The Price Curve

If a SKU's elasticity is −2, a 1% raise loses about 2% of its units. At −0.5 it loses about half a percent. The sign is always negative: a higher price sells fewer units. The size is the whole question.

Why it decides the price

Revenue is price times units. Profit is contribution times units, and contribution moves faster than price: a dollar added to the price lands in the margin whole, less the referral fee, while landed cost and the fulfilment fee stay where they were. So a raise that loses units can still make more money. Elasticity says how many units, and so which way to move.

Three kinds of SKU

• Between 0 and −1: inelastic. A raise loses a smaller share of units than it adds to the price. Profit keeps rising with the price, so there is no best price to find: walk up, and measure each step.
• Below −1: elastic. There is one price where profit peaks. Lesson 4 finds it.
• Near −1: the hard case. The best price could be a little above today's or several times it. Lesson 3 says when the data cannot tell, and what it tells you instead.

Where the numbers are

On Amazon, Business Reports has "Detail Page Sales and Traffic by Child Item". Set a date range and it gives each SKU's units ordered and ordered product sales. On Shopify, the Sales by product report gives the same two numbers for any range. The average price for a period is sales divided by units, so coupons and promotions are already in it.

One row per period

Half-months work well: long enough to sell, short enough to catch a change. For each period type the days in it, the units and the average price. Twelve half-months, six months of history, is a good start.

Units a day, not units a period

A 16-day period sells more than a 15-day one at the same price. If the long periods happen to fall at one price, a fit reads the extra day as that price's effect. Hubricon's engine measured the trap on calendar half-months: more than a whole unit of elasticity, wrong. The sheet divides units by days before anything else.

Two rules before any fit

At least 5 periods. And the price has to have moved: at least 2% variation, the standard deviation of the prices over their mean. Below either, the sheet says so and fits nothing, because a price that never moved cannot tell you what a different one would do. These are the engine's own rules.

Periods to leave out

• Stockouts: the units were capped by stock, not by price.
• Big event deals: a different customer arrives on Prime Day.
• A launch, or a listing change: the product was not the same one.

Leave those rows blank rather than adjusting them. A period with no sales is ignored.

The line

Take the log of each period's price and of its units a day, and draw the straight line that fits them best. Its slope is the elasticity: on logs, a slope is a percentage per percentage. Drawn back on ordinary axes, the line becomes the demand curve.

An invented listing · Not a client · Not a result

How sure

Every period is a noisy draw, so the slope is too. Its standard error says how far another twelve periods like these could put it. The sheet computes it the way Hubricon's engine does, HC3, which lets a period at an unusual price carry more noise than the rest instead of quietly steering the line. On this history that is 0.28, against 0.22 the simple way: the honest interval is the wider one.

With few periods, a 95% interval is not plus or minus two standard errors. On 12 periods it is plus or minus 2.23; on five it is plus or minus 3.18. The sheet uses the right multiplier for your count.

What one history can and cannot tell you

Hubricon's engine ran this exact price test 200 times on a listing whose true elasticity is −1.6, each time with fresh noise. The median standard error was 0.28; one run in ten came out under 0.18 and one in ten over 0.45. In 50% of the runs the interval sat too close to −1 to name a price at all. In 95% of them, the direction came out right: up.

That is the shape of pricing one SKU from its own history. The direction is usually knowable long before the distance is.

Too close to −1

Lesson 4's formula multiplies by elasticity ÷ (1 + elasticity), and that factor runs away near −1: at −1.15 it is already 7.7. An interval from −1.66 to −0.64 covers best prices from a few percent above today's to no limit at all. A price printed to the cent there would be a guess with a decimal point on it.

So the sheet, like the engine, names no best price when the interval includes −1, when the estimate sits within two standard errors of −1, or when one standard error would move the best price by more than 50%. Then it works the formula at the estimate alone. If that puts the best price above today's, the direction is up: near −1 the top runs off to the right, and the estimate agrees. If it puts the best price below today's, the estimate says down while the range cannot rule out far up, so the sheet says hold. The engine forces no direction there either.

Split the fees the way they move

Amazon's referral fee is a percentage of the price, so it grows when the price does. The fulfilment fee is fixed per unit by size and weight, and storage is fixed per unit too. Treat the fixed fees like landed cost, and gross the whole thing up by the percentage. Lumping every fee into one percentage moves the best price the wrong way.

On Shopify the split is the same. Shopify Payments takes a share of the price, 2.9% on the Basic plan, plus $0.30 an order, on a standard card (Shopify's pricing); shipping and packing are fixed per unit. The share goes where the referral rate goes, and the fixed costs, the per-order fee included, go with the fulfilment fee.

