Question

Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by \(C_{1}=60 Q_{1}\) and \(C_{2}=60 Q_{2}\), where \(Q_{1}\) is the output of Firm 1 and \(Q_{2}\) the output of Firm 2. Price is determined by the following demand curve: $$P=300-Q$$ where \(Q=Q_{1}+Q_{2}\) a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium. b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm's profit. c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm \(1^{\prime}\) s profit differ from that found in part (b) above? d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm's profits?

Short answer

a) Cournot-Nash equilibrium is Q1=Q2=100, price is 100 and each firm's profit is 4000. b) In the cartel, each firm produces 120 widgets and makes no profit. c) As a monopoly, Firm 1 produces 240 widgets and makes no profit. d) If Firm 2 cheats by increasing production, its profits will increase while Firm 1's profits will decrease.

Step-by-step solution

Cournot-Nash Equilibrium

In the Cournot-Nash Equilibrium, each firm maximizes its profit given the other's output. Since both firms are identical, they have symmetrical reactions. So, Firm 1's and Firm 2's output reaction functions are \(Q_{1}=Q_{2}=(300-Q)/2\). Solving these equations simultaneously gives \(Q_{1}=Q_{2}=100\). The price at this equilibrium is \(P=300-Q=300-200=100\). The profit of each firm is \(\Pi=(P*C)-Cost=(100*100)-(60*100)=4000\).

Cartel Profits

When the two firms form a cartel to maximize joint profits, they act as a monopoly and produce where marginal cost equals marginal revenue. The total quantity produced by the cartel is \(Q=300-P\). Solving for Q gives \(Q=240\) and, because the firms are identical, they each produce half: \(Q_{1}=Q_{2}=120\). The price at this output level is \(P=300-Q=60\). The profit of each firm is \(\Pi=(P*C)-Cost=(60*120)-(60*120)=0\). Here, profits are zero because the firms are essentially producing at cost in a perfectly competitive market.

Monopoly Profits

If Firm 1 were the only firm in the industry, it would act as a monopoly and produce where marginal cost equals marginal revenue. Solving for Q gives \(Q=240\). The price at this output level is \(P=300-Q=60\). The profit of the firm is \(\Pi=(P*C)-Cost=(60*240)-(60*240)=0\). Again, profits are zero because the firm is essentially producing at cost in a perfectly competitive market.

Cheating within the Cartel

If Firm 1 abides by the agreement but Firm 2 increases production, the new joint quantity is higher and consequently, the price is lower. While the exact quantity Firm 2 will produce depends on a number of factors including the demand elasticity and the cost structure, it is clear that Firm 2's profit will increase while Firm 1's profit will decrease.

Key concepts

Duopoly

In a duopoly, two companies dominate the market. This type of market structure is significant because it allows firms to influence market prices and outputs with their decisions. In the exercise, both firms are identical and produce widgets. A typical model used to study duopolies is the Cournot-Nash model. Here, each company selects its output level based on what it expects its rival will produce.

One hallmark of a duopoly is interdependence. Each firm's decisions depend on the actions of the other firm. If one firm increases its output, the total supply of widgets in the market increases, which can lead to a lower market price. Similarly, if one firm decreases its production, the market price can go up. The exercise shows this interdependence when both firms calculate their best response functions. This process leads to a Cournot-Nash equilibrium where neither firm can benefit by changing its output unilaterally.

Cartel

A cartel forms when competing firms in an industry decide to work together to control prices and outputs. This collective action is designed to maximize joint profits, as firms effectively act as a monopoly. However, the formation of cartels is often illegal or heavily regulated since they tend to reduce competition which can harm consumers through higher prices.

In the provided exercise, the two firms consider forming a cartel to act like a single monopolistic entity. By producing only the quantity where total marginal revenue equals total marginal cost, they increase their joint profits. However, in a real-world scenario, cartels are often unstable. There is a temptation for individual firms to "cheat" by producing more than the agreed amount to capture higher profits, as seen later in the exercise.

Monopolistic Competition

Monopolistic competition is characterized by many firms that sell products which are similar but not identical. In such a market, each firm has a certain degree of market power, allowing them to set prices above marginal cost. Unlike in pure monopoly, no individual firm can control the entire market. This market structure leads to product differentiation and competitive pricing strategies.

Although not directly a focus in the exercise, the concept of monopolistic competition helps to distinguish it from the duopoly scenario. When firms consider abandoning their coordinated actions in a cartel, the market could shift towards a more competitive structure with unique products or services being emphasized, thus moving closer to monopolistic competition.

Profit Maximization

Profit maximization is a key goal for firms regardless of market structure. It involves determining the output level that generates the highest possible profit by balancing revenue and costs. In the context of the exercise, both firms in the duopoly and the potential cartel aim to maximize their profits through strategic output decisions.

For the duopoly, each firm independently chooses its output to maximize profit, taking into account the production level of the other firm. This results in the Cournot-Nash equilibrium. When forming a cartel, the combined firm aims to increase joint profits by acting as a monopoly, producing where total marginal revenue equals total marginal cost. Understanding these strategies and their implications provides insight into how firms operate to achieve their economic objectives.