Question

Assume that a company has received an order for 200 units of its product and wants to distribute its production between two of its plants, plant 1 and plant 2. Let q1 and q2 be the productions of plants 1 and 2, respectively, and assume that the Total cost function is given by c = f (q₁ q₂) = 2q₁² + q₁ q₂ + q₂² + 200. How should production be distributed to minimize costs? The answers are q₁=50 and q₂=150 explain the procedure.

Answer 1

Total output is 200 units. Let q1 and q2 be the productions of plants 1 and 2, respectively, and assume that the Total cost function is given by c = f (q₁ q₂) = 2q₁² + q₁ q₂ + q₂² + 200. How should production be distributed to minimize costs? The answers are q₁=50 and q₂=150

Find the partial derivatives of the cost function. This will give the equations of Marginal cost of two plants and then keep them equal to each other so that a single equation is determined.

dc(q1) = MC1 = 4q1 + q2,

dc(q2) = MC2 = q1 + 2q2.

This gives MC1 = MC2

4q1 + q2 = q1 + 2q2

The final equation is 3q1 – q2 = 0.

Use this and q1 + q2 = 200

q1 + 3q1 = 200 .................. (q2 = 3q1 taken from the equation 3q1 – q2 = 0)

4q1 = 200. We therefore get q1 = 200/40 = 50*

and q2 = 200 – q1

= 200 – 50

= 150 units.

This shows that q1 = 50 and q2 = 150.