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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.
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Solution
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
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