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Reaction functions are crucial in understanding Cournot competition. They describe how one firm's output decision depends on the output level of the competitor. In a Cournot duopoly, each firm is trying to maximize its own profit given the quantity produced by the other firm. To derive the reaction function for each firm, we take the derivative of the profit function concerning its own output and set it equal to zero. This allows the firm to find its optimal response to the competitor's output level.

For example, in our exercise with firms facing a market demand of \( P = 200 - q_1 - q_2 \), Firm 1's reaction function is computed as \( q_1 = 90 - \frac{q_2}{2} \). This means Firm 1 will adjust its output to be less as Firm 2 increases its output. Similarly, Firm 2's reaction function is \( q_2 = 90 - \frac{q_1}{2} \).

Understanding these reaction functions helps visualize how output levels interact in a competitive setting, forming a foundational part of reaching market equilibrium.

Market equilibrium in Cournot competition occurs when both firms have no incentive to change their output, given the other firm's choice. It is a point where the reaction functions intersect, providing each firm's optimal quantity produced based on the competitor's actions.

In our example, substituting Firm 2’s reaction function into Firm 1’s, or vice versa, leads us to solve for both \( q_1 \) and \( q_2 \) simultaneously. We find that in equilibrium, \( q_1 = q_2 = 60 \). At this output, the price in the market, determined by substituting these quantities into the demand equation, is \( P = 80 \).

The benefit of reaching market equilibrium is that both firms can predictably optimize their output, knowing the results of maintaining this balance in competitiveness.

Marginal cost (MC) is the cost of producing one additional unit of output. It plays a significant role in profit maximization strategies of firms in any market, including Cournot competition. Constant marginal cost simplifies the analysis, as it means the cost per additional unit remains the same no matter the production level.

For both firms in this scenario, the constant marginal and average total costs are $20 per unit. This number is critical in calculating the profit for both firms as they decide their optimal production levels. When Firm 1 considers investing in technology to reduce its marginal cost to $15, the impact on profitability becomes significant: it allows for a new profit-maximizing quantity and price, altering the competition landscape.

In Cournot competition, managing and minimizing marginal costs can lead to a competitive advantage, allowing a firm to either increase profits or lower prices to gain market share.

Profit maximization is the ultimate goal for any firm, including those in Cournot competition. It involves setting a firm’s production level such that it cannot increase profit by changing its output given the output of competitors. In our example, the profit functions for the firms help them determine where equilibrium is achieved concerning their competitors.

For instance, with the initial marginal cost of \(20, both firms aim to maximize profit by producing where \( \pi_1 = (80 - 20) \times 60 = 3600 \) for Firm 1. If Firm 1 chooses to further reduce its marginal cost to \)15 through investment in technology, it needs to evaluate how much it should invest based on the expected change in profit. The optimal profit increases beyond initial expectations, to $4225 when they solely access new technology.

Effective profit maximization in a Cournot setting involves strategic thinking about cost management and understanding competitive responses, ideally leading to greater profitability under the prevailing market conditions.