document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169746>Question 1169746</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4)	The equations defining a model of two trading nations are given by <BR>\n" );
document.write( "𝑌              𝑌  𝐶1 = 0.7𝑌1 + 50                        𝐶2 = 0.8𝑌2 + 100 𝐼                                      𝐼  𝑀1 = 0.3𝑌1                                 𝑀2 = 0.1𝑌2 <BR>\n" );
document.write( "a.	Express this system in matrix form and hence find the values of  𝑌1 and 𝑌2. <BR>\n" );
document.write( "b.Calculate the balance of payments between these countries<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851305 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851305\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**Understanding the Model**\r<BR>\n" );
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document.write( "We are given a model of two trading nations, with the following equations:\r<BR>\n" );
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document.write( "* **Country 1:**<BR>\n" );
document.write( "    * Y1 = C1 + I1 + X1 - M1<BR>\n" );
document.write( "    * C1 = 0.7Y1 + 50<BR>\n" );
document.write( "    * M1 = 0.3Y1<BR>\n" );
document.write( "* **Country 2:**<BR>\n" );
document.write( "    * Y2 = C2 + I2 + X2 - M2<BR>\n" );
document.write( "    * C2 = 0.8Y2 + 100<BR>\n" );
document.write( "    * M2 = 0.1Y2\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* Y = National Income<BR>\n" );
document.write( "* C = Consumption<BR>\n" );
document.write( "* I = Investment (we'll assume I1 and I2 are exogenous, so we'll treat them as constants)<BR>\n" );
document.write( "* X = Exports<BR>\n" );
document.write( "* M = Imports\r<BR>\n" );
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document.write( "We also know that:\r<BR>\n" );
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document.write( "* X1 = M2 (Exports of country 1 are imports of country 2)<BR>\n" );
document.write( "* X2 = M1 (Exports of country 2 are imports of country 1)\r<BR>\n" );
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document.write( "**a) Express the System in Matrix Form and Find Y1 and Y2**\r<BR>\n" );
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document.write( "1.  **Substitute the Equations:**\r<BR>\n" );
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document.write( "    * For Country 1:<BR>\n" );
document.write( "        * Y1 = (0.7Y1 + 50) + I1 + M2 - 0.3Y1<BR>\n" );
document.write( "        * Y1 = 0.4Y1 + 50 + I1 + 0.1Y2<BR>\n" );
document.write( "        * 0.6Y1 - 0.1Y2 = 50 + I1<BR>\n" );
document.write( "    * For Country 2:<BR>\n" );
document.write( "        * Y2 = (0.8Y2 + 100) + I2 + M1 - 0.1Y2<BR>\n" );
document.write( "        * Y2 = 0.7Y2 + 100 + I2 + 0.3Y1<BR>\n" );
document.write( "        * -0.3Y1 + 0.3Y2 = 100 + I2\r<BR>\n" );
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document.write( "2.  **Matrix Form:**\r<BR>\n" );
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document.write( "    We can write this system of equations in matrix form as:\r<BR>\n" );
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document.write( "    ```<BR>\n" );
document.write( "    [ 0.6  -0.1 ] [ Y1 ] = [ 50 + I1 ]<BR>\n" );
document.write( "    [ -0.3  0.3 ] [ Y2 ] = [ 100 + I2 ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "    Let's represent this as:\r<BR>\n" );
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document.write( "    ```<BR>\n" );
document.write( "    A * Y = B<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "    Where:\r<BR>\n" );
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document.write( "    ```<BR>\n" );
document.write( "    A = [ 0.6  -0.1 ]<BR>\n" );
document.write( "        [ -0.3  0.3 ]\r<BR>\n" );
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document.write( "    Y = [ Y1 ]<BR>\n" );
document.write( "        [ Y2 ]\r<BR>\n" );
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document.write( "    B = [ 50 + I1 ]<BR>\n" );
document.write( "        [ 100 + I2 ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "3.  **Solve for Y (Y1 and Y2):**\r<BR>\n" );
