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Homework Help: Leontief model - anyone familiar with this?

  1. Sep 25, 2012 #1
    Suppose a company produces three products, X, Y , and Z. Manufacturing a
    unit of X uses up 0.1 units of X, and 0.2 units of Y . Manufacturing a unit of Y uses up 0.1 units of X and 0.5 units of Y . Manufacturing a unit of Z uses 0.2 units of Z.

    This is an application of the leontief model.

    Write down the internal demand matrix for this company.

    Is the internal demand matrix
    0.1 0.2 0

    0.2 0.5 0

    0 0 0.2

    Sorry Idk how to enter a matrix on here. The first row is 0.1, 0.2 and 0
    second row is 0.2, 0.5, 0
    3rd row is 0, 0, 0.2
  2. jcsd
  3. Sep 25, 2012 #2

    Ray Vickson

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    You could enter something like D = [[0.1, 0.2, 0],[0.2,0.5,0],[0,0,0.2]], and perhaps explain that this means [row1,row2,row3]. Or, you could use LaTeX and apply the "pmatrix" command: you could say "[t e x] \p m a t r i x{ 0.1 & 0.2& 0\\0.2& 0.5&0\\0 & 0 & 0.2} [/ t e x]" (remove the spaces in [..] and in the word 'pmatrix'). Here, the '&' separates the different items in the row--and you must use it, not a comma--- while the \\ starts a new row. This produces
    [tex] \pmatrix{0.1&0.2&0\\0.2&0.5&0\\0&0&0.2}[/tex]
    If you don't like the rounded brackets you can use the commands '\left[ \b e g i n{a r r a y}{ccc} put the rows in here just like above \e n d {a r r a y} \right]', so you could enter
    "[t e x] \left[ \b e g i n {a r r a y}{ccc} 0.1&0.2&0\\0.2 & 0.5 & 0\\ 0 & 0 & 0.2 \e n d {a r r a y} \right] [/ t e x]". Here, the {ccc} after the first 'array' statement is a column justifier, so in this case all three columns would be centered. (If you want column 1 right-justified, column 2 centered and column 3 left-justified, you would say {rcl}.) Note that there are no brackets { and } around the rows. The '\left[' and '\right]' statements specify the bracket style on the left and the right; every '\left' statement must be accompanied by a '\right' statement, but they don't have to be of the same type. For example, I might want [ on the left and ) on the right, or ( on the left and nothing on the right (which would be entered as '\right.' (that is, as \ right period). So, you can have
    [tex]\left[ \begin{array}{ccc} 0.1&0.2&0\\0.2&0.5&0\\0&0&0.2 \end{array} \right] \text{ or }
    \left( \begin{array}{ccc} 0.1&0.2&0\\0.2&0.5&0\\0&0&0.2 \end{array} \right| \text{ or }
    \left[ \begin{array}{ccc}0.1&0.2&0\\0.2&0.5&0\\0&0&0.2 \end{array} \right. [/tex]

  4. Sep 25, 2012 #3


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    Interesting. Either of these will give a matrix with rounded brackets, but if you change to bmatrix (b instead of p) for square brackets, only the first one works.

    Code (Text):

    $$\begin{pmatrix}  0.1 & 0.2& 0\\0.2& 0.5&0\\0 & 0 & 0.2\end{pmatrix}$$

    $$\pmatrix{0.1 & 0.2& 0\\0.2& 0.5&0\\0 & 0 & 0.2}$$
    $$\begin{pmatrix} 0.1 & 0.2& 0\\0.2& 0.5&0\\0 & 0 & 0.2\end{pmatrix}$$
    $$\begin{bmatrix} 0.1 & 0.2& 0\\0.2& 0.5&0\\0 & 0 & 0.2\end{bmatrix}$$
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