForm"], ")"}], "2"], TraditionalForm]]], " and the Key Term ", StyleBox["(", "InlineFormula"], Cell[BoxData[ FormBox[ RowBox[{\(\[Mu]\^i\), "-", StyleBox["RP", FontSlant->"Italic"]}], TraditionalForm]]], StyleBox[")/", "InlineFormula"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(\[Tau]\^i\), "TraditionalForm"], ")"}], "2"], TraditionalForm]]], ".The moments have been reported in the ", StyleBox["notebook file ", "Text"], StyleBox["infgauss1.nb", "Text", FontWeight->"Bold"], StyleBox[" and are loaded. As before", "Text"], ", the number of signals is ", StyleBox["nSig", FontWeight->"Bold"], " and the number of investors ", StyleBox["iInv", FontWeight->"Bold"], " to prevent ", StyleBox["Mathematica", FontSlant->"Italic"], " from confusing these variables with internal variables. Define ", Cell[BoxData[ \(TraditionalForm\`E[\((\[Mu] - RP)\)/\[Tau]\^2]\)]], " as ", StyleBox["expNewsSubscribers", FontWeight->"Bold"], " and ", Cell[BoxData[ \(TraditionalForm\`V[\((\[Mu] - RP)\)/\[Tau]\^2]\)]], " as ", StyleBox["varNewssubscribers", FontWeight->"Bold"], " for News Subscribers, and similarly for Price Watchers. 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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)\))\))\)\^\(1/3\))\)}, {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[{ StyleBox["The ", "Text"], "cutoff of ", StyleBox["the prior payoff variance ", "Text"], Cell[BoxData[ \(\[Tau]\^2\)]], ", denoted", StyleBox[" ", "Text"], StyleBox["tausq", FontWeight->"Bold"], ", at which ", StyleBox["the pre-posterior variance of the normalized expected excess \ return", "Text"], " ", Cell[BoxData[ \(\(1\/2\) elasttauNewsSubscribers + \(1\/2\) elastvarNewsSubscribers\)]], " turns positive is" }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", StyleBox["tausq", FontWeight->"Bold"]}], StyleBox["+", FontWeight->"Bold"], FractionBox[ RowBox[{\(\[Gamma]\^2\), " ", \((1 - 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\ \[Sqrt]\((\(-4\)\ iInv\^2\ nSig\ \[Gamma]\^2\ \[Lambda]\ \((\(-1\) + \ \[Lambda]\^2)\)\ \[Sigma]\^6\ \[Omega]\^2\ \((iInv\^2\ nSig\ \[Lambda] + \ \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2)\)\^3\ \((iInv\^2\ nSig\ \[Lambda]\^2 + \ \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2)\) + \((iInv\^6\ nSig\^3\ \[Lambda]\^4\ \ \[Sigma]\^2 + 3\ iInv\^4\ nSig\^2\ \[Gamma]\^2\ \[Lambda]\^4\ \[Sigma]\^4\ \ \[Omega]\^2 + iInv\^2\ nSig\ \[Gamma]\^4\ \[Lambda]\ \((\(-3\) + \[Lambda]\ \ \((5 + \[Lambda])\))\)\ \[Sigma]\^6\ \[Omega]\^4 + \[Gamma]\^6\ \((\(-1\) + 2\ \ \[Lambda])\)\ \[Sigma]\^8\ \[Omega]\^6)\)\^2)\))\)\)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Also note that the pre-posterior variance of the normalized \ expected excess return for News Subscribers monotonically decreases in the \ prior payoff variance ", "Text"], Cell[BoxData[ \(\[Tau]\^2\)]], ", denoted", StyleBox[" ", "Text"], StyleBox["tausq", FontWeight->"Bold"], StyleBox[". 