# Estimation of Demand and Substitution in Case of Multinomial Logit Model

In MNL model, given the different alternative choices, we estimate the probability of choice across different options.

E.g. using the fishing data, we can use the following calculations to estimate the probability across 4 alternatives for each individual choices.

Fish <- mlogit.data(Fishing, shape = "wide", varying = 2:9, choice = "mode")

mlogit.model1 <- mlogit(mode ~ 1 | income, data=Fish)
result <- as.data.frame(fitted(mlogit.model1, outcome = FALSE))


The probabilities are as follows:

My question is, 1) How to calculate the substitution probabilities across each item among each other, given the probabilities of each item?

2) How to obtain the total demand using MNL model?

If someone can help me to answer these 2 questions, it will be of great help. I came across this article and similar others. But nowhere they explicitly defined how to solve these 2 questions.

Regarding Q2) I think the MNL is built to predict shares of different choices. A very useful book is: Train "Discrete Choice Methods with Simulation" which is available here: https://eml.berkeley.edu/books/train1201.pdf

On p.38 the formula for calcularing the overall share of a certain choice j is given by:

$\hat{N}_j=\sum_{i=1}^n \hat{P}_{ij}$

where $\hat{P}_{ij}$ is the probability of subject i choosing alternative j. If you now would go one step further and calculate the overall demand you should calculate

$\sum_j \hat{N}_j$

but recall that this will yield allways N - the number of subjects in your sample- since the probabilities of each subject among all choices sum up to 1.

Regarding Q1) I'm not quite shure what you mean. If you want to calculate the chance of choosing alternative j over k for subject i, this is simply $\frac{P_{ij}}{P_{ik}}$ This formula can be found on p.54

• Thanks for sharing the answer. But just 1 more question. The substitution you suggested is if 2 items are IIA (based on Train's book). How will it change if IIA condition does not fulfil? – Beta Sep 22 '17 at 11:04
• @ Beta: Let's start with an equivalent method, that manifests the IIA: cross elasticities of alternative k with respect to a variable - take the price of the alternative - of alternative j: $\frac{\partial P_{ik}}{\partial p_{j} }=P_{ik} P_{ij} \beta_{p}$. This version of the IIA means, that if the attribute price of alternative j changes, all other alternatives are affected. Recall that the change of the prob. of alternative j does depend only on the alternative with the canged price and alternaitve j. This might lead to weired substiutution patterns. But now comes the point:... – Jogi Sep 22 '17 at 11:28
• ...according to the simple MNL model, the substitution pattern is defined as shown before. So there is no change of anything in the MNL world. What you need to do, to get reasonable elasticities, is to switch to a nested or mixed logit model, which do have different formulas for the choice prob. and also different versions of the elasticities. – Jogi Sep 22 '17 at 11:29