(* 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[ 133539, 4082] NotebookOptionsPosition[ 126295, 3848] NotebookOutlinePosition[ 126633, 3863] CellTagsIndexPosition[ 126590, 3860] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["A Revised Notebook for the Brand Stretching Case", "Subtitle", CellChangeTimes->{{3.418479055762384*^9, 3.418479057711632*^9}}], Cell[BoxData[ RowBox[{"Off", "[", RowBox[{"General", "::", "spell1"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Off", "[", RowBox[{"General", "::", "spell"}], "]"}]], "Input"], Cell["\<\ Assume that there are n firms in the market and that the brand-stretcher \ enters only if its brand strength is at least 1/n. Brand strength is \ uniformly distributed on [0, 1]. We assume that the market grows by some \ factor \[Delta] between the initial and the second, final period. We further \ assume that the brand-stretcher enters exactly at one incumbent's location, \ while we can call location 0. This allows us to calculate expected profit \ for an incumbent in the initial and in the final period. Consider first \ profits in the final period. \ \>", "Text", CellChangeTimes->{{3.418479090401107*^9, 3.41847921734968*^9}, 3.418479305527835*^9}], Cell[CellGroupData[{ Cell["1.\tFinal Period Profit for an Incumbent", "Subsubtitle", CellChangeTimes->{{3.418479325058659*^9, 3.418479336824534*^9}}], Cell["\<\ Incumbent firm 0 survives only if the entrant has brand strength \[Alpha] < \ 1/n. Its expected profit is:\ \>", "Text", CellChangeTimes->{{3.4184793500758677`*^9, 3.418479387337579*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"es0", "=", RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"1", "/", "n"}]], RowBox[{ FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"2", " ", SuperscriptBox["n", "2"]}]], RowBox[{"\[DifferentialD]", "a"}]}]}]}]], "Input", CellChangeTimes->{ 3.41847939478885*^9, {3.418479445401721*^9, 3.41847944573925*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"2", " ", SuperscriptBox["n", "3"]}]]], "Output", CellChangeTimes->{3.4184794005652037`*^9, 3.418479446718582*^9}] }, Open ]], Cell["The nearest neighbor has expected profit", "Text", CellChangeTimes->{3.418479441650764*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"es1", "=", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "n"]], RowBox[{ FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"2", " ", SuperscriptBox["n", "2"]}]], 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Expected profit for an incumbent \ in the final period is then:\ \>", "Text", CellChangeTimes->{{3.418480651843449*^9, 3.418480680191794*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"esf", "=", RowBox[{"Simplify", "[", RowBox[{ FractionBox["es0", "n"], "+", RowBox[{ FractionBox["2", "n"], " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "1"}], "m"], RowBox[{"es", "[", "j", "]"}]}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.418480698246381*^9, 3.418480749193679*^9}, { 3.4184809033322563`*^9, 3.418480948355755*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"d", " ", "r", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "m"}], "+", RowBox[{ SuperscriptBox["m", "2"], " ", "r"}]}], ")"}]}], RowBox[{"2", " ", SuperscriptBox["n", "4"]}]]], "Output", CellChangeTimes->{3.4184807550151463`*^9, 3.418480950431966*^9}] }, Open ]], Cell["Substitute for m:", "Text", CellChangeTimes->{{3.4184809617189703`*^9, 3.418480962329908*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"esf", "=", RowBox[{"Simplify", "[", RowBox[{"esf", "/.", RowBox[{"m", "\[Rule]", RowBox[{ RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}], "/", "2"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4184809687056*^9, 3.4184809739804583`*^9}, { 3.4184810045816717`*^9, 3.418481008293542*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"d", " ", "r", " ", RowBox[{"(", RowBox[{"n", "+", RowBox[{ FractionBox["1", "4"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "n"}], ")"}], "2"], " ", "r"}]}], ")"}]}], RowBox[{"2", " ", SuperscriptBox["n", "4"]}]]], "Output", CellChangeTimes->{3.4184810115991373`*^9}] }, Open ]], Cell["Aggregate profit over the two periods is then:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"esagg", "=", RowBox[{"Simplify", "[", RowBox[{"esf", "+", FractionBox[ RowBox[{"d0", " ", "r"}], RowBox[{"2", " ", SuperscriptBox["n", "2"]}]]}], "]"}]}]], "Input", CellChangeTimes->{3.418484941382316*^9}], Cell[BoxData[ FractionBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ RowBox[{"4", " ", "d0", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"d", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "n", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "r"}], ")"}]}], "+", "r", "+", RowBox[{ SuperscriptBox["n", "2"], " ", "r"}]}], ")"}]}]}], ")"}]}], RowBox[{"8", " ", SuperscriptBox["n", "4"]}]]], "Output", CellChangeTimes->{3.418484962112195*^9}] }, Open ]], Cell["where d0 is consumer density in the first period.", "Text"], Cell["\<\ Now suppose that n is even = 2m+2. The aggregate profit is as above plus the \ profit expected by the firm furthest from the entrant - firm m+1. For this \ firm profit if all other firms are eliminated is a \"diamond\". This firm's \ expected profit is then\ \>", "Text", CellChangeTimes->{{3.41848167314894*^9, 3.418481697920252*^9}, { 3.418482510873917*^9, 3.418482526020105*^9}, {3.418484710181077*^9, 3.418484729879475*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"esm1", "=", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{ FractionBox[ RowBox[{"r", " ", "m"}], "n"], "+", FractionBox["1", "n"]}]], RowBox[{ FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"2", " ", SuperscriptBox["n", "2"]}]], RowBox[{"\[DifferentialD]", "a"}]}]}], "+", RowBox[{ FractionBox["d", RowBox[{"2", " ", "r"}]], RowBox[{"(", RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ FractionBox[ RowBox[{"r", " ", "m"}], "n"], "+", FractionBox["1", "n"]}], RowBox[{ FractionBox["r", "2"], "+", FractionBox["1", "n"]}]], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ FractionBox["r", "2"], "+", FractionBox["1", "n"], "-", "a"}], ")"}], "2"], RowBox[{"\[DifferentialD]", "a"}]}]}], ")"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4184847858198566`*^9, 3.41848482170774*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "6"], " ", "d", " ", "r", " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ FractionBox["1", "2"], "-", FractionBox["m", "n"]}], ")"}], "3"], " ", "r"}], "+", FractionBox[ RowBox[{"3", "+", RowBox[{"3", " ", "m", " ", "r"}]}], SuperscriptBox["n", "3"]]}], ")"}]}]], "Output", CellChangeTimes->{3.418484826307637*^9}] }, Open ]], Cell["Substitute for m = n/2 - 1", "Text", CellChangeTimes->{{3.4184848651515923`*^9, 3.418484875655306*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"esm1", "=", RowBox[{"Simplify", "[", RowBox[{"esm1", "/.", RowBox[{"m", "->", RowBox[{ FractionBox["n", "2"], "-", "1"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4184848830922813`*^9, 3.418484912531089*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"d", " ", "r", " ", RowBox[{"(", RowBox[{"6", "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"3", " ", "n"}]}], ")"}], " ", "r"}]}], ")"}]}], RowBox[{"12", " ", SuperscriptBox["n", "3"]}]]], "Output", CellChangeTimes->{3.4184849139717407`*^9}] }, Open ]], Cell["This profit occurs with probability 1/n.", "Text"], Cell["\<\ We can now define a series of functions defining aggregate profit. In doing \ so we assume that the market growth rate is \[Delta], i.e. that d = \[Delta] \ d0 with \[Delta] >= 1. Note further that with this substitution all profits \ are homogeneous of degree one in d0, so that we can eliminate d0 by measuring \ all fixed costs as fixed costs per initial consumer, f/d0.\ \>", "Text", CellChangeTimes->{3.418485128232991*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"esagg", "/.", RowBox[{"{", RowBox[{ RowBox[{"d0", "\[Rule]", "1"}], ",", RowBox[{"d", "\[Rule]", "\[Delta]"}]}], " ", "}"}]}], "]"}]], "Input"], Cell[BoxData[ FractionBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "n", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "r"}], ")"}], " ", "\[Delta]"}], "+", RowBox[{"r", " ", "\[Delta]"}], "+", RowBox[{ SuperscriptBox["n", "2"], " ", RowBox[{"(", RowBox[{"4", "+", RowBox[{"r", " ", "\[Delta]"}]}], ")"}]}]}], ")"}]}], RowBox[{"8", " ", SuperscriptBox["n", "4"]}]]], "Output", CellChangeTimes->{3.418485048094791*^9}] }, Open ]], Cell["Expected profit when n is odd and greater than 2 is:", 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Then profit to the entrants is \ d0r/2n^2 in the first period and \[Delta]d0r/2n^2 in the second. 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