(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 18775, 447]*) (*NotebookOutlinePosition[ 21762, 540]*) (* CellTagsIndexPosition[ 21647, 532]*) (*WindowFrame->Normal*) Notebook[{ Cell["1", "ChapterLine", CellTags->"1.1"], Cell[CellGroupData[{ Cell["Derivation and Discussion of Equilibria", "Title", CellTags->"1.1"], Cell[CellGroupData[{ Cell["1.1 Proposition 1: Autarky Equilibrium", "Section", CellTags->"1.1"], Cell[TextData[StyleBox["Let's label one region's variables with a 1 at the \ end, and the other region's variables with a 2. To derive the autarky \ equilibrium for region 1, we can make the agricultural good the numeraire. \n\ However, the derivations as saved in the current notebook file have been \ calculated by not executing the following line.", "Text"]], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[BoxData[ \(\(Px1 = 1;\)\)], "Input"], Cell[TextData[{ StyleBox["To keep this notebook file compatible with earlier versions of \ the paper, the following simultaneous equation system includes the \ possibility of capital accumulation under adjustment cost. The equilibrium as \ reported in proposition 1 of the most recent version of the paper results by \ setting Tobin's ", "Text"], StyleBox["q", "Text", FontSlant->"Italic"], StyleBox[" to one (", "Text"], StyleBox["q1=1", "Text", FontWeight->"Bold"], StyleBox[") and the depreciation rate of capital to zero (so that ", "Text"], StyleBox["\[Delta]=0", "Text", FontWeight->"Bold"], StyleBox["). Then the equilibrium savings rate is zero (", "Text"], StyleBox["s1=0", "Text", FontWeight->"Bold"], StyleBox["), the rate of change of Tobin's ", "Text"], StyleBox["q", "Text", FontSlant->"Italic"], StyleBox[" is zero (", "Text"], StyleBox["gq1=0", "Text", FontWeight->"Bold"], StyleBox["), and the capital stock is constant (so that ", "Text"], StyleBox["gk1=0", "Text", FontWeight->"Bold"], StyleBox["). \nSo, to obtain the most recent reduced equilibrium we can set \ the variables below. ", "Text"] }], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[BoxData[ \(\(\(\ \)\(\(\(\[Delta] = 0\) \)\(;\)\(\(q1 = 1\) \)\(;\)\(\ \)\)\)\)], "Input"], Cell[TextData[StyleBox["The remaining variables are defined just as in the \ most recent version of the paper. The equilbrium satisfies the following \ equation system.", "Text"]], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Solve[\[IndentingNewLine]{\(\[Theta]*\((1 - s1)\)*Y1\)\/\(N1\ *p1\) \ \[Equal] \[Alpha]\/\(1 - \[Alpha]\)*A1* L0, \[IndentingNewLine]\((1 - \[Theta])\)*\((1 - s1)\)* Y1\/Px1 \[Equal] X1, \[IndentingNewLine]w1 \[Equal] \(\[Gamma]\ *\ Px1\ *\ \ X1\)\/\(\((1 - \[Lambda]1)\)\ *\ L1\), \[IndentingNewLine]r1\ \[Equal] \ \(1\/q1\) \(\((1 - \[Gamma])\)\ *\ Px1\ *\ X1\)\/K1 + gq1 + \(1\/\(2 \[Chi]\)\) \((q1 - 1)\)\^2\/q1 - \[Delta], \ \[IndentingNewLine]p1\ \[Equal] \ w1\/\(\[Alpha]\ *\ A1\), \[IndentingNewLine]q1 \[Equal] 1 + \[Chi]\ *I1\/K1, \[IndentingNewLine]Y1 \[Equal] \ w1\ *\ L1\ + \ r1\ *\ K1, \[IndentingNewLine]p1* N1*\[Alpha]\/\(1 - \[Alpha]\)*A1*L0 \[Equal] \ w1\ *\[Lambda]1* L1, \[IndentingNewLine]I1 + \(\[Chi]\/2\) I1\^2\/K1 \[Equal] s1\ *\ Y1, \[IndentingNewLine]gk1 \[Equal] \ \(I1 - \[Delta]\ \ K1\)\/K1, \[IndentingNewLine]Px1*X1\ + \ \[CapitalOmega] \[Equal] \ w1\ *\((1 - \[Lambda]1)\)\ *\ L1\ + \ r1\ *\ K1}, \[IndentingNewLine]{I1, s1, \[Lambda]1, N1, p1, w1, r1, Y1, \ gk1, gq1, \ \[CapitalOmega]}]]\)], "Input"], Cell[BoxData[ \({{\[CapitalOmega] \[Rule] 0, gk1 \[Rule] 0, gq1 \[Rule] 0, r1 \[Rule] \(-\(\(Px1\ X1\ \((\(-1\) + \[Gamma])\)\)\/K1\)\), I1 \[Rule] 0, N1 \[Rule] \(L1\ \((\(-1\) + \[Alpha])\)\ \[Theta]\)\/\(\(-L0\)\ \ \[Gamma] + L0\ \((\(-1\) + \[Gamma])\)\ \[Theta]\), \[Lambda]1 \[Rule] \ \[Theta]\/\(\[Gamma] + \[Theta] - \[Gamma]\ \[Theta]\), s1 \[Rule] 0, p1 \[Rule] \(Px1\ X1\ \[Gamma] + Px1\ X1\ \[Theta] - Px1\ X1\ \ \[Gamma]\ \[Theta]\)\/\(A1\ L1\ \[Alpha] - A1\ L1\ \[Alpha]\ \[Theta]\), w1 \[Rule] \(Px1\ X1\ \((\[Gamma]\ \((\(-1\) + \[Theta])\) - \ \[Theta])\)\)\/\(L1\ \((\(-1\) + \[Theta])\)\), Y1 \[Rule] \(-\(\(Px1\ X1\)\/\(\(-1\) + \[Theta]\)\)\)}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["To clear the values set above, execute the following \ line.", "Text"]], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[BoxData[ \(\(Clear[Px1, \[Delta], q1];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["1.2 Proposition 2: Free Trade Equilibrium", "Section", CellTags->"1.2"], Cell[TextData[{ "After trade liberalization, the two regions face a new equilibrium. We \ could", StyleBox[" make the agricultural good the numeraire again, but now \ world-wide. \nSo, to obtain the most recent reduced equilibrium we can \ execute the following line. However, the derivations as saved in the current \ notebook file have been calculated disregarding this line.", "Text"] }], "Text", CellTags->"1.2"], Cell[BoxData[ \(\(Px = 1;\)\)], "Input"], Cell["\<\ The present version of the paper is based on a model of no capital \ accumulation, and the equation system below accounts for