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\[Lambda])\)\ \[Sigma]\^4\ \[Omega]\^2\ \ \((iInv\^10\ nSig\^5\ \[Lambda]\^6\ \((1 + \[Lambda])\)\ \((nSig\ tausq\ \ \[Sigma] + \[Sigma]\^3)\)\^2 + 2\ iInv\^8\ nSig\^4\ \[Gamma]\^2\ \[Lambda]\^4\ \[Sigma]\^4\ \ \((nSig\ tausq + \[Sigma]\^2)\)\ \((nSig\ tausq\ \((1 + \[Lambda]\ \((4 + \((\ \(-1\) + \[Lambda])\)\ \[Lambda])\))\) + \((2 + 2\ \[Lambda] + \[Lambda]\^3)\)\ \[Sigma]\^2)\)\ \ \[Omega]\^2 + iInv\^6\ nSig\^3\ \[Gamma]\^4\ \[Lambda]\^2\ \[Sigma]\^6\ \((2\ \ nSig\^2\ tausq\^2\ \[Lambda]\ \((6 + 3\ \[Lambda] + \[Lambda]\^3)\) + 2\ nSig\ tausq\ \((2 + 11\ \[Lambda] + 7\ \[Lambda]\^3)\)\ \[Sigma]\^2 + \((5 + \[Lambda]\ \ \((5 + \[Lambda]\ \((3 + 7\ \[Lambda])\))\))\)\ \[Sigma]\^4)\)\ \[Omega]\^4 + 2\ iInv\^4\ nSig\^2\ \[Gamma]\^6\ \[Sigma]\^8\ \((nSig\^2\ \ tausq\^2\ \[Lambda]\^2\ \((9 + \[Lambda]\ \((\(-1\) + 2\ \[Lambda])\))\) + nSig\ tausq\ \[Lambda]\ \((6 + \[Lambda]\ \((3 + \ \[Lambda]\ \((10 + \[Lambda])\))\))\)\ \[Sigma]\^2 + \((1 + \[Lambda])\)\ \ \((1 + \[Lambda]\^2\ \((3 + \[Lambda])\))\)\ \[Sigma]\^4)\)\ \[Omega]\^6 + iInv\^2\ nSig\ \[Gamma]\^8\ \[Sigma]\^10\ \((nSig\^2\ tausq\^2\ \ \[Lambda]\ \((10 + \((\(-1\) + \[Lambda])\)\ \[Lambda])\) + 2\ nSig\ tausq\ \((1 + \[Lambda])\)\ \((2 + 3\ \[Lambda])\)\ \[Sigma]\^2 + \((3 + \[Lambda]\ \ \((3 + 4\ \[Lambda])\))\)\ \[Sigma]\^4)\)\ \[Omega]\^8 + 2\ \[Gamma]\^10\ \[Sigma]\^12\ \((nSig\ tausq + \ \[Sigma]\^2)\)\^2\ \[Omega]\^10)\))\)/\((\((iInv\^2\ nSig\ \[Lambda]\^2\ \ \((nSig\ tausq + \[Sigma]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((nSig\ tausq\ \ \[Lambda] + \[Sigma]\^2)\)\ \[Omega]\^2)\)\^2\ \((iInv\^4\ nSig\^2\ \[Lambda]\ \^2\ \((nSig\ tausq + \[Sigma]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \ \[Sigma]\^2\ \((2\ nSig\ tausq\ \[Lambda] + \((1 + \[Lambda]\^2)\)\ \ \[Sigma]\^2)\)\ \[Omega]\^2 + \[Gamma]\^4\ \[Sigma]\^4\ \((nSig\ tausq + \ \[Sigma]\^2)\)\ \[Omega]\^4)\)\^2)\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Then subtract the denominator from the numerator to infer whether \ the term changes less or more than -1 in ", "Text"], StyleBox["tausq", FontWeight->"Bold"], StyleBox[".", "Text"] }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[\(-\((\((iInv\^2\ nSig\ \[Lambda]\^2\ \((nSig\ tausq + \ \[Sigma]\^2)\) + \[Gamma]\^2\ \[Sigma]\^2\ \((nSig\ tausq\ \[Lambda] + \ \[Sigma]\^2)\)\ \[Omega]\^2)\)\^2\ \((iInv\^4\ nSig\^2\ \[Lambda]\^2\ \((nSig\ \ tausq + \[Sigma]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Sigma]\^2\ \((2\ \ nSig\ tausq\ \[Lambda] + \((1 + \[Lambda]\^2)\)\ \[Sigma]\^2)\)\ \[Omega]\^2 \ + \[Gamma]\^4\ \[Sigma]\^4\ \((nSig\ tausq + \[Sigma]\^2)\)\ \ \[Omega]\^4)\)\^2)\)\) + \[Gamma]\^2\ \((1 - \[Lambda])\)\ \[Sigma]\^6\ \ \[Omega]\^2\ \((iInv\^10\ nSig\^5\ \[Lambda]\^6\ \((1 + \[Lambda])\)\ \((nSig\ \ tausq + \[Sigma]\^2)\)\^2 + 2\ iInv\^8\ nSig\^4\ \[Gamma]\^2\ \[Lambda]\^4\ \[Sigma]\^2\ \ \((nSig\ tausq + \[Sigma]\^2)\)\ \((nSig\ tausq\ \((1 + \[Lambda]\ \((4 + \((\ \(-1\) + \[Lambda])\)\ \[Lambda])\))\) + \((2 + 2\ \[Lambda] + \[Lambda]\^3)\)\ \[Sigma]\^2)\)\ \ \[Omega]\^2 + iInv\^6\ nSig\^3\ \[Gamma]\^4\ \[Lambda]\^2\ \[Sigma]\^4\ \((2\ \ nSig\^2\ tausq\^2\ \[Lambda]\ \((6 + 3\ \[Lambda] + \[Lambda]\^3)\) + 2\ nSig\ tausq\ \((2 + 11\ \[Lambda] + 7\ \[Lambda]\^3)\)\ \[Sigma]\^2 + \((5 + \[Lambda]\ \ \((5 + \[Lambda]\ \((3 + 7\ \[Lambda])\))\))\)\ \[Sigma]\^4)\)\ \[Omega]\^4 + 2\ iInv\^4\ nSig\^2\ \[Gamma]\^6\ \[Sigma]\^6\ \((nSig\^2\ \ tausq\^2\ \[Lambda]\^2\ \((9 + \[Lambda]\ \((\(-1\) + 2\ \[Lambda])\))\) + nSig\ tausq\ \[Lambda]\ \((6 + \[Lambda]\ \((3 + \ \[Lambda]\ \((10 + \[Lambda])\))\))\)\ \[Sigma]\^2 + \((1 + \[Lambda])\)\ \ \((1 + \[Lambda]\^2\ \((3 + \[Lambda])\))\)\ \[Sigma]\^4)\)\ \[Omega]\^6 + iInv\^2\ nSig\ \[Gamma]\^8\ \[Sigma]\^8\ \((nSig\^2\ tausq\^2\ \ \[Lambda]\ \((10 + \((\(-1\) + \[Lambda])\)\ \[Lambda])\) + 2\ nSig\ tausq\ \((1 + \[Lambda])\)\ \((2 + 3\ \[Lambda])\)\ \[Sigma]\^2 + \((3 + \[Lambda]\ \ \((3 + 4\ \[Lambda])\))\)\ \[Sigma]\^4)\)\ \[Omega]\^8 + 2\ \[Gamma]\^10\ \[Sigma]\^10\ \((nSig\ tausq + \ \[Sigma]\^2)\)\^2\ \[Omega]\^10)\)]\)], "Input"], Cell[BoxData[ \(\(-\((iInv\^2\ nSig\ \[Lambda]\^2\ \((nSig\ tausq + \[Sigma]\^2)\) + \ \[Gamma]\^2\ \[Sigma]\^2\ \((nSig\ tausq\ \[Lambda] + \[Sigma]\^2)\)\ \ \[Omega]\^2)\)\^2\)\ \((iInv\^4\ nSig\^2\ \[Lambda]\^2\ \((nSig\ tausq + \ \[Sigma]\^2)\) + iInv\^2\ nSig\ \[Gamma]\^2\ \[Sigma]\^2\ \((2\ nSig\ tausq\ \ \[Lambda] + \((1 + \[Lambda]\^2)\)\ \[Sigma]\^2)\)\ \[Omega]\^2 + \[Gamma]\^4\ \ \[Sigma]\^4\ \((nSig\ tausq + \[Sigma]\^2)\)\ \[Omega]\^4)\)\^2 + \ \[Gamma]\^2\ \((1 - \[Lambda])\)\ \[Sigma]\^6\ \[Omega]\^2\ \((iInv\^10\ nSig\ \^5\ \[Lambda]\^6\ \((1 + \[Lambda])\)\ \((nSig\ tausq + \[Sigma]\^2)\)\^2 + 2\ iInv\^8\ nSig\^4\ \[Gamma]\^2\ \[Lambda]\^4\ \[Sigma]\^2\ \ \((nSig\ tausq + \[Sigma]\^2)\)\ \((nSig\ tausq\ \((1 + \[Lambda]\ \((4 + \((\ \(-1\) + \[Lambda])\)\ \[Lambda])\))\) + \((2 + 2\ \[Lambda] + \[Lambda]\^3)\)\ \[Sigma]\^2)\)\ \ \[Omega]\^2 + iInv\^6\ nSig\^3\ \[Gamma]\^4\ \[Lambda]\^2\ \[Sigma]\^4\ \((2\ \ nSig\^2\ tausq\^2\ \[Lambda]\ \((6 + 3\ \[Lambda] + \[Lambda]\^3)\) + 2\ nSig\ tausq\ \((2 + 11\ \[Lambda] + 7\ \[Lambda]\^3)\)\ \[Sigma]\^2 + \((5 + \[Lambda]\ \ \((5 + \[Lambda]\ \((3 + 7\ \[Lambda])\))\))\)\ \[Sigma]\^4)\)\ \[Omega]\^4 + 2\ iInv\^4\ nSig\^2\ \[Gamma]\^6\ \[Sigma]\^6\ \((nSig\^2\ \ tausq\^2\ \[Lambda]\^2\ \((9 + \[Lambda]\ \((\(-1\) + 2\ \[Lambda])\))\) + nSig\ tausq\ \[Lambda]\ \((6 + \[Lambda]\ \((3 + \[Lambda]\ \ \((10 + \[Lambda])\))\))\)\ \[Sigma]\^2 + \((1 + \[Lambda])\)\ \((1 + \ \[Lambda]\^2\ \((3 + \[Lambda])\))\)\ \[Sigma]\^4)\)\ \[Omega]\^6 + iInv\^2\ nSig\ \[Gamma]\^8\ \[Sigma]\^8\ \((nSig\^2\ tausq\^2\ \ \[Lambda]\ \((10 + \((\(-1\) + \[Lambda])\)\ \[Lambda])\) + 2\ nSig\ tausq\ \((1 + \[Lambda])\)\ \((2 + 3\ \[Lambda])\)\ \[Sigma]\^2 + \((3 + \[Lambda]\ \((3 \ + 4\ \[Lambda])\))\)\ \[Sigma]\^4)\)\ \[Omega]\^8 + 2\ \[Gamma]\^10\ \[Sigma]\^10\ \((nSig\ tausq + \[Sigma]\^2)\)\^2\ \ \[Omega]\^10)\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["So, the the last term in the solution to ", "Text"], Cell[BoxData[ \(\((\(1\/2\) elasttauNewsSubscribers + \(1\/2\) elastvarNewsSubscribers)\)\)]], StyleBox[" changes less strongly than -1 and the overall effect of ", "Text"], StyleBox["tausq", FontWeight->"Bold"], StyleBox[" on ", "Text"], Cell[BoxData[ \(\((\(1\/2\) elasttauNewsSubscribers + \(1\/2\) elastvarNewsSubscribers)\)\)]], StyleBox[" is negative.", "Text"] }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2.4 Derivation of Terms in Table 2 (Utility Responses to Signal \ Acquisition, Appendix H)\ \>", "Section", CellTags->"1.1"], Cell[TextData[StyleBox["Combine the results to find ", "Text"]], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[" for News Subscribers", "BulletListItem", CellTags->"1.2"], Cell[CellGroupData[{ Cell[BoxData[{ \(focEtermNewsSubscribers = FullSimplify[\(1\/nSig\) \(\(tausqNewsSubscribers\ \ expNewsSubscribers\^2\)\/\(1 + \(tausqNewsSubscribers\/\(1 + R\)\) varNewsSubscribers\)\) \((\(1\/2\) elasttauNewsSubscribers + elastexpNewsSubscribers)\)]\), "\[IndentingNewLine]", \(focVartermNewsSubscribers = FullSimplify[\(1\/nSig\) \(\(\(tausqNewsSubscribers\/\(1 + R\)\) varNewsSubscribers\)\/\((1 + \(tausqNewsSubscribers\/\(1 + \ R\)\) varNewsSubscribers)\)\^2\) \((\(1\/2\) elasttauNewsSubscribers + \(1\/2\) elastvarNewsSubscribers)\)]\), "\[IndentingNewLine]", \(focDelfactorNewsSubscribers = FullSimplify[\((1 + R)\) + tausqNewsSubscribers\ varNewsSubscribers - 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