Question

A monopolist has a demand curve x = 106 – 2p and average cost curve AC = 5 + `x/50`, where p is the price per unit output and x is the number of units of output. If the total revenue is R = px, determine the most profitable output and the maximum profit.

Answer 1

x = 106 – 2p

(or) 2p = 106 – x

p = `1/2`(106 – x)

Revenue, R = px

= `1/2`(106 – x) x

= 53x – `x^2/2`

Average Cost, AC = `5 + x/50`

Cost C = (AC)x

= `(5 + x/50)x`

= `5x + x^2/50`

Profit (P) = Revenue – Cost

`"dP"/"dx" = 48 - (13(2x))/25`

`"dP"/"dx"` = 0 gives

`48 - (13(2x))/25` = 0

`48 = (13 xx 2x)/25`

x = `(48 xx 25)/(13 xx 2)`= 46.1538 = 46 (approximately)

Also `("d"^2"P")/"dx"^2 = 0 - (13)^2/25`, negative since `("d"^2"P")/"dx"^2` is negative, profit is maximum at x = 46 units.

Profit = `48x – 13/25` x²

When x = 46,

Profit = `48 × 46 - 13/25` × 46 × 46

`= 2208 - 27508/25`

= 2208 – 1100.32

= ₹ 1107.68