Willingness to Pay (WTP)

1. What Is WTP?

Willingness to Pay (WTP) answers a simple question: "How much more would consumers pay for this feature?"

For example, if you run a conjoint study about cars and want to know how much extra people would pay for an SUV compared to a Sedan, WTP gives you a concrete Euro amount.

Choicery offers three WTP methods on the WTP tab of the Market Simulator. Each one has its own sub-tab directly under the tab heading: Simulation WTP, Robust WTP and Pseudo-WTP. Only one sub-tab is shown at a time, and each sub-tab keeps its own result.

2. Prerequisites: What You Need Before Starting

Before you can use WTP, you need:

3. Using the Simulation WTP Sub-Tab

The Simulation WTP sub-tab computes WTP against the fixed set of products that you define. The panel description states it plainly: Considers all products defined under PRODUCTS. Price is adjusted until simulated preference share returns to its baseline level (competitive context). It is more accurate than Pseudo-WTP because it considers the full competitive market. For an estimate that does not depend on a single competitive scenario, see Section 4: Robust WTP.

3.1 Setting Up Products

Simulation WTP requires at least two products defined in the PRODUCTS panel on the left side of the Market Simulator. These represent your focal product and its competitors.

3.2 Running the Calculation

With products defined, go to the WTP tab and stay on the Simulation WTP sub-tab (it is the first one and active by default). The selection row has four controls:

The same selection row is shared with the Robust WTP sub-tab, so switching between the two keeps your focal product and attribute change.

The shared selection row on the WTP tab: pick the focal product, the attribute to change, and the target level. The Current level is read-only and mirrors the focal product's setting in the PRODUCTS panel.

The Include "None" toggle sits in the same row, to the right of Change to. It adds the None option (outside good) to the choice set used for the aggregate break-even search, and it applies to the Robust WTP sub-tab as well. It appears only when the study actually has a None option. It does not affect the respondent-level summary described in Section 3.3, which is a utility-equality calculation without None.

Click Calculate. Choicery automatically:

1. Computes baseline market shares for all products using individual-level HB utilities.
2. Changes the focal product's attribute to the upgraded level.
3. Runs a bracketed search: it first looks for a price window that contains the break-even point, starting 3 tested price ranges away from the original price and widening the window in growing steps (at most 24 times, never further than 10 times the tested price range), then bisects inside that window (at most 50 steps) until the focal share is within 0.01 share points of the baseline. If no window is found, the result is a boundary estimate.
4. Reports the WTP as the price increase that was needed.

3.3 Interpreting Results

The result card is headed Simulation WTP and carries an Excel download button (plus a segment chip when a filter is active). It shows the monetary amount that the focal product's price could increase before losing preference share, labelled Aggregate Market WTP, followed by a compact meta bar:

Reading the meta bar:

• Base is the focal product's preference share before the attribute change, at its current price. B/E is the share at the break-even price, by construction it should match Base almost exactly.
• Price shows the search result: original price → break-even price. WTP = break-even price − original price.
• iter is the number of binary-search steps that were needed.
• N is the number of respondents behind the respondent-level WTP summary below: people with a finite individual indifference price and, with Exclude outliers on, after the P5 to P95 trim. It can be smaller than the pool used in the break-even search. The Notes list how many respondents were excluded from that pool because of unusual price responses or a segment filter.

Longer explanations do not sit in the card body. They collapse behind a small Notes toggle at the bottom of the result, with the number of notes in brackets. Two of them matter most for interpretation:

• Unusual price responses excluded: respondents with positive or near-zero price coefficients are removed by default because they make the share-versus-price curve non-monotonic (see Analysis Settings above to change this).
• Break-even price outside the observed price range: the search had to raise the price beyond the highest price shown in the survey. The linear price utility is only estimated inside the design's price range. Beyond it, the number is a straight-line extrapolation, not an in-range estimate. Treat such WTP values as directional at best (see Section 3.4).

If no finite estimate exists at all, the card replaces the amount with Not estimable or Not estimable (boundary only). This is also the case when the search reaches the end of its price window (or a price of 0 when negative prices are off) without the share returning to the baseline: the price where the search stopped is a search limit, not a willingness to pay. The message under Not estimable names that price and says that the WTP is above (or below) the difference; the PDF and the Excel export show the same status.