Best price = (landed cost + fixed fees per unit) ÷ (1 − referral rate) × elasticity ÷ (1 + elasticity)

Valid for elasticity below −1. It is the price where one more cent loses exactly as much in units as it adds in margin.

An invented listing · Not a client · Not a result

The lesson in the flat top

At the best price this listing makes +$25 a month more than today, on $5,593. Near the peak, a few percent either way costs little. Far from it, or on a discount, the curve falls away fast. That is why the next two lessons matter more than the cent.

When there is no best price

An inelastic SKU, between 0 and −1, has no peak: profit rises with the price until something the model does not see stops it, like a competitor, the Buy Box on Amazon, or conversion on Shopify. And a SKU too close to −1 has a peak nobody can locate. In both cases the sheet says "none" and points up; lesson 7 is how to walk.

What the formula assumes

That the fees per unit stay the same between today's price and the new one. Amazon's fulfilment fee changes where a price crosses $10 or $50. If a move crosses one, the sheet says so: price both sides with The Fee Staircase.

A raise makes the same money if the units it loses are exactly paid for by the margin it adds. That gives the break-even in one line, and it needs no elasticity at all:

Units can fall by = 1 − contribution per unit today ÷ contribution per unit at the new price

Why a raise is usually safer than it feels

The fear prices a raise in revenue: 5% more per unit against some share of the units. The break-even prices it in margin, and every dollar of a raise lands in the margin. The thinner the margin, the more units a raise can lose and still pay, because the same dollar is a bigger share of a small contribution.

And the asymmetry

Read the table the other way. A 10% cut on this listing has to sell 24.8% more units just to stand still; its elasticity says it sells 21.6% more. The cut costs −$141 a month. A raise has to be badly wrong to lose money. A cut has to be right to break even.

Extra units it must sell = contribution per unit today ÷ contribution per unit on the discount − 1

Add what the deal itself costs

A coupon or a deal can carry its own fee per unit sold. The sheet has a cell for it: it comes straight off the contribution, so it raises the units the sale must sell.

What the sheet does not price

• Rank. A sale that lifts your sales rank can sell more at full price afterwards. That is a bet on rank. It may be a good one, but it is a separate bet, and it should be made knowing what the sale itself cost.
• Stock that is about to age. Sometimes a discount is cheaper than the storage it saves: The Fee Staircase, lesson 7, prices the 271-day cliff.

At most 5% at a time

The fit describes prices near the ones you have tried, not prices far beyond them. On Amazon the Buy Box weighs your price against other sellers'; on Shopify every shopper weighs it against the tabs they have open. So a step is never more than 5%, and the next one waits for the measurement. The invented listing's best price is $26.84; its next step stops at $26.24.

Called before, measured after

1. Before it goes live, write down three things: the new price, the units a day you expect, and the change a month the sheet expects. Writing it first is what stops a result from being explained after the fact.
2. While it is live, watch your Buy Box share on Amazon, or your conversion rate on Shopify. If either drops, reverse the step: the model cannot see that, and you can.
3. After two weeks, add the period to Your history. The fit moves, the interval tightens, and the next step starts from the new reading.

Each step is also an experiment. A history whose price never moved becomes one that can be read.

What one sheet cannot do

Three things lie beyond a single SKU's spreadsheet. Each is in Hubricon's engine, and each is worth knowing about even if you never use it:

• Borrowing strength. A SKU with a thin history is pulled toward its catalogue's common elasticity, by exactly as much as its own estimate is noisy. A single SKU has nothing to be pulled toward.
• Prices set because demand moved. If you raised a price because it was selling fast, the history mixes your decision with the market's, and the fit reads demand too steep or too flat. That has to be tested for and corrected.
• A range on the profit. The sheet gives one number for the change. The engine draws the elasticity, the volume and the fees thousands of times and sizes the step so that the bad end stays inside a budget: double the uncertainty, and the step shrinks.

1. Fit each SKU. Run its history through Your history, one SKU at a time, and copy four readings into Catalogue: the elasticity, its standard error and the interval's two ends.
2. Add its economics. Price, units a month, landed cost, referral rate, and the fixed fees per unit.
3. Read the ranking. The sheet names each SKU's best price where it can, its direction, the next step, and what that step is worth a month, ranked.

Where to start

At the top of the ranking. Then the SKUs marked "Too close to −1" whose direction reads up: a 5% step is the cheapest way to learn the distance. Those that read hold, leave until more history says which way.

Where to stop

At SKUs without five periods or with a price that never moved. They are not wrong; they are unread. Moving them on purpose, one 5% step at a time, is how they get a reading.

How often

Every two weeks: one step for each SKU at the top, one new period into each history, the ranking read again. A catalogue walked this way finds its prices, and the record of each step is the proof of what pricing earned.