ApolloChoiceModelling forum

Dear Stephane,

I have looked in the literature (including the Hensher textbook and your recent articles on WTP) and cannot find answers to my questions below. I was hoping you might be able to shed some light on them given your expertise and the fact this can be done in Apollo so might be relevant to others too.

I understand that WTP is a marginal rate of substitution and is simply calculated as the ratio of two attributes for example:

WTP = Beta(k) / Beta(c) , where k is an attribute of interest and Beta(c) is cost attribute

In my DCE, I calculate Willingness-to-wait similar to the above i.e., WTW = Beta(k) / Beta(t) , where t is a time attribute.

I have used the Delta Method in Apollo to calculate the SE around the mean estimate of WTW and it is very large. This is expected, we know from the Beta's and can see through choice probability analysis that people's sensitivity to wait time varies based on a few key attributes (this is fine). However, I also note that most published DCE's also report WTP with a large SE. Which made me ask why do analysts often just report the mean WTP as this doesn't provide a very good (nuanced) picture for interpretation. It would be useful to depict WTP in scenarios, similar to how we present scenarios in choice probability analysis. This would be particularly useful in my paper where WTW is a very important outcome.

On this, in the Hensher textbook (section 8.4.3 on Willingness to pay) it states the following:

Another popular transformation [of the WTP equation] is to include an interaction between two attributes, such as between xk and x1, where the latter could be a socio-economic characteristic or another attribute of an alternative.The WTP equation becomes:

WTP = Beta(k)*x1/Beta(c).

Questions:
[1] My first question is why is it so uncommon to use interactions in WTP estimation? I cannot see any examples in the literature where they go beyond presenting the mean WTP. At face value, it seems like a convenient way to show WTP within different groups or scenarios?

[2] My second question is, what we do see in the literature (sometimes) is subgroup analysis of WTP by effectively “splitting” a sample up - e.g. using beta’s from a model only run on a sub-group with certain characteristics. Is there any reason why, other than ease of use, this would be used over the approach of using interactions in the WTP equation? Do you have any strong opinions on this type of subgroup analysis for WTP? We wouldn’t split the sample in this way to calculate model predictions, so it seems like a second-best approach to looking at WTP estimates?

[3] My third question relates to the additive nature of utility functions and whether my understanding is correct. If a utility function is linear in attributes and coefficients, in the WTP equation, is it valid to add the marginal utility of a main effect + any associated utility/ disutility of an interaction effect with the main effect and use this in calculation of WTP? E.g.:

In my experiment, instead of using WTW = Beta(k) / Beta(t). I would include a scenario variables (e.g. Quality of Life) that were presented as constant between alternatives, but change between choice tasks - in this respect they’re not quite demographics but Hensher states you can interact with attributes. If my interpretation of the Hensher explanation is correct, then my equation to work out the WTW of someone with “poor QOL” becomes:

WTW(for someone with poorQOL) = (Beta(Certainty) + Beta(Certainty*PoorQol)) / (Beta(wait) + Beta(wait*PoorQOL)

**Note here I add the Beta of the interaction with PoorQol with Certainty and Wait on the numerator and the denominator respectively because we know that wait for someone with PoorQOL is not the same so without this the ratio between the (added) coefficients wouldn’t be a fair comparison?

Is this approach correct? It gives a very reasonable WTP estimate that we would expect (after looking at other Betas and predictions etc.) for people with poorQOL compared to the mean.

Thanks in advance for any advice it is much appreciated!

Rob

Hi

if you include interactions in the utilities, e.g. with socio-demographics, then you will get different WTP for different subgroups of people, and you can calculate an estimated WTP and a standard error for each group. I wouldn't use the term mean, as this relates to a distribution of wtp.

the WTP is the ratio of partial derivatives, and if the utility includes interactions, then so will WTP.

You say

[1] My first question is why is it so uncommon to use interactions in WTP estimation? I cannot see any examples in the literature where they go beyond presenting the mean WTP. At face value, it seems like a convenient way to show WTP within different groups or scenarios?

Many studies use interactions in WTP, so I'm not sure what to say

[2] My second question is, what we do see in the literature (sometimes) is subgroup analysis of WTP by effectively “splitting” a sample up - e.g. using beta’s from a model only run on a sub-group with certain characteristics. Is there any reason why, other than ease of use, this would be used over the approach of using interactions in the WTP equation? Do you have any strong opinions on this type of subgroup analysis for WTP? We wouldn’t split the sample in this way to calculate model predictions, so it seems like a second-best approach to looking at WTP estimates?

If you have interactions, you get different WTP. You can calculate WTP for each group, or you'll need to use sample enumeration across the sample to get a mean WTP. This is easy to do in Apollo

[3] My third question relates to the additive nature of utility functions and whether my understanding is correct. If a utility function is linear in attributes and coefficients, in the WTP equation, is it valid to add the marginal utility of a main effect + any associated utility/ disutility of an interaction effect with the main effect and use this in calculation of WTP? E.g.:

In my experiment, instead of using WTW = Beta(k) / Beta(t). I would include a scenario variables (e.g. Quality of Life) that were presented as constant between alternatives, but change between choice tasks - in this respect they’re not quite demographics but Hensher states you can interact with attributes. If my interpretation of the Hensher explanation is correct, then my equation to work out the WTW of someone with “poor QOL” becomes:

WTW(for someone with poorQOL) = (Beta(Certainty) + Beta(Certainty*PoorQol)) / (Beta(wait) + Beta(wait*PoorQOL)

**Note here I add the Beta of the interaction with PoorQol with Certainty and Wait on the numerator and the denominator respectively because we know that wait for someone with PoorQOL is not the same so without this the ratio between the (added) coefficients wouldn’t be a fair comparison?

Is this approach correct? It gives a very reasonable WTP estimate that we would expect (after looking at other Betas and predictions etc.) for people with poorQOL compared to the mean.

This is in line with what we say above - you just need to not think about WTP as being a ratio of two coefficients, but a ratio of partial derivatives of the utility

Stephane & David

Hello,

I saw this and thought I might chime in.

A reference/example I have been using lately in my classes that illustrates the point about how there can be many conditional values of WTP (due to, e.g., interactions) is the paper by Axhausen, et al. [Transport Policy 15 (2008) 173–185] "Income and distance elasticities of values of travel time savings: New Swiss results."

In this case WTP is value of travel time savings, which can vary along many dimensions (e.g., income, trip distance, etc.). This paper estimates many interaction effects and eschews random coefficients. It also eschews the approach where the sample is completely broken up into sub-samples. A main feature of this application is that it includes a nonlinear elasticity expression that parsimoniously captures the effect of, e.g., income.

Given your questions, I think you should check out this paper.

-- David Bunch