Consider the extent to which the analysis depends upon the...

Consider the extent to which the analysis depends upon the demand function $D$ taking the specific form $D(p)=\alpha-p .$ Suppose that $D$ is any function for which $D(p) \geq 0$ for all $p$ and there exists $\bar{p}>c$ such that $D(p)>0$ for all $p \leq \bar{p} .$ Is $(c, c)$ still a Nash equilibrium? Is it still the only Nash equilibrium?

Key Concepts

Demand Function

The demand function describes the relationship between the price set by firms and the quantity of the product demanded by consumers. Its characteristics—such as being non-negative and remaining positive up to a particular price—are central in determining how consumers react to price changes. This, in turn, affects the profitability and strategic decisions of the firms.

Equilibrium Uniqueness and Robustness

Uniqueness of equilibrium means that the Nash equilibrium found is the only stable outcome of the game, where no alternative strategy profiles can also be equilibria. Robustness, in this context, involves analyzing whether the equilibrium (like both firms setting price equal to cost) persists when the specific form of the demand function is relaxed to a more general form with basic properties maintained. This examination is crucial in understanding if the equilibrium result is dependent on the form of the demand function or if it holds under broader conditions.

Nash Equilibrium

A Nash equilibrium is a set of strategies, one for each player in a game, such that no player can increase their payoff by unilaterally changing their own strategy given the strategies of the others. In economic models of competition, it represents a state where each firm’s pricing strategy is optimal, taking into account the pricing choices of its competitors.

Profit Maximization and Best Response

In competitive market models, firms are assumed to choose prices to maximize their profits. The concept of the best response refers to the optimal pricing decision a firm makes in reaction to the price chosen by its competitor. When the equilibrium is such that firms make zero profit from any deviation (for example, pricing at marginal cost), no unilateral change in price would yield a better outcome.

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