document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199304>Question 1199304</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Matrices<BR>\n" );
document.write( "Suppose that three companies dominate the market for a certain product and are competing <BR>\n" );
document.write( "against each other for a large share of the market. Suppose, further, that market surveys have <BR>\n" );
document.write( "shown that buyers shift from one brand to another during a given year according to the <BR>\n" );
document.write( "following schedule:<BR>\n" );
document.write( " C gains 20% of the customer of A and 40% remain to be the user of A. <BR>\n" );
document.write( " C gains 30% of its customer from B, retain 50% of its customer and loses 40% to B <BR>\n" );
document.write( " The customers of B have never shifted to A. <BR>\n" );
document.write( " It is found that the three companies have an equal share of the market today <BR>\n" );
document.write( "a) Present the state vector and the transition matrix for the above case.<BR>\n" );
document.write( "b) What share of the market will each possess two years from now?<BR>\n" );
document.write( "c) Calculate the long run proportions that will be possessed by the three companies.<BR>\n" );
document.write( "d) Company A wants to launch an advertising campaign that will cost Birr 200,000. Given the <BR>\n" );
document.write( "result in (c) above, what would you advise them?<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 #848201 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848201\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this, we use **Markov Chains**. Here's the breakdown:\r<BR>\n" );
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document.write( "### Part (a): State Vector and Transition Matrix\r<BR>\n" );
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document.write( "#### State vector:<BR>\n" );
document.write( "Let the current market shares of companies \( A \), \( B \), and \( C \) be represented as:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\mathbf{x}_0 = \begin{bmatrix} 1/3 \\ 1/3 \\ 1/3 \end{bmatrix}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Since the companies initially have an equal market share.\r<BR>\n" );
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document.write( "#### Transition Matrix:<BR>\n" );
document.write( "Based on the problem:<BR>\n" );
document.write( "1. Company \( A \): Retains 40% of its customers, loses 20% to \( C \), and loses 40% to \( B \).<BR>\n" );
document.write( "2. Company \( B \): Retains 50%, loses 40% to \( C \), and no customers shift to \( A \).<BR>\n" );
document.write( "3. Company \( C \): Retains 50%, gains 20% from \( A \), and gains 30% from \( B \).\r<BR>\n" );
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document.write( "The transition matrix \( P \) is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P = \begin{bmatrix}<BR>\n" );
document.write( "0.4 & 0 & 0.2 \\<BR>\n" );
document.write( "0.4 & 0.5 & 0.3 \\<BR>\n" );
document.write( "0.2 & 0.5 & 0.5<BR>\n" );
document.write( "\end{bmatrix}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Part (b): Market Share Two Years from Now\r<BR>\n" );
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document.write( "We calculate the state vector after 2 years:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\mathbf{x}_2 = P^2 \cdot \mathbf{x}_0<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Let's compute \( P^2 \) and then multiply it by \( \mathbf{x}_0 \).\r<BR>\n" );
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document.write( "The results are:\r<BR>\n" );
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document.write( "- Transition matrix after 2 steps (\( P^2 \)):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P^2 = \begin{bmatrix}<BR>\n" );
document.write( "0.2 & 0.1 & 0.18 \\<BR>\n" );
document.write( "0.42 & 0.4 & 0.38 \\<BR>\n" );
document.write( "0.38 & 0.5 & 0.44<BR>\n" );
document.write( "\end{bmatrix}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "- Market shares after 2 years (\( \mathbf{x}_2 \)):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\mathbf{x}_2 = \begin{bmatrix}<BR>\n" );
document.write( "0.16 \\<BR>\n" );
document.write( "0.40 \\<BR>\n" );
document.write( "0.44<BR>\n" );
document.write( "\end{bmatrix}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Thus:<BR>\n" );
document.write( "- Company \( A \) has 16% of the market.<BR>\n" );
document.write( "- Company \( B \) has 40% of the market.<BR>\n" );
document.write( "- Company \( C \) has 44% of the market.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Part (c): Long-Run Market Shares<BR>\n" );
document.write( "In the long run, the market shares reach a steady state. This occurs when:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\mathbf{x} = P \cdot \mathbf{x}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "where \( \mathbf{x} \) is the steady-state vector. We solve this using the eigenvector corresponding to eigenvalue \( 1 \).\r<BR>\n" );
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document.write( "The long-run market shares are:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\mathbf{x} = \begin{bmatrix}<BR>\n" );
document.write( "0.3333 \\<BR>\n" );
document.write( "0.3333 \\<BR>\n" );
document.write( "0.3333<BR>\n" );
document.write( "\end{bmatrix}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Thus, in the long run, each company will possess **33.33% of the market**, maintaining equal market shares.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Part (d): Advertising Campaign for Company \( A \)\r<BR>\n" );
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document.write( "Since the long-run proportions suggest that all companies will converge to equal shares regardless of initial conditions or changes, the advertising campaign will not significantly alter the equilibrium market shares in the long term.\r<BR>\n" );
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document.write( "#### Recommendation:<BR>\n" );
document.write( "Company \( A \) should consider whether the \( 200,000 \, \text{Birr} \) advertising investment is justified by short-term gains or whether it could invest the funds more effectively elsewhere. If the goal is to improve short-term market share, the campaign might be useful; otherwise, it would likely not affect the long-run proportions. </SPAN>\n" );
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