Source: NYU Introduction to Microeconomics, Class Notes (pp. 318, 361)
Difficulty: Intermediate | Prerequisites: Monopoly pricing, marginal cost, marginal revenue, basic algebra.
You already know that a monopolist sets MR = MC to maximise profit. This topic extends that in two directions. First, how do economists measure how much market power a firm actually has? The Lerner Index, the four-firm concentration ratio, and the Herfindahl-Hirschman Index each answer that question differently. Second, what happens when a monopolist operates more than one factory with different cost structures? The profit-maximisation rule still applies, but the firm must allocate output across plants so that marginal cost is equalised. This is a classic exam problem.
Market power is measured by how far a firm can push price above marginal cost (Lerner Index), by the combined market share of the largest firms (concentration ratios), and by the sum of squared market shares (HHI). When a monopolist runs two plants with different cost functions, it maximises profit by producing where MR = MC in each plant, which means equalising marginal cost across plants.
Lerner Index
A measure of market power defined as (P - MC) / P, where P is the firm's price and MC is its marginal cost. The index ranges from 0 (perfectly competitive, price equals MC) to approaching 1 (extreme markup). Think of it as the percentage of the price that is pure markup.
Four-firm concentration ratio (CR4)
The sum of the market shares of the four largest firms in an industry. Expressed as a percentage. In simple terms, it tells you how much of the market the top four players control. A CR4 near 100% signals a highly concentrated market.
Herfindahl-Hirschman Index (HHI)
The sum of the squared market shares of all firms in the industry, where shares are expressed as whole numbers (e.g., 20 for 20%). Ranges from near 0 (many tiny firms) to 10,000 (pure monopoly). Think of it as a concentration measure that gives extra weight to larger firms. Squaring the shares means a market dominated by one firm scores much higher than one split evenly, even if the CR4 is the same.
Dominant-firm model
A market structure in which one large firm sets the price and a competitive fringe of smaller firms takes that price as given. The dominant firm faces a residual demand curve: market demand minus the fringe supply.
Two-plant (two-factory) problem
A scenario in which a single firm operates two production facilities with different marginal cost functions. The firm must decide how to split total output across plants to minimise costs and maximise profit.
Lerner Index
Formula: L = (P - MC) / P.
A perfectly competitive firm charges P = MC, so L = 0.
A monopolist with significant markup will have L close to 1.
Useful for evaluating individual firm behaviour, but requires data on marginal cost, which is often hard to observe.
Four-firm concentration ratio (CR4)
Add the market shares of the four largest firms.
Limitation: it does not distinguish between distributions. A market with shares of 97, 1, 1, 1 has the same CR4 (100) as one with shares of 25, 25, 25, 25. The competitive dynamics of those two markets are very different.
Herfindahl-Hirschman Index (HHI)
Square each firm's market share (as a whole number), then sum.
Example from the lecture: two firms each with sales of $10 and one firm with sales of $30. Total market = $50. Shares are 20%, 20%, 60%. HHI = 20² + 20² + 60² = 400 + 400 + 3,600 = 4,400.
HHI captures both the number of firms and the inequality of their sizes. That is its advantage over CR4.
U.S. antitrust guidelines use HHI thresholds: below 1,500 is unconcentrated, 1,500 to 2,500 is moderately concentrated, above 2,500 is highly concentrated.
One large firm, many small "fringe" firms.
The fringe firms are price-takers: they supply along their individual supply curves at whatever price the dominant firm sets.
The dominant firm's residual demand = market demand minus fringe supply.
The dominant firm then sets MR (from residual demand) = its own MC to find its profit-maximising quantity and price.
A monopolist operates two plants.
Plant 1: MC₁ = 6Q₁
Plant 2: MC₂ = 2Q₂
Market demand: P = 150 - 3Q, where Q = Q₁ + Q₂.
Step 1: Express total quantity in terms of one variable.
Equalise marginal costs across plants (profit maximisation requires MC₁ = MC₂):
6Q₁ = 2Q₂, so Q₂ = 3Q₁.
Total quantity: Q = Q₁ + Q₂ = Q₁ + 3Q₁ = 4Q₁.
Step 2: Find MR and set MR = MC.
Substitute Q = 4Q₁ into the demand function:
P = 150 - 3(4Q₁) = 150 - 12Q₁.
Total revenue = P × Q = (150 - 12Q₁)(4Q₁). But the quicker route: MR in terms of Q is 150 - 6Q (double the slope of the inverse demand). Substituting Q = 4Q₁:
MR = 150 - 6(4Q₁) = 150 - 24Q₁.
Alternatively, MC₁ = 6Q₁. Set MR = MC₁:
150 - 24Q₁ = 6Q₁, so 30Q₁ = 150, giving Q₁ = 5.
Step 3: Solve for Q₂, price, and verify.
Q₂ = 3Q₁ = 15. Total Q = 20.
MC₁ = 6 × 5 = 30. MC₂ = 2 × 15 = 30. Marginal costs are equal. ✓
MR = 150 - 6(20) = 150 - 120 = 30. MR = MC = 30. ✓
Price: P = 150 - 3(20) = 90.
