Question

A commodity was produced by using 3 units of labour and 2 units of capital, the total cost is ₹ 62. If the commodity had been produced by using 4 units of labour and one unit of capital, the cost is ₹ 56. What is the cost per unit of labour and capital? (Use determinant method)

Answer 1

Let the cost per unit of labour be ₹ x and cost per unit of capital be ₹ y.

Labour	Capital	Total cost
3 units	2 units	62
4 units	1 unit	56

So 3x + 2y = 62

4x + y = 56

`Delta = |(3, 2),(4, 1)|`

= 3 – 8

= – 5 ≠ 0

Hence the system has a unique solution which can be found by Cramer's rule

`Delta_x = |(62, 2),(56, 1)|`

= 62 – 112

= – 50

`Delta_y = |(3, 62),(4, 56)|`

= 168 – 248

= – 80

 ∴ x = `Delta_x/Delta = (- 50)/(-5)` = 10

y = `Delta_y/Delta = (- 80)/(-5)` = 16

Hence the cost per unit of labour is ₹ 10 and cost per per unit of capital is ₹ 16