Question

Consider an industry with three firms, with demand curves as given below. Q1 = 140 -12P1 + 3P2 + 3P3 Q2 = 130 – 10P2 + 3P1 + 2P3 Q3 = 135 -9P3 + 2P1 + 2P2.Each firm has a marginal cost of 4.1. Calculate the pre-merger prices.2. Assume that firms 1 and 2 merge, to form a new firm “M”. Assume also that the merger allows M to reduce marginal cost from 4 to 2. Firm 3’s marginal cost remains at 4. What are the new equilibrium prices?

Answer 1

1. Pre-merger prices.

Each firm will try to maximize their profit i.e. MR=MC for each firm will determine the equilibrium prices.For each firm MC=4

For firm 1,

TR=P₁ × Q₁= 140P₁-12(P₁​​​​​) ^{​​​​​2​​​​​​}+3P₂P₁​​​​​​+3P₃​​​​​P₁

MR=dTR/dP₁​​​​​​= 140-24P₁​​​​​​+3P₂​​​​​​+3P₃=4(MC)

P₁​​​​​​=(136+3P₂​​​​​​+3P₃​​​​​) /24 equation (1)

For firm 2,

TR=P₂ ×Q₂​​​​​​= 130P₂​​​​​​-10(P₂​​​​​) ² +3P₁​​​​​P₂+2P₃P₂

MR=MC

P₂​​​​​​=(126+3P₁​​​​​​+2P₃​​​​​) /20 equation (2)

For firm 3,

TR=P₃​​​​​​×Q₃= 135P₃​​​​​​- 9(P₃​​​​​) ²+ 2P₁P₃​​​​​​+2P₂P₃

MR=MC

P₃=(131+2P₂+2P₁​​​​​) /18 equation (3)

Solving for equation 1,2,&3,we get equilibrium prices as follows:

P₁​​​​​​=7.84

P₂​​​​​​=8.38

P₃​​​​​​= 9.08

2. When firms 1&2 merge, to form firm M then their total output, Q=Q₁​​​​​​+Q₂​​​​​​ and prices P₁​​​​​​=P₂​​​​​​=P

Therefore, for firm M,

Q=(140-12P+3P+3P₃​​​​​) +(130-10P+3P+2P₃​​​​​) = 270-16P+5P₃

MC=2

TR= P × Q = 270P-16P²+5P₃P

MR= 270 - 32P +5P_³​​​​​​= 2(MC)

P= (268 + 5P_{3^{​​​​​​}}​​​​​​) /32 equatiin (4)

For firm 3,

TR=P₃ × Q₃​​​​​​= 135P₃​​​​​​- 9(P₃​​​​​) ²​​​​​​+4PP₃

MR=MC

135-18P₃​​​​​​+4P = 4

P₃​​​​​​=(131+4P)/18 equation (5)

Solving for equations 4&5,we get, new equilibrium prices,

P= 9.85

P₃=9.46