document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169744>Question 1169744</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The general linear supply and demand for one-commodity market model are given by <BR>\n" );
document.write( "P = a𝑄𝑠+b      (a>0, b>0) <BR>\n" );
document.write( "P = - c𝑄𝐷+d     (c>0, d>0) <BR>\n" );
document.write( "a. Show that in matrix notation the equilibrium price, P, and quantity, Q satisfy. <BR>\n" );
document.write( "	 	 	[1	−𝑎]  [𝑃]=[𝑏] <BR>\n" );
document.write( "1	𝑐	𝑄	𝑑<BR>\n" );
document.write( "b.	Solve this system to express P and Q in terms of a, b, c and d. <BR>\n" );
document.write( "c.	Write down the multiplier for Q due to changes in b and deduce that an increase in b lends to an decrease in Q. <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 #851306 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=851306\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's solve this supply and demand problem.\r<BR>\n" );
document.write( "\n" );
document.write( "**Understanding the Model**\r<BR>\n" );
document.write( "\n" );
document.write( "We have a one-commodity market with linear supply and demand equations:\r<BR>\n" );
document.write( "\n" );
document.write( "* Supply: P = aQs + b (a > 0, b > 0)<BR>\n" );
document.write( "* Demand: P = -cQd + d (c > 0, d > 0)\r<BR>\n" );
document.write( "\n" );
document.write( "Where:\r<BR>\n" );
document.write( "\n" );
document.write( "* P = Price<BR>\n" );
document.write( "* Qs = Quantity supplied<BR>\n" );
document.write( "* Qd = Quantity demanded\r<BR>\n" );
document.write( "\n" );
document.write( "At equilibrium, Qs = Qd = Q.\r<BR>\n" );
document.write( "\n" );
document.write( "**a) Show the Matrix Notation**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Rewrite the Equations:**\r<BR>\n" );
document.write( "\n" );
document.write( "    * Supply: P - aQ = b<BR>\n" );
document.write( "    * Demand: P + cQ = d\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Matrix Form:**\r<BR>\n" );
document.write( "\n" );
document.write( "    We can represent these equations in matrix form:\r<BR>\n" );
document.write( "\n" );
document.write( "    ```<BR>\n" );
document.write( "    [ 1  -a ] [ P ] = [ b ]<BR>\n" );
document.write( "    [ 1   c ] [ Q ] = [ d ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
document.write( "\n" );
document.write( "    Thus, we have shown the required matrix equation.\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Solve for P and Q**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Matrix Equation:**\r<BR>\n" );
document.write( "\n" );
document.write( "    Let:\r<BR>\n" );
document.write( "\n" );
document.write( "    ```<BR>\n" );
document.write( "    A = [ 1  -a ]<BR>\n" );
document.write( "        [ 1   c ]\r<BR>\n" );
document.write( "\n" );
document.write( "    X = [ P ]<BR>\n" );
document.write( "        [ Q ]\r<BR>\n" );
document.write( "\n" );
document.write( "    B = [ b ]<BR>\n" );
document.write( "        [ d ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
document.write( "\n" );
document.write( "    We have AX = B, so X = A⁻¹B.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Inverse of A:**\r<BR>\n" );
document.write( "\n" );
document.write( "    * det(A) = (1 * c) - (-a * 1) = c + a<BR>\n" );
document.write( "    * A⁻¹ = (1 / (c + a)) * [ c   a ]<BR>\n" );
document.write( "                           [ -1  1 ]\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Solve for X (P and Q):**\r<BR>\n" );
document.write( "\n" );
document.write( "    ```<BR>\n" );
document.write( "    [ P ] = (1 / (c + a)) * [ c   a ] [ b ]<BR>\n" );
document.write( "    [ Q ]                 [ -1  1 ] [ d ]<BR>\n" );
document.write( "    ```\r<BR>\n" );
document.write( "\n" );
document.write( "    ```<BR>\n" );
document.write( "    P = (1 / (c + a)) * (cb + ad) = (cb + ad) / (c + a)<BR>\n" );
document.write( "    Q = (1 / (c + a)) * (-b + d) = (d - b) / (c + a)<BR>\n" );
document.write( "    ```\r<BR>\n" );
document.write( "\n" );
document.write( "    Therefore:\r<BR>\n" );
document.write( "\n" );
document.write( "    * P = (cb + ad) / (c + a)<BR>\n" );
document.write( "    * Q = (d - b) / (c + a)\r<BR>\n" );
document.write( "\n" );
document.write( "**c) Multiplier for Q due to Changes in b**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Q in terms of b:**\r<BR>\n" );
document.write( "\n" );
document.write( "    * Q = (d - b) / (c + a)\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Multiplier:**\r<BR>\n" );
document.write( "\n" );
document.write( "    The multiplier for Q due to changes in b is the derivative of Q with respect to b:\r<BR>\n" );
document.write( "\n" );
document.write( "    * dQ/db = -1 / (c + a)\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Deduction:**\r<BR>\n" );
document.write( "\n" );
document.write( "    Since a > 0 and c > 0, (c + a) > 0. Therefore, -1 / (c + a) is always negative.\r<BR>\n" );
document.write( "\n" );
document.write( "    This means that an increase in b (the y-intercept of the supply curve) leads to a decrease in Q (the equilibrium quantity).\r<BR>\n" );
document.write( "\n" );
document.write( "**Final Answers**\r<BR>\n" );
document.write( "\n" );
document.write( "* **(a)** The matrix notation is: [ 1 -a ] [ P ] = [ b ]<BR>\n" );
document.write( "                                    [ 1 c ] [ Q ] = [ d ]<BR>\n" );
document.write( "* **(b)** P = (cb + ad) / (c + a)<BR>\n" );
document.write( "           Q = (d - b) / (c + a)<BR>\n" );
document.write( "* **(c)** The multiplier for Q due to changes in b is -1 / (c + a). Since this is negative, an increase in b leads to a decrease in Q.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );