Solving Cartel, Bertrand, Cournot & Stackelberg Models

This section covers product differentiation and location competition models. When products are differentiated, demand for each firm depends on both its own price and the competitor's price: q1 = a1 - p1 + p2/2, q2 = a2 - p2 + p1/2. This captures consumer loyalty—when Budweiser raises prices, some consumers switch to Kingfisher, but not all. Hotelling's model describes location competition on a beach of length L with uniformly distributed consumers. The only Nash equilibrium is both vendors locating at the midpoint (L/2). If one vendor moves away from the midpoint, it loses customers to the competitor who remains at the midpoint. If vendors are at different positions, both have incentives to move toward the competitor. This explains why competing businesses cluster at the same location—they each want to be closer to the competitor to capture more customers.

This lecture introduces the foundational models of static competition in industrial organization, beginning with Cournot's 1838 quantity-setting oligopoly model where firms simultaneously choose outputs, leading to price exceeding marginal cost and deadweight loss; then progressing to Bertrand's 1883 price-setting model where any number of firms achieves perfect competition; followed by Hotelling's 1929 spatial differentiation model where consumers are distributed along a line and firms compete on both price and location, producing linear demand curves with markups proportional to the differentiation parameter; and finally introducing modern multidimensional utility models where consumers have independent idiosyncratic preferences for each product, showing how the Herfindahl-Hirschman Index measures industry concentration and how the number of firms affects markups differently depending on the assumed distribution of consumer tastes.

This comprehensive section covers three fundamental oligopoly models. In Bertrand oligopoly with uncertain costs, firms face incomplete information about competitors' cost structures, creating a trade-off between winning market share and maintaining profit margins. The equilibrium price is P = 1 + NC/(1+N), higher than marginal costs. In Cournot oligopoly with asymmetric costs, firms maximize profit by choosing quantities, with equilibrium quantity Q = (NA - ΣC_i)/(B(N+1)). The Lerner Index (P-MC)/P equals market share divided by demand elasticity, showing market power depends on concentration and elasticity. Product differentiation allows firms to charge above marginal cost by creating unique products. The degree of differentiation is measured by γ/β, where γ is the cross-effect parameter. When γ > 0, products are substitutes; when γ < 0, products are complements. Strategic complements occur when one firm's action makes another's action more attractive, while strategic substitutes occur when one firm's action makes another's action less attractive.

This video explains key oligopoly models including the Hotelling spatial differentiation model (where firms position farther apart to reduce competition), the Solow model showing price approaching marginal costs as competitors increase, and the Bertrand model with vertical differentiation where high-quality firms earn higher profits. The content emphasizes that Cournot (quantity competition) generally yields higher profits than Bertrand (price competition), making firms prefer output competition. Product differentiation reduces competition intensity, and firms strategically choose competition modes based on fixed costs and quality differences.

This section provides detailed analysis of three major oligopoly models. Cournot model: firms compete by choosing output quantities simultaneously, assuming competitor's output is fixed. Equilibrium output is n/(n+1) of perfect competition level, with price between monopoly and perfect competition. Individual firm output is 1/(n+1) of total. Bertrand model: firms compete by setting prices, assuming competitor's price is fixed. With homogeneous products and zero MC, equilibrium is P=MC (perfect competition outcome). With differentiated products, equilibrium price equals average cost. Stackelberg model: one leader moves first and knows follower's reaction function, incorporating it into profit maximization. Leader produces more than Cournot, earns higher profits. Comparison: Cournot (quantity competition, downward sloping reaction curves), Bertrand (price competition, upward sloping reaction curves), Stackelberg (leader-follower, first-mover advantage).

Antitrust screens are proactive statistical tools used to identify markets and firms suspected of collusive behavior, with two main types: structural screens (based on industry characteristics that facilitate collusion, such as market concentration and product homogeneity) and behavioral screens (based on pricing and bidding patterns, such as stable prices, unusual bid distributions, and price change patterns). These screens serve as red flags for collusive behavior and can be used alongside reactive methods like leniency programs. Effective screening requires high-quality data, collaboration between authorities, and understanding that screens should be combined with other evidence for conclusive proof of cartel activity.

Merger simulation is an economic analysis method used by competition authorities to assess the competitive effects of mergers by estimating demand functions (using logit or nested logit models with cross-price elasticities), calculating marginal costs from pre-merger equilibrium data, and comparing pre- and post-merger Nash equilibrium prices to predict whether prices will increase significantly; if prices rise substantially, the merger may be blocked or approved with divestment remedies.

