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 \(Q_{E^{\prime}} Q_{D^{\prime}}\) 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 \(Q_{E^{\prime}} Q_{D^{\prime}}\) 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 \(Q_{E^{\prime}}\) \(Q_{D^{\prime}}\) and \(P ?\) What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of \(Q_{E^{\prime}} Q_{D^{\prime}}\) and \(P ?\) What are each firm's profits?

Short answer

a. Q_E=Q_D=45, P=55, Profit=992.5, b. Q_E=Q_D=30, P=40, Profit=800, c. Q_E=36, Q_D=27, P=37, Profit_Everglow=704, Profit_Dimlit=569, d. Q_E=Q_D=30, P=40, Profit=800

Step-by-step solution

Perfect Competition

Since Everglow and Dimlit are behaving as short-run perfect competitors, they set marginal cost equal to marginal revenue to maximize profits: MC=MR. The marginal cost using the cost function for each firm, \(MC=10+Q_i\). The marginal revenue is given by the derivative of Price times quantity, \(MR=P+Q\cdot(-1)=100-Q\). Setting these equal gives \(Q = 45\) for each Q_E, Q_D. The equilibrium price is \(P = 55\) (As P=100-Q and Q = Q_E+Q_D). Each firm's profit will be \(\pi= PQ_i - C_i = 45*55 - (10*45 + 0.5*45^2)= 992.5 \)

Oligopoly - Cournot Competition

In Cournot competition, each firm decides its own output, considering the output of the competitor as given. Based on the reaction function of firm i to the other firm j \( Q_i = (100 - Q_j -10)/2)\), the equilibrium quantity for both firms can be found: equating \( Q_D = (100 - Q_E -10)/2\) and \( Q_E = (100 - Q_D -10)/2\) with each other. Solving the system of equations results in \( Q_E = Q_D = 30 \) and \(P = 40\). Each firm's profit will be \(\pi= PQ - C = 40*30 - (10*30 + 0.5*30^2)= 800 \)

Stackelberg Competition

In a Stackelberg competition, one firm (the leader, Everglow here) moves first and the other firm (the follower, Dimlit here) reacts. Everglow will anticipate Dimlit's reaction when choosing its quantity. Applying the reaction function of Dimlit in Everglow's profit function ends up with best response function of Everglow: \(Q_E= (100 - 10) / 2.5 = 36\). Dimlit then reacts to Everglow's quantity: \(Q_D = (100 - 36 -10) / 2 = 27\). The price will be \(P = 37\) and the profits will be \(\pi_E = 704\) and \(\pi_D = 569\)

Collusion

In case of collusion, companies maximize their combined profit. The combined quantity will be \(Q = (100 - 10)/1.5 = 60\) and hence each firm produces \(Q_E=Q_D = Q / 2 = 30\). The price will be \(P = 40\) and the profits will be \(\pi= 800 \) for each

Key concepts

Perfect Competition

Perfect competition is a market structure characterized by a large number of small firms, identical products, and easy entry and exit. In this case, firms are price takers, meaning that they cannot influence the market price. Each company maximizes its profit by setting its output where marginal cost (MC) equals marginal revenue (MR).

In the scenario provided, Everglow and Dimlit operate under such conditions. They determine their production quantities to balance the MC and MR, using their cost structure to calculate these values. The MC for each is derived from their cost function, which, in this example, is given by the formula:

• \( MC = 10 + Q_i \)

The MR, derived from the demand function, is:

• \( MR = 100 - Q \)

By equating MC and MR, the equilibrium point for each firm is found. This results in equal output levels, \( Q_E = Q_D = 45 \), and an equilibrium price of \( P = 55 \). Allowing these parameters gives equal profits for each firm as calculated.

Cournot Competition

Cournot competition depicts a scenario where firms compete on the quantity of output produced, believing their rival's quantity is fixed. This strategy occurs in an oligopoly situation, where each firm decides its strategy based on the actions of the others. Here, the firms aim to maximize profits by rationally anticipating the competitor's output.

Everglow and Dimlit, recognizing the oligopolistic setting, act according to Cournot principles. They use reaction functions to predict their competitor's behavior:

• \( Q_i = \frac{100 - Q_j - 10}{2} \)

By realizing these in tandem, the equilibrium quantities are resolved as \( Q_E = Q_D = 30 \) with a price of \( P = 40 \). Thus, each firm benefits from the conjectural variations, ending with the same profit calculated using this refined equilibrium strategy.

Stackelberg Competition

The Stackelberg model allows for a leader-follower dynamic within an oligopoly. One firm, the leader, sets its quantity before others in the market, whereas the follower firms respond accordingly. This sequential move allows the leader to achieve a strategic advantage.

In this instance, Everglow takes the position of the leader by predicting Dimlit's response and setting its output strategically. Everglow uses Dimlit's reaction function within its calculations to optimize its own profit:

• Everglow's best response: \( Q_E = \frac{100 - 10}{2.5} = 36 \)
• Dimlit's reaction: \( Q_D = \frac{100 - 36 - 10}{2} = 27 \)

The resulting market price is \( P = 37 \), and through this planned leadership position, Everglow can enhance its profits compared to equal competition strategies.

Collusion

Collusion occurs when firms in an oligopoly cooperate, rather than competing against each other, to maximize their joint profits. This usually leads to outcomes similar to monopoly behavior, where the total industry output is reduced to increase prices and therefore profits.

In the provided example, if Everglow and Dimlit collude, they will jointly determine the quantity to maximize combined profits. This can result in each firm producing the same amount, effectively reducing competition:

• \( Q = \frac{100 - 10}{1.5} = 60 \)
• Each firm produces: \( Q_E = Q_D = \frac{Q}{2} = 30 \)

The price remains at \( P = 40 \), and each firm's profit remains high due to the decreased competition and aligned strategy focused on mutual profitability rather than market share dominance.

Equilibrium

Equilibrium in economics represents a state where market supply and demand balance each other, and as a result, prices become stable. In competitive market structures, Nash Equilibrium plays a crucial role, representing a situation where no participant can gain by unilaterally changing their strategy.

In this exercise, equilibrium values were calculated under several market interactions. For perfect competition, firms reach equilibrium by matching MC with MR. In Cournot competition, firms anticipate competitors' strategies to reach their equilibrium outputs. Meanwhile, under Stackelberg competition, one firm assumes leadership, setting the pace for equilibrium. Through collusion, both firms reach equilibrium by mutual agreement. In all scenarios, achieving equilibrium is key for optimizing each firm’s outcomes regardless of the competitive model employed.

Profit Maximization

Profit maximization is the chief objective for firms, achieved by setting production levels where marginal revenue equals marginal cost. This ensures that every single unit produced adds positively to the overall profit, with no over-production or costs outweighing returns.

Everglow and Dimlit’s differing strategies directly reflect their profit maximization goals across diverse competitive scenarios. Whether in perfect or Cournot competition or adopting Stackelberg leadership or colluding, each strategy hinges on adjusting output to perfect this economic fundamental. Using the provided cost functions and market parameters, they navigate through to define these conditions and ascertain the most advantageous production quantities that align with their profit targets.