Imperfect Competition: Monopoly, Cournot and Bertrand Models

Iterated elimination of strictly dominated strategies refines the set of plausible actions in strategic settings. When a strategy survives this process, it can be part of a dominant‑strategy equilibrium, which is also a pure‑strategy Nash equilibrium played with probability 1. Stronger assumptions about rationality and higher‑order knowledge sharpen predictions, narrowing the gap between dominant‑strategy, Nash, and rationalizable outcomes.

Imperfect Competition Overview

Imperfect competition occupies the middle of the spectrum between perfect competition and monopoly. Perfect competition features many small firms, price‑taking behavior, and homogeneous products. Monopoly concentrates market power in a single firm with extreme pricing ability. Imperfect competition—often an oligopoly—contains a few large firms that enjoy market power because of barriers such as patents (pharmaceuticals), advantageous locations (coffee shops), or product differentiation (sports leagues like the NFL and MLB). These sources allow firms to influence price and output without the full dominance of a monopoly.

Demand Modeling

Economics traditionally places price on the vertical axis and quantity on the horizontal axis. Individual consumer demands aggregate into market demand, expressed as a demand function (Q(P)). The inverse demand function (P(Q)) is the mathematical inverse of (Q(P)) and is useful for profit maximization because it directly relates price to the quantity a firm chooses to sell.

Monopoly Problem

A monopolist faces a trade‑off: raising price increases profit per unit but reduces the total number of units sold. Profit equals (\pi(p) = (p - c)\times Q(p)). By differentiating (\pi(p) = (p-c)(1-p)) with respect to (p) and setting the derivative to zero, the optimal monopoly price emerges as

[ p^{*} = \frac{1 + c}{2}. ]

Geometrically, this price maximizes the area of a rectangle with a fixed perimeter, forming a square. The monopolist therefore charges a markup over marginal cost.

Models of Competition

Cournot Quantity Competition (1838)

In Cournot competition each firm chooses a quantity, and the market clears at a price determined by the aggregate supply. The best‑response function for firm (i) given rival output (q_j) is

[ BR_i(q_j) = \frac{1 - q_j - c}{2}. ]

Solving the simultaneous best‑response equations for (n) symmetric firms yields the equilibrium quantity per firm

[ q^{*} = \frac{1 - c}{n + 1}. ]

As the number of firms grows, the equilibrium price falls, and the market outcome converges to the efficient price of perfect competition when (n \to \infty).

Bertrand Price Competition (1883)

In Bertrand competition firms set prices directly. If one firm charges a higher price than its rival, all consumers purchase from the lower‑priced firm. This creates a powerful incentive to undercut the competitor by the smallest possible amount. The only stable outcome is where price equals marginal cost,

[ p = c, ]

so efficiency is achieved with just two firms. The undercutting motive is so strong that any price above marginal cost is quickly eliminated.

Key Takeaways

• Iterated elimination of strictly dominated strategies links dominant‑strategy equilibria, Nash equilibria, and rationalizable strategies, providing progressively tighter predictions.
• Imperfect competition lies between perfect competition and monopoly, with market power stemming from barriers such as patents, location advantages, and product differentiation.
• A monopolist maximizes profit by setting price (p^{*} = (1 + c)/2), balancing higher per‑unit margins against lower sales volume.
• In the Cournot model, each firm produces (q^{*} = (1‑c)/(n+1)); as the number of firms rises, price drops and the outcome approaches perfect‑competition efficiency.
• The Bertrand model shows that with only two firms, price undercutting drives the equilibrium price down to marginal cost, delivering efficient outcomes faster than Cournot competition.