Notation: Q = Aggregate Quantity and q = individual firm quantity Suppose we hav
ID: 1189707 • Letter: N
Question
Notation: Q = Aggregate Quantity and q = individual firm quantity Suppose we have two firms in the market selling differentiated product. Firm 1 faces the following demand function: q_1 = 12 - 2p_1 + p_2, while Firm 2 faces the following demand function: q_2 = 12- 2p_2 + p_1. The cost to produce products for both firms are the same and are given by C_i = 10q)i, where i = 1,2. Calculate the profit maximizing price and quantity for both firms. Now, suppose the two firms collude so that P_1 = p_2 = p. Calculate the profit maximizing price and quantity for both firms.Explanation / Answer
(1)
Marginal cost of both firms = dC / dq = 10
q1 = 12 - 2p1 + p2
q2 = 12 - 2p2 + p1
Revenue of firm 1, TR1 = q1 x p1 = 12p1 - 2p12 + p1p2
Revenue of firm 2, TR2 = q2 x p2 = 12p2 – 2p22 + p1p2
Profit of firm 1, Z1 = TR1 – TC
= 12p1 - 2p12 + p1p2 – 10q1
= 12p1 - 2p12 + p1p2 – 10 (12 - 2p1 + p2)
= 12p1 - 2p12 + p1p2 – 120 + 20p1 – 10p2
= 32p1 - 2p12 + p1p2 – 10p2 – 120
Profit is maximized when dZ1 / dp1 = 0
32 – 4p1 + p2 = 0
4p1 – p2 = 32 ..... (1) [Firm 1’s response function]
Profit of firm 2, Z2 = TR2 – TC
= 12p2 – 2p22 + p1p2 – 10q2
= 12p2 – 2p22 + p1p2 – 10 (12 - 2p2 + p1)
= 12p2 – 2p22 + p1p2 – 120 + 20p2 – 10p1
= 32p2 - 2p22 + p1p2 – 10p1 – 120
Profit is maximized when dZ2 / dp2 = 0
32 – 4p2 + p1 = 0
4p2 – p1 = 32 ..... (2) [Firm 2’s response function]
We need to solve (1) and (2).
4p1 – p2 = 32 ..... (1)
(2) x 4 gives us: 16p2 – 4p1 = 128 (3)
(1) + (3) gives,
15p2 = 160
p2 = 10.67
p1 = 4p2 – 32 [From (2)]
= (4 x 10.67) – 32 = 10.67
q1 = 12 - 2p1 + p2 = 12 – 21.34 + 10.67 = 1.33
q2 = 12 - 2p2 + p1 = 12 – 21.34 + 10.67 = 1.33
(2) p1 = p2 = p
q1 = 12 - 2p + p = 12 – p
p = 12 – q1
q2 = 12 – p
p = 12 – q2
TR1 = 12q1 – q12 & TR2 = 12q2 – q22
MR1 = 12 – 2q1 & MR2 = 12- 2q2
Equating MR1 = MC,
12 – 2q1 = 10
2q1 = 2
q1 = 1
equating MR2 = MC,
12 – 2q2 = 10
2q2 = 2
q2 = 1
p = 12 – q1 (Or 12 – q2) = 12 – 1 = 11
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