Question

Two firms, Acme and Roadco, produce anvils, and compete with each other as Cournot oligopolists (i.e. they compete in quantities). The (inverse) demand for anvils is given by P(Q)=500-3Q. Both firms have constant marginal costs of MC=50 and no fixed costs. Hint: the partial derivative of (c-bX-bY)X with respect to X is c-2bX-bY.

1. What is the equilibrium consumer and producer surplus in the market? (5 points)

Answer 1

Answer 2

To find the equilibrium consumer and producer surplus in the market, we first need to determine the Cournot equilibrium quantity and price for the anvils.

Cournot Equilibrium: In a Cournot duopoly, firms simultaneously choose their quantities to maximize their profits, taking into account the output of their competitor. The Cournot equilibrium occurs when both firms' quantities are such that they have no incentive to change their output given their competitor's output.

Step 1: Determine the reaction functions of the firms. Let Q_A be the quantity produced by firm Acme, and Q_R be the quantity produced by firm Roadco.

The profit function for each firm is given by: π_A(Q_A, Q_R) = (P(Q_A+Q_R) - MC) * Q_A π_R(Q_A, Q_R) = (P(Q_A+Q_R) - MC) * Q_R

Plugging in the demand function P(Q) = 500 - 3Q and MC = 50, we get: π_A(Q_A, Q_R) = (500 - 3(Q_A + Q_R) - 50) * Q_A π_R(Q_A, Q_R) = (500 - 3(Q_A + Q_R) - 50) * Q_R

Step 2: Set the partial derivatives of each firm's profit function with respect to their own quantity to zero to find their reaction functions. For firm Acme (A): ∂π_A / ∂Q_A = 0 500 - 3(Q_A + Q_R) - 50 - 6Q_A = 0 450 - 6Q_A - 3Q_R = 0 450 - 6Q_A = 3Q_R Q_R = (450 - 6Q_A) / 3 Q_R = 150 - 2Q_A

For firm Roadco (R): ∂π_R / ∂Q_R = 0 500 - 3(Q_A + Q_R) - 50 - 6Q_R = 0 450 - 6Q_A - 3Q_R = 0 450 - 6Q_A = 3Q_R Q_R = (450 - 6Q_A) / 3 Q_R = 150 - 2Q_A

Step 3: Equate the two firms' reaction functions to find the Cournot equilibrium quantity (Q*) and then calculate the market price (P*) using the demand function.

Q_A = Q_R 150 - 2Q_A = Q_A 2Q_A = 150 Q_A = 75 Q_R = 150 - 2Q_A Q_R = 150 - 2(75) Q_R = 150 - 150 Q_R = 0

Cournot equilibrium quantity: Q* = Q_A + Q_R = 75 + 0 = 75

Now, we can calculate the equilibrium price (P*) using the demand function P(Q) = 500 - 3Q:

P* = 500 - 3Q* = 500 - 3(75) = 500 - 225 = 275

Equilibrium price: P* = 275

Step 4: Calculate consumer surplus and producer surplus at the Cournot equilibrium.

Consumer Surplus: Consumer surplus is the area between the demand curve and the price line up to the equilibrium quantity.

Consumer surplus = (1/2) * (P* - MC) * Q* = (1/2) * (275 - 50) * 75 = (1/2) * 225 * 75 = 16875

Producer Surplus: Producer surplus is the area between the price line and the marginal cost curve up to the equilibrium quantity.

Producer surplus = (1/2) * (P* - MC) * Q* = (1/2) * (275 - 50) * 75 = (1/2) * 225 * 75 = 16875

The equilibrium consumer surplus and producer surplus in the market are both 16875.