Part-worth utilities become money through the price coefficient; a logit share simulator then turns that money into a price.
Willingness to pay from a conjoint study is the utility gain of a feature divided by the utility lost per unit of price. In an illustrative choice-based study where a 3-year warranty adds 0.46 of utility and price costs 0.0305 per unit, the warranty is worth 15.08 a unit to the average buyer. That is the premium that leaves market share unchanged, not the premium to charge: in a share simulation the profit-maximising price rises by only 7.72 over the best price without the warranty.
Worked in full in Pricing Strategy by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Conjoint is the method pricing teams reach for when they need a number for a feature that has no market price yet. The conversion from utilities to money is one line of arithmetic. The step that is usually skipped is the second one: turning that money figure into a price decision, which needs a simulator, a cost and a competitor.
An industrial product sold at a list price of 140 against one main competitor. A choice-based conjoint has been run and estimated as a multinomial logit; these are the average part-worth utilities.
| Attribute | Level | Utility |
|---|---|---|
| Price | 120 / 140 / 160 | 0.60 / 0.00 / −0.62 |
| Brand | Ours / Competitor | 0.25 / 0.00 |
| Warranty | 1 year / 3 years | 0.00 / 0.46 |
| Delivery | 5 days / Next day | 0.00 / 0.30 |
| Choose neither | 0.40 |
For the simulation: 10,000 buyers a year in the segment, a unit cost of 90.00, and an expected 4.00 a unit of extra cost for the longer warranty. The competitor sells at 140 with a 3-year warranty and 5-day delivery; we sell at 140 with a 1-year warranty and 5-day delivery.
Price utilities are almost linear across the tested range, so fit one slope: the utility lost per unit of price.
β = (−0.62 − 0.60) ÷ (160 − 120) = −1.22 ÷ 40 = −0.0305 per unit
Excel, prices in A2:A4 and utilities in B2:B4: =SLOPE(B2:B4,A2:A4)
WTP = Δ utility ÷ |β|
3-year warranty: 0.46 ÷ 0.0305 = 15.08
Next-day delivery: 0.30 ÷ 0.0305 = 9.84
Our brand over the competitor's: 0.25 ÷ 0.0305 = 8.20
Each figure reads as: the average buyer is indifferent between the product with the feature at 140 + WTP and the product without it at 140. Because the logit model is additive, the figures also add: warranty and next-day delivery together are worth 24.92 on these utilities.
Do not confuse this with attribute importance, the range of each attribute's utilities as a share of the total. Here price has 54.7 per cent of the importance and the warranty 20.6 per cent. Those shares depend on which price levels the study happened to test, and they carry no currency unit. Only the ratio to the price coefficient does.
The share of buyers choosing each option is the logit formula: eU divided by the sum of eU over our product, the competitor and "neither".
Ours at 140, 1-year warranty: U = 0.25. Competitor: U = 0.46. Neither: 0.40.
eU = 1.284, 1.584 and 1.492, so our share = 1.284 ÷ (1.284 + 1.584 + 1.492) = 29.5%, 2,945 units, a contribution of (140 − 90) × 2,945 = 147,253.
Excel: =EXP(U_ours)/(EXP(U_ours)+EXP(U_comp)+EXP(U_none))
Add the 3-year warranty and charge exactly the WTP, 155.08. Our utility is unchanged, so our share stays at 29.5 per cent by construction; the profit rises to 179,890 because each unit now earns 61.08. That is the share-neutral price. It is a ceiling for keeping volume, not the best price.
| Price | Our share | Units | Unit margin | Contribution |
|---|---|---|---|---|
| 140.00 | 39.8% | 3,980 | 46.00 | 183,103 |
| 145.00 | 36.2% | 3,621 | 51.00 | 184,690 |
| 150.00 | 32.8% | 3,277 | 56.00 | 183,513 |
| 155.08 | 29.5% | 2,945 | 61.08 | 179,890 |
| 160.00 | 26.4% | 2,643 | 66.00 | 174,456 |
| 165.00 | 23.6% | 2,358 | 71.00 | 167,386 |
Contribution peaks at 145.26, at a 36.0 per cent share, for 184,694. Without the warranty the best price would be 137.54 for 147,532. So the warranty is worth a price premium of 7.72 and 37,162 a year of contribution, against a WTP of 15.08. The rest of the WTP is better taken as volume: the feature lets us win share from the competitor and from "neither" at a price only slightly higher.
A check on the optimum. In a logit model the profit-maximising unit margin is 1 ÷ (|β| × (1 − share)). At 36.0 per cent: 1 ÷ (0.0305 × 0.640) = 51.25, and 145.26 − 94.00 = 51.26. The rule shows why the optimal premium is smaller than the WTP: as our share rises the margin the model supports rises too, but only by a fraction of the utility gained.
WTP = utility gain ÷ |price coefficient|: 15.08 for the warranty here. Then run the share simulator with cost and competitor: the profit-maximising premium is 7.72. Conjoint WTP sits beside the other two ways to put a number on value, economic value to the customer and the Van Westendorp price sensitivity meter; what one point of price is worth to profit is in the free workbook for this case.
WTP equals the difference in part-worth utility between two levels divided by the absolute value of the price coefficient. If prices from 120 to 160 span 1.22 of utility, the coefficient is 0.0305 per unit, and a warranty worth 0.46 of utility has a WTP of 0.46 / 0.0305 = 15.08. The result is only valid inside the tested price range.
Usually not. WTP is the premium at which share stays unchanged. In the example, charging 155.08 for the warranty keeps a 29.5 per cent share and earns 179,890, but the simulator finds a better price at 145.26, earning 184,694. The profit-maximising premium is 7.72, about half the WTP, with the rest taken as volume.
No. Importance is each attribute's utility range as a share of the total, here 54.7 per cent for price and 20.6 per cent for the warranty, and it moves with the price levels a study chooses to test. It has no currency unit. Willingness to pay comes from dividing a utility difference by the price coefficient.
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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