Question

This exercise extends the analysis of the Stackelberg duopoly game (from Chapter 15) to include fixed costs of production. The analysis produces a theory of limit quantity, which is a quantity the incumbent firm can produce that will induce the potential entrant to stay out of the market. Suppose two firms compete by selecting quantities \(q_{1}\) and \(q_{2}\), respectively, with the market price given by \(p=1000-3 q_{1}-3 q_{2}\). Firm 1 (the incumbent) is already in the market. Firm 2 (the potential entrant) must decide whether or not to enter and, if she enters, how much to produce. First the incumbent commits to its production level, \(q_{1}\). Then the potential entrant, having seen \(q_{1}\), decides whether to enter the industry. If firm 2 chooses to enter, then it selects its production level \(q_{2}\). Both firms have the cost function \(c\left(q_{i}\right)=100 q_{i}+F\), where \(F\) is a constant fixed cost. If firm 2 decides not to enter, then it obtains a payoff of 0 . Otherwise, it pays the cost of production, including the fixed cost. Note that firm \(i\) in the market earns a payoff of \(p q_{i}-c\left(q_{i}\right)\). (a) What is firm 2's optimal quantity as a function of \(q_{1}\), conditional on entry? (b) Suppose \(F=0\). Compute the subgame perfect Nash equilibrium of this game. Report equilibrium strategies as well as the outputs, profits, and price realized in equilibrium. This is the Stackelberg or entryaccommodating outcome. (c) Now suppose \(F>0\). Compute, as a function of \(F\), the level of \(q_{1}\) that would make entry unprofitable for firm 2. This is called the limit quantity. (d) Find the incumbent's optimal choice of output and the outcome of the game in the following cases: (i) \(F=18,723\), (ii) \(F=8,112\), (iii) \(F=1,728\), and (iv) \(F=108\). It will be easiest to use your answers from parts (b) and (c) here; in each case, compare firm 1's profit from limiting entry with its profit from accommodating entry.

Short answer

The optimal strategy involves comparing profits from limiting entry versus accommodating entry for each fixed cost and choosing the most profitable option. For given \(F\) values, firm 1 adjusts \(q_1\) to either deter or accommodate firm 2.

Step-by-step solution

Determine Firm 2's Reaction Function

When Firm 2 enters the market and produces quantity \(q_2\), its payoff is given by \(\pi_2 = (1000 - 3q_1 - 3q_2)q_2 - (100q_2 + F)\). To maximize \(\pi_2\), we differentiate with respect to \(q_2\): \(\frac{d\pi_2}{dq_2} = 1000 - 6q_2 - 3q_1 - 100 = 0\). Solving for \(q_2\), we find the reaction function: \(q_2 = \frac{900 - 3q_1}{6}\).

Subgame Perfect Nash Equilibrium with F=0

For \(F = 0\), Firm 2 enters if profits are non-negative. Taking Firm 2's reaction function, substitute \(q_2 = \frac{900 - 3q_1}{6}\) into its profit equation: \(\pi_2 = (1000 - 3q_1 - 3\times\frac{900 - 3q_1}{6})\times\frac{900 - 3q_1}{6} - 100\times\frac{900 - 3q_1}{6}\). Simplifying yields \(q_2 = 100 - \frac{q_1}{2}\). Firm 1 maximizes its payoff by selecting \(q_1\) observing \(q_2\), results in \(q_1 = 225\), \(q_2 = 62.5\). The price is \(p = 1000 - 3\times 225 - 3\times 62.5 = 375\).

Determine Limit Quantity for F > 0

Firm 2 will not enter if its profit is less than zero. Using its reaction function, calculate the condition: \( (1000 - 3q_1 - 3q_2)q_2 - (100q_2 + F) = 0\). Simplify and solve this condition for limit quantity \(q_1^l\) replacing \(q_2 = \frac{900 - 3q_1^l}{6}\). Solving, \(q_1^l = 300 - \frac{3F}{500}\).

Compute Incumbent's Optimal Output and Profits for Different F Values

For given values of \(F\), compare profits for entry and limit scenarios. With \(F=18,723\), \(q_1 = 288.554\); with \(F=8,112\), \(q_1 = 272.784\); with \(F=1,728\), entry is profitable. For \(F=108\), \(q_1 = 299.784\). Profits under limiting entry are calculated: \(\pi_1 = (1000 - 3q_1)q_1 - 100q_1\). Compare these profits with those when accommodating entry by calculating: \(\pi_1' = (price after firm 2 enters) \times q_1 - cost\). Choose the best strategy based on the maximum profits for \(q_1\).

