91Ó°ÊÓ

(Repeated Bertrand duopoly) Consider Bertrand's model of duopoly (Section 3.2) in the case that each firm's unit cost is constant, equal to \(c\). Let \(\Pi(p)=(p-c) D(p)\) for any price \(p\), and assume that \(\Pi\) is continuous and is uniquely maximized at the price \(p^{m}\) (the "monopoly price"). a. Let \(s\) be the strategy for the infinitely repeated game that charges \(p^{m}\) in the first period and subsequently as long as the other firm continues to charge \(p^{m}\), and punishes any deviation from \(p^{m}\) by the other firm by choosing the price \(c\) for \(k\) periods, then reverting to \(p^{m}\). Given any value of \(\delta\), for what values of \(k\) is the strategy pair \((s, s)\) a Nash equilibrium of the infinitely repeated game? b. Let \(s\) be the strategy for the infinitely repeated game defined as follows: \- in the first period charge the price \(p^{m}\) \- in every subsequent period charge the lowest of all the prices charged by the other firm in all previous periods. Is the strategy pair \((s, s)\) a Nash equilibrium of the infinitely repeated game for any discount factor less than 1 ?

Step by step solution

01

Understand the Bertrand Model

In the Bertrand duopoly model, two firms compete by setting prices simultaneously. The firm with the lower price captures the entire market. If both prices are the same, they split the market. The profit for each firm is given by \(\Pi(p) = (p - c) D(p) \) where \( p \) is the price, \( c \) is the constant marginal cost, and \( D(p) \) is the market demand at price \( p \). The monopoly price \( p^m \) maximizes monopoly profit.

02

Define the Strategy

For part (a), the strategy \( s \) involves both firms charging \( p^m \) in the first period and continuing to do so as long as the other firm also charges \( p^m \). If one firm deviates, the other firm punishes by charging price \( c \) for \( k \) periods and then reverts to \( p^m \).

03

Calculate the Discounted Profits

To find the values of \( k \) for which \( (s,s) \) is a Nash equilibrium, compare the discounted profits if both firms cooperate versus if one firm deviates. The condition for Nash equilibrium is that the present value of profits from deviating must not exceed the present value of profits from cooperating. The profit from cooperation per period is \( \Pi(p^m) = (p^m - c) D(p^m) \). The profit from deviation in the first period is \( \Pi(p^d) = (p^d - c) D(p^d) \), where \( p^d \) is lower than \( p^m \).

04

Set Up the Profit Inequality

The present value of cooperating forever is \[ PV_{coop} = \frac{\Pi(p^m)}{1 - \delta} \] where \( \delta \) is the discount factor. If a firm deviates, it earns \( \Pi(p^d) \) immediately, but then earns zero due to punishment for \( k \) periods and then reverts to earning \( \Pi(p^m) \). The present value of deviating is: \[ PV_{dev} = \Pi(p^d) + \delta^{k+1} \frac{\Pi(p^m)}{1 - \delta} \] To find the values of \( k \) for which \( (s,s) \) is a Nash equilibrium, solve: \[ \frac{\Pi(p^m)}{1 - \delta} \geq \Pi(p^d) + \delta^{k+1} \frac{\Pi(p^m)}{1 - \delta} \] Rearrange to find: \[ k \geq \frac{\log(\delta(\Pi(p^d) - \delta \Pi(p^m))(1 - \delta) \log(\delta)}{-}\]

05

Analyze Part (b) Strategy

For part (b), analyze the strategy where each firm charges the monopoly price \( p^m \) initially and then charges the minimum price ever charged by the other firm in subsequent periods. The incentive to deviate is eliminated because any deviation will be matched or bettered by the competing firm, leading to lower future profits.

06

Prove Nash Equilibrium for Part (b)

To be a Nash equilibrium, neither firm should benefit by unilaterally deviating. With strategy \( s \) defined in part (b), any deviation leads to permanently lower prices, ensuring all future profits are reduced. For any discount factor \( \delta < 1 \), the short-term gain from deviation is outweighed by the infinite discounted loss from reduced prices.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

game theory

Game theory studies strategic interactions between decision-makers. In Bertrand duopoly, firms decide pricing strategies to outdo each other. Their choices create outcomes affecting each competitor. This field helps predict behaviors in competitive markets.
In game theory, each player aims to maximize their payoff, anticipating competitors' responses. For example, considering how a firm’s price drop affects the rival's decision. Understanding these dynamics helps in constructing optimal strategies and achieving equilibrium in competitive settings.
Game theory is essential in economics for analyzing markets like duopolies, where few entities dominate.

Nash equilibrium

A Nash equilibrium occurs when each player's strategy is optimal, given the other players' strategies. In Bertrand duopoly, each firm sets prices, assuming the rival's price remains unchanged.
Here, a Nash equilibrium ensures neither firm benefits from altering their price, as both are already playing best-responses. For example, if both firms charge the monopoly price, deviating would lead to zero profits due to punishment strategies.
Understanding Nash equilibrium helps determine stable outcomes where no firm has an incentive to change its strategy unilaterally.

repeated games

Repeated games analyze interactions occurring over numerous periods. In Bertrand duopoly, firms repeatedly choose prices each period, considering past behaviors.
Repetition introduces strategies involving history, like punishment for deviations. For instance, firms charging the monopoly price initially and punishing deviations fosters long-term cooperation.
Repeated games provide insights into sustaining collusive agreements and ensuring firms maintain mutually beneficial prices.

monopoly pricing

Monopoly pricing is setting a price maximizing single-firm profit, assuming no competitors. In Bertrand duopoly, firms might collude to charge monopoly prices, maximizing joint profits.
This price yields maximum revenue from consumers, considering their willingness to pay. For duopolies, maintaining the monopoly price requires strategies deterring price cuts, like punishment rules.
Monopoly pricing highlights the importance of strategies supporting collusion, avoiding price wars that diminish profits.

price competition

Price competition involves firms competing by lowering prices, aiming to capture market share. In Bertrand duopoly, this forms the core competition mechanism.
The firm with the lowest price earns all market sales, pressuring rivals to match or undercut prices. This often drives prices down to marginal cost levels where profits vanish.
Understanding price competition in duopolies showcases how firms strategically set prices, balancing market capture and profit sustainability.