Markup and elasticity: the Lerner index
Learning goals
After this page, you can:
- Compute the Lerner index from a price and a marginal cost, and estimate it from a 10-K as the gross margin — and say why that is only an estimate.
- Use the rule Lerner = 1 ÷ |ε| in both directions: from a markup to the implied elasticity, and from an elasticity to the markup and the price.
- Explain with numbers why a profit-maximizer always prices where demand is elastic, and compare the margin-based elasticity with your Week 1 estimate.
The idea
Two companies each sell $1 of product. One keeps 70 cents after paying for what went into that dollar. The other keeps 20 cents. You have not met a single customer, but you already know whose customers are more price-sensitive. And the 10-K tells you which company is which.
The Lerner index. The Lerner index is the markup as a share of the price: (P − MC) ÷ P. In plain words, it is the share of each dollar of price that sits above the cost of one more unit. A price taker prices at marginal cost, so its Lerner index is 0. A seller whose extra unit costs almost nothing, like a software download, is close to 1. Ridgeline Cable (fictional) charges $75 a month, and one more subscriber costs $25 to serve. Its Lerner index is (75 − 25) ÷ 75 = 0.67 (two-thirds): 67 cents of every dollar is markup.
The rule at the best price: Lerner = 1 ÷ |ε|. When a seller is at its profit-maximizing price, its markup and its customers’ price sensitivity are tied together. The more price-sensitive the customers, the smaller the markup they allow. Ridgeline’s two-thirds means |ε| = 1 ÷ (2/3) = 1.5. You can check this on the demand line itself: on a straight line P = a − b·Q, the elasticity at price P is |ε| = P ÷ (a − P). At $75 that is 75 ÷ 50 = 1.5 — the same number, seen from the other side.
The rule works in both directions:
- From a markup to an elasticity: |ε| = 1 ÷ Lerner.
- From an elasticity to a price: P = MC ÷ (1 − 1 ÷ |ε|). With |ε| = 4 and MC = $30, the Lerner index is 0.25, so P = 30 ÷ 0.75 = $40. Not $37.50: the markup is a share of the price, not of the cost.
Why a profit-maximizer never prices where demand is inelastic. Recall the revenue rule from elasticity and revenue: when demand is inelastic (|ε| < 1), a price rise raises revenue. It also means you sell fewer units, so you save the cost of making them. More revenue and less cost: the price rise wins twice. So a seller sitting in the inelastic range can always do better by raising its price. That is why the best price is always in the elastic range (|ε| > 1). On a straight demand line, that is the upper half — above the midpoint price a ÷ 2. Below the midpoint, demand is inelastic and MR is negative.
Reading the markup off a 10-K. You cannot see a company’s marginal cost. But its annual report shows two lines: revenue and cost of revenue — the costs tied directly to making what was sold. The gross margin is (revenue − cost of revenue) ÷ revenue. It is the closest public number to the Lerner index. A software company might keep 70 cents of each dollar after cost of revenue; a grocery chain closer to 25.
It is only an estimate, for three reasons:
- Cost of revenue is not marginal cost. It can include some fixed costs, like depreciation of a network. It can also leave out some variable costs that are booked elsewhere, like sales commissions.
- The margin mixes products. One number covers every product the company sells.
- The rule holds only at the best price. If the company is not at its profit-maximizing price, Lerner and 1 ÷ |ε| need not match.
Week 1 vs. Week 5. In Week 1 you estimated your company’s elasticity from two price points. Now the margin gives a second estimate of the same thing. If the two are close, good. If they are far apart, the gap is worth one clear sentence: which of the reasons above, or a price change that was not a clean experiment, best explains it? That is Playbook P5 prompt 2.
Lerner = (P − MC) ÷ P, between 0 (price taker) and 1
At the best price: Lerner = 1 ÷ |ε| → |ε| = 1 ÷ Lerner and P = MC ÷ (1 − 1 ÷ |ε|)
10-K version: gross margin = (revenue − cost of revenue) ÷ revenue ≈ Lerner
Straight demand P = a − b·Q: |ε| = P ÷ (a − P) — elastic above a ÷ 2, inelastic below it
Try it
Part A: the elastic-range explorer. Green is the elastic half of the demand line; red is the inelastic half, where MR is negative. The lower chart shows profit at every price.
