Question

The demand and cost functions of a firm are x = 6000 – 30p and C = 72000 + 60x respectively. Find the level of output and price at which the profit is maximum.

Answer 1

We know that profit is maximum when marginal Revenue (MR) = Marginal Cost (MC)

The demand function, x = 6000 – 30p

30p = 6000 – x

p = `1/30` (6000 – x)

p = 200 - `x/30`  ....(1)

Revenue, R = px

`= (200 - x/30)x`

`= 200x - x^2/30`

Marginal Revenue (MR) = `"dR"/"dx"`

`= "d"/"dx" (200 x - x^2/30)`

`= (200 "d")/"dx" ("x") - 1/30 "d"/"dx" (x^2)`

`= 200(1) - 1/30 (2x)`

`= 200 - x/15`

Cost function, C = 72000 + 60x

Marginal cost, `"dC"/"dx" = "d"/"dx"`(72000 + 60x)

= 0 + 60(1)

= 60

But marginal revenue = marginal cost

`200 - x/15 = 60`

`- x/15 = 60-200`

`- x/15 = - 140`

-x = – 140 × 15

x = 140 × 15 = 2100

The output is 2100 units.

By (1) we have p = `200 - x/30`

When x = 2100,

Profit, p = `200 -2100/30` = 200 - 70 = 130

p = ₹ 130