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Flashcards: Finite Mathematics

It is suggested (but not necessary) that you use a calculator with matrix capability to work this problem.

A village is famous for being the most prolific producer in the world of ambrosia, butterbeer, and silphium. Every year, they try to meet the world's demand for 20,000 kg of ambrosia, 30,000 kg of butterbeer, and 15,000 kg of silphium. However, the production of these items requires the consumption of these items as well:

The production of 1 kg of ambrosia requires the use of 0.05 kg of butterbeer and 0.08 kg of silphium;

The production of 1 kg of butterbeer requires the use of 0.04 kg of ambrosia and 0.09 kg of silphium; and,

The production of 1 kg of silphium requires the use of 0.06 kg of ambrosia and 0.07 kg of butterbeer.

Use a Leontief model to determine how much silphium (nearest hundred kilograms) must be produced to meet the demand.

$18,700 \textup{ kg }$

$18,500 \textup{ kg }$

$18,800 \textup{ kg }$

$18,600 \textup{ kg }$

$18,400 \textup{ kg }$

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Flashcards: Finite Mathematics

It is suggested (but not necessary) that you use a calculator with matrix capability to work this problem.

The production of 1 kg of ambrosia requires the use of 0.05 kg of butterbeer and 0.08 kg of silphium;

The production of 1 kg of butterbeer requires the use of 0.04 kg of ambrosia and 0.09 kg of silphium; and,

The production of 1 kg of silphium requires the use of 0.06 kg of ambrosia and 0.07 kg of butterbeer.

Use a Leontief model to determine how much silphium (nearest hundred kilograms) must be produced to meet the demand.

Your answer:

$18,500 \textup{ kg }$

Your answer:

$18,800 \textup{ kg }$

Your answer:

$18,600 \textup{ kg }$

Your answer:

$18,400 \textup{ kg }$

Correct answer:

$18,700 \textup{ kg }$

Explanation:

The Leontief model for an input-output system requires the following steps.

Let be the technology matrix for the system, the entries of which are the amounts of each product consumed in the manufacture of each product. Letting the rows/columns represent, in order, ambrosia, butterbeer, and silphium,

$A = \begin{bmatrix} 0 & 0.05 & 0.08 \\ 0.04 & 0 & 0.09 \\ 0.06 & 0.07 & 0\end{bmatrix}$

Let be the external demand vector, which gives the amount of each product demanded by outside consumers:

$D= \begin{bmatrix} 20,000 \\ 30,000 \\ 15,000 \end{bmatrix}$

The output vector , which gives the amount of each product that must be produced to meet the demand, can be obtained through the matrix equation

$X = (I- A )^{-1}D$

$= \left ( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} - \begin{bmatrix} 0 & 0.05 & 0.08 \\ 0.04 & 0 & 0.09 \\ 0.06 & 0.07 & 0\end{bmatrix} \right )^{-1} \begin{bmatrix} 20,000 \\ 30,000 \\ 15,000 \end{bmatrix}$

$= \begin{bmatrix} 1 &-0.05 & -0.08 \\ -0.04 & 1 & -0.09 \\ -0.06 & -0.07 & 1 \end{bmatrix} ^{-1} \begin{bmatrix} 20,000 \\ 30,000 \\ 15,000 \end{bmatrix}$

$\approx \begin{bmatrix} 1.007394521 &0.056366243 & 0.085664523 \\0.046025673 & 1.008915193 & 0.094484421 \\ 0.063665468 & 0.074006038 &1.011753781 \end{bmatrix} \begin{bmatrix} 20,000 \\ 30,000 \\ 15,000 \end{bmatrix}$

$\approx \begin{bmatrix} 23,123.85 \\ 32,605.24\\ 18,669.80 \end{bmatrix}$

Silphium is represented by the bottom entry; the amount the village must produce rounds to 18,700 kg.

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