- Start at $50 (red zone). Read the “Raise the price $5” line. Who wins from the price rise?
- Move the price until Lerner equals 1 ÷ |ε|. What price is it? Which zone is it in?
- Set the price to $62.50, where |ε| = 1 and revenue is highest. Is profit highest there?
Part B: the Lerner calculator. It opens with a fictional company’s 10-K numbers.
- Read the gross margin, the Lerner index, the implied |ε|, and the Week 1 vs. Week 5 bars.
- Switch the mode to From a price and type $75 and $25. Do you get Ridgeline’s 1.5?
- Press Use my numbers to load your company’s revenue, cost of revenue, and Week 1 elasticity. Then Save to My Numbers and Export PNG for Playbook P5.
Worked example
Part 1. From a price. Ridgeline Cable charges $75 with MC $25. Lerner = (75 − 25) ÷ 75 = 50 ÷ 75 = 0.67 (two-thirds). Implied |ε| = 1 ÷ (2/3) = 1.5 — elastic, as it must be at the best price. On the line, |ε| at $75 = 75 ÷ (125 − 75) = 1.5.
Part 2. Why never inelastic. At $50, the line P = 125 − 0.025·Q gives 3,000 subscribers, and |ε| = 50 ÷ 75 = 0.67. Raise the price to $55: subscribers fall to 2,800. Revenue goes from $150,000 to $154,000 (+$4,000). Serving 200 fewer homes saves 200 × $25 = $5,000. Profit rises $9,000.
Part 3. From a 10-K. A fictional company, Valley Cable, reports revenue of $840 million and cost of revenue of $378 million (programming fees and line maintenance). Gross margin = (840 − 378) ÷ 840 = 462 ÷ 840 = 0.55. Read it as the Lerner index: implied |ε| = 1 ÷ 0.55 = 1.82 (ε = −1.82). If its basic plan sells for $77, cost of revenue per subscriber is about 77 × (1 − 0.55) = $34.65 — a rough marginal-cost estimate.
Part 4. Week 1 vs. Week 5. Last year Valley Cable raised its basic plan from $70 to $77, and subscribers fell from 50,000 to 46,000. The Week 1 midpoint formula gives %ΔQ = −4,000 ÷ 48,000 = −8.3% and %ΔP = 7 ÷ 73.5 = +9.5%, so |ε| = 0.0833 ÷ 0.0952 = 0.875 (inelastic). The two numbers are about (1.82 − 0.875) ÷ 0.875 = 108% apart. Possible reasons:
- Cost of revenue includes some fixed costs (network depreciation). True MC is lower, so the true Lerner index is higher and the true |ε| lower than 1.82.
- The margin mixes internet, TV, and phone service.
- A 0.875 elasticity says the old $70 price was below the profit-maximizing level. The firm was still climbing toward the elastic range.
Part 5. The reverse direction. A firm at its best price faces |ε| = 4, with MC $30. Lerner = 1 ÷ 4 = 0.25. P = 30 ÷ (1 − 0.25) = 30 ÷ 0.75 = $40. Check: (40 − 30) ÷ 40 = 0.25. (Not $37.50 — that would be a markup of 25% of the cost.)
Check yourself
A company’s 10-K shows revenue of $840 million and cost of revenue of $378 million. Treating gross margin as the Lerner index, which conclusion follows?
Gross margin = (revenue − cost of revenue) ÷ revenue ≈ Lerner, and |ε| = 1 ÷ Lerner.
On demand P = 125 − 0.025·Q with marginal cost $25, a seller charges $50 (3,000 subscribers; |ε| ≈ 0.67). What would raising the price to $55 (2,800 subscribers) do?
Where demand is inelastic, a price rise raises revenue and lowers cost, so a profit-maximizer never stays there.
A student’s Week 1 estimate from two price points is |ε| = 0.9; the 10-K gross margin implies |ε| = 1.8. Which is the most likely reason for the gap?
The margin-based elasticity is an estimate; state why it may differ from your price-point estimate.
A firm at its profit-maximizing price faces |ε| = 4, and its marginal cost is $30. What is its price?
P = MC ÷ (1 − 1/|ε|): the markup is a share of the price.
A product has a Lerner index of 0.30. What does that mean?
Lerner = (P − MC) ÷ P, the markup as a share of the price.
Video
Video coming soon.