Consider The Market For Steel Made Of Two Identical

Consider The Market For Steel Which Is Made Of Two Identical Firms U

Consider the market for steel which is made of two identical firms (U.S. Steel and Bethlehem) which have identical marginal costs of $5. The demand for steel is given by Q_D = 80 − 4P. The assignment involves analyzing this market under three different competitive scenarios: Cournot competition, collusive cartel, and Bertrand competition, to determine the equilibrium price and quantity in each case.

Paper For Above instruction

The market for steel, characterized by two identical firms—U.S. Steel and Bethlehem—presents an interesting case for examining different competitive behaviors. Both firms have identical marginal costs of $5, and the demand function is Q_D = 80 − 4P, where Q_D is the quantity demanded at price P. The analysis discusses three distinct competition models: Cournot, collusive cartel, and Bertrand, to determine their respective equilibrium outcomes.

1. Cournot Competition

In the Cournot model, each firm chooses its quantity to maximize its profit, assuming the other firm's quantity is fixed. The equilibrium occurs where both firms' quantities are best responses to each other. To find the Cournot equilibrium, we start by deriving the inverse demand function:

P = (80 − Q) / 4, where Q = q₁ + q₂.

The revenue for each firm is:

R_i = P × q_i = ((80 − (q_i + q_j)) / 4) × q_i.

The profit function for firm i is:

π_i = R_i − MC × q_i = ((80 − (q_i + q_j)) / 4) × q_i − 5q_i.

To find the best response functions, differentiate π_i with respect to q_i and set to zero:

∂π_i/∂q_i = (1/4)(80 − 2q_i − q_j) − 5 = 0.

Simplify and solve for q_i:

80 − 2q_i − q_j = 20.

2q_i + q_j = 60.

Since the firms are symmetric, the best response functions are identical, so at equilibrium q_i = q_j = q^*.

Substitute q_j = q: 2q + q = 60, leading to 3q = 60, so q^* = 20.

Thus, total quantity Q_total = 2 × 20 = 40, and the equilibrium price is:

P = (80 − 40)/4 = 10.

Each firm's equilibrium quantity: 20 units; equilibrium price: $10.

2. Collusive Cartel

When firms act as a collusive cartel, they coordinate to maximize joint profits, effectively acting as a monopoly with combined output. The combined profit maximization problem is:

Maximize Π = (P − MC) × Q_total

Subject to Q_total = 80 − 4P, or inversely, P = (80 − Q_total)/4.

Profit as a function of Q_total: Π = ( (80 − Q_total)/4 − 5) × Q_total.

Simplify the profit function:

Π(Q_total) = (20 − Q_total/4 − 5) × Q_total = (15 − Q_total/4) × Q_total.

Maximize Π by differentiating with respect to Q_total: dΠ/dQ_total = 15 − (Q_total/2) = 0.

Solve for Q_total: Q_total = 30.

The collusive price is:

P = (80 − 30)/4 = 12.5.

Each firm produces half of the total quantity: 15 units each, at a price of $12.50.

3. Bertrand Competition

In Bertrand competition, firms simultaneously choose prices. Assuming both firms have identical costs and compete aggressively, the equilibrium is characterized by price equal to marginal cost, provided that firms can meet customer demands at this price.

Because both firms have the same cost of $5, the Nash equilibrium in Bertrand competition occurs at:

P_Bertrand = MC = $5.

At this price, if either firm raises its price, the other captures the entire market. If both firms set prices at $5, they share the market equally, resulting in quantities:

Q_total = Q_D at P = $5:

Q_D = 80 − 4(5) = 80 − 20 = 60.

Thus, each firm's quantity is 30 units, and the price is $5, which equals their marginal cost and results in zero economic profit in the long run.

Conclusion

In summary, the equilibrium outcomes vary significantly across the three models. Under Cournot competition, each firm produces 20 units at a price of $10. Collusive behavior leads to a joint output of 30 units at a higher price of $12.50, maximizing joint profits as a cartel. Lastly, Bertrand competition drives prices down to marginal cost ($5), with each firm selling 30 units, resulting in zero economic profits but a highly competitive market price. These models underscore how strategic interactions among firms influence market outcomes and consumer prices in the steel industry.

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