(* 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[ 59787, 1582] NotebookOptionsPosition[ 57753, 1513] NotebookOutlinePosition[ 58181, 1530] CellTagsIndexPosition[ 58138, 1527] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "An Analytically Solvable Core-Periphery model\n", StyleBox["by Rickard Forslid and Gianmarco Ottaviano", "Subtitle"] }], "Title"], Cell["\<\ Pour r\[EAcute]aliser un mod\[EGrave]le th\[EAcute]orique, il faut partir d' \ un mod\[EGrave]le de base. Il faut rep\[EAcute]rer les \[EAcute]quations \ principales, dans le mod\[EGrave]le de forslid et ottaviano les salaires sont \ essentiels pour expliquer les migrations. Nous allons donc partir de l'\ \[EAcute]quation 14. Dans cette \[EAcute]quation les revenus Y \ dp\[EAcute]endent des salaires, on commence donc par \[EAcute]valuer cette \ expression\ \>", "Text", CellChangeTimes->{{3.541104296609236*^9, 3.541104435201236*^9}, { 3.541104557520836*^9, 3.541104582574436*^9}, {3.541106787684636*^9, 3.5411067992910357`*^9}}, FontSize->14], Cell[BoxData[{ RowBox[{ RowBox[{"Y1", "=", RowBox[{"(", RowBox[{ RowBox[{"h1", "*", "w1"}], "+", RowBox[{"L", "/", "2"}]}], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Y2", "=", RowBox[{"(", RowBox[{ RowBox[{"h2", "*", "w2"}], "+", RowBox[{"L", "/", "2"}]}], ")"}]}], ";"}]}], "Input"], Cell[TextData[{ "on \[EAcute]value ensuite 14 sous la forme suivante pour pouvoir \ r\[EAcute]soudre un syst\[EGrave]me (pour simplifier l'\[EAcute]criture nous \ notons b=", Cell[BoxData[ FormBox[ FractionBox["\[Mu]", "\[Sigma]"], TraditionalForm]]], ")" }], "Text", CellChangeTimes->{{3.541104296609236*^9, 3.541104435201236*^9}, { 3.541104612089636*^9, 3.541104642150836*^9}, 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r\[EAcute]solvable, qui nous donne l'equation 16 (on ne \ la reconnait pas facilement car elle est arrang\[EAcute]e \ diff\[EAcute]remment)\ \>", "Text", CellChangeTimes->{{3.541104296609236*^9, 3.541104435201236*^9}, { 3.541104612089636*^9, 3.541104668624036*^9}, {3.541104720072836*^9, 3.541104739026836*^9}, {3.541104781320436*^9, 3.5411048237992363`*^9}, { 3.5411048595232363`*^9, 3.5411048840780363`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"eqns", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox[ StyleBox["eq", FormatType->StandardForm], "1"], StyleBox["[", FormatType->StandardForm], RowBox[{ StyleBox[ RowBox[{ StyleBox["w", FormatType->StandardForm], "1"}]], ",", "w2"}], StyleBox["]", FormatType->StandardForm]}], StyleBox["==", FormatType->StandardForm], StyleBox["0", FormatType->StandardForm]}], StyleBox[",", FormatType->StandardForm], RowBox[{ RowBox[{ SubscriptBox[ StyleBox["eq", FormatType->StandardForm], "2"], StyleBox["[", FormatType->StandardForm], 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Pour savoir pour quel \ degre d'ouverture \[Phi] l'un des deux effets dominent, les auteurs \ d\[EAcute]rivent le salaire relatif par rapport \[AGrave] h et \ r\[EAcute]solvent pour \[Phi].\ \>", "Text", CellChangeTimes->{{3.541105141245036*^9, 3.5411052742662363`*^9}, { 3.541105326526236*^9, 3.541105391983836*^9}, {3.541105551259836*^9, 3.541105581149436*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"c", "=", RowBox[{ SubscriptBox["\[PartialD]", "h1"], RowBox[{"(", FractionBox[ RowBox[{"1", "-", RowBox[{"h1", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "\[Phi]"}], ")"}], "2"]}], "+", SuperscriptBox["\[Phi]", "2"], "-", RowBox[{"b", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "h1"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[Phi]", "2"]}], ")"}]}]}], RowBox[{ RowBox[{"2", " ", "\[Phi]"}], "+", RowBox[{"h1", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "\[Phi]"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "b", "+", "\[Phi]", "+", RowBox[{"b", " ", "\[Phi]"}]}], ")"}]}]}]], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Solve", "[", RowBox[{ RowBox[{"c", "\[Equal]", "0"}], ",", "\[Phi]"}], "]"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.541105412778636*^9, 3.541105454103036*^9}, { 3.541105599229836*^9, 3.541105603270236*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", FractionBox[ RowBox[{"1", "-", "b"}], RowBox[{"1", "+", "b"}]]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", "b"}], RowBox[{"1", "+", "b"}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.541105416023436*^9, 3.541105445710236*^9}}] }, Open ]], Cell[TextData[{ "La troisi\[EGrave]me racine est celle obtenu par les auteurs (puisque b=", Cell[BoxData[ FormBox[ FractionBox["\[Mu]", "\[Sigma]"], TraditionalForm]]], "). Apr\[EGrave]s ce point de lib\[EAcute]ralisation la force d'agglom\ \[EAcute]ration market access domine la force de dispersion.