The respondent-level WTP summary

Below the meta bar sits a collapsible block titled Respondent-level WTP summary (per-person utility-equivalence). Expand it to see Respondent Mean, Respondent Median and Respondent N, plus a CSV button that downloads the per-respondent WTP values. These are the means and medians of individual indifference prices, which is a different quantity from the Aggregate Market WTP above: that one is a market-share break-even, not an average of individuals. The respondent block ignores the Include "None" setting by design.

The Excel export adds a Respondents sheet with one row per participant: the individual WTP, a status, an Extrapolated column and the price-response class. The individual WTP is a price difference, so it is not cut at a price of 0. If balancing the change would need a price below 0, the value is still shown and marked as extrapolated, like every value whose balancing price lies outside the tested prices. The respondent summary on the card uses the same values. The status says in plain words whether the value entered the summary, lies outside the range kept by the outlier trim, or why there is no value: the participant prefers higher prices or barely reacts to price, or the balancing price lies beyond the search limit of 10 times the tested price range. Each value is one point estimate per person from that participant's estimated utilities; it is not a confidence interval.

A worked example:

3.4 Why the Starting Price Matters (Position Dependence)

Simulation WTP answers a positional question: "Given where my product currently sits in this market, what premium exactly offsets the attribute upgrade?" The answer genuinely depends on the focal product's starting price, and the direction of that dependence often surprises people.

Why this happens. In a market where every respondent had the same price sensitivity, the WTP would be identical at both starting prices (the utility gain divided by the price coefficient, see the proof in Section 10.3). With heterogeneous respondents, the break-even search is governed by whoever sits at the choice margin:

• Strong starting position (low price, high base share): the marginal buyers are ordinary, averagely price-sensitive respondents. A moderate premium pushes exactly enough of them back to the competitors. The break-even price usually stays inside the observed price range, the estimate is trustworthy.
• Weak starting position (high price, low base share): at baseline, only the most price-tolerant respondents still choose the focal product. After the upgrade, the search must push the share back down to that small baseline, which means pricing out everyone except the least price-sensitive tail. Those respondents barely react to price, so the search runs far beyond the design's price range. The result is a mechanical straight-line extrapolation, flagged by the out-of-range warning.

What to do instead:

1. Price the focal product mid-range (for a 45 to 65 EUR design: around 50 to 55 EUR), so the break-even search stays inside the range where the price coefficient was actually estimated.
2. Cross-check with the respondent-level WTP summary (the collapsible panel under the result). It reports per-person utility-equivalence WTP, which does not depend on the starting price or the competitive set, if the aggregate WTP moves a lot with the starting price but the respondent-level summary stays put, the discrepancy is positional, not preference-driven.
3. Prefer the Robust WTP sub-tab for reporting. It re-runs the calculation across hundreds of random competitive scenarios (optionally also varying the focal price) and reports the median with a confidence interval, which removes the dependence on one particular market position.
4. Without a None option, be extra careful at high prices. If the simulation has no outside option, respondents must choose some product, so the focal share can never fall below its die-hard floor, which further inflates out-of-range break-even prices.

4. Using the Robust WTP Sub-Tab (SOS Bootstrap)

The Robust WTP sub-tab is described in the app as the recommended WTP approach. It extends Simulation WTP (Section 3) by removing the dependency on a specific set of competitor products. Instead of computing WTP against one fixed competitive scenario, it runs hundreds of Simulation WTP calculations, each with a different randomly generated competitive landscape. The result is a median WTP with bootstrap confidence intervals, a robust, scenario-independent estimate.

The panel description on the sub-tab reads: Sampling of Scenarios (SOS, Orme 2021), the recommended WTP approach. Does not use the products you set up under PRODUCTS. Instead, every scenario generates its own random competitive set and re-runs Simulation WTP for the selected attribute change. This stress-tests the WTP across many possible markets, so it does not depend on which competitors you happened to configure. The one exception is the Sample from products mode described in Section 4.3, which does draw from the PRODUCTS list.

SOS stands for Sampling of Scenarios (Orme, 2021). The idea: rather than relying on one particular competitive context (which may be arbitrary), sample many possible competitive contexts and report the central tendency.

