There are only a few firms that produce At-Home Back Waxing Kits. Demand for these kits
is given by P = 50-2Q. There are 3 firms in the market, and they compete according to
Bertrand Competition. Find the Market Price, quantity sold by each firm, quantity traded in
the market, consumer surplus, and producer surplus for each firm under the following cost
scenarios:
a) MC1 = $10, MC2 = $10, MC3 = $15
To find the equilibrium price and quantity sold by each firm under Bertrand competition, we first need to calculate the Marginal Cost (MC) for each firm, and then set it equal to the market price to find the equilibrium quantity produced by each firm.
Given:
Demand: P = 50 - 2Q
Firm 1: MC1 = $10
Firm 2: MC2 = $10
Firm 3: MC3 = $15
Setting MC1 = P:
$10 = 50 - 2Q1
2Q1 = 40
Q1 = 20
Setting MC2 = P:
$10 = 50 - 2Q2
2Q2 = 40
Q2 = 20
Setting MC3 = P:
$15 = 50 - 2Q3
2Q3 = 35
Q3 = 17.5
So, the equilibrium quantity sold by each firm is:
Q1 = 20
Q2 = 20
Q3 = 17.5
Since Q1 = Q2, the market price will be $10.
Market quantity traded = Q1 + Q2 + Q3
= 20 + 20 + 17.5
= 57.5
To find consumer surplus and producer surplus for each firm under this price:
Consumer Surplus = 1/2 * (Pmax - P) * Q
Producer Surplus = (P - MC) * Q
For each firm:
Consumer Surplus = 1/2 * (50 - 10) * Q
= 20 * Q
= 20 * 20 (for Firm 1 and Firm 2)
= $400
Consumer Surplus = 1/2 * (50 - 10) * Q
= 20 * Q
= 20 * 17.5 (for Firm 3)
= $350
Producer Surplus = (P - MC) * Q
= ($10 - $10) * 20 (for Firm 1)
= $0
Producer Surplus = (P - MC) * Q
= ($10 - $10) * 20 (for Firm 2)
= $0
Producer Surplus = (P - MC) * Q
= ($10 - $15) * 17.5 (for Firm 3)
= -$87.5
In this scenario, both Firm 1 and Firm 2 have consumer surplus of $400 and producer surplus of $0, while Firm 3 has a consumer surplus of $350 and a negative producer surplus of -$87.5.