Question

Use the following news clip to work. Bud and Wise are the only two producers of aniseed beer, a New Age product designed to displace root beer. Bud and Wise are trying to figure out how much of this new beer to produce. They know (i) If they both produce 10,000 gallons a day, they will make the maximum attainable joint economic profit of \(\$ 200,000\) a day, or \(\$ 100,000\) a day each. (ii) If either firm produces 20,000 gallons a day while the other produces 10,000 gallons a day, the one that produces 20,000 gallons will make an economic profit of \(\$ 150,000\) and the other will incur an economic loss of \(\$ 50,000\). (iii) If both produce 20,000 gallons a day, each firm will make zero economic profit. Find the Nash equilibrium of the game that Bud and Wise play.

Short answer

The Nash equilibrium is for both Bud and Wise to produce 20,000 gallons each.

Step-by-step solution

Understand What Nash Equilibrium Is

Nash equilibrium occurs when no player can benefit by changing their strategy while the other player keeps their strategy unchanged. Both players choose strategies that are best responses to each other.

Identify the Payoffs for Each Strategy

The payoffs for each firm are given based on their production choices. We have the following payoff matrix: - If both produce 10,000 gallons: Both earn \( 100,000 \) each. - If Bud produces 20,000 gallons and Wise produces 10,000 gallons: Bud earns \( 150,000 \) and Wise incurs a loss of \( 50,000 \). - If Bud produces 10,000 gallons and Wise produces 20,000 gallons: Bud incurs a loss of \( 50,000 \) and Wise earns \( 150,000 \). - If both produce 20,000 gallons: Both earn zero profit.

Evaluate Each Firm’s Best Response

For Bud and Wise: - If Wise produces 10,000 gallons, Bud would be tempted to produce 20,000 gallons to earn \( 150,000 \) rather than \( 100,000 \). - If Wise produces 20,000 gallons, Bud would produce 20,000 gallons to avoid a loss and break even.Symmetrically for Wise: - If Bud produces 10,000 gallons, Wise would produce 20,000 gallons to earn \( 150,000 \) instead of \( 100,000 \). - If Bud produces 20,000 gallons, Wise would also produce 20,000 gallons to avoid a loss and break even.

Find the Nash Equilibrium

Both Bud and Wise have the incentive to produce 20,000 gallons regardless of the other's choice to avoid incurring losses or earning less. When both produce 20,000 gallons, neither can improve their payoff by changing their production, as changing would lead to a loss or zero profit. Therefore, the strategy (20,000 gallons, 20,000 gallons) is the Nash equilibrium.

Key concepts

Game Theory

Game theory is the study of strategic interaction among rational decision-makers. In the world of economics, it helps explain how competing firms make decisions considering the reactions of their competitors. For instance, Bud and Wise, two firms in the same market, need to decide how much beer to produce. They must consider not only their own benefits and losses but also anticipate how the other will react. Their decisions are interconnected, creating a 'game' they both play.

Payoff Matrix

A payoff matrix is a tool used in game theory to represent the possible outcomes of different strategies chosen by the players. The matrix displays the payoffs received by each player for every possible combination of strategies. In the case of Bud and Wise:
- If both produce 10,000 gallons: They each earn \(100,000\).
- If Bud produces 20,000 gallons and Wise produces 10,000 gallons: Bud earns \(150,000\) and Wise incurs a loss of \(50,000\).
- If Wise produces 20,000 gallons and Bud produces 10,000 gallons: Wise earns \(150,000\) and Bud incurs a loss of \(50,000\).
- If both produce 20,000 gallons: Both earn zero profit.

Best Response

The term 'best response' in game theory refers to the strategy that yields the highest payoff for a player, given the strategy chosen by the other player. For Bud and Wise:
- If Wise chooses 10,000 gallons, Bud’s best response is 20,000 gallons as it yields \(150,000\) instead of \(100,000\).
- If Wise chooses 20,000 gallons, Bud’s best response remains 20,000 gallons to avoid a loss and break even.
- The same logic applies symmetrically for Wise.
This interdependence in choosing strategies ensures that both players are always opting for their best response.

Economic Profit

Economic profit is the difference between total revenue and total costs, including opportunity costs. It tells us how much a firm gains or loses by operating. When Bud and Wise both produce 10,000 gallons each, they achieve the maximum economic profit of \(100,000\) each.
If one increases their production to 20,000 gallons while the other stays at 10,000 gallons, the firm producing more gains \(150,000\) while the other incurs a loss of \(50,000\).
If both produce 20,000 gallons, their economic profit is zero, demonstrating the competitive nature of oligopolies and the crucial role of strategic decision-making.