The midpoint formula, a control region and a contribution check, worked on one price test, and why an elastic reading can still mean the rise paid.
Price elasticity is the percentage change in volume divided by the percentage change in price, each measured on the midpoint of the two observations, with the volume change taken against a control group rather than against last month. In the worked test below, a rise from 40.00 to 44.00 took weekly units from 1,000 to 880, a raw elasticity of −1.34. Net of the 3.0 per cent fall in the control region it is −1.02, and the rise added 13.4 per cent to contribution.
Worked in full in Pricing Strategy by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Elasticity is the number pricing decisions are argued over, and it is usually computed in the one way that guarantees an argument: two prices, two volumes, a simple percentage, no control. Each of those shortcuts moves the answer, and the last one moves it most.
| Input | Before | After |
|---|---|---|
| Price in the test region | 40.00 | 44.00 |
| Weekly units, test region | 1,000 | 880 |
| Weekly units, control region (price unchanged) | 1,200 | 1,164 |
| Variable cost per unit | 24.00 | 24.00 |
The control region is a set of comparable stores, or customers, whose price did not move. Its 3.0 per cent fall is the seasonality and the market drift that would have hit the test region anyway.
Step 1: the counterfactual. Without the price change, the test region would have fallen with the control: 1,000 × (1 − 3.0%) = 970 units a week. That, not 1,000, is the volume the price rise is measured against.
Step 2: the midpoint changes. Divide each change by the average of the two values, not by the starting value.
E = [(Q1 − Q0) ÷ ((Q1 + Q0) ÷ 2)] ÷ [(P1 − P0) ÷ ((P1 + P0) ÷ 2)]
=((Q1-Q0)/AVERAGE(Q1,Q0))/((P1-P0)/AVERAGE(P1,P0))Without the control, the same formula on 1,000 and 880 gives a volume change of −12.77% on an average of 940, and an elasticity of −1.34. Of the 120 units the test region appears to have lost, 30 were the market, not the price.
| Weekly, test region | Without the rise | With the rise | Change |
|---|---|---|---|
| Units | 970 | 880 | −90 |
| Price | 40.00 | 44.00 | |
| Revenue | 38,800 | 38,720 | −0.2% |
| Unit contribution | 16.00 | 20.00 | |
| Contribution | 15,520 | 17,600 | 2,080 |
Demand is almost exactly unit elastic, so revenue barely moves. Contribution rises by 2,080 a week, 13.4 per cent, because each of the 90 lost units took only 16.00 of margin with it while each of the 880 retained units now carries 4.00 more. A pricing committee looking at revenue would call this test a wash. It was a clear success.
The elasticity that matters for profit is not −1, it is the break-even. Measured the simple way it is −1 ÷ (margin + price change): at a 40 per cent margin and a 10 per cent rise, −1 ÷ 0.50 = −2.00, a volume loss of 20.0 per cent. Demand has to be twice as elastic as unit elasticity before this rise costs money. The same break-even, stated as a volume loss rather than an elasticity, is worked in how much volume a price increase can lose before EBITDA falls.
| Elasticity | Price −10% | Price −5% | Price +5% | Price +10% |
|---|---|---|---|---|
| −0.5 | −20.9% | −10.2% | +9.8% | +19.2% |
| −1.0 | −16.7% | −7.9% | +7.1% | +13.6% |
| −1.5 | −12.2% | −5.5% | +4.6% | +8.3% |
| −2.0 | −7.4% | −3.0% | +2.0% | +3.3% |
| −2.5 | −2.4% | −0.5% | −0.4% | −1.5% |
| −3.0 | +2.9% | +2.1% | −2.8% | −6.1% |
Read across a row and the asymmetry of price is plain. At any elasticity between −0.5 and −2.0, a cut loses contribution and a rise gains it. A rise stops paying somewhere between −2.0 and −2.5, and a cut starts paying only between −2.5 and −3.0. Very few established products, measured properly against a control, show elasticities that high for moves of this size.
No control group. Here it turns −1.02 into −1.34, enough to move a product from "raise" to "hold" in most pricing reviews. If volume was falling anyway, the price gets the blame; if it was rising, the price gets the credit.
The simple percentage. Measured from the starting point, the raw test reads −12.0% ÷ 10.0% = −1.20 on the way up and +13.64% ÷ −9.09% = −1.50 on the way back down. Same two points, two answers. The midpoint formula removes the direction.
Extrapolating the curve. The tempting next step is the textbook optimal price, cost × E ÷ (E + 1). At the measured −1.02 it returns 1,224.00, and at −1.5 it returns 72.00, nearly double today's price. An elasticity measured on a 10 per cent move says nothing reliable about a move five times larger. Test the next step, measure again, and move in increments.
The free discount calculator workbook on the Pricing Strategy companion page computes the break-even volume of a price cut and of a rise at any margin, and what one point of price is worth against one point of volume. Before measuring elasticity on list price, check what price customers actually pay: see how to calculate pocket price from list price.
Because the simple percentage method gives a different answer depending on direction. A move from 40.00 to 44.00 that takes volume from 1,000 to 880 reads -1.20 going up and -1.50 coming back down. The midpoint, or arc, formula divides each change by the average of the two points and gives -1.34 either way, so the elasticity no longer depends on which price you started from.
No. Elasticity above 1 in absolute value means revenue falls, but profit depends on contribution. At a 40 per cent contribution margin a 10 per cent rise breaks even at a simple elasticity of -2.00, a 20.0 per cent volume loss. Demand would have to be twice as elastic as unit elasticity before the rise destroyed contribution.
Measured the simple way, it is minus one divided by the contribution margin plus the price change. At a 40 per cent margin and a 10 per cent increase that is -1/(0.40 + 0.10) = -2.00. Any measured elasticity closer to zero than that means the increase adds contribution; any further from zero means it removes it.
This article is one calculation from Pricing Strategy. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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