\nCeci est un r\ \[EAcute]sultat interm\[EAcute]diaire car pour comprendre les migrations il \ ne faut pas regarder seulement les salaires nominaux mais les salaires r\ \[EAcute]els, ou plus exactement de l'utilit\[EAcute] indirecte qui repr\ \[EAcute]sente le bien etre dans les deux r\[EAcute]gions" }], "Text", CellChangeTimes->{{3.541105614174636*^9, 3.541105688539836*^9}, { 3.541105771594236*^9, 3.541105775899836*^9}, {3.541108011053836*^9, 3.5411081061358356`*^9}, {3.541108180703836*^9, 3.541108198831036*^9}}], Cell["\<\ En partant des utilit\[EAcute]s indirectes, les auteurs se demandent jusqu'\ \[AGrave] quel niveau d'ouverture la dispersion est stable. pour \ r\[EAcute]pondre \[AGrave] cette question ils d\[EAcute]rivent le diff\ \[EAcute]rentiel d'utilit\[EAcute] indirecte pour une situation de dispersion \ h1=1/2. Ici nous d\[EAcute]rivons le rapport des utilit\[EAcute]s indirecte \ (ce qui revient au m\[EHat]me comme nous l'avons vu en cours)\ \>", "Text", CellChangeTimes->{{3.541105614174636*^9, 3.541105688539836*^9}, { 3.541105771594236*^9, 3.541105775899836*^9}, {3.541105882011036*^9, 3.5411059743318357`*^9}, {3.541106874124236*^9, 3.541106874592236*^9}, { 3.541108117555036*^9, 3.5411081770534363`*^9}, {3.541108213588636*^9, 3.541108299544636*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"h1", "=."}], ";", RowBox[{"\[Phi]", "=."}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"w1", "=", RowBox[{"-", FractionBox[ RowBox[{"b", " ", "L", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "h1", " ", "\[Phi]"}], "+", RowBox[{"h2", " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[Phi]", "2"], "+", RowBox[{"b", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[Phi]", "2"]}], ")"}]}]}], ")"}]}]}], ")"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "b"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["h1", "2"], " ", "\[Phi]"}], "+", RowBox[{ SuperscriptBox["h2", "2"], " ", "\[Phi]"}], "+", RowBox[{"h1", " ", "h2", " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[Phi]", "2"], "+", RowBox[{"b", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[Phi]", "2"]}], ")"}]}]}], ")"}]}]}], ")"}]}]]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"w2", "=", RowBox[{"-", FractionBox[ RowBox[{"b", " ", "L", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "h2", " ", "\[Phi]"}], "+", RowBox[{"h1", " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[Phi]", "2"], "+", RowBox[{"b", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[Phi]", "2"]}], ")"}]}]}], ")"}]}]}], ")"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "b"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["h1", "2"], " ", "\[Phi]"}], "+", RowBox[{ SuperscriptBox["h2", "2"], " ", "\[Phi]"}], "+", RowBox[{"h1", " ", "h2", " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[Phi]", "2"], "+", RowBox[{"b", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[Phi]", "2"]}], ")"}]}]}], ")"}]}]}], ")"}]}]]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"break", "=", RowBox[{ SubscriptBox["\[PartialD]", "h1"], RowBox[{"(", RowBox[{ RowBox[{"(", FractionBox["w1", SuperscriptBox[ RowBox[{"(", RowBox[{"h1", "+", RowBox[{"\[Phi]", "*", "h2"}]}], ")"}], RowBox[{"\[Mu]", "/", RowBox[{"(", RowBox[{"1", "-", "\[Sigma]"}], ")"}]}]]], ")"}], "/", RowBox[{"(", FractionBox["w2", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"\[Phi]", "*", "h1"}], "+", "h2"}], ")"}], RowBox[{"\[Mu]", "/", RowBox[{"(", RowBox[{"1", "-", "\[Sigma]"}], ")"}]}]]], ")"}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"h1", "=", RowBox[{"1", "/", "2"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Simplify", "[", "break", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{"break", "\[Equal]", "0"}], ",", "\[Phi]"}], "]"}], "]"}]}], "Input", CellChangeTimes->{{3.541105997934636*^9, 3.5411059984806356`*^9}, { 3.541106038323036*^9, 3.541106041396236*^9}, {3.541106122126236*^9, 3.541106124559836*^9}, {3.541106231451036*^9, 3.541106294880636*^9}, { 3.541106452035036*^9, 3.541106468259036*^9}}, FontSize->14], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "b"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "\[Mu]", "-", "\[Sigma]"}], ")"}]}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "b"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "\[Mu]", "+", "\[Sigma]"}], ")"}]}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.541106004549036*^9, 3.541106043393036*^9, 3.541106126478636*^9, 3.541106253400236*^9, {3.541106288203836*^9, 3.541106301276636*^9}, 3.541106471129436*^9}] }, Open ]], Cell["\<\ La deuxi\[EGrave]me racine est celle obtenu par les auteurs \[AGrave] l'eq \ 26. Donc entre 0 et ce point de lib\[EAcute]ralisation la dispersion est \ stable. Par contre apr\[EGrave]s ce point l'agglom\[EAcute]ration peut \ dominer, les auteurs analysent donc ensuite le sustain point qui est le point \ pour lequel l'agglom\[EAcute]ration est stable. 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