Question

A monopolist operates under two plants, A and B. The marginal costs of these plants are given by 360 — 14x — 2x2and 310 — 15x — x2 with x representing units of output produced by each plant. If the price of the product is given by 396 — 4x2 , calculate the overall marginal cost and determine profit maximizing output in each plant.

Answer 1

Plant 1: Mc₁ $=360-14x-2x^2$

Plant 2: Mc₂ $=310 - 15x -x^2$

C₁ = $\intop$Mc₁ dx = $360x-7x-2x^3/3$

C₂ = $\int$Mc₂ dx = $310x-15x^2/2-x^3/3$

Total lost of the monopolist $=$ C₁ +C₂

$360x-7x^2-2x^3/3+310x-15x^2/2-x^3/3$

$670x-7x^2-15x^2/2-2x^3/3-x^3/3$

$670x-(14x^2-15x^2)/2-x^3$

$670x-(29x^2)/2-x^3$

Overall MC $= 670-29x-3x^2$

For monopolist profit will be max.

$MR=MC$

$TR=P×Q$

$TR=(396-4x^2)x=396x-4x^3$

$MR=dTR/dx=396-12x^2$

in Plant 1 profit will be max

$MR=MC$₁

$396-12x^2=360-14x-2x^2$

$36=-14x+10x^2$

hence

$-10x^2+14x+36=0$

$10x^2-14x-36=0$

$5x^2-7x-18=0$

$x=2.722$

hence $x=3$ units

in Plant 2

$MR=MC$₂

$396-12x^2=310-15x-x^2$

$86=-15x+11x^2$

$-11x^2+15x+86=0$

$11x^2-15x-86=0$

$x=3.5598$

hence $x=4$ units