Key concepts

Subgame Perfect Nash Equilibrium

In the context of game theory, the Subgame Perfect Nash Equilibrium (SPNE) is a powerful concept used to predict the outcome of strategic interactions in dynamic games. A Subgame Perfect Nash Equilibrium ensures that players make optimal decisions at every point in the game, thereby preventing any threats that lack credibility.
A game is dynamic if players make decisions one after another and base their choices on prior actions. For instance, in the Stackelberg duopoly problem, the incumbent firm first decides how much to produce. Firm 2, the potential entrant, observes this choice and subsequently decides its production quantity or opts out of the market entirely.
By solving backwards, starting from the last decision-making point and moving to earlier ones, you determine the strategies that players will realistically follow. This backward induction process aids in finding the Subgame Perfect Nash Equilibrium by ensuring that each response is optimal, given the other players' strategies.

• With no fixed costs (F=0), Firm 2 will enter and produce based on Firm 1's output. It optimally reacts using a calculated response or reaction function.
• The equilibrium reflects optimal production quantities that maximize profits for both firms within this strategy framework.

Understanding SPNE is crucial for analyzing scenarios where parties have sequential decision-making power.

Limit Quantity Theory

Limit Quantity Theory provides significant insights into how incumbent firms deter entry in an oligopoly. This theory suggests determining a specific production level, known as the limit quantity, which discourages potential competitors from entering the market to avoid loss. Essentially, the incumbent produces at a level high enough to make it unprofitable for the newcomer.
Consider a market where the incumbent (Firm 1) might increase production to reduce the market price, aiming to limit the profits that a new entrant (Firm 2) could potentially make. If Firm 2 can't cover its costs, including fixed expenses, it will choose not to enter.
Mathematically speaking, the incumbent calculates this limit quantity using the potential entrant's reaction function. The formula, derived from the condition of non-profitable entry, is used to find the precise amount of production that effectively deters entry.

• For example, if fixed costs are high, the incumbent must produce more to maintain a suitable limit price.
• The limit quantity is versatile, changing as fixed costs change, showcasing strategic flexibility.

This concept is strategic for incumbents aiming to protect their market position without waiting for competition.

Reaction Function

In economics, a reaction function is a pivotal concept, especially in oligopolistic markets where firms strategically react to one another. It's an equation that shows how one firm's optimal output level depends on the outputs of other firms in the market. The idea is simple: each firm chooses its output based on what it believes its competitors will produce.
In a Stackelberg duopoly, Firm 2 observes the quantity chosen by Firm 1 and then decides the best production response. The reaction function of Firm 2 is derived by maximizing its profit function, given Firm 1's output.
For example, Firm 2's reaction function might look like this:
\[ q_2 = \frac{900 - 3q_1}{6} \]
Here, it clearly shows the inverse relationship with Firm 1's output. If Firm 1 increases its quantity, Firm 2 will decrease its quantity, maintaining market balance and profitability.

• Reaction functions underline how rivalry in quantity decision-making shapes market outcomes.
• They offer critical insights into competitive strategies in real-world markets.

Knowing your competitor's likely reaction function enables better informed and strategic decision-making for a firm.

Fixed Costs of Production

Fixed costs of production are vital in understanding market entry and competition. These costs are expenses that do not change with the level of goods produced. They must be incurred regardless of output levels and can include rent, salaries, and machinery costs.
In the context of Stackelberg duopoly, fixed costs play a crucial role in deciding whether a firm enters a market or at what production level it operates. Bigger fixed costs mean a new entrant needs a higher market share or price to break even.
If a new entrant has fixed costs, the incumbent's response could be adjusted to produce at a level that makes it impossible for the new entrant to cover those costs, hence staying out of the market.

• High fixed costs can be a barrier to entry, providing market protection for incumbents.
• Firms factor these costs into strategic decision-making to anticipate and manipulate competitor behavior.

By leveraging knowledge of fixed costs, firms can strategically position themselves to deter new competition or bolster their market standing.