Question

Demand for light bulbs can be characterized by \(Q=100-P,\) where \(Q\) is in millions of boxes of lights sold and \(P\) is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: \\[ \begin{array}{c} C_{i}=10 Q_{i}+\frac{1}{2} Q_{i}^{2}(i=E, D) \\ Q=Q_{E}+Q_{D} \end{array} \\] a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of \(Q_{E}, Q_{D},\) and \(P ?\) What are each firm's profits? b. Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of \(Q_{E}\) \(Q_{D},\) and \(P ?\) What are each firm's profits? c. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of \(Q_{E}\) \(Q_{D},\) and \(P ?\) What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of \(Q_{E}, Q_{D},\) and \(P ?\) What are each firm's profits?

Short answer

Under perfect competition, the equilibrium values are \(Q_{E}=Q_{D}=30, P=40\) and each firm's profits are 800. Under Cournot competition, the equilibrium values are \(Q_{E}=Q_{D}=20, P=60\) and each firm's profits are 600. Under Stackelberg, with Everglow as the leader, the equilibrium values are \(Q_{E}=25, Q_{D}=12.5, P=62.5\), with profits of respectively 756.25 and 281.25. Under collusion, the equilibrium values are \(Q_{E}=Q_{D}=20, P=60\) and each firm's profits are 600.

Step-by-step solution

Perfect Competition

Under perfect competition, each firm takes the market price as given and equates marginal cost to price. The marginal cost for each firm \(i\) is \(MC_{i}=10+Q_{i}\). We equate this to the price derived from the market demand: \(10+Q_{i}=100-Q\), where \(Q=Q_{E}+Q_{D}\). Solving these equations, we get \(Q_{E}=Q_{D}=30\) and \(P=40\). Profits for each firm are \(\pi = P*Q_{i} - C_{i} = 800\).

Cournot competition

Under Cournot, each firm assumes the output of the competitor is fixed and chooses its own output to maximize its profit. We first write down the profit function for each firm: \(\pi_{i} = (100 - Q)*Q_{i} - (10*Q_{i} + 0.5*Q_{i}^{2})\). Taking first order conditions and solving the subsequent system of equations yields \(Q_{E}=Q_{D}=20\) and \(P=60\). Profits for each firm are \(\pi = 600\).

Stackelberg competition

Under Stackelberg, one firm (Everglow) becomes the leader and assumes Dimlit will adjust its output according to the output of Everglow. We first determine the reaction function of Dimlit (firm 2) from the Cournot setup, and then maximize Everglow's profit with respect to this reaction function. Solving the appropriate equations yields \(Q_{E}=25\), \(Q_{D}=12.5\), and \(P=62.5\). Profits for each firm are \(\pi_{E} = 756.25\) and \(\pi_{D} = 281.25\).

Collusion

Under collusion, the two firms act as a single monopoly maximizing its joint profit. The total marginal cost is \(MC = MC_{E} + MC_{D} = 20 + Q\), equating this to the price we find \(Q=40\) and \(P=60\). Given that the firms share the total output, we have \(Q_{E}=Q_{D}=20\). Profits for each firm are \(\pi = 600\).

Key concepts

Cournot competition

Imagine two firms, like Everglow and Dimlit, who are trying to sell light bulbs. Each firm believes that the other firm will keep its production steady. This scenario is known as Cournot competition, where companies choose how much to produce based on what they think other companies will do. They don't talk to each other, but each tries to pick the best output level to maximize its own profit. It's like a game: each firm tries to guess what the other will do and act accordingly.

In Cournot competition, each firm responds to demand by setting production so that their costs are balanced by the expected price. They calculate potential profits and adjust accordingly, but both firms end up with similar decisions because they operate under the same assumptions. For instance, with Everglow and Dimlit, both firms will decide to produce 20 million boxes of light bulbs each, resulting in a market price of $60 per box. At these levels, they each make a profit of $600 million, which might seem profitable, but still less than if they collaborated.

Overall, Cournot competition reflects a balanced yet competitive marketplace where firms do not get into price wars, but also do not capitalize fully on potential cooperative opportunities.

Stackelberg competition

In Stackelberg competition, one firm is the market leader and the other follows. Think of it as a leader and follower scenario. This is what happens when Everglow thinks it can anticipate Dimlit's actions and decides to take the role of leader.

Everglow, as the leader, picks a production level first, and Dimlit responds with its output based on what Everglow does. Here, Everglow would analyze Dimlit's likely reaction and choose a level of production that would maximize its own profits while considering Dimlit's response. This gives Everglow a strategic advantage by setting higher expectations for its profits.

As a result, Everglow ends up producing 25 million boxes while Dimlit scales back to 12.5 million boxes. The price of light bulbs rises to $62.50 per box, benefitting both firms in terms of pricing. Everglow could achieve higher profits of $756.25 million, while Dimlit makes a smaller profit of $281.25 million.

Stackelberg competition shows us how crucial market leadership can be in shaping the outcomes of competition within oligopolies.

Collusion

In an ideal world for companies, they might think about colluding to act as a single enterprise. Collusion is when firms agree to coordinate their actions, typically production levels, to maximize their profits, behaving like a monopoly.

By cooperating rather than competing, Everglow and Dimlit can both agree to produce less and keep prices high. In this case, both firms produce 20 million boxes each. This reduces the total output to benefit from a higher price point of $60 per box. Under this strategy, both achieve the same profit of $600 million, mirroring the benefits they obtain during the Cournot-shaped equilibrium but with a structured agreement in place valued through strategic collaboration.

However, collusion often leads to legal and ethical issues, and is generally avoided through competition laws. Real markets are regulated to maintain fair practices and protect consumers from artificially high prices resulting from such agreements.

Perfect competition

Under perfect competition, firms operate where they cannot influence prices and must accept market conditions. Each firm takes the market price for granted and only focuses on minimizing costs to increase efficiency.

With Everglow and Dimlit acting as perfect competitors, each firm aims to produce at a level where their marginal cost, which is the cost of producing one more unit, equals the market price. Here, both Everglow and Dimlit believe they can't change prices by their own actions. Hence, they produce as much as they can while still making a profit.

Following this mode of operation, both companies end up producing 30 million boxes, working with a market price of $40 per box. This strategy ensures lower profits, amounting to $800 million for each firm, but upholds fairness in market dynamics since neither can exert unilateral market control.

• Firms accept given market prices.
• Established equilibrium dictates production decisions.
• Competition ensures efficiency and limits monopoly-like profit ranges.