4.1 How It Works (Overview)

Robust WTP is a wrapper around the Simulation WTP procedure. For each of N iterations:

1. Generate K random competitor products (or sample them from your product list).
2. Run standard Simulation WTP: upgrade the focal product's attribute, then raise its price until the market share returns to baseline.
3. Record the WTP value from this scenario.

After all iterations, the median WTP across all successful scenarios is reported together with two different measures of uncertainty:

• Respondent bootstrap 95% CI. Respondents are resampled with replacement (500 draws by default, configurable in the server limits). Every draw is evaluated on the very same scenarios, and the break-even search runs for all draws at once. For each draw the median over scenarios is taken; the 2.5th and 97.5th percentiles of those medians form the interval. This is sampling uncertainty over people, the same idea as Sawtooth's SOS confidence interval, and it does not shrink just because you request more scenarios.
• Scenario band P5 to P95. The 5th and 95th percentiles of the scenario WTP values themselves. This is how much the competitive context moves the estimate; a wide band means the WTP depends strongly on who the competitors are.

The respondent bootstrap is used whenever the price attribute is linear, with or without the focal-product variations. Only when the run stores the price utility in a format that cannot be evaluated as a function of price does Choicery fall back to a resample of the scenario values, a Monte-Carlo interval of the scenario median that narrows with more iterations; the card then labels the row Scenario-resample 95% interval.

Exclude outliers works differently here than on the Simulation WTP sub-tab. With the setting on, scenario WTP values outside the P5 to P95 range of the scenario distribution are removed before the median, mean, confidence interval and SE are computed; more than 10 successful scenarios are required, otherwise the trim is skipped and the Notes say so. The Scenarios sheet of the Excel export marks such scenarios as not used and names the range.

4.2 The Role of the Focal Product

The focal product is the product whose attribute you want to change, exactly as in Simulation WTP. You pick it in the shared Focal Product dropdown, and its levels come from the PRODUCTS panel. The competitors are always replaced per scenario; the focal product is only varied if you ask for it.

What the focal product determines:

• Starting configuration: All attribute levels of the focal product define the “before” state.
• Attribute to change: You select which attribute to change and the target level (the same selection row as Simulation WTP).
• Starting price: The price of the focal product is the reference point. WTP = how much this price can increase after the change while maintaining preference share.

4.3 Competitor Generation Modes

The Mode dropdown controls how competitors are generated for each iteration:

4.4 Running the Calculation

Click the Robust WTP sub-tab. The selection row stays where it is, the Calculate button is replaced by the SOS controls, and the panel description switches to the SOS text quoted above. Nothing is collapsed: Iterations, Competitors, Mode and Run SOS sit directly under the selection row, with the two focal-product checkboxes and the CSV checkbox below them.

The SOS controls on the Robust WTP sub-tab. They reuse the focal product and attribute change selected in the row above; the competitive set is regenerated per iteration.

Click Run SOS to start. A progress bar shows the running count of iterations and the percentage complete, and a Cancel button appears next to the run button while the job is running.

4.5 Interpreting Results

The result card is headed Robust WTP (SOS) and carries an Excel button (and a CSV button when the per-iteration CSV was requested). The headline figure is labelled Median WTP for Change (SOS), next to two small histograms: the bootstrap distribution with the 95% CI marked, and the distribution of the scenario values with the P5-P95 band marked. Everything else sits in the meta row below it:

Warnings, exclusion notices and the CSV note collapse behind the same Notes toggle as on the Simulation WTP sub-tab. If no scenario produced a usable estimate, the card shows the failure message instead of a value, and the CSV button stays available, because the per-iteration file is exactly what explains why everything failed. When at least one successful scenario placed the break-even price outside the tested price range, the Notes say how many of the scenarios did so. Those WTP values are extrapolations of the linear price effect; they stay in the median (Sawtooth reports the same share of extrapolated draws), the Excel Summary counts them, and the Scenarios sheet flags each of them.

The Excel export of a Robust WTP result has no Respondents sheet, because the headline is not a summary of individual values. In each scenario it is the price change at which the focal product’s share across all used respondents returns to its level before the change, and the card reports the median over the scenarios. People who hesitate between the focal product and its competitors and react to price move that share the most; people who choose it anyway, never choose it or barely react to price move it little. Neither the median nor the mean of the individual WTP values reproduces the headline, so the per-person values stay in the Simulation WTP export. The confidence interval on the card describes the median of the sample, not a single participant.