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document.write( "    To find Y, we need to calculate A⁻¹ and multiply it by B:\r<BR>\n" );
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document.write( "    ```<BR>\n" );
document.write( "    Y = A⁻¹ * B<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "    First, find the determinant of A:\r<BR>\n" );
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document.write( "    * det(A) = (0.6 * 0.3) - (-0.1 * -0.3) = 0.18 - 0.03 = 0.15\r<BR>\n" );
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document.write( "    Now, find the inverse of A:\r<BR>\n" );
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document.write( "    ```<BR>\n" );
document.write( "    A⁻¹ = (1 / det(A)) * [ 0.3   0.1 ]<BR>\n" );
document.write( "                           [ 0.3   0.6 ]\r<BR>\n" );
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document.write( "    A⁻¹ = (1 / 0.15) * [ 0.3   0.1 ]<BR>\n" );
document.write( "                           [ 0.3   0.6 ]\r<BR>\n" );
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document.write( "    A⁻¹ = [ 2   2/3 ]<BR>\n" );
document.write( "           [ 2   4 ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "    Now, multiply A⁻¹ by B:\r<BR>\n" );
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document.write( "    ```<BR>\n" );
document.write( "    Y = [ 2   2/3 ] [ 50 + I1 ]<BR>\n" );
document.write( "        [ 2   4   ] [ 100 + I2 ]\r<BR>\n" );
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document.write( "    Y1 = 2(50 + I1) + (2/3)(100 + I2)<BR>\n" );
document.write( "    Y1 = 100 + 2I1 + 200/3 + (2/3)I2<BR>\n" );
document.write( "    Y1 = 500/3 + 2I1 + (2/3)I2\r<BR>\n" );
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document.write( "    Y2 = 2(50 + I1) + 4(100 + I2)<BR>\n" );
document.write( "    Y2 = 100 + 2I1 + 400 + 4I2<BR>\n" );
document.write( "    Y2 = 500 + 2I1 + 4I2<BR>\n" );
document.write( "    ```\r<BR>\n" );
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document.write( "**b) Calculate the Balance of Payments Between These Countries**\r<BR>\n" );
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document.write( "The balance of payments is the difference between exports and imports.\r<BR>\n" );
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document.write( "* Balance of Payments for Country 1 = X1 - M1 = M2 - M1 = 0.1Y2 - 0.3Y1<BR>\n" );
document.write( "* Balance of Payments for Country 2 = X2 - M2 = M1 - M2 = 0.3Y1 - 0.1Y2\r<BR>\n" );
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document.write( "Let's calculate the balance of payments for Country 1:\r<BR>\n" );
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document.write( "* Balance of Payments 1 = 0.1Y2 - 0.3Y1<BR>\n" );
document.write( "* Balance of Payments 1 = 0.1(500 + 2I1 + 4I2) - 0.3(500/3 + 2I1 + (2/3)I2)<BR>\n" );
document.write( "* Balance of Payments 1 = 50 + 0.2I1 + 0.4I2 - 50 - 0.6I1 - 0.2I2<BR>\n" );
document.write( "* Balance of Payments 1 = -0.4I1 + 0.2I2\r<BR>\n" );
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document.write( "Let's calculate the balance of payments for Country 2:\r<BR>\n" );
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document.write( "* Balance of Payments 2 = 0.3Y1 - 0.1Y2<BR>\n" );
document.write( "* Balance of Payments 2 = 0.3(500/3 + 2I1 + (2/3)I2) - 0.1(500 + 2I1 + 4I2)<BR>\n" );
document.write( "* Balance of Payments 2 = 50 + 0.6I1 + 0.2I2 - 50 - 0.2I1 - 0.4I2<BR>\n" );
document.write( "* Balance of Payments 2 = 0.4I1 - 0.2I2\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "* Y1 = 500/3 + 2I1 + (2/3)I2<BR>\n" );
document.write( "* Y2 = 500 + 2I1 + 4I2<BR>\n" );
document.write( "* Balance of Payments for Country 1 = -0.4I1 + 0.2I2<BR>\n" );
document.write( "* Balance of Payments for Country 2 = 0.4I1 - 0.2I2\r<BR>\n" );
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document.write( "The balance of payments for Country 2 is the negative of the balance of payments for Country 1, meaning that if one country has a surplus, the other has a deficit.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );