Question

One unit of commodity X is produced by combining 1 unit of land, 2 units of labour and 5 units of capital. One unit of commodity Y is produced by 2 units of land, 3 units of labour and 1 unit of capital. Similarly, 1 unit of commodity Z is produced by 1 unit of labour and 2 units of capital and 3 units of land. If the prices of 3 commodities X, Y and Z are Rs.20, Rs.15 & Rs.22 respectively, find the rent, wages and rate of interest. Using matrix method.

Answer 1

Given : One unit of commodity X is produced by combining 1 unit of land, 2 units of labour and 5 units of capital.  

To find : the rent, wages and rate of interest.

Solution:

Land = A   => Rent

Labour = B  => Wages

Capital = C  =>  Rate of interest

X  = A  +  2B  + 5C

Y  = 2A  + 3B  + C

Z =   3A  + B   +  2C

X = 20    Y  = 15  Z  = 22

$\left[\begin{array}{ccc}1&2&5\\2&3&1\\3&1&2\end{array}\right]$   $\left[\begin{array}{ccc}A\\B\\C\end{array}\right]$   $= \left[\begin{array}{ccc}20\\15\\22\end{array}\right]$

1(6 - 1) - 2(4 - 3)  + 5(2-9)

= 5 - 2 -35

= -32

$\left[\begin{array}{ccc}A\\B\\C\end{array}\right]$     $= \frac{-1}{32} \left[\begin{array}{ccc}5&1&-13\\-1&-13&9\\-7&5&1\end{array}\right]$  $\left[\begin{array}{ccc}20\\15\\22\end{array}\right]$

A = (-1/32)( 100 + 15  - 286) = 171/32

B =  (-1/32)(-20 - 195  +198) = 17/32

C  = (-1/32)( -140  + 75 - 22)  = 87/32

Rent = 171/32

Wages = 17/32

Rate of interest = 87/32

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