Elasticity with random parameter in mixed logit.

[Niranjan]

I have a question regarding the utility equation for understanding the elasticity in mixed logit model.

I can think of two ways of including elasticity, suppose i have a utility equation with time and cost and income.

1) U = B_time * Travel time + [([B_cost + sigma_cost * normal ) * (income)^B_inc] * Travel cost

2) U = B_time * Travel time + [B_Cost * (income)^B_inc + sigma_Cost *normal] * Travel cost

Are both of these methods correct? especially the second one where the sensitivity is observed just for the mean parameter while variance observed will be same compared to a MMNL model without income parameter (my guess is the second method should still work with change in interpretation). Or is it that only the first method is correct where variance also changes according to income. Is there any published work where i can see the interpretation of parameter for Mixed logit case?

Additionally in the first method how can i check for the significance of the whole term (B_cost + sigma_cost * normal ) * (income)^B_inc (like delta method) as there is a variable income in between the parameters.

[stephanehess]

Hi

my first observation needs to be that you really should not use a Normal distribution for a cost coefficient.

That aside, if you want to capture an income elasticity on the sensitivity to cost, then the income effect should be multiplying the cost attribute, implying that each unit of the cost attribute matters less for someone with higher income. That is in line with your specification 1

Regarding significance testing of the whole term, this would be complicated and would require you to manually apply the delta method

Stephane

[Niranjan]

Thankyou for your valuable insights, i wanted to add a quick simple theoretical question (assuming the cost attribute is still normal),
Does addition of the term to capture an income elasticity as mentioned by doing [([B_cost + sigma_cost * normal ) * (income)^B_inc] means we are changing the distribution of cost parameter [([B_cost + sigma_cost * normal ) * (income)^B_inc] from simple normal distribution as well?

[dpalma]

Hi,

In the approach you describe: bCost = ([B_cost + sigma_cost * normal ) * (income)^B_inc, bCost would remain normal.
A normal random variable remains normal under linear transformations, and what you are doing is scaling the normal variable by a factor of the income (assuming B_inc is not random), so bCost remains normal. This is true for each individual.

Now, at the sample level, you have that all individuals have normal random coefficients for the cost, but each o those coefficients have different mean and s.d. (because the income varies from one individual to the next). Notwithstanding, the central limit theorem guarantees that the average of all those coefficients will also be normally distributed.

Cheers
David

[Niranjan]

Yes that does makes perfect sense. Thank you

[martinf]

Dear Professor

Thank you for providing us the opportunity to work with your software Appolo !
I want to implement a RPL model in order to calculate the WTP of renewable energies (wind power, solar and biogas).
In particular, I would like to add the price elasticity to income, i.e. to measure how WTP may vary according to variations of income.
Looking at your example number 3 on your website, I would like to know if this equation could be the right equation?

with b_price = -exp((b_price_x) + sigma_log_price*draws_price) and other terms follow a normal distribution.

U = ASC + b_price*(b_photo*photo +b_eol*eol+b_meth*meth + price) + b_price*((income/mean(income)^b_elasticitypriceincome )*price

Best regards,

Martin

[stephanehess]

Hi Martin

not quite, as you're using the price attribute twice there

instead, it should be:

U = ASC + b_price*(b_photo*photo +b_eol*eol+b_meth*meth + (income/mean(income)^b_elasticitypriceincome*price)

Stephane