WTP for mixed logit with random parameters

[JvB]

Dear Michiel,

I have a question on the calculation of willingness to pay in mixed logit:

Is it appropriate to randomise the price attribute as well as the other two attributes of my experiment AND then use the random parameter averages to calculate the WTP (mu_Attribute/mu_price)?
I am not sure if it might make more sense to not randomise the price due to the fact that it might be imprecise to devide randomised parameter by randomised parameter (even if the sigma of the price is significant and therefore indicates that there is heterogeneity in price).

Could you give any insights on this?

Thank you in advance and best wishes,
J.

[Michiel Bliemer]

For computing WTP in mixed logit where coefficients are random, you can do the following:

1. Assume a fixed coefficient for cost
2. Assume a probability distribution with non-zero mass at zero (such as lognormal or constrained triangular, but NOT normal)
3. Estimate parameters in WTP-space instead of preference-space, where the cost coefficient is normalised to -1 and a scale parameter lambda is estimated, i.e. V = lambda * (b1*X1 + b2*X2 - 1*cost), where b1 and b2 are directly the WTP distributions

In cases 1 and 3 you can easily compute mean(WTP) directly. In case 2 you will need to simulate both distributions by taking draws from both distributions, dividing them, and averaging them. It is not sufficient to simply divide the means in this case.

This is a relevant paper to look at:
https://link.springer.com/article/10.1007/s11116-011-9331-3

If you would like to compute the standard error of WTP and test for statistical significance, please look at these papers:

https://www.sciencedirect.com/science/article/pii/S0191261513001707
https://www.sciencedirect.com/science/article/pii/S0191261522001886

Michiel

[JvB]

Thank you very much for your valuable insights.

Regarding option 2:
You mean assuming the cost coefficient being e.g. lognormal distributed and the other attributes being normally distributed?

Best,
J.

[Michiel Bliemer]

Yes that is what I mean, the cost coefficient cannot have a non-zero probability mass at 0 to avoid division by zero, but other coefficients in the numerator can have any distribution.

[JvB]

Okay, so if my cost coefficient is lognormally distributed and my other attributes are normally distributed, I can compute WTP according to papers mentioned in option 2?

[Michiel Bliemer]

Yes you can.
The last paper is the best method using the Delta method, but you can also use the Krinsky & Robb method (which requires more extensive simulation).

[JvB]

With regard to option 1:
Is it appropriate here to compute the WTP by calculating
mu_Attribute/fixed price parameter
or is it also necessary to use the distribution of the random attribute parameter instead of the mean and divide it by the fixed price parameter?

Your advice is highly appreciated.
Thank you very much.
J.

[Michiel Bliemer]

With option 1, the random parameter of your attribute, beta, will have a mean of mu and a standard deviation of sigma, E(beta)=mu and Var(beta) = sigma^2. If you divide this with a fixed price coefficient, gamma, then the WTP will have a distribution with mean E(beta/gamma) = mu/gamma and Var(beta/gamma) = (sigma/gamma)^2. So you can simply divide both mu and sigma by the fixed price parameter to get the distribution.

If you want to compute the confidence interval of the WTP distribution, then you need to account for the standard errors of mu, sigma, and gamma using Delta method in previously mentioned papers.

Michiel