Question

Consider the following market game: An incumbent firm, called firm 3 , is already in an industry. Two potential entrants, called firms 1 and 2, can each enter the industry by paying the entry cost of 10 . First, firm 1 decides whether to enter or not. Then, after observing firm 1's choice, firm 2 decides whether to enter or not. Every firm, including firm 3, observes the choices of firms 1 and 2. After this, all of the firms in the industry (including firm 3) compete in a Cournot oligopoly, where they simultaneously and independently select quantities. The price is determined by the inverse demand curve \(p=12-Q\), where \(Q\) is the total quantity produced in the industry. Assume that the firms produce at no cost in this Cournot game. Thus, if firm \(i\) is in the industry and produces \(q_{i}\), then it earns a gross profit of \((12-Q) q_{i}\) in the Cournot phase. (Remember that firms 1 and 2 have to pay the fixed cost 10 to enter.) (a) Compute the subgame perfect equilibrium of this market game. Do so by first finding the equilibrium quantities and profits in the Cournot subgames. Show your answer by designating optimal actions on the tree and writing the complete subgame perfect equilibrium strategy profile. [Hint: In an \(n\)-firm Cournot oligopoly with demand \(p=12-Q\) and 0 costs, the Nash equilibrium entails each firm producing the quantity \(q=12 /(n+1) .]\) (b) In the subgame perfect equilibrium, which firms (if any) enter the industry?

Short answer

Firm 1 does not enter, Firm 2 enters, and Firm 3 competes with Firm 2 in equilibrium.

Step-by-step solution

Understanding the Cournot Equilibrium

In the Cournot competition phase, we are given that the price is determined by the inverse demand curve \( p = 12 - Q \), where \( Q \) is the total quantity produced. In an \( n \)-firm Cournot oligopoly, each firm produces \( q = \frac{12}{n+1} \) and earns a profit \( \pi_i = q_i \times (12 - Q) \). The firms produce at no cost, but firms 1 and 2 incur a fixed entry cost of 10 if they choose to enter.

Subgame Analysis: Cournot Equilibrium Given Different Entrants

Firms can enter the market and face different environments based on how many firms are in the market:- **Three Firms (Firm 1, 2, and 3):** Profits for each firm \( \pi = \left(\frac{12}{4}\right)^2 = 9 \). Firms 1 and 2 would incur a net profit of -1 (since they must pay an entry cost of 10).- **Two Firms (Firm 1 or 2 and Firm 3):** Profits for each firm \( \pi = \left(\frac{12}{3}\right)^2 = 16 \). Entering firm has net profit 6; Firm 3 has profit 16.- **One Firm (Only Firm 3):** Firm 3 earns \( \pi = \left(\frac{12}{2}\right)^2 = 36 \).

Deciding Optimal Entry Actions for Firms 1 and 2

- **Firm 1's Decision:** If Firm 1 enters, Firm 2 observes and knows it would also enter, leading both Firms 1 and 2 to have -1 net profit. Hence, Firm 1 gains more if it does not enter (0 profit) rather than entering. - **Firm 2's Decision:** If Firm 1 does not enter, Firm 2 compares entering and earning net 6 or not entering for 0. It chooses to enter.

Determining the Subgame Perfect Equilibrium Strategy

Considering the best outcomes for each firm's decisions: - **Firm 1:** Chooses `Not Enter` to avoid negative profit. - **Firm 2:** Chooses `Enter` when Firm 1 does not enter since it earns net profit 6. - Firm 3 remains and reacts accordingly in both scenarios. In subgame perfect equilibrium, Firm 1 does not enter, Firm 2 enters, and Firm 3 competes in a Cournot duopoly with Firm 2.

Key concepts

Cournot Oligopoly

In the Cournot oligopoly model, firms compete by setting quantities rather than prices. This model is used to describe industries where firms decide how much of a product to produce independently and at the same time. Each firm's decision affects the market price, which in turn affects the profits of all firms.

In the context of the given exercise, when the firms enter the market, they engage in Cournot competition. The market price is determined by the inverse demand curve, expressed as \( p = 12 - Q \), where \( Q \) is the total output produced by all firms. Here, each firm produces at a quantity that maximizes its profit, taking into consideration the production quantities of the other firms.

Specifically, the Nash equilibrium for an \( n \)-firm Cournot oligopoly with zero costs and a demand of \( p = 12 - Q \) results in each firm producing:

• \( q = \frac{12}{n+1} \)

Understanding the Cournot equilibrium is crucial because it illustrates how firms' production choices will ultimately determine the distribution of profits in the market.

Subgame Perfect Equilibrium

Subgame perfect equilibrium (SPE) is a refinement of Nash equilibrium, applicable in dynamic games where players make decisions at various stages. SPE requires that strategies constitute a Nash equilibrium in every subgame, ensuring credibility and consistency of the strategy across all stages of the game.

In the exercise, the decision process of firms 1 and 2 can be broken down into subgames. Firm 1 moves first, deciding whether to enter the market, while firm 2 observes this decision and chooses its action accordingly. This setup creates multiple subgames depending on firm 1's action.

Finding the SPE involves identifying optimal strategies for each decision point:

• Firm 1 decides not to enter to avoid negative profits, knowing that entry would lead to a poor outcome when firm 2 also enters.
• Firm 2 will enter if firm 1 does not, because entering alone with firm 3 in the market produces a positive net profit.

The outcome of SPE ensures that each firm's strategy is a best response given the subsequent actions in the subgames, leading to firm 1 staying out and firm 2 entering.

Market Entry Strategy

Deciding market entry is a strategic choice that firms must carefully assess by weighing costs and potential profits. The concept of a market entry strategy is about determining whether to enter an industry given the competitive landscape and the fixed costs associated with entry.

In the scenario provided, firms 1 and 2 must each decide if paying the fixed entry cost of 10 is justified by the prospective profits in the Cournot competition phase. Each firm evaluates its strategy based on:

• Potential profits from entering the market.
• The expected actions of other firms, particularly following the game theory models demonstrated by Nash equilibrium and subgame perfect equilibrium.
• The effect of their decision on market dynamics, including the total number of competitors and market output.

Firm 1 opts out of entering to avoid a negative profit scenario when both firms enter, while firm 2 rationalizes entry when firm 1 does not, maximizing its profit outcome with fewer competitors.

Nash Equilibrium

Nash equilibrium, named after John Nash, refers to a situation where each player's strategy is optimal given the strategies of all other players. In this state, no player has anything to gain by unilaterally changing their own strategy.

In the market game described, Nash equilibrium occurs during the Cournot competition phase where each firm selects a quantity to produce. The absence of production costs and the given demand curve \( p = 12 - Q \) allow firms to compute their equilibrium strategies.

At Nash equilibrium:

• Each firm's output choice maximizes its profit, considering competitors' quantities.
• The derived production levels ensure that no single firm can improve its profit by altering its quantity unilaterally.

In this exercise, Nash equilibrium for production, with firm 1 not entering and firm 2 and 3 competing, leads to a stable output distribution where both firms produce equal, profit-maximizing quantities.