I need help with my microeconomics homework

Question

Three identical firms face the following market demand curve P=130-Q. Assume that each of the firms has a marginal cost MC=10.

a) Calculate each firms’ equilibrium quantity under Cournot game. Calculate market price and consumer surplus.

b) Now assume that these three firms collude and form a monopoly. Calculate the monopoly price and quantity. Then calculate the consumer surplus.

c) Now assume that one of the firm shuts down. There are two firms and one of them chooses its quantity before the other firm. Calculate quantity choice of the firms. Calculate market price and consumer surplus. d) Assume that there are two firms again, but they compete over prices. Calculate the equilibrium price and consumer surplus under the case.

Answer 1

Cournot Competition with Three Identical Firms

Given:

Market Demand: ( P = 130 - Q )

Marginal Cost for each firm: ( MC = 10 )

Objective:

1. Calculate each firm's equilibrium quantity under Cournot game.

2. Calculate market price.

3. Calculate consumer surplus.

Solution:

Step 1: Reaction Functions

Each firm's reaction function in a Cournot game can be given by: Ri = (P - MC)/3 = ((130 - Q) - 10)/3

After simplification:

Ri = (120 - Q)/3 Where Q = q1 + q2 + q3

Step 2: Equilibrium Quantities

At equilibrium, (q_1 = q_2 = q_3). Using the reaction functions:

q1 = (120 - (q2 + q3))/3

q2 = (120 - (q1 + q3))/3

q3 = (120 - (q1 + q2))/3

Adding all three gives:

q1 + q2 + q3 = (360 - (q1 + q2 + q3))/3

Solving gives q1 = q2 = q3 = 30

Step 3: Market Price

[ P = 130 - Q = 130 - 90 = 40 ]

Step 4: Consumer Surplus

Consumer Surplus is given by the area of the triangle formed by the demand curve and the market price:

[ CS = 0.5 * (130 - 40) * 90 = 0.5 * 90 * 90 = 4050 ]

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