(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 105713, 2357] NotebookOptionsPosition[ 100901, 2197] NotebookOutlinePosition[ 101240, 2212] CellTagsIndexPosition[ 101197, 2209] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Welfare Maximization with Brand-Stretching - Revised", "Subtitle", CellChangeTimes->{{3.421254375620006*^9, 3.421254398472725*^9}}], Cell[TextData[{ "The purpose of this notebook is to consider the welfare-maximizing \ configuration of \"no-name\" and brand-stretching firms. The assumption is \ that the social planner can determine the initial number of entrants and the \ number that survive into the second period, given that there is \ brand-stretching entry by a firm in period 2. The nature of the \ brand-stretcher is not known a priori to the planner.\n\nThe objective of the \ planner is to minimize total cost net of the additional consumer surplus that \ the brand-stretching firm generates.\n\nWe know that the efficient allocation \ by the social planner is the allocation that would be determined by \ competition between the brand-stretcher and the incumbents. So an incumbent \ survives if and only if it can sell at a positive price given the entrant's \ brand-strength. \n\nIn this notebook we assume that the brand-stretcher \ enters exactly on an incumbent - call this incumbent 1. \n\nIf the entrant's \ brand strength lies in the interval \[Alpha] \[Element] [", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "n"], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"j", "-", "1"}], ")"}], "r"}], "n"]}], ",", " ", RowBox[{ FractionBox["1", "n"], "+", FractionBox[ RowBox[{"j", " ", "r"}], "n"]}]}], "]"}], " ", "then", " ", "the", " ", "incumbents", " ", "j", " ", RowBox[{"survive", "."}]}], TraditionalForm]]], " The market radius of the brand-stretcher is then:" }], "Text", CellChangeTimes->{{3.421254430224311*^9, 3.4212547111015797`*^9}, { 3.421254761910919*^9, 3.4212547737238617`*^9}, 3.421255485951507*^9, { 3.4212555925800037`*^9, 3.42125559274164*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Off", "[", RowBox[{"General", "::", "spell"}], "]"}], ";", RowBox[{"Off", "[", RowBox[{"General", "::", "spell1"}], "]"}], ";"}]], "Input", CellChangeTimes->{{3.421258005940263*^9, 3.421258019028982*^9}}], Cell["\<\ Solve for the market radius of the brand-stretcher given that the nearest \ incumbents to survive are nearest neighbors j.\ \>", "Text", CellChangeTimes->{{3.421423620325778*^9, 3.421423657143681*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sbj", "=", RowBox[{"s", "/.", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"r", " ", "s"}], "-", RowBox[{"(", RowBox[{"\[Alpha]", "-", FractionBox["1", "n"]}], ")"}]}], "\[Equal]", RowBox[{"r", RowBox[{"(", RowBox[{ FractionBox["j", "n"], "-", "s"}], ")"}]}]}], ",", "s"}], "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.421254777051455*^9, 3.421254845509636*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"j", " ", "r"}], "+", RowBox[{"n", " ", "\[Alpha]"}]}], RowBox[{"2", " ", "n", " ", "r"}]]], "Output", CellChangeTimes->{3.421254846665119*^9, 3.42142366419571*^9, 3.422025585678698*^9}] }, Open ]], Cell["\<\ The market radius for the jth incumbent firm on the side nearer to the \ entrant is:\ \>", "Text", CellChangeTimes->{{3.421255428701523*^9, 3.421255478475974*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sij", "=", RowBox[{"Simplify", "[", RowBox[{ FractionBox["j", "n"], "-", "sbj"}], "]"}]}]], "Input", CellChangeTimes->{{3.421255471597756*^9, 3.421255511202366*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"1", "+", RowBox[{"j", " ", "r"}], "-", RowBox[{"n", " ", "\[Alpha]"}]}], RowBox[{"2", " ", "n", " ", "r"}]]], "Output", CellChangeTimes->{3.421255513416786*^9, 3.4214236687666597`*^9, 3.422025588684464*^9}] }, Open ]], Cell["The net cost given that nearest firms j survive is then:", "Text", CellChangeTimes->{{3.4212556178466682`*^9, 3.421255630251878*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ncj", "=", RowBox[{"Simplify", "[", RowBox[{"d", RowBox[{"(", RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ FractionBox["1", "n"], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"j", "-", "1"}], ")"}], "r"}], "n"]}], RowBox[{ FractionBox["1", "n"], "+", FractionBox[ RowBox[{"j", " ", "r"}], "n"]}]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"r", " ", SuperscriptBox["sbj", "2"]}], "+", RowBox[{"r", " ", SuperscriptBox["sij", "2"]}], "+", RowBox[{ RowBox[{"(", RowBox[{"n", "-", RowBox[{"2", " ", "j"}]}], ")"}], FractionBox["r", RowBox[{"4", " ", SuperscriptBox["n", "2"]}]]}], "-", RowBox[{"2", RowBox[{"(", RowBox[{"\[Alpha]", "-", FractionBox["1", "n"]}], ")"}], "sbj"}]}], ")"}], " ", RowBox[{"\[DifferentialD]", "\[Alpha]"}]}]}], ")"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.421255661987997*^9, 3.4212558311037807`*^9}, { 3.421255908067191*^9, 3.421255908629074*^9}, {3.421256104105389*^9, 3.4212561042930813`*^9}, {3.421256537445456*^9, 3.421256585957552*^9}, 3.421257069047028*^9}], Cell[BoxData[ FractionBox[ RowBox[{"d", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"6", " ", "j"}], "-", RowBox[{"12", " ", SuperscriptBox["j", "2"]}], "+", RowBox[{"3", " ", "n"}]}], ")"}], " ", SuperscriptBox["r", "2"]}], RowBox[{"12", " ", SuperscriptBox["n", "3"]}]]], "Output", CellChangeTimes->{3.421256619857957*^9, 3.421257070951995*^9, 3.421424136913999*^9, 3.4220255948587513`*^9}] }, Open ]], Cell["\<\ The first two terms are the customization costs of the entrant and of the two \ surviving jth nearest neighbors on the side nearest the incumbent. The third \ term is customization costs for the surviving firms plus those for the jth \ neighbors on the opposite side to the entrant. The fourth term is the \ additional surplus that the entrant generates. \ \>", "Text", CellChangeTimes->{{3.421256830607424*^9, 3.421256959369063*^9}, { 3.421256991568767*^9, 3.421257012718103*^9}, {3.42142369282021*^9, 3.421423693118824*^9}, {3.421423759356985*^9, 3.421423759819168*^9}}], Cell[CellGroupData[{ Cell["1\tOdd Number of Incumbents: n = 2m + 1.", "Subsubtitle", CellChangeTimes->{{3.421257085921973*^9, 3.421257101406241*^9}, { 3.42125720304626*^9, 3.421257205307192*^9}}], Cell["\<\ Assuming that n is odd we sum ncj over j = 1 to m. This implies that firms m \ (the two firms furthest from the entrant) survive.\ \>", "Text", CellChangeTimes->{{3.421257105129301*^9, 3.421257109619031*^9}, { 3.42142376815672*^9, 3.421423821668696*^9}, {3.4214240908580627`*^9, 3.421424130219982*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"nc1", "=", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "1"}], "m"], "ncj"}], "/.", RowBox[{"m", "\[Rule]", RowBox[{ RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}], "/", "2"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4212571123915873`*^9, 3.421257155694984*^9}}], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{"d", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"8", " ", "n"}], "-", RowBox[{"9", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"2", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", SuperscriptBox["r", "2"]}], RowBox[{"48", " ", SuperscriptBox["n", "3"]}]]}]], "Output", CellChangeTimes->{3.421257160194232*^9, 3.421424141039732*^9, 3.422025619499004*^9}] }, Open ]], Cell["\<\ To this we add net cost when the mth nearest firms are forced to exit and the \ incumbent monopolizes the market:\ \>", "Text", CellChangeTimes->{{3.421257264874875*^9, 3.4212572964570847`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ncm1", "=", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"d", RowBox[{"(", RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ FractionBox["1", "n"], "+", FractionBox[ RowBox[{"m", " ", "r"}], "n"]}], "1"], RowBox[{ RowBox[{"(", RowBox[{ FractionBox["r", "4"], "-", RowBox[{"(", RowBox[{"\[Alpha]", "-", FractionBox["1", "n"]}], ")"}]}], ")"}], RowBox[{"\[DifferentialD]", "\[Alpha]"}]}]}], ")"}]}], "/.", RowBox[{"m", "\[Rule]", RowBox[{ RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}], "/", "2"}]}]}], "]"}], " "}]], "Input",\ CellChangeTimes->{{3.421257299279085*^9, 3.421257464085579*^9}, { 3.422025652117392*^9, 3.422025652328053*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"d", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"2", " ", "n"}], "-", "r"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "r"}], ")"}]}], RowBox[{"8", " ", SuperscriptBox["n", "2"]}]]], "Output", CellChangeTimes->{{3.421257435617635*^9, 3.421257471098193*^9}, 3.4214241545457*^9, 3.422025725671155*^9}] }, Open ]], Cell[TextData[{ "We also add the total cost of the incumbents given that the entrant does \ not enter because its brand strength is less than 1/n: there are n incumbents \ each with total cost r/4", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["n", "2"], "."}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.4212575550006943`*^9, 3.421257587670144*^9}, { 3.421424167721146*^9, 3.421424210644123*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"nc01", "=", RowBox[{"Simplify", "[", RowBox[{"d", RowBox[{"(", RowBox[{ SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "n"]], RowBox[{ 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This gives the first-order condition:\ \>", "Text", CellChangeTimes->{{3.421258490600039*^9, 3.421258506798213*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"focodd", "=", RowBox[{"Simplify", "[", RowBox[{ SubscriptBox["\[PartialD]", "n"], " ", "ncodd1"}], "]"}]}]], "Input", CellChangeTimes->{{3.4212585742105083`*^9, 3.42125859275881*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"48", " ", "f", " ", SuperscriptBox["n", "4"]}], "-", RowBox[{"3", " ", SuperscriptBox["r", "2"], " ", "\[Delta]"}], "+", RowBox[{"4", " ", "n", " ", RowBox[{"(", RowBox[{"12", "-", RowBox[{"6", " ", "r"}], "+", SuperscriptBox["r", "2"]}], ")"}], " ", "\[Delta]"}], "-", RowBox[{"3", " ", SuperscriptBox["n", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"4", " ", "r"}], "+", RowBox[{"16", " ", "\[Delta]"}], "-", RowBox[{"4", " ", "r", " ", "\[Delta]"}], "+", RowBox[{ SuperscriptBox["r", "2"], " ", "\[Delta]"}]}], ")"}]}]}], RowBox[{"48", " ", SuperscriptBox["n", "4"]}]]], "Output", 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