How this instrument works
This instrument solves the textbook monopoly-pricing problem: given a straight-line demand curve, P = a − bQ, and a constant marginal cost, what single price maximizes profit? It is the tool a product manager reaches for after estimating demand from two test prices for a differentiated product — a specialty subscription tier, a patented tool, a single regional supplier with no close substitute — where 'what the market will bear' is a slope and an intercept, not a guess.
The formula is shaped the way it is because of a quirk of straight-line demand: the marginal revenue line shares the same intercept, a, but falls at twice the slope, so MR = a − 2bQ. Setting MR equal to marginal cost, c, and solving for Q, then feeding that quantity back into the demand line, collapses to a strikingly plain result — the profit-maximizing price sits exactly halfway between a and c. Notice what drops out: the slope b never appears in P* at all. It fully determines the quantity sold at that price, but not the price itself.
That halfway rule only holds while the estimated line stays straight across the range being priced; real demand often bends, especially far from where a and b were measured, and a curve fit from two points can be noisy. The formula also assumes the seller faces the whole demand curve alone, with no rival free to undercut, and it ignores fixed costs entirely — a $60 price that clears $800 over marginal cost still needs checking against overhead before anyone actually charges it.
- Enter Demand curve intercept, a (in P = a − bQ) — the price at which quantity demanded would fall to zero.
- Enter Demand curve slope, b — how many dollars price must drop to sell one more unit.
- Enter Marginal cost, $ — what producing or serving one additional unit actually costs.
- Read Profit-maximizing price, $ — the price where marginal revenue and marginal cost meet.
- Compare that price against the intercept and cost you entered to see how much slack the demand curve leaves above marginal cost.
Worked example — intercept 100, cost 20
Set Demand curve intercept, a (in P = a − bQ) to 100, Demand curve slope, b to 2, and Marginal cost, $ to 20 — a specialty product with no close substitute, priced against a demand line the seller estimated from two trial prices. The instrument computes Profit-maximizing price, $ as (100 + 20) ⁄ 2, which comes to exactly $60.
Feed that $60 back into the demand line and the implied quantity is Q = (100 − 20) ⁄ (2 × 2) = 20 units, matching P = 100 − 2(20) = 60. Revenue at that point is 20 × $60 = $1,200, and the gap over marginal cost is $40 per unit — $800 in total — the most any single price on this line can extract before fixed costs are even counted. Raise the price past $60 and volume falls faster than the extra margin makes up for; cut it and the added units sell for less than they are worth giving up.
Questions
Why does the optimal price not depend on the demand slope, b?
Because b cancels out algebraically once marginal revenue is set equal to marginal cost and the result is substituted back into the demand line — the halfway point between a and c stays fixed regardless of how steep the curve is. The slope still matters enormously for how many units sell at that price: a flatter curve (small b) means the same $60 price moves far more volume than a steep one, even though the price itself is unchanged.
Who actually estimates a and b before using a formula like this?
Pricing analysts and product managers who have run a real test — offering two price points to comparable customer segments and recording the quantity sold at each — then solve two linear equations for the intercept and slope. Economics students working monopoly-pricing problem sets use the same two numbers straight from a textbook prompt. Either way, the formula only ever sees a straight line; it never sees the raw sales data that produced it.
Does a positive result here mean the price is actually profitable to charge?
Not by itself. This figure sets marginal revenue equal to marginal cost, which maximizes the contribution over variable cost — it says nothing about rent, salaries, or other fixed costs sitting above that line. A seller still has to check that the resulting contribution, price minus marginal cost times quantity, covers fixed costs before the product is worth running at all.
How is this different from a markup or margin calculation?
Markup and margin both start from a known cost and apply a chosen percentage on top — the price is set by policy, not by what customers will actually buy. This instrument starts from the demand curve itself and derives the price the market supports at the profit-maximizing quantity, with no target percentage assumed anywhere in the formula.
What if the real demand curve is not a straight line?
Then the halfway rule is only a local approximation, accurate near the price range where a and b were actually measured and less reliable far from it. Many real demand curves bend — flattening at low prices, steepening near a ceiling — and a formula built for a straight line will misprice a market that curves, especially at the extremes.
How does this differ from checking pricing power with a Lerner index?
A Lerner index measures pricing power after the fact, from a price and marginal cost that already exist, and reports how far above cost that price sits. This instrument works in the other direction — it derives the price itself from an assumed demand curve, before any unit has been sold, rather than describing one that was already chosen.
References
- U.S. Small Business Administration — Manage Your Business guide
- MIT OpenCourseWare — Principles of Microeconomics (14.01SC)
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.