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March 8th, 2018, 12:13 AM   #1
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Systems of Linear Equations

Hi all, need help in this. I am so confused.

Suppose that in a fixed population of food consumers, there are just three brands X Y Z that share the market. Suppose that each year,
10% of brand X buyers shift to brand Y and 20% to brand Z and
20% of brand Y buyers shift to brand X and 20% to brand Z
10% of brand Z buyers shift to brand X and none to brand Y
Suppose that it is found that despite all this shifting, the market shares of these brands are nevertheless unchanged, what are these market shares?

TIA

Last edited by skipjack; March 8th, 2018 at 01:09 AM.
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March 8th, 2018, 01:53 AM   #2
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You need to assume that each consumer uses just one brand at a time.

During each year, for every 5 consumers of brand Y, 1 moves to Brand X and 1 moves to brand Z, so there must have been 20 consumers of brand X originally, 2 of which shift to brand Y.

I'll leave you to show by similar means that the corresponding number of consumers of brand Z was 50.

Hence brand Z has 2/3 of the market, brand X has 4/15 of the market, and brand Y has the remaining 1/15 of the market.
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April 5th, 2018, 11:19 AM   #3
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Let $x_n$ be the number of buyers of brand X in the nth year, $y_n$ be the number of buyers of brand Y, and $z_n$ be the number of buyers of brand Z.

"10% of brand X buyers shift to brand Y and 20% to brand Z"
"20% of brand Y buyers shift to brand X and 20% to brand Z"
"10% of brand Z buyers shift to brand X and none to brand Y"
So $X_{n+1}= .7X_n+ .2Y_n+ .1Z_n$
$Y_{n+1}= .1X_n+ .6Y_n$
$Z_{n+1}= .2X_n+ .2Y_n+ .9Z_n$

The "market share change" for each is
$X_{n+1}- X_n= .3X_n- .2Y_n- .1Z_n$
$Y_{n+1}- Y_n= -.1X_n+ .4Y_n$
$Z_{n+1}- Z_n= -.2X_n- 2Y_n+ .1Z_n$

Saying that "the market shares are unchanged" means that those changes are all 0.

Solve $.3X_n- .2Y_n- .1Z_n= 0$
$-.1X_n+ .4Y_n= 0$ and
$-.2X_n- .2Y_n+ .1Z_n= 0$.

An obvious solution is the "trivial solution" $X_n= Y_n= Z_n= 0$. The question is whether there at e other, non-trivial, solutions and, if so, what are they?
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