L = \frac{P - MC}{P}CR_4 = s_1 + s_2 + s_3 + s_4HHI = \sum_{i=1}^{N} s_i^2 \quad \text{(shares as whole numbers, e.g. 20 for 20\%)}Two-plant profit-maximisation rule:
MR = MC_1 = MC_2The firm sets marginal revenue equal to each plant's marginal cost. This automatically means the marginal costs are equalised across plants.
Antitrust regulators (the U.S. Department of Justice, the FTC, the European Commission) use the HHI to screen proposed mergers. If a merger would push the HHI above 2,500 or increase it by more than 200 points, it triggers closer scrutiny. The two-plant problem is the logic behind how multinational manufacturers decide which factory produces how many units: a car company with plants in two countries allocates production so that marginal cost is the same at each site, adjusting for transport and tariffs.
Students often assume that the four-firm concentration ratio tells you everything about market structure. It does not. Two very different distributions (97/1/1/1 vs 25/25/25/25) produce the same CR4 of 100. The HHI captures the difference.
A frequent error in the two-plant problem is optimising each plant independently. The plants share the same demand curve and the same marginal revenue. You cannot find each plant's output without using the combined MR.
Students sometimes forget to double the slope coefficient when deriving MR from a linear inverse demand function. If P = a - bQ, then MR = a - 2bQ, not a - bQ.
Some students confuse the Lerner Index with profit margin. The Lerner Index measures the markup over marginal cost relative to price; profit margin accounts for all costs including fixed costs. A firm can have a high Lerner Index and still earn zero economic profit if fixed costs are large enough.
⚠️ HHI calculations are a near-certainty on the exam. Practise squaring market shares quickly and remember to use whole numbers (20, not 0.20).
⚠️ The two-plant problem is a favourite for a worked-problem question. Expect to be given two MC functions and a demand curve, then asked to find Q₁, Q₂, total Q, price, and sometimes profit.
⚠️ You may be asked to compare the Lerner Index and HHI as measures of market power: know which one measures firm-level power (Lerner) and which measures industry-level concentration (HHI/CR4).
Fill in the blank: The Lerner Index is calculated as (P - ______) / P.
MC (marginal cost).
True or false: An HHI of 10,000 indicates a pure monopoly.
True. One firm with 100% share: 100² = 10,000.
Fill in the blank: In a two-plant problem, the profit-maximising firm produces where MC₁ = MC₂ = ______.
MR (marginal revenue).
True or false: A four-firm concentration ratio of 80 always means the market is less competitive than one with a CR4 of 60.
Not necessarily. CR4 ignores how shares are distributed among those four firms and does not account for the remaining firms.
Fill in the blank: When computing HHI, market shares are expressed as ______ numbers, not decimals.
Whole.
Q: An industry has five firms with market shares of 40%, 25%, 20%, 10%, and 5%. Calculate the HHI.
A: HHI = 40² + 25² + 20² + 10² + 5² = 1,600 + 625 + 400 + 100 + 25 = 2,750. This is above 2,500, so the market is highly concentrated by U.S. antitrust standards.
Q: Two industries both have a CR4 of 100%. Industry A has firm shares of 97, 1, 1, 1. Industry B has shares of 25, 25, 25, 25. Calculate the HHI for each and explain why CR4 alone is misleading.
A: Industry A: HHI = 97² + 1² + 1² + 1² = 9,409 + 3 = 9,412. Industry B: HHI = 4 × 25² = 2,500. Despite identical CR4 values, Industry A is far more concentrated. CR4 cannot distinguish a near-monopoly from an evenly split oligopoly.
Q: A monopolist has two plants with MC₁ = 6Q₁ and MC₂ = 2Q₂. Market demand is P = 150 - 3Q. Find the profit-maximising output for each plant and the market price.
A: Equalise marginal costs: 6Q₁ = 2Q₂, so Q₂ = 3Q₁. Total Q = 4Q₁. MR = 150 - 6Q = 150 - 24Q₁. Set MR = MC₁: 150 - 24Q₁ = 6Q₁, giving 30Q₁ = 150, Q₁ = 5. Then Q₂ = 15, total Q = 20. MC₁ = MC₂ = 30 = MR. Price: P = 150 - 3(20) = 90.
Q: What is the Lerner Index for a firm that charges $90 with a marginal cost of $30?
A: L = (90 - 30) / 90 = 60/90 = 2/3, or approximately 0.67. This indicates substantial market power.
The Lerner Index ties directly back to monopoly pricing: a monopolist with a steeper demand curve (less elastic demand) can sustain a higher markup and a higher Lerner Index. The two-plant problem extends the standard MR = MC framework you learned in the monopoly unit. It also previews cost-minimisation logic that reappears in more advanced courses when firms choose across multiple inputs or locations. The concentration measures connect forward to antitrust policy and regulation topics, where you will see HHI thresholds used in real merger reviews.
Market power, Lerner Index, markup, price-cost margin, four-firm concentration ratio, CR4, Herfindahl-Hirschman Index, HHI, market concentration, industry concentration, antitrust, merger review, dominant-firm model, price leadership, competitive fringe, residual demand, two-plant monopoly, multi-plant monopoly, two-factory problem, marginal cost equalisation, MR = MC, profit maximisation with multiple plants, output allocation.