The partial collusion model assumes each firm maximizes its own profit plus κ times the profit of other firms, where κ ranges from 0 (pure competition) to 1 (full collusion). After a merger, firms may coordinate their pricing behavior with κ > 0, meaning they partially internalize each other's profits. This framework allows researchers to test whether observed market outcomes require assuming some degree of collusion or can be explained by static competition alone. The 2008 Miller-Coors merger analysis used this approach to examine whether post-merger price increases could be explained by unilateral incentives, cost savings, or collusion effects. The random coefficients logit model estimates demand by allowing different consumers to have different sensitivities to product characteristics and prices, recognizing that some consumers are highly price-sensitive while others are not.

The lecture examines real-world cartel cases: (1) Late 1800s oil and railroad cartels that repeatedly broke down, solved by forming 'trusts' with public commitment mechanisms; (2) Movie industry cartels where production companies bought theaters to control distribution, broken up by federal antitrust action; (3) 2004 British Airlines-Virgin Atlantic fuel surcharge cartel that collapsed when one party reported the other; (4) NFL as an explicit cartel of 32 teams selling TV rights as a monopoly product, exempted from antitrust laws by Congress. These examples illustrate how cartels form, break down, and how governments attempt to regulate them.

Standard economic models show that any merger between competitors increases prices in the absence of efficiencies. This creates a general presumption against horizontal mergers. Required efficiency gains to offset competitive harm increase substantially in concentrated markets, making such efficiencies unlikely. Traditional thresholds (four-to-three concentration increases) should evolve based on changing market conditions—what was acceptable 20 years ago causes greater harm today given already-higher margins. A tiered approach is appropriate: safe harbor for unconcentrated industries, strong presumption for super-dominant firms (over 60-70% market share), and intermediate analysis for cases in between.

Limit pricing is a strategic pricing mechanism where an incumbent firm sets its price below the monopoly price but above the competitive price to prevent new competitors from entering the market. The firm sacrifices some short-term profits to ensure that potential entrants find the market unattractive due to low expected profitability. This strategy works because if the price falls below the limit price, new entrants would face losses, thereby deterring them from entering. The concept is particularly relevant in oligopoly markets where firms face constant threats of competition.

Limit pricing involves setting prices below the static monopoly level to deter entry. A classic signaling explanation involves asymmetric information about costs: if a firm has low costs, it can credibly signal this by pricing below what a high-cost firm would optimally choose. High-cost firms cannot mimic this low pricing without losing profits. This creates a separating equilibrium where low-cost firms signal their type through aggressive pricing, while high-cost firms reveal their type by pricing normally. If entry would occur regardless of information, pooling equilibria may emerge where all types price low to prevent learning that would induce entry.

Advanced oligopoly models explain strategic interactions: (1) Stackelberg model (1934) is sequential quantity-setting where one firm (leader) moves first and the other (follower) moves second. The leader anticipates the follower's reaction and optimizes accordingly, gaining first-mover advantage. The equilibrium is found where the leader's iso-profit curve is tangent to the follower's reaction curve. The model requires one sophisticated and one naive firm for a determinate solution. (2) Kinked Demand Curve hypothesis (1939) explains price rigidity: if a firm raises prices, competitors do not follow (elastic demand); if a firm lowers prices, competitors match (inelastic demand). This creates a kink at the current price, making firms reluctant to change prices. (3) Limit pricing theory explains how incumbents set prices just above competitive levels to prevent new entrants, creating barriers to entry through strategic pricing.

Traditional theory considers entry of new firms, but game theoretic approaches distinguish between actual entrants and potential entrants. Incumbent firms use strategic pricing practices: limit pricing (setting prices low to deter entry), predatory pricing (setting prices so low that rivals must leave the market), and market penetration (setting low prices initially to grab market share, then raising prices later). In monopoly and most oligopoly cases, entry is blocked in the short run and only possible in the long run. These strategic behaviors were not considered in the traditional theory, which assumed perfect information and rational behavior.

In asymmetric Bertrand competition, when firms have different marginal costs, the firm with the lower marginal cost can implement a limit pricing strategy by setting its price equal to the higher marginal cost firm's cost level, thereby eliminating the competitor's potential profits and capturing the entire market; this contrasts with symmetric Bertrand competition where both firms have identical marginal costs and end up earning zero economic profit.