Question

Consider an industry in which two firms compete by simultaneously and independently selecting prices for their goods. Firm 1's product and firm 2's product are imperfect substitutes, so that an increase in one firm's price will lead to an increase in the quantity demanded of the other firm's product. In particular, each firm \(i\) faces the following demand curve: $$ q_{i}=\max \left\\{0,24-2 p_{i}+p_{j}\right\\} $$ Consider an industry in which two firms compete by simultaneously and independently selecting prices for their goods. Firm 1's product and firm 2's product are imperfect substitutes, so that an increase in one firm's price will lead to an increase in the quantity demanded of the other firm's product. In particular, each firm \(i\) faces the following demand curve: $$ q_{i}=\max \left\\{0,24-2 p_{i}+p_{j}\right\\} $$ where \(q_{i}\) is the quantity that firm \(i\) sells, \(p_{i}\) is firm \(i\) 's price, and \(p_{j}\) is the other firm's price. (The "maximum" operator is needed to ensure that quantity demanded is never negative.) Suppose that the maximum possible price is 20 and the minimum price is zero, so the strategy space for each firm \(i\) is \(S_{i}=[0,20]\). Both firms produce with no costs. (a) What is firm \(i\) 's profit as a function of the strategy profile \(\left(p_{1}, p_{2}\right)\) ? Let each firm's payoff be defined as its profit. (b) Show that for each firm \(i\), prices less than 6 and prices greater than 11 are dominated strategies, whereas prices from 6 to 11 are not dominated, so \(B_{i}=[6,11]\). (c) Consider a reduced game in which firms select prices between \(x\) and \(y\), where \(x

Short answer

The profit is \( \pi_i = p_i \times \max \{0, 24 - 2p_i + p_j\} \). Prices less than 6 or greater than 11 are dominated. Between 6 and 11, strategies are undominated.

Step-by-step solution

Define the Profit Function

Each firm's profit function, given its price and the competitor's price, is defined by the profit from selling its product, since there are no costs. Thus, Firm i's profit can be expressed as \( \pi_i = p_i \times q_i \). Substitute the demand equation into the profit function: \( \pi_i = p_i \times \max \{0, 24 - 2p_i + p_j\} \). This is the profit function for both Firm 1 and Firm 2, with respective substitutions.

Analyze Dominated Strategies

A strategy is dominated if there's another strategy that guarantees a higher payoff regardless of the competitor's choice. Calculate profits for boundary cases: - If \( p_i < 6 \), then \( q_i = \max \{0, 24 - 2p_i + p_j\} \) approaches zero as prices drop.- If \( p_i > 11 \), \( q_i = 24 - 2p_i + p_j \) could become negative, hence unprofitable.Thus, these price ranges are dominated as they either provide lower profit or risk negative demands.

Confirm Undominated Strategies

For prices \( p_i \) between 6 and 11, the product remains substitute as demand stays positive and profits do not fall into zero or negative levels. These strategies are not dominated because no single alternative in the range offers universally higher profits.

Rationalizable Strategies in Reduced Game

If strategies are selected between \( x \) and \( y \) (\( x < y \)), mimic the thoughtful analysis from earlier steps: - Ensure \( 6 \leq x < y \leq 11 \). - The selected range must guarantee demand remains positive, and each firm cannot guarantee better payoff by shifting entirely outside of [6,11], given its reduced space overlap.

Compute Rationalizable Strategies in Original Game

Considering the results from the reduced game, rational strategies in the original game mirror the undominated range which ensures mutually positive profits for firms. Therefore, both firms naturally rationalize to stay between prices 6 and 11 to maximize their chances of positive and non-dominated profits.

Key concepts

Nash Equilibrium

Nash Equilibrium is a fundamental concept in game theory, where each player's strategy is optimal given the strategies of the other players. In the context of our exercise, it refers to the pricing strategies adopted by the two firms such that neither can benefit by unilaterally changing their strategy.

Let's break this down with an example from our scenario:

• Each firm sets its price based on the competitor's price.
• Given the demand functions provided, both firms aim to set a price that maximizes their quantity sold and thus maximizes their profit.
• A Nash Equilibrium price strategy would be where Firm 1 and Firm 2 set prices between 6 and 11, after considering each other's potential pricing decisions.

In this range, neither firm has an incentive to deviate, as moving out of this bracket could lead to reduced profits. Thus, the firms have found a mutually stable strategy.

Dominated Strategies

Dominated strategies are choices that result in lower payoffs for a player, no matter what the other players do. They are essentially inferior strategies that can be disregarded in order to refine strategic decision-making.

In relation to our problem, the firms have distinctly defined dominated strategies:

• Prices less than 6 are dominated because they tend to yield very low or even zero demand, translating to low profits.
• Prices higher than 11 have the potential of resulting in negative demand values, which is sensibly unprofitable.

As a result, these inferior price strategies can be dismissed from consideration, allowing the firms to focus on more viable options.

Rationalizable Strategies

Rationalizable strategies are those strategies that remain plausible when players think about what everyone else might do. This concept looks at what strategies can withstand methodical and logical analysis by the players.

In our original exercise:

• The rationalizable strategies encompass the undominated range from prices 6 to 11.
• This range reflects prices that do not threaten zero or negative demand, so firms can rationalize these strategies as sensible choices.
• Both firms realize that by choosing a price within this interval, they are maximizing their strategic efficacy and ensuring viable profits.

Through logical reasoning, each firm identifies this pricing range as the best way to ensure competitive yet profitable interactions within the market.