Mathematics, 11.08.2021 15:00 pennygillbert

(b) An economy has an agricultural industry and a textile industry. Each unit of agricultural output requires 0.4 unit of agricultural input and 0.1 unit of textiles input. Each unit of textiles output requires 0.1 unit of agricultural input and 0.2 unit of textiles input. (i) Write the technology matrix for this economy. [2 marks]

(ii) If surpluses of 5 units of agricultural products and 195 units of textiles are desired, find the gross production of each industry

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Answer from: nick5442

Leontief input output model (technology matrix) is an economic model that shows the quantitative relationship and sectorial interdependency in a national economy

The responses with regards to the question are;

(i) The technology matrix for the economy is presented as follows;

$\mathbf{ A} =\left[\begin{array}{ccc}Agric&&Textile\\0.4&&0.1\\&&\\0.1&&0.2\end{array}\right] \begin{array}{ccc}\mathbf{Per \ Unit}\\Agriculture\\\\Textile\end{array}\right]$

(ii) The required gross production of each industry to meet the desired surplus are;

50 units of agriculture and 250 units of textile

The reason the above values are correct is as follows:

(i) The given parameters are;

The industries in the economy = Agricultural industry and textile industry

Units of agricultural input required per unit of agricultural output = 0.4

Units of textile input required per unit of agricultural output = 0.1

Units of agricultural input required per unit of textile output = 0.1

Units of textile input required per unit of textile output = 0.2

Let X represent agriculture, and let Y represent textile, we have;

$Agric \ for \ agric = \dfrac{0.4 \ units \ of \ agriculture}{1\ unit \ of \ agric \ produced} \times X \ Agric \ produced= 0.4 \cdot X$

$Agric \ for \ textile = \dfrac{0.1 \ units \ of \ agriculture}{1\ unit \ of \ textile \ produced} \times Y \ textile \ produced= 0.1 \cdot Y$

We also have;

Textile for agriculture = 0.1·X

Textile for textile = 0.2·Y

Therefore;

X = 0.4·X + 0.1·Y

Y = 0.1·X + 0.2·Y

Therefore;

The technology matrix for the economy is presented as follows;

$\mathbf{Technology \ matrix, A} =\left[\begin{array}{ccc}Agric&&Textile\\0.4&&0.1\\&&\\0.1&&0.2\end{array}\right] \begin{array}{ccc}\mathbf{Per \ Unit}\\Agriculture\\\\Textile\end{array}\right]$

(ii) Let P represent the production vector, and let d represent the demand vector, we have;

$P = \left[\begin{array}{c}X \\Y\end{array}\right]$ , $d = \left[\begin{array}{c}5 \\195\end{array}\right]$

P = A·P + d

∴ P - A·P = d

Therefore;

$P = \mathbf{ \dfrac{d}{(I - A)}}$

Where I = The 2 by 2 identity matrix

We get;

$I - A =\left[\begin{array}{ccc}1&&0\\&&\\0&&1\end{array}\right] - \left[\begin{array}{ccc}0.4&&0.1\\&&\\0.1&&0.2\end{array}\right] = \mathbf{\left[\begin{array}{ccc}0.6&&-0.1\\&&\\-0.1&&0.8\end{array}\right]}$

With the use of a graphing calculator, we have;

$P =\left[\begin{array}{c}X \\Y\end{array}\right] = \dfrac{\left[\begin{array}{c}5 \\195\end{array}\right]}{\left[\begin{array}{ccc}0.6&&-0.1\\&&\\-0.1&&0.8\end{array}\right]} = \left[\begin{array}{ccc}50\\\\\ 250\end{array}\right]$

The required gross product of agriculture, X = 50 units

The required gross product of textile, Y = 250 units

Learn more about the Leontief input output model here:

Answer from: Jenifermorales101

We have that he technology matrix for this economy and the the gross production of each industry are

a) $X= \begin{vmatrix}0.4 & 0.1 \\0.1 & 0.2\end{vmatrix}$

b) $\begin{vmatrix}A\\T\end{vmatrix}=\begin{vmatrix}50\\250\end{vmatrix}$

From the Question we have told that

Each unit of agricultural output requires 0.4 unit of agricultural input

Each unit of agricultural output requires 0.1 unit of textiles input.

Each unit of textiles output requires 0.1 unit of agricultural input

Each unit of textiles output requires 0.2 unit of textiles input.

Generally the technology matrix for this economy is given below

With

X =Agricultural industry Gross output

Y= Textile industry Gross Output

Therefore

$X= \begin{vmatrix}0.4 & 0.1 \\0.1 & 0.2\end{vmatrix}$

From the Question we are told that

Surpluses of 5 units of agricultural products and 195 units of textiles are desired.

Therefore, we have Desired surplus matrix of

$D= \begin{vmatrix}5\\195\end{vmatrix}$

Generally the Technology equation is mathematically given as

$(I-X)\phi=D$

Where

X =Agricultural industry Gross output

I=A Unit matrix

\phi=Matrix of gross production

Therefore

$\begin{vmatrix}1 & 0\\0 & 1\end{vmatrix}-(\begin{vmatrix}0.4 & 0.1 \\0.1 & 0.2\end{vmatrix}))\begin{vmatrix}A\\T\end{vmatrix}=\begin{vmatrix}5\\195\end{vmatrix}$