(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 1707042, 34581]*) (*NotebookOutlinePosition[ 1707791, 34607]*) (* CellTagsIndexPosition[ 1707747, 34603]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Calculations to accompany \"Transfer Fee Regulations and Development of \ Talent\" Marko Tervi\[ODoubleDot]\ \>", "Subsection"], Cell["Updated April 18th, 2005", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Helpful Functions/Definitions", "Subsection"], Cell[BoxData[{ \(\(Dashed = Dashing[{0.01, 0.01}];\)\), "\n", \(\(Dotted = Dashing[{0.003, 0.01}];\)\), "\[IndentingNewLine]", \(DiscreteMax[funk_, xmin_, xmax_, xsteps_] := Module[{tabx, tempfunk}, tabx = Table[{xi, funk[xi]}, {xi, xmin, xmax, \((xmax - xmin)\)/xsteps}]; tempfunk = Interpolation[tabx]; gtab = tabx; NMaximize[{tempfunk[x], xmin \[LessEqual] x \[LessEqual] \ xmax}, {x}, MaxIterations \[Rule] 1000, PrecisionGoal \[Rule] 10, AccuracyGoal \[Rule] Infinity]]\), "\[IndentingNewLine]", \(Off[General::"\"]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Uniform distributions", "Section"], Cell["\<\ Small market: b uniform in [1,\[Beta]] Large market: b uniform in [1,\[Beta]+\[Delta]] Talent: \[Theta] uniform in [0,\[Alpha]]\ \>", "Text"], Cell[BoxData[{ \(\(assums = {\[Beta] > 1, \[Delta] > 0, \[Alpha] > 0, 1 > i > 0, 1 > z > 2/3};\)\), "\[IndentingNewLine]", \(Simp[x_] := FullSimplify[x, assums]\)}], "Input"], Cell[CellGroupData[{ Cell["Profile, density, and CDF of club size in autarky", "Subsubsection"], Cell[BoxData[{ \(b[i_, \[Beta]_] := 1 + i \((\[Beta] - 1)\)\), "\n", \(g[b_, \[Beta]_] := 1/\((\[Beta] - 1)\)\), "\[IndentingNewLine]", \(Ng[b_, \[Beta]_] := If[b \[LessEqual] \[Beta], 1/\((\[Beta] - 1)\), 0]\), "\[IndentingNewLine]", \(G[b_, \[Beta]_] := \((b - 1)\)/\((\[Beta] - 1)\)\), "\[IndentingNewLine]", \(NG[b_, \[Beta]_] := If[b \[LessEqual] \[Beta], \((b - 1)\)/\((\[Beta] - 1)\), 0]\)}], "Input"], Cell[BoxData[ \(bmean[\[Beta]_] := \(1 + \[Beta]\)\/2\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["The Integrated market", "Subsubsection"], Cell[BoxData[{ \(GI[b_, \[Beta]_, \[Delta]_] := \((1/2)\) \((Min[1, G[b, \[Beta]]] + Min[1, G[b, \[Beta] + \[Delta]]])\)\), "\n", \(ihat[\[Beta]_, \[Delta]_] := GI[\[Beta], \[Beta], \[Delta]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simp[ihat[\[Beta], \[Delta]]]\)], "Input"], Cell[BoxData[ \(\(\(-2\) + 2\ \[Beta] + \[Delta]\)\/\(2\ \((\(-1\) + \[Beta] + \ \[Delta])\)\)\)], "Output"] }, Open ]], Cell["Solving for the profile:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(FullSimplify[ b /. 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\(da[i_, z_, \[Alpha]_] := If[i \[LessEqual] z, 0, \[Alpha]\/z]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Analysis", "Subsection"], Cell[TextData[{ "Equilibrium z maximizes total surplus Y[z] = ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Integral]\_0\%1 a[j | z] b[j] \[DifferentialD]j\)\(=\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Theta]\&_\)]], Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%z b[ j] \(\(\[DifferentialD]\)\(j\)\(\ \)\)\)]], "+", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Integral]\_z\%1 a[j | z] b[j] \[DifferentialD]j\)\)\)]] }], "Text"], Cell[CellGroupData[{ Cell["Autarky", "Subsubsection"], Cell["Total surplus:", "Text"], Cell[BoxData[ \(Y[z_, \[Alpha]_, \[Beta]_] := amean[\[Alpha]] Integrate[b[i, \[Beta]], {i, 0, z}] + Integrate[ b[i, \[Beta]] \[Theta][1 - \(1 - i\)\/z, \[Alpha]], {i, z, 1}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simp[D[Y[z, \[Alpha], \[Beta]], z]]\)], "Input"], Cell[BoxData[ \(\(\[Alpha]\ \((2 + \[Beta] + z\^2\ \((\(-9\) + 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" }], "Text"], Cell[BoxData[ \(foc[z_, \[Beta]_] := 2 + \[Beta] + z\^2\ \((\(-9\) + 7\ z + 3\ \[Beta] - 7\ z\ \[Beta])\)\)], "Input"], Cell["\<\ First-order condition is a cubic, without a useful closed form solution.\ \>", "Text"], Cell["\<\ Using the implicit function theorem to solve for the slope of z[\[Beta]+\ \[Delta]]:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(dFdz = FullSimplify[D[foc[z, \[Beta] + \[Delta]], z], assums];\)\), "\[IndentingNewLine]", \(\(dFd\[Delta] = FullSimplify[D[foc[z, \[Beta] + \[Delta]], \[Delta]], assums];\)\), "\[IndentingNewLine]", \(dZLd\[Delta] = FullSimplify[\(-dFd\[Delta]\)/dFdz, assums]\)}], "Input"], Cell[BoxData[ \(\(1 + \((3 - 7\ z)\)\ z\^2\)\/\(3\ z\ \((\(-2\)\ \((\(-3\) + \[Beta] + \ \[Delta])\) + 7\ z\ \((\(-1\) + \[Beta] + \[Delta])\))\)\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Integrated market (sufficiently small ", StyleBox["\[Delta]", FontSlant->"Italic"], " so that", StyleBox[" z"Italic"], ")" }], "Subsubsection"], Cell[TextData[{ "Note that z must be at least ", Cell[BoxData[ FormBox[ FractionBox["1", StyleBox[\(\(2 - F\[Theta][amean[\[Alpha]], \[Alpha]]\)\()\)\), FontSlant->"Italic"]], TraditionalForm]]], StyleBox["=", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`2\/3\)]], " while ", StyleBox["ihat[\[Beta],0]=1", FontSlant->"Italic"], "." }], "Text"], Cell["Total Surplus:", "Text"], Cell[BoxData[ \(Yint[z_, \[Alpha]_, \[Beta]_, \[Delta]_] := amean[\[Alpha]] Integrate[B1[i, \[Beta], \[Delta]], {i, 0, z}] + Integrate[\[Theta][1 - \(1 - i\)\/z, \[Alpha]] B1[i, \[Beta], \[Delta]], {i, z, ihat[\[Beta], \[Delta]]}] + Integrate[\[Theta][1 - \(1 - i\)\/z, \[Alpha]] B2[i, \[Beta], \[Delta]], {i, ihat[\[Beta], \[Delta]], 1}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(Simp[Yint[z, \[Alpha], \[Beta], \[Delta]]]\), "\[IndentingNewLine]", \(Simp[ D[Yint[z, \[Alpha], \[Beta], \[Delta]], z]]\), "\[IndentingNewLine]", \(Collect[%*\((24 z\^2/\[Alpha])\), z]\)}], "Input"], Cell[BoxData[ \(\((\[Alpha]\ \((\(-28\)\ z\^3\ \((\(-1\) + \[Beta])\)\ \((\(-1\) + \ \[Beta] + \[Delta])\)\^2 + 6\ z\ \((\(-1\) + \[Beta] + \[Delta])\)\ \((\(-2\) + 2\ \[Beta] + \[Delta])\)\ \((6 + 2\ \[Beta] + \[Delta])\) - \((\(-2\) + 2\ \[Beta] + \[Delta])\)\ \((4\ \((\(-2\) + \[Beta] + \ \[Beta]\^2)\) + 6\ \((1 + \[Beta])\)\ \[Delta] + \[Delta]\^2)\) + 24\ z\^2\ \((\(-1\) + \[Beta] + \[Delta])\)\ \((3 - 2\ \[Delta] + \[Beta]\ \((\(-4\) + \[Beta] + \[Delta])\))\ \))\))\)/\((24\ z\ \((\(-1\) + \[Beta] + \[Delta])\)\ \((\(-2\) + 2\ \[Beta] + \[Delta])\))\)\)], "Output"], Cell[BoxData[ \(\((\[Alpha]\ \((\(-56\)\ z\^3\ \((\(-1\) + \[Beta])\)\ \((\(-1\) + \ \[Beta] + \[Delta])\)\^2 + \((\(-2\) + 2\ \[Beta] + \[Delta])\)\ \((4\ \((\(-2\) + \[Beta] + \ \[Beta]\^2)\) + 6\ \((1 + \[Beta])\)\ \[Delta] + \[Delta]\^2)\) + 24\ z\^2\ \((\(-1\) + \[Beta] + \[Delta])\)\ \((3 - 2\ \[Delta] + \[Beta]\ \((\(-4\) + \[Beta] + \[Delta])\))\ \))\))\)/\((24\ z\^2\ \((\(-1\) + \[Beta] + \[Delta])\)\ \((\(-2\) + 2\ \[Beta] + \[Delta])\))\)\)], "Output"], Cell[BoxData[ \(\(-\(\(56\ z\^3\ \((\(-1\) + \[Beta])\)\ \((\(-1\) + \[Beta] + \ \[Delta])\)\)\/\(\(-2\) + 2\ \[Beta] + \[Delta]\)\)\) + \(4\ \((\(-2\) + \[Beta] + \ \[Beta]\^2)\) + 6\ \((1 + \[Beta])\)\ \[Delta] + \[Delta]\^2\)\/\(\(-1\) + \ \[Beta] + \[Delta]\) + \(24\ z\^2\ \((3 - 2\ \[Delta] + \[Beta]\ \((\(-4\) + \ \[Beta] + \[Delta])\))\)\)\/\(\(-2\) + 2\ \[Beta] + \[Delta]\)\)], "Output"] }, Open ]], Cell["\<\ Again, \[Alpha] canceled out. 