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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[ 59394, 1341]*) (*NotebookOutlinePosition[ 63405, 1459]*) (* CellTagsIndexPosition[ 63287, 1451]*) (*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 Derivation of Terms in Table 2 (Utility Responses to Signal \ Acquisition, Appendix H)\ \>", "Section", CellTags->"1.1"], Cell[TextData[{ StyleBox["To derive the terms in Table 2, combine results from the previous \ sections 2.1 and 2.2 in 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[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 - tausqNewsSubscribers\ expNewsSubscribers\^2]\)}], "Input"], Cell[BoxData[ \(\(-\(\((\((1 + R)\)\ 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\ 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This is done below so that the value of ", "Text"], Cell[BoxData[ \(TraditionalForm\`\(x\&_\)\_0\%2\)], FontWeight->"Bold"], StyleBox[", as reported in appendix G, can be derived as", "Text"] }], "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\)\/\(1 + \(\[Gamma]\^4\ \ \[Sigma]\^6\ \[Tau]\^2\ \[Omega]\^4\ \((iInv\^2\ nSig\ \[Lambda]\^2 + \ \[Gamma]\^2\ \[Sigma]\^2\ \[Omega]\^2)\)\)\/\(\((1 + R)\)\ \((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]", \(FullSimplify[ Limit[xbarsq /. 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R)\)\ \[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)\) + \(\((1 + \ R)\)\ \[Gamma]\^6\ \[Sigma]\^8\ \[Omega]\^6\ \((iInv\^2\ nSig\ \[Lambda]\^2\ \ \((\[Sigma]\^2 + nSig\ \[Tau]\^2)\) + \[Gamma]\^2\ \[Sigma]\^4\ \ \[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)\)\)\/\(1 + \(\[Gamma]\^4\ \[Sigma]\^6\ \ \[Tau]\^2\ \[Omega]\^4\ \((iInv\^2\ nSig\ 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