The Scenarios sheet lists every scenario: its WTP, whether it entered the median and, in plain words, why not, whether the break-even price lies outside the tested prices, the focal product’s share before the change, the starting and the break-even price, and one column per competitor. With active filters a Group column names the filter group. When the focal product is varied, share and prices are averages over the variations, and the Extrapolated column says in how many of them the break-even price lies outside the tested prices.

5. Using the Pseudo-WTP Sub-Tab

Click the Pseudo-WTP sub-tab. It uses a simple formula to convert utility differences into monetary amounts. It does not consider competitor products, it only looks at the utilities from the HB estimation. It is the sub-tab Choicery falls back to when the price-based methods are unavailable.

A thin bar at the top of the sub-tab shows the respondent count (N) and an info icon with the formula:

5.1 Attribute Change Mode

The default mode, selected with the left half of the pill toggle above the form (Attribute Change / Product Comparison). Select an attribute, the current level and the target level. Choicery computes the WTP for this single attribute change, holding everything else constant.

Click Calculate to see the WTP result across all respondents.

5.2 Product Comparison Mode

Switch the pill toggle to Product Comparison. There is nothing to select here: the note in the form states that it compares Product 1 → Product 2 from the Products panel, and that all attribute values except price are taken from there automatically. Choicery then computes the total WTP difference between the two profiles, summing the utility differences across all attributes. Edit the two products in the PRODUCTS panel if you want to compare a different pair.

5.3 Interpreting Results

The headline figure is labelled Median Pseudo-WTP whenever the WTP divides by each respondent's own price coefficient, that is for a linear and for a categorical price (see the box below for why). Only with a manual conversion factor, which applies one factor to everyone, is it the Mean Pseudo-WTP. It sits next to a small spread graphic marked respondent spread, P5-P95. Underneath it, three stat tiles (the other central statistic, Std Dev, P5-P95) and a context line naming the attribute change; the N in the formula bar is the number of respondents that entered the statistics:

As on the other sub-tabs, exclusion and trimming notices collapse behind a Notes toggle. With Include unusual price responses on, the Notes say how many respondents who seem to prefer higher prices or barely react to price were kept; their values can be very large or have the opposite sign. If the price attribute was not estimated as linear and you entered a conversion factor by hand, the result additionally carries a Manual Mode badge.

6. Which Method Should I Use?

7. Best Practices & Common Pitfalls

Do

• Run HB with linear price coding and the price attribute set to Negative (<= 0) under Sign Constraints.
• Configure Price Setup in the simulator before opening the WTP tab.
• Always report the median WTP, not just the mean.
• For industry studies: define realistic competitor products in the PRODUCTS panel.
• When competitors are uncertain, use the Robust WTP sub-tab to get a scenario-independent WTP with confidence intervals.
• Set the focal product's price mid-range so the break-even search stays inside the price range that was actually shown in the survey (see Section 3.4).
• Cross-check the aggregate result against the respondent-level WTP summary, it is independent of the starting price and the competitive set.
• Consider the WTP distribution, a wide spread means consumers disagree significantly about the feature's value.
• Include a None option in your conjoint design to reduce hypothetical bias.

Don't

• Don't report Pseudo-WTP as "the" WTP when you have competitor products defined, use Simulation WTP or Robust WTP instead.
• Don't report a WTP whose break-even price lies outside the observed price range, the app flags this explicitly; the value is a straight-line extrapolation, not a market estimate.
• Don't compare Simulation WTP values computed at different focal starting prices or baseline shares, the measure is positional by construction (Section 3.4).
• Don't trust a break-even search that starts from a very low baseline share, the result then hinges on the few most price-insensitive respondents.
• Don't ignore extreme values. If mean WTP is 50,000 EUR but median is 3,000 EUR, you have respondents who ignored price. Report the median.
• Don't use WTP without a price attribute. Without price, there is no β_price to divide by.
• Don't compare WTP values across different studies without accounting for hypothetical bias, sample composition, and competitive context.

8. Key Definitions

These concepts are related but distinct (Orme, 2020, Section 9.6). WTP measures consumer valuation. Price sensitivity measures consumer reaction. Price elasticity measures market-level demand response. MVAI measures firm pricing power.

9. Pseudo-WTP: Formula & Known Problems

The simplest and most widely used WTP method in conjoint analysis. It converts utility differences into monetary units by dividing by the price coefficient. Allenby et al. (2014) explicitly term this "pseudo-WTP" to distinguish it from the formal welfare-theoretic WTP measure.

9.1 Formula

With Two Price Levels (Categorical Price Coding)

With Linear Price Coefficient

Where βprice is the linear price coefficient (a single parameter). Using linear price coding yields a consistent conversion factor regardless of how many price levels exist in the design.

With three or more categorical price levels, Choicery does not pick two levels for Equation 9.1. It fits a least-squares line through all (price value, utility) points of each respondent and uses that respondent's slope as βprice in Equation 9.3. With exactly two levels the line passes through both points and the two formulas coincide. This is the same per-respondent line that Reservation Price uses.

9.2 Worked Example

9.3 Known Problems

Orme (2020) and Allenby et al. (2014) identify several fundamental problems, ranked by severity:

9.4 Mitigations

9.5 When Is Pseudo-WTP Appropriate?

Despite its limitations, the simple ratio method remains the standard approach in academic research when the research question is about consumer preferences rather than firm pricing. It is appropriate when:

• The goal is to measure consumer valuation of a feature, not to optimize firm pricing.
• There is no meaningful competitive set (e.g., policy research).
• Individual-level HB estimates are available and the median is reported.
• The price attribute uses linear coding with sign constraints.

10. Simulation-Based WTP (Single Scenario): Algorithm & Proof

Originally proposed by Orme (2020, Section 9.6; Orme & Chrzan, 2017, Ch. 13) as the preferred method. Instead of dividing utilities by a price coefficient, this method uses the market simulator to find the price increase that exactly offsets a feature improvement's share gain. Section 11 extends this to multiple randomized scenarios (SOS Bootstrap).

10.1 Simple Version (Two Products)

1. Simulate Product A (with feature, current price) and Product B (without feature, same price).
2. Both start at 50% share (identical except for the focal feature).
3. Raise Product A's price until both products return to 50% share.
4. WTP = price increase needed to equalize shares.

10.2 Competitive Version (Strongly Recommended)

This is Orme's strongly preferred approach. It accounts for the full competitive landscape. The underlying algorithm performs these steps automatically:

Why Simulation-Based Is Superior

• Accounts for competition, addresses the fundamental flaw of the ratio method.
• References specific products, not an abstract "average product."
• Focuses on relevant customers, via individual-level utilities, considers current customers and likely switchers.
• Reveals cross-effects and substitution patterns between competitors.
• Captures segment-specific price sensitivity and interaction effects.

10.3 When Identical Competitors Collapse to Pseudo-WTP

Proof sketch: In a logit model, the choice probability for product j is:

With N identical competitors (each having utility V_c) and one focal product with utility V_f:

The simulated WTP procedure finds the price change Δp such that the focal product's share returns to its baseline after adding a feature with utility δ. Setting the shares equal:

Since the structure is identical on both sides, the N cancels, and the solution is simply Δp = δ / βp, exactly the pseudo-WTP formula. This means that for studies where competition is identical or undefined, the simple ratio method is mathematically equivalent.

10.4 Position Dependence, Heterogeneity & Extrapolation

The identical-competitors proof above has an important corollary: in a homogeneous logit market, the break-even price increase is Δp = δ / βp regardless of the focal product's starting price or baseline share. The starting position cancels out exactly the way N did.

With heterogeneous respondents this invariance breaks. The aggregate Share of Preference is a mixture of individual logit shares:

The break-even search solves S(p₀ + Δp | upgraded) = S(p₀ | baseline). Because each respondent's share contribution saturates at a different rate, the solution Δp depends on which respondents are near their choice margin at p₀:

• At a moderate baseline share, many respondents sit on the steep part of their individual logit curves. The mixture behaves approximately like an average-β market and Δp ≈ δ / β̄_p.
• At an extreme baseline share (very low or very high), the respondents still in play are a selected tail. For a low baseline share, they are the respondents with the smallest |β_p,i|, the aggregate share then responds very weakly to further price increases, and Δp is driven toward δ / min|β_p,i| rather than δ / β̄_p. The measured WTP inflates accordingly.

Two practical consequences follow:

1. Simulation WTP is a positional measure. Reporting it without stating the focal price and baseline share is incomplete; two runs of the same upgrade at different starting positions can legitimately differ by an order of magnitude (see the worked example in Section 3.4).
2. Out-of-range break-even prices are extrapolations. The linear price coefficient is estimated from choices inside the design's price range. When the search leaves that range, utility is extended linearly into territory the data never observed. Choicery flags this case explicitly; such values should not be reported as market estimates. The absence of a None option amplifies the problem, because the focal share is bounded below by the respondents whose competitor options are all unattractive, without an outside option they cannot leave the market at any price.

This is the same class of aggregation phenomenon that Ofek & Srinivasan (2002) address with MVAI (Section 12): monetizing an attribute change at the market level requires weighting respondents by their marginal impact on share, and that weighting depends on the market position at which it is evaluated. SOS Bootstrap (Section 11) mitigates, though does not remove, position dependence by averaging over many competitive contexts and, optionally, focal configurations.

11. SOS Bootstrap: Sampling of Scenarios (Orme, 2021)

Standard Simulation WTP (Section 10) computes the price a firm could charge for an attribute improvement in a specific competitive scenario. But this raises a question: how sensitive is the WTP estimate to the particular competitors chosen?

Orme (2021) addresses this with Sampling of Scenarios (SOS): instead of computing WTP in one fixed competitive context, run the procedure across many randomly sampled competitive contexts and report the median WTP as a robust, scenario-independent estimate.

11.1 Motivation

In practice, researchers often face uncertainty about which competitors to include. Different competitor configurations can produce materially different WTP estimates. SOS addresses this by:

• Marginalizing over competitive uncertainty, the median WTP is robust across many plausible competitive landscapes.
• Quantifying estimation uncertainty, a respondent bootstrap gives a confidence interval for the SOS median, and the spread of the scenario values shows the sensitivity to the competitive context.
• Reducing researcher degrees of freedom, no single competitor definition drives the result.

11.2 Algorithm

11.3 Why the Median?

The median is used instead of the mean for two reasons:

1. Robustness to outlier scenarios: Some randomly generated competitive scenarios may produce extreme WTP values (e.g., when all competitors happen to have very low utility, making the focal product dominant). The median is unaffected by such outliers.
2. Consistency with Orme's recommendation: Orme (2021) and Orme & Chrzan (2017, Ch. 13) consistently recommend the median for WTP distributions, whether across respondents or across scenarios.

11.4 Bootstrap Percentile Intervals

The confidence interval is estimated using the percentile bootstrap method (Efron & Tibshirani, 1993). The procedure:

1. Draw B = 500 resamples of the respondents with replacement (each of size n).
2. Evaluate every scenario for every resample and take, per resample, the median over the successful scenarios.
3. The 2.5th and 97.5th percentiles of these B medians form the 95% CI.
4. Their standard deviation is the bootstrap standard error.

Only when the respondent bootstrap is unavailable does Choicery resample the scenario values instead (B = 2,000, labelled Scenario-resample 95% interval).

11.5 Handling Design Constraints

When generating random competitors in “Random from levels” mode, Choicery respects the study's design constraints:

• Prohibitions: Impossible attribute combinations (e.g., “Budget brand at 50,000 EUR”) are rejected. The generator resamples until a valid combination is found (up to 500 attempts per competitor).
• Conditional pricing: If the study shows brand-specific amounts for its price levels (e.g., “Low” is 9 EUR for one brand and 12 EUR for another), each randomly generated competitor keeps the shared relative price level that HB estimated. The amounts shown in the survey are not substituted, so WTP on this relative axis is not a brand-specific amount in euros.

11.6 Boundary Estimates and Failed Iterations

Not every scenario yields a valid WTP. Two types of non-results are tracked:

If all iterations fail, Choicery reports “All iterations failed. Check your product configuration.”, typically indicating a problem with the price attribute setup or the focal product definition.

11.7 Relationship to Other Methods

12. MVAI: Market Value of an Attribute Improvement

Developed by Ofek & Srinivasan (2002), MVAI provides a theoretically grounded, closed-form expression for the market's value of a product attribute improvement. It answers: "How much more could a firm charge for an attribute improvement while keeping market share constant?"