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\((1 - i)\)/z, \[Alpha]] B1[i, \[Beta], \[Delta]], {i, z, ihat[\[Beta], \[Delta]]}] + Integrate[\[Theta][1 - \((1 - i)\)/z, \[Alpha]] B2[i, \[Beta], \[Delta]], {i, ihat[\[Beta], \[Delta]], 1}], Integrate[B1[i, \[Beta], \[Delta]], {i, 0, ihat[\[Beta], \[Delta]]}] amean[\[Alpha]] + Integrate[ B2[i, \[Beta], \[Delta]] amean[\[Alpha]], {i, ihat[\[Beta], \[Delta]], z}] + Integrate[\[Theta][1 - \((1 - i)\)/z, \[Alpha]] B2[i, \[Beta], \[Delta]], {i, z, 1}]]\)], "Input"], Cell["\<\ To avoid any problems caused by the kink point at ihat, the optimization \ checks for maxima on both sides of ihat separately\ \>", "Text"], Cell[BoxData[{ \(ZBigStar[\[Alpha]_, \[Beta]_, \[Delta]_] := z /. \(NMaximize[{NY[z, \[Alpha], \[Beta], \[Delta]], 1 > z > 2/3}, z, MaxIterations \[Rule] 1000]\)[\([2]\)]\), "\[IndentingNewLine]", \(ZBigStarRobust[\[Alpha]_, \[Beta]_, \[Delta]_] := Module[{z1, z2, ix = ihat[\[Beta], \[Delta]]}, If[ix\ \[LessEqual] 2/3, ZBigStar[\[Alpha], \[Beta], \[Delta]], z1 = z /. \(NMaximize[{NY[z, \[Alpha], \[Beta], \[Delta]], ix \[GreaterEqual] \ z > 2/3}, z, MaxIterations \[Rule] 1000]\)[\([2]\)]; \[IndentingNewLine]z2 = z /. \(NMaximize[{NY[z, \[Alpha], \[Beta], \[Delta]], 1 > \ z \[GreaterEqual] \ ix}, z, MaxIterations \[Rule] 1000]\)[\([2]\)]; \[IndentingNewLine]If[ NY[z1, \[Alpha], \[Beta], \[Delta]] > NY[z2, \[Alpha], \[Beta], \[Delta]], z1, z2]]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\[Beta]x = 2; \[Delta]x = 1;\), "\[IndentingNewLine]", \(Plot[{b[i, \[Beta]x + \[Delta]x], B[i, \[Beta]x, \[Delta]x], b[i, \[Beta]x]}, {i, 0, 1}, PlotStyle \[Rule] {Black, Dashed, Red}, PlotRange \[Rule] All, PlotLabel \[Rule] "\"]\), "\[IndentingNewLine]", \(ihat[\[Beta]x, \[Delta]x]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font 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\((\(-1\) + 2\ i)\ \)\ \((\[Beta] + \[Delta])\)\^\(1 - \[Eta]\))\)\^\(1\/\(1 - \[Eta]\)\)\), "\ \[IndentingNewLine]", \(B[i_, \[Eta]_, \[Beta]_, \[Delta]_] := If[i \[LessEqual] \ ihat[\[Eta], \[Beta], \[Delta]], B1[i, \[Eta], \[Beta], \[Delta]], B2[i, \[Eta], \[Beta], \[Delta]]]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Talent", "Subsubsection"], Cell[BoxData[{ \(\[Theta][i_, \[Xi]_, \[Alpha]_] := b[i, \[Xi], \[Alpha]]\), "\[IndentingNewLine]", \(a[i_, z_, \[Xi]_, \[Alpha]_] := If[i < z, amean[\[Alpha], \[Xi]], \[Theta][ 1 - \(1 - i\)\/z, \[Xi], \[Alpha]]]\), "\[IndentingNewLine]", \(f\[Theta][a_, \[Xi]_, \[Alpha]_] := g[a, \[Xi], \[Alpha]]\), "\[IndentingNewLine]", \(F\[Theta][a_, \[Xi]_, \[Alpha]_] := G[a, \[Xi], \[Alpha]]\), "\[IndentingNewLine]", \(amean[\[Xi]_, \[Alpha]_] := bmean[\[Xi], \[Alpha]]\), "\[IndentingNewLine]", \(zmin[\[Xi]_, \[Alpha]_] := 1\/\(2 - F\[Theta][amean[\[Xi], \[Alpha]], \[Xi], \[Alpha]]\)\)}], \ "Input"], Cell[CellGroupData[{ Cell[BoxData[ 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