Question

Differentiated goods Rather than identical goods, now the two firms are producing differentiated goods, with each behaves as the competitor to the other. Specifically, two goods have following market demand functions: 91 = D, (P1, P2) = 110 – p, + 2p2 92 = D2(p1,P2) = 55 – 2p2 + P1 %3D Also, two firms have following marginal costs: MC, = 10, MC, = 5 Please calculate what is the equilibrium price and quantity for each firm.

Answer 1

Equilibrium prices are p1 = 150 and p2 = 100, while equilibrium quantities are q1 = 150 and q2 = 5

Step-by-step explanation:

To provide solution to this question, the two market demand functions to be restated correctly as follow:

q1 = D1(P1,P2) = 110 – p1 + 2p2 .................................... (1)

q2 = D2(p1,P2) = 55 – 2p2 + P1 ................................... (2)

Since, total revenue revue (TR) is the multiplication of price, p, and quantity, q, the TRs for q1 (TR1) and for q2 (TR2) are obtained by multiplying equation (1) by p1 and equation (2) by p2 as follows:

TR1 = p1*q1 = p1(110 – p1 + 2p2)

TR1 = p110 – p1^2 + 2p1p2 .................................... (3)

TR2 = p2*q2 = p2(55 – 2p2 + p1)

TR2 = p2(255) – 2p2^2 + p1p2 ................................... (4)

Marginal revenue for q1 (MR1) and for q2 (MR2) are obtained by partially differentiating equation (3) with respect to p1 and equation (2) with respect to p2 and then solve for p1 and p2 as follows:

MR1 = dTR1/dp1 = 110 – 2p1 + 2p2 .................................... (5)

MR2 = dTR2/dp2 = 55 – 4p2 + p1 ................................... (6)

In monopolistic competitive market with differentiated goods, equilibrium occurs where MR = MC. Since,

MC1 = 10 ..................................................................................... (7)

MC2 = 5 ...................................................................................... (8)

We will therefore equate equations (5) with equation (7) and also equate equation (6) with equation (8), and then solve for p1 and p2 as follows:

For MR1 = MC1:

110 – 2p1 + 2p2 = 10

2p1 = 110 - 10 + 2p2

p1 = (100 + 2p2)/2

p1 = 50 + 2p2 ....................................................................... (9)

For MR2 = MC2:

255 – 4p2 + p1 = 5

4p2 = 55 - 5 + p1

p2 = (50 + p1)/4

p2 = 12.5 + p1/4 ................................................................ (10)

Now, substitute equation (10) for p2 in equation (9) and solve for p1 as follows:

p1 = 50 + 2(12.5 + p1/4)

p1 = 50 + 25 + 0.5p1

p1 - 0.5p1 = 75

p1 = 75/0.5

p1 = 150 .......................................................................... (11)

substitute equation (11) into equation (10) for p1 and solve for p2 as follows, we have:

p2 = 62.5 + 150/4

p2 = 62.5 + 37.5

p2 = 100 ............................................................... (12)

The p1 and p2 in equations (11) and (12) are the equilibrium prices for q1 and q2 respectively.

To get equilibrium quantity, substitute p1 = 150 and p2 = 100 into equations (1) and (2) as follows:

q1 = 110 – 150 + 2(100)

q1 = – 50 + 200

q1 = 150 .................................... (13)

q2 = 55 – 2(100) + 150

q2 = 55 + 150 - 200

q2 = 5 ....................................... (5)

Therefore, equilibrium prices are p1 = 150 and p2 = 100, while equilibrium quantities are q1 = 150 and q2 = 5.