12.1 Core Insight

MVAI is defined as the ratio of the change in market share due to an attribute improvement to the change in market share due to a price increase (Ofek & Srinivasan, 2002, Eq. 7):

In words: the incremental price that maintains constant market share after an attribute improvement. This is equivalent to the Simulated WTP concept (Section 10.2) but derived analytically.

12.2 Closed-Form Expression (Logit Model)

Using the multinomial logit framework with individual-level parameters, Ofek & Srinivasan (2002, Eq. 14) derive:

where the customer weights ai are:

Here:

• wik = individual i's utility weight for attribute k
• wip = individual i's price sensitivity
• θji = individual i's probability of choosing firm j's product
• μi = individual i's logit scale parameter (inversely related to error variance)
• qi = individual i's maximum purchase quantity

12.3 The Critical Weighting Factor: θ(1 − θ)

12.4 MVAI vs. Simple Averaging of Individual Ratios

The standard practice (Orme, 2001; Wyner, 1997) is to average individual WTP ratios wik/wip across respondents. Ofek & Srinivasan (2002) show this differs substantially from MVAI:

13. Equilibrium Profits (Allenby et al.)

Allenby, Brazell, Howell, and Rossi (2014) argue that the only sensible metric for assessing a product feature's value to a firm is its impact on profits. They define the economic value of a feature enhancement as:

where A* is the market with the enhanced product and A is the market without it, both evaluated at their respective Nash equilibrium prices.

13.1 Requirements

1. Cost information, marginal production cost per unit for each product.
2. Complete competitive set, all major competitors with their attributes and prices.
3. Nash equilibrium computation, iterative algorithm to find equilibrium prices where no firm can improve profits by unilaterally changing price.
4. Outside option, a "none" alternative for category demand modeling.

13.2 Why Allenby et al. Criticize Pseudo-WTP

Allenby et al. (2014) make two key critiques:

1. It ignores stochastic utility: The logit model includes random utility errors. Even if the deterministic utility of product A is higher, a competitor may be chosen due to the random component. Pseudo-WTP ignores this entirely.
2. It is scale-invariant in a way that profits are not: Pseudo-WTP is invariant to which product the feature is applied to. But the actual economic value of adding a feature depends heavily on the product's current market position.

Their formal WTP measure, derived from the indirect utility function (Eq. 2.12):

14. The “Gilligan's Island” Problem

Both Orme (2020) and Orme & Chrzan (2017) use this parable to illustrate the fundamental flaw of WTP without competition:

15. Hypothetical Bias

Respondents consistently spend "survey dollars" more readily than real ones. This makes price utilities too small (respondents appear less price-sensitive than they actually are), which inflates WTP (Orme & Chrzan, 2017, Ch. 2 and 13).

Remedies

16. Attribute Non-Attendance (ANA)

When respondents completely ignore the price attribute during choice tasks, their estimated price coefficient approaches zero. Dividing by a near-zero price coefficient produces enormous WTP values (approaching infinity). This is a major source of WTP outliers (Orme, 2001; Hensher, Rose, & Greene, 2005).

Remedies

1. Zero out non-attended attributes and re-estimate, culling respondents who ignored both the focal attribute and price (Orme, 2001).
2. Use median WTP rather than mean across respondents, this effectively trims extreme outliers.
3. Fixed-price parameter (Revelt & Train, 1998): Estimate feature utilities as random coefficients but price as a fixed coefficient. Prevents huge WTP from price-ignoring respondents. Caveat: This assumes the same price sensitivity for all respondents, which is "almost certainly false" (Orme & Chrzan, 2017).
4. Sign constraints on price (Sablotny-Wackershauser et al., 2024): Constrain the price coefficient to be negative during HB estimation. This prevents reversed signs but may "pile up" probability mass near zero.

17. Respondent Heterogeneity

WTP can be estimated at three levels of granularity:

The field is moving toward individual-level WTP, considering the full distribution rather than a single point estimate. Sampling from HB posterior draws gives a more complete picture (Orme & Chrzan, 2017, Ch. 13). Best practice is to report both the median (robust to outliers) and the distribution (histogram or kernel density).