Let Demand For Car Batteries Be Such That Q=10-2p

Let Demand For Car Batteries Be Such That Q 10 2p Assumeconsta

Let demand for car batteries be such that Q = 10 – 2P. Assume constant marginal costs of 3. Compute the equilibrium price, quantity, consumer surplus, and producer surplus for (a) a perfectly competitive firm; (b) a monopoly; (c) two firms engaged in Cournot competition; (d) an oligopoly where one firm has a marginal cost of 3 and the other has a different cost; (e) the Pareto frontier of cartellike agreements with a discount factor of 0.99.

Additionally, assume a linear market demand curve and a concave average cost curve. Analyze how an incumbent can prevent market entry by threatening to produce a large quantity, why this may be irrational if facing an entrant, and how purchasing additional capacity—though never used—can make such behavior rational. Define what is meant by additional capacity in practical terms.

Paper For Above instruction

The analysis of market structures and strategic behaviors in oligopoly and perfectly competitive markets reveals complex interactions influencing prices, output, and welfare. Beginning with the basic demand function, Q = 10 – 2P, and a constant marginal cost of 3, we examine different market scenarios to understand equilibrium outcomes, consumer and producer surpluses, and strategic considerations related to entry deterrence and capacity investments.

Equilibrium in Perfect Competition

In perfect competition, firms are price takers, and equilibrium occurs where market supply equals demand. The supply curve is horizontal at the marginal cost of 3, assuming many firms with identical costs supply the market. Setting marginal cost equal to demand price:

Q = 10 – 2P

Rearranged to find P:

P = (10 – Q) / 2

At equilibrium, P = MC = 3, so:

3 = (10 – Q) / 2

⇒ 6 = 10 – Q

⇒ Q = 4

Then, the equilibrium price is P = (10 – 4)/2 = 3, matching the marginal cost. Consumer surplus is the area of the triangle between demand and price level up to the equilibrium quantity, and producer surplus is the rectangle at marginal cost.

Consumer surplus: ½ (maximum willingness to pay – price) quantity = ½ [(P at Q=0) – 3] 4. When Q=0, P=5 (from Q=10 – 2P ⇒ P= (10 – Q)/2 ⇒ P=5 when Q=0).

Thus, consumer surplus = ½ (5 – 3) 4 = ½ 2 4 = 4.

Producer surplus: since price equals marginal cost, producer surplus is zero.

Monopoly Equilibrium

The monopolist maximizes profit where marginal revenue (MR) equals marginal cost. Derive MR from the demand function:

TR = P Q = ( (10 – Q)/2 ) Q = (10Q – Q^2)/2

MR = d(TR)/dQ = (10 – 2Q)/2 = 5 – Q

Set MR = MC (which is 3):

5 – Q = 3 ⇒ Q = 2

The monopoly price is:

P = (10 – Q)/2 = (10 – 2)/2 = 4

Consumer surplus is calculated as the area between demand and price up to Q = 2:

Consumer surplus = ½ (P max – P monopoly) Q = ½ (5 – 4) 2 = 1

Producer surplus or monopoly profit is:

π = (P – MC) Q = (4 – 3) 2 = 2

Cournot Duopoly Equilibrium

For two firms, each chooses quantity to maximize profit, taking the other’s quantity as given. Let q₁ and q₂ denote quantities of firms 1 and 2.

Total quantity: Q = q₁ + q₂

Price: P = (10 – Q)/2

Each firm's profit function:

π_i = (P – MC) q_i = [(10 – q₁ – q₂)/2 – 3] q_i

Optimizing with respect to q_i, set the derivative to zero:

∂π_i/∂q_i = [(–1/2) q_i – (1/2) q_j + (10/2) – 3] – (1/2) * q_i = 0

Simplified best response functions lead to symmetry: q₁ = q₂ = q*

Solving yields q = 2, total quantity Q = 4, and price P* = (10 – 4)/2 = 3, matching the competitive outcome. Both firms earn zero economic profit, indicating a Cournot equilibrium coinciding with perfect competition in this case.

Oligopoly with Asymmetric Costs

When one firm has marginal costs of 3 and the other has a different cost, say c₂, the equilibrium shifts. The firm with higher costs produces less, and the equilibrium must be determined via best response functions considering the different cost structures. Without specific values, the outcome results from strategic decision-making accounting for each firm’s cost and response functions, leading to asymmetric equilibria.

Typically, the firm with the lower marginal cost has a competitive advantage, producing more and setting prices accordingly, while the higher-cost firm produces less or exits the market under certain conditions.

Cartel Formation under Repeated Interaction

With a discount factor of 0.99, firms valuing future profits highly, collusion becomes sustainable if the gains from cheating are outweighed by the discounted losses in future cooperation. The Pareto frontier of agreements includes colluding prices and quantities that maximize joint profits but are individually stable because deviations are deterred by the threat of punishment or renegotiation. Under repeated game setups, firms can sustain cartel agreements that maximize joint welfare, provided the discount factor exceeds the critical value necessary to prevent profitable deviations (Fudenberg & Tirole, 1990).

Entry Deterrence Through Quantity Competition

In markets with linear demand and concave average costs, incumbents can threaten to produce a large quantity to discourage entrants, as the potential reduction in market profitability deters new firms. This strategic threat acts as a barrier, exploiting the entrant’s expectation of market suppression (Tirole, 1988). However, this behavior may be irrational if the incumbent’s marginal cost exceeds that of the entrant or if the threat leads to unnecessary overproduction, reducing overall profits.

Purchasing additional capacity—regardless of use—can make this threat credible, as the incumbent commits to rapid, large-scale output if challenged. Additional capacity is practically interpreted as infrastructure, machinery, or infrastructure investments that expand an incumbent's capacity to flood the market if necessary.

This strategic capacity investment can be rationalized under models of strategic entry deterrence, where the cost of capacity is offset by the strategic advantage gained by deterring new entrants (Besanko et al., 2013). Such investments act as a commitment device, enhancing credibility in threats, even if the capacity remains unused in equilibrium.

Conclusion

The market analysis demonstrates the interconnectedness of strategic behaviors, market structure, cost structures, and future expectations. Firms' ability to influence prices and outputs depends on their strategic interactions, cost advantages, and capacity investments. Understanding these dynamics provides insights into anti-competitive practices, optimal market regulation, and the strategic planning necessary to maintain market dominance or encourage competition.

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