that. The \ equilibrium becomes\ \>", "Text", CellTags->"1.2"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\[IndentingNewLine]{p1\/p2 \[Equal] \ \((A2\/A1)\)\^\(1 - \ \[Alpha]\), \[IndentingNewLine]\((p1*N1*A1\ + \ p2*N2*A2)\)*\[Alpha]\/\(1 - \[Alpha]\)* L0 \[Equal] \ \[Theta]*\((Y1 + Y2)\), \[IndentingNewLine]\((1 - \[Theta])\)* Y1\/Px + \((1 - \[Theta])\)*Y2\/Px \[Equal] X1 + X2, \[IndentingNewLine]\[IndentingNewLine]w1 \[Equal] \(\ \[Gamma]\ *\ Px\ *\ X1\)\/\(\((1 - \[Lambda]1)\)\ *\ L1\), \ \[IndentingNewLine]r1\ \[Equal] \(\((1 - \[Gamma])\)\ *\ Px\ *\ X1\)\/K1, \ \[IndentingNewLine]p1\ \[Equal] \ w1\/\(\[Alpha]\ *\ A1\), \[IndentingNewLine]Y1 \[Equal] \ w1\ *\ L1\ + \ r1\ *\ K1, \[IndentingNewLine]p1* N1*\[Alpha]\/\(1 - \[Alpha]\)*A1*L0 \[Equal] \ w1*\[Lambda]1*L1, \[IndentingNewLine]Px*X1\ \[Equal] \ w1\ *\((1 - \[Lambda]1)\)\ *\ L1\ + \ r1\ *\ K1, \[IndentingNewLine]\[IndentingNewLine]w2 \[Equal] \(\ \[Gamma]\ *\ Px\ *\ X2\)\/\(\((1 - \[Lambda]2)\)\ *\ L2\), \ \[IndentingNewLine]r2\ \[Equal] \(\((1 - \[Gamma])\)\ *\ Px\ *\ X2\)\/K2, \ \[IndentingNewLine]p2\ \[Equal] \ w2\/\(\[Alpha]*A2\), \[IndentingNewLine]Y2 \[Equal] \ w2*L2 + \ r2*\ K2, \[IndentingNewLine]p2* N2*\[Alpha]\/\(1 - \[Alpha]\)*A2*L0 \[Equal] \ w2*\[Lambda]2*L2, \[IndentingNewLine]Px*X2\ \[Equal] \ w2*\((1 - \[Lambda]2)\)*L2\ + \ r2*K2}, \[IndentingNewLine]\[IndentingNewLine]{\[Lambda]1, N1, p1, w1, r1, Y1, \[Lambda]2, N2, p2, w2, r2, Y2}]\)], "Input"], Cell[BoxData[ \({{r1 \[Rule] \(-\(\(Px\ X1\ \((\(-1\) + \[Gamma])\)\)\/K1\)\), r2 \[Rule] \(-\(\(Px\ X2\ \((\(-1\) + \[Gamma])\)\)\/K2\)\), Y1 \[Rule] Px\ X1 - Px\ X1\ \[Gamma] + \(L1\ Px\ \((X1 + X2)\)\ \((\(-\[Gamma]\) - \ \[Theta] + \[Gamma]\ \[Theta])\)\)\/\(\((L1 + \((A2\/A1)\)\^\[Alpha]\ L2)\)\ \ \((\(-1\) + \[Theta])\)\), Y2 \[Rule] Px\ X2 - Px\ X2\ \[Gamma] + \(\((A2\/A1)\)\^\[Alpha]\ L2\ Px\ \((X1 + \ X2)\)\ \((\(-\[Gamma]\) - \[Theta] + \[Gamma]\ \[Theta])\)\)\/\(\((L1 + \((A2\ \/A1)\)\^\[Alpha]\ L2)\)\ \((\(-1\) + \[Theta])\)\), N1 \[Rule] \(\((\(-1\) + \[Alpha])\)\ \((\(-\((A2\/A1)\)\^\[Alpha]\)\ \ L2\ X1\ \[Gamma] + L1\ X2\ \[Gamma] + L1\ X1\ \[Theta] + L1\ X2\ \[Theta] + \ \((A2\/A1)\)\^\[Alpha]\ L2\ X1\ \[Gamma]\ \[Theta] - L1\ X2\ \[Gamma]\ \ \[Theta])\)\)\/\(L0\ \((X1 + X2)\)\ \((\(-\[Gamma]\) - \[Theta] + \[Gamma]\ \ \[Theta])\)\), N2 \[Rule] \(\((A2\/A1)\)\^\(-\[Alpha]\)\ \((\(-1\) + \[Alpha])\)\ \ \((\((A2\/A1)\)\^\[Alpha]\ L2\ X1\ \[Gamma] - L1\ X2\ \[Gamma] + \((A2\/A1)\)\ \^\[Alpha]\ L2\ X1\ \[Theta] + \((A2\/A1)\)\^\[Alpha]\ L2\ X2\ \[Theta] - \ \((A2\/A1)\)\^\[Alpha]\ L2\ X1\ \[Gamma]\ \[Theta] + L1\ X2\ \[Gamma]\ \ \[Theta])\)\)\/\(L0\ \((X1 + X2)\)\ \((\(-\[Gamma]\) - \[Theta] + \[Gamma]\ \ \[Theta])\)\), \[Lambda]1 \[Rule] 1 - \(\((L1 + \((A2\/A1)\)\^\[Alpha]\ L2)\)\ X1\ \[Gamma]\ \ \((\(-1\) + \[Theta])\)\)\/\(L1\ \((X1 + X2)\)\ \((\(-\[Gamma]\) - \[Theta] + \ \[Gamma]\ \[Theta])\)\), \[Lambda]2 \[Rule] 1 - \(\((A2\/A1)\)\^\(-\[Alpha]\)\ \((L1 + \((A2\/A1)\)\^\[Alpha]\ \ L2)\)\ X2\ \[Gamma]\ \((\(-1\) + \[Theta])\)\)\/\(L2\ \((X1 + X2)\)\ \((\(-\ \[Gamma]\) - \[Theta] + \[Gamma]\ \[Theta])\)\), p1 \[Rule] \(Px\ \((X1 + X2)\)\ \((\(-\[Gamma]\) - \[Theta] + \ \[Gamma]\ \[Theta])\)\)\/\(A1\ \((L1 + \((A2\/A1)\)\^\[Alpha]\ L2)\)\ \ \[Alpha]\ \((\(-1\) + \[Theta])\)\), p2 \[Rule] \(\((A2\/A1)\)\^\[Alpha]\ Px\ \((X1 + X2)\)\ \ \((\(-\[Gamma]\) - \[Theta] + \[Gamma]\ \[Theta])\)\)\/\(A2\ \((L1 + \ \((A2\/A1)\)\^\[Alpha]\ L2)\)\ \[Alpha]\ \((\(-1\) + \[Theta])\)\), w2 \[Rule] \(\((A2\/A1)\)\^\[Alpha]\ Px\ \((X1 + X2)\)\ \ \((\(-\[Gamma]\) - \[Theta] + \[Gamma]\ \[Theta])\)\)\/\(\((L1 + \((A2\/A1)\)\ \^\[Alpha]\ L2)\)\ \((\(-1\) + \[Theta])\)\), w1 \[Rule] \(Px\ \((X1 + X2)\)\ \((\(-\[Gamma]\) - \[Theta] + \ \[Gamma]\ \[Theta])\)\)\/\(\((L1 + \((A2\/A1)\)\^\[Alpha]\ L2)\)\ \((\(-1\) + \ \[Theta])\)\)}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["To clear the numeraire value set above, execute the \ following line.", "Text"]], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[BoxData[ \(\(Clear[Px];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["1.3 Corollary 2.3: Complete Specialization in Agriculture", "Section", CellTags->"1.2"], Cell[TextData[{ "One region may completely specialize in agriculture after trade \ liberalization. \nAgain, we can", StyleBox[" make the agricultural good the numeraire but we need not. The \ specialization equilibrium below has been calculated ignoring the following \ line.", "Text"] }], "Text", CellTags->"1.2"], Cell[BoxData[ \(\(Px = 1;\)\)], "Input"], Cell["\<\ By adjusting the first equation in the below system accordingly, \ the complete specialization equilibrium becomes\ \>", "Text", CellTags->"1.2"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\[IndentingNewLine]{\(\[Theta]*\((Y1 + Y2)\)\)\/\(N2\ *p2\) \ \[Equal] \[Alpha]\/\(1 - \[Alpha]\)*A2* L0, \[IndentingNewLine]\((1 - \[Theta])\)* Y1\/Px + \((1 - \[Theta])\)*Y2\/Px \[Equal] X1 + X2, \[IndentingNewLine]\[IndentingNewLine]w1 \[Equal] \(\ \[Gamma]\ *\ Px\ *\ X1\)\/\(\((1 - \[Lambda]1)\)\ *\ L1\), \ \[IndentingNewLine]r1\ \[Equal] \(\((1 - \[Gamma])\)\ *\ Px\ *\ X1\)\/K1, \ \[IndentingNewLine]Y1 \[Equal] \ w1\ *\ L1\ + \ r1\ *\ K1, \[IndentingNewLine]Px*X1 \[Equal] \ w1\ *\((1 - \[Lambda]1)\)\ *\ L1\ + \ r1\ *\ K1, \[IndentingNewLine]\[IndentingNewLine]w2 \[Equal] \(\ \[Gamma]\ *\ Px\ *\ X2\)\/\(\((1 - \[Lambda]2)\)\ *\ L2\), \ \[IndentingNewLine]r2\ \[Equal] \(\((1 - \[Gamma])\)\ *\ Px\ *\ X2\)\/K2, \ \[IndentingNewLine]p2\ \[Equal] \ w2\/\(\[Alpha]\ *\ A2\), \[IndentingNewLine]Y2 \[Equal] \ w2\ *\ L2 + \ r2\ *\ K2, \[IndentingNewLine]p2* N2*\[Alpha]\/\(1 - \[Alpha]\)*A2*L0 \[Equal] \ w2\ *\[Lambda]2*L2, \[IndentingNewLine]Px*X2\ \[Equal] \ w2\ *\((1 - \[Lambda]2)\)\ *\ L2\ + \ r2\ *\ K2}, \[IndentingNewLine]\[IndentingNewLine]{\[Lambda]1, w1, r1, Y1, \[IndentingNewLine]\[Lambda]2, N2, p2, w2, r2, Y2}]\)], "Input"], Cell[BoxData[ \({{r1 \[Rule] \(-\(\(Px\ X1\ \((\(-1\) + \[Gamma])\)\)\/K1\)\), r2 \[Rule] \(-\(\(Px\ X2\ \((\(-1\) + \[Gamma])\)\)\/K2\)\), \ \[Lambda]1 \[Rule] 0, Y1 \[Rule] Px\ X1, Y2 \[Rule] \(\(-Px\)\ X2 - Px\ X1\ \[Theta]\)\/\(\(-1\) + \[Theta]\), N2 \[Rule] \(\(-L2\)\ X1\ \[Theta] - L2\ X2\ \[Theta] + L2\ X1\ \ \[Alpha]\ \[Theta] + L2\ X2\ \[Alpha]\ \[Theta]\)\/\(L0\ \((\(-X2\)\ \[Gamma] \ - X1\ \[Theta] - X2\ \[Theta] + X2\ \[Gamma]\ \[Theta])\)\), w1 \[Rule] \(Px\ X1\ \[Gamma]\)\/L1, \[Lambda]2 \[Rule] \(\(-X1\)\ \ \[Theta] - X2\ \[Theta]\)\/\(\(-X2\)\ \[Gamma] - X1\ \[Theta] - X2\ \[Theta] \ + X2\ \[Gamma]\ \[Theta]\), p2 \[Rule] \(Px\ \((\(-X2\)\ \[Gamma] - X1\ \[Theta] - X2\ \[Theta] + \ X2\ \[Gamma]\ \[Theta])\)\)\/\(A2\ L2\ \[Alpha]\ \((\(-1\) + \[Theta])\)\), w2 \[Rule] \(Px\ \((\(-X2\)\ \[Gamma] - X1\ \[Theta] - X2\ \[Theta] + \ X2\ \[Gamma]\ \[Theta])\)\)\/\(L2\ \((\(-1\) + \[Theta])\)\)}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["To clear the numeraire value set above, execute the \ following line.", "Text"]], "Text", TaggingRules:>{"IndexingCellTag" -> "i:1", "IndexEntries" -> {{ "decentralization", "", ""}, {"mobility", "", ""}}}, CellTags->{"i:1", "1.1"}], Cell[BoxData[ \(\(Clear[Px];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["1.4 Table 2: Numerical Welfare Analysis and Simulations", "Section", CellTags->"1.2"], Cell[TextData[{ "To better understand the welfare implications of the model, we can \ simulate it for various levels of ", StyleBox["\[Gamma]", FontWeight->"Bold"], " and ", StyleBox["\[Theta]", FontWeight->"Bold"], ". 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