Answer to Question #287924 in Financial Math for Likenaw

Question

Two duopolists produce and quantities of a homogenous product . The market demand of the product is given by Q= 240-2p where Q= Qa +Qb and the price of the product. The total functions of the duopolists are given by: C(Qa)= 60+4Qa and C(Qb)= 50+0.625Qb^2

a) Find the level of output that maximizes the profit of each firm and the corresponding profit

and price.

b) Find the level of output that maximizes their profit if the two firms corporate and the

corresponding profit and price.

Answer 1

$a)Q=240-2p\\c(Q_a)=60+4Q_a\\c(Q_b)=50+0.625Q_b,where \\Q=Q_a+Q_b,from~Q=240-2p,p=120-0.5Q\\ \pi_a=P.Q-c(Q_a)\\=(120-0.5Q)Q_a-(60+4Q_a)\\=120Q_a-0.5(Q_a)^2-0.5Q_aQ_b-60-4Q_a\\ {\pi_a}'=120-Q_a-0.5Q_b-4\\ {\pi_a}'=0\\ \implies 120-Q_a-0.5Q_b-4=0\\ Q_a=116-0.5Q_b......equ(1)\\ \pi_b=P.Q-c(Q_b)\\=(120-0.5Q)Q_b-(50+0.625Q_b^2)\\=120Q_b-0.5Q_aQ_b-0.5Q_b^2-50-0.625Q_b^2\\= 120Q_b-0.5Q_aQ_b-50-1.125Q_b^2\\{\pi_b}'=120-0.5Q_a-2.25Q_b\\ {\pi_b}'=0\\ \implies 120-0.5Q_a-2.25Q_b=0\\ Q_b=53.3-0.22Q_a......equ(2)\\ \text{from equation 1 and 2}\\ Q_b=53.3-0.22(116-0.5Q_b)\\ Q_b=53.3-25.52+0.11Q_b\\ Q_b=31.21.~~Also,\\Q_a=116-0.5Q_b\\=116-0.5(31.21)=100.4\\ Then~we~have\\ \pi_a=120Q_a-0.5(Q_a)^2-0.5Q_aQ_b-60-4Q_a\\ \pi_a=120(100.4)-0.5(100.4)^2-0.5(100.4)(31.21)-60-4(100.4)\\ \pi_a=4979.578\\ \pi_b=120(31.21)-0.5(100.4)(31.21)-50-1.125(31.21)^2\\ \pi_b=1032.64\\ p=120-0.5(Q_a+Q_b)\\ p=120-0.5(100.4+31.21)\\ p=54.195\\ b)Q=240+2p\\ c(Q)=c(Q_a)+c(Q_b)=60+4Q_a+50+0.0625Q_b^2\\ =110+4Q_a+0.625Q_b^2\\ p=120-0.5Q\\ \pi = P.Q-c(Q)\\ \pi =(120-0.5Q)Q- 110-4Q-0.625Q^2\\ \pi =120Q-0.5Q^2- 110-4Q-0.625Q^2\\\pi '=120-Q-4-1.25Q=0\\ 2.25Q=116, ~Q=51.56\\ p=120-0.5Q\\ p=120-0.5(51.56)\\ p=94.22\\ \pi= 116Q-1.125Q^2- 110\\ \pi=2880.22$