Why Two Ways of Estimating Willingness to Pay Can Tell the Same Story

Researchers often estimate the value of a faster journey by studying choices between alternatives with different prices and travel times. Such estimates can inform transport appraisal, but their interpretation depends on what the model assumes about how people respond to cost and time.

In a theoretical note, Stephane Hess of the University of Leeds brings together earlier results and develops further examples to clarify two ways of writing these models: preference space and willingness-to-pay, or WTP, space. The paper is an arXiv preprint submitted on 5 November 2026; the landing page does not indicate journal acceptance. It offers mathematical arguments, not new estimates of what travellers actually pay.

Two descriptions of a choice

In preference space, a model estimates separate coefficients describing how cost, travel time and other attributes affect the attractiveness, or utility, of an option. The value of a small reduction in travel time is obtained by dividing the time coefficient by the cost coefficient. In WTP space, the same utility can be written using the cost coefficient and a parameter representing the monetary value of time.

Hess's central test is whether the two specifications, with the same assumptions about unexplained choice variation, imply the same alternative-specific constants and the same marginal effects of every attribute, apart from normalisations needed for identification. When coefficients vary randomly across people, the corresponding joint distributions must match. Matching the ratio of time and cost effects alone is not enough: their absolute sizes also influence predicted choices.

For models without random coefficients, and with a nonzero cost coefficient, the paper restates an established result: equivalent specifications have the same maximum likelihood, WTP estimates and asymptotic standard errors. Calculating a valuation after estimation in preference space, with a standard error from the delta method, is not inherently inferior to estimating it directly in WTP space. Small numerical differences can arise, but larger discrepancies may signal different specifications or failure to converge.

There is a computational trade-off. Rewriting utility in WTP space makes it nonlinear in its parameters. Hess notes that this can make optimisation more sensitive to starting values, even where the underlying model has the same optimum. The cost coefficient is still estimated in the usual WTP-space formulation; the apparent one in front of cost inside the expression is not a constraint fixing that coefficient to one.

When the reported parameter is not everyone's value

The paper extends the discussion to observable differences between people. Suppose cost sensitivity varies with income while sensitivity to travel time does not. Then the value of time changes with income because it is the ratio of those two sensitivities. A WTP-space model can represent this relationship, but its directly estimated valuation may apply only at a reference income; other values must be calculated from the relevant marginal effects.

That is different from making income multiply the entire monetised part of utility. In that case, the time and cost effects change together and their ratio stays the same. The distinction matters for interpreting whether income affects willingness to pay or instead changes the scale of that part of utility.

Nonlinear cost creates another complication. If utility depends on the logarithm of cost, the marginal effect of cost varies with the price of the alternative, so the value of time varies with it too. Hess shows how to normalise the expression so that a valuation parameter corresponds directly to a chosen reference price. That reference-point value is not necessarily the sample average. Nor is a marginal value automatically the payment for a substantial time saving: with nonlinear cost, the latter requires a separate calculation of the amount that keeps utility unchanged.

The hidden role of distributions

Models often allow preferences to vary for reasons not captured by observed characteristics. Here, choosing convenient distributions in each space can produce genuinely different models. A better fit then reflects those distributional assumptions, not an inherent advantage of one set of coordinates.

Hess shows that a latent class model, which assigns people probabilistically to groups with their own coefficient vectors, can be written equivalently in either space when each class has a nonzero cost coefficient and corresponding restrictions match. A class with zero cost sensitivity cannot be represented by a finite WTP ratio.

For continuously distributed coefficients, equivalence requires matching the joint distributions implied by the transformation. One example is fixed-sign, jointly lognormal coefficients with an unrestricted covariance matrix: the corresponding model in the other space can represent the same distribution. Independence restrictions generally do not carry over. For example, assuming cost sensitivity and the value of time are independent in WTP space can imply that more cost-sensitive people are also more sensitive to travel time.

The practical lesson is to inspect both the implied utility sensitivities and the implied WTP distribution, rather than treating independence as a neutral default. These mathematical equivalences do not establish which assumptions best describe a particular population; that remains a question for empirical modelling and data.