Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (c)

MATH152 April 2022

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Question B2 (c)
This is a continuation of B2 (a). (c) In the long run, what will the share of each type of apple be?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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Hint
If the share of each apple stabilizes in the long run, how should the transition matrix act on the vector of shares?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Suppose that, in the long run, $v_{1},v_{2},v_{3}$ are the proportions of buyers who buy Ambrosia, Braeburn and Cameo apples, respectively. Then the transition matrix $A$ should leave the vector $\left[{\begin{array}{c}v_{1}\\v_{2}\\v_{3}\end{array}}\right]$ fixed. In other words, we are looking for an eigenvector with eigenvalue 1, or an element of the null-space of the matrix $$A-\mathrm {Id} =\left[{\begin{array}{ccc}-0.3&0.2&0.2\\0.2&-0.4&0.4\\0.1&0.2&-0.6\end{array}}\right].$$ We can find the elements of the null-space by row-reducing. Again, we indicate the row operations as in B1 (a), but we also use $T_{i,j}$ for the operation "swap rows $i$ and $j$". $${\begin{aligned}\left[{\begin{array}{ccc}-0.3&0.2&0.2\\0.2&-0.4&0.4\\0.1&0.2&-0.6\end{array}}\right]\xrightarrow {D_{\text{all}}(10),\ T_{1,3}} \left[{\begin{array}{ccc}1&2&-6\\2&-4&4\\-3&2&2\end{array}}\right]\xrightarrow {L_{2,1}(-2),\ L_{3,1}(3)} \left[{\begin{array}{ccc}1&2&-6\\0&-8&16\\0&8&-16\end{array}}\right]\xrightarrow {L_{3,2}(1),\ D_{2}(1/8),L_{1,2}(2)} \left[{\begin{array}{ccc}1&0&-2\\0&-1&2\\0&0&0\end{array}}\right].\end{aligned}}$$ We can easily see from the row-reduced matrix above that eigenvectors of $A$ with eigenvalue 1 are of the form $\left[{\begin{array}{c}v\\v\\{\frac {v}{2}}\end{array}}\right].$ If we want the entries of this vector to be proportions of people buying the three different kinds of apples, then the entries of the eigenvector must sum up to 1. The equation $v+v+{\frac {v}{2}}=1$ implies that $v=0.4$, so we have that the long-run shares of each apple variety are $$v_{1}=0.4,\qquad v_{2}=0.4,\qquad v_{3}=0.2.$$