Disutility costs

One form for a discrete choice model is the multinomial logit (MNL):

\[\begin{split}U_{in} & = V_{in} + \varepsilon_{in} = \beta^\prime x_{in} + \varepsilon_{in} \\ P(i | C_n) & = P(U_{in} = \max_{j \in C_n}{U_{jn}}) = \frac{\exp{\mu V_{in}}}{\sum_{j \in C_n}{\exp{\mu V_{jn}}}}\end{split}\]

wherein:

Symbol	Description
\(C_n\)	Choice set of alternatives for agent \(n\)
\(U_{in}\)	Agent \(n\)’s utility of alternative \(i\)
\(\varepsilon_{in}\)	Random part of utility \(\sim \text{ExtremeValue}(0, \mu)\)
\(V_{in}\)	Systematic part of utility, a function of observables
\(x_{in}\)	Vector of observables for alternative \(i\) and agent \(n\)
\(\beta\)	Vector of parameters
\(P(i | C_n)\)	Probability that agent \(n\) chooses alternative \(i\)

Some points about this general formulation:

• The particular random distribution selected for \(\varepsilon_{in}\) is what makes this a logit as opposed to probit or other kind of model.

• Commonly \(C_n\) is the same for all \(n\).

• When individual data are used to estimate the model (i.e. \(\beta\)), then \(n\) may enumerate every individual.

  On the other hand, when the estimated model is used to predict the behaviour of 1 or more representative agents, then \(n\) enumerates those agents: e.g. single representative agent for each country in a model; or a single representative agent for each of 2 or more consumer groups within each such country.

• Some elements of \(x_{in}\) may be 0 for a given \(i\) and for all \(n\). This allows the concept of “different functional forms,” so long as all forms are linear.

  For example: suppose \(C_n\) includes the alternatives bus and walk, for which the systematic utilities are:

  \[\begin{split}V_{\text{bus},n} & = a \cdot \text{speed}_n + b \cdot \text{ticket price}_n \\ V_{\text{walk},n} & = a \cdot \text{speed}_n + c \cdot \text{temp}_n\end{split}\]

  The observable \(\text{ticket price}\) is only meaningful for the bus alternative, and not for walking. Likewise, the outdoor temperature \(\text{temp}\) is only meaningful for the walking alternative, but not for riding the (climate controlled) bus.

  These functional forms appear different, but both are linear, so they can be written as:

  \[\begin{split}V_{in} & = \beta^\prime x_{in} \\ \beta & = \begin{bmatrix}a & b & c\end{bmatrix} \\ x_{in} & = \begin{bmatrix}\text{speed}_{in} & \text{ticket price}_{in} & \text{temp}_in\end{bmatrix}\end{split}\]

  Now when \(\text{ticket price}_{\text{bus},n} = 0 \forall n\), i.e. the third element is always 0 when the alternative is bus; and likewise \(\text{temp}_{\text{walk},n} = 0 \forall n\).

  This method also covers the possibility the agents are differently sensitive to the speeds of the bus and of walking. Then instead of \(\text{speed}\), define two distinct observables \(\text{speed}^\text{walk}\) and \(\text{speed}^\text{bus}\), with distinct corresponding coefficients in \(\beta\), and treat them the same way.

Cost equivalents

Suppose \(C_n\) is a set of some vehicle types: internal combustion engine vehicle (ICEV), electric vehicle (EV), etc.

\[\begin{split}x^{A\prime}_{in} & = \begin{bmatrix} \text{purchase price} \\ \text{variable price} \\ \text{fuel/energy price} \\ \text{charging station availability} \\ \text{technology novelty} \\ \text{peer social influence} \\ \end{bmatrix}_{in} \\ \beta^A & = \begin{bmatrix} b_1 & b_2 & b_3 & b_4 & b_5 & b_6\end{bmatrix}\end{split}\]

In this model, which we call the ‘full’ model:

• Some observables (the first three) are costs, i.e. they are measured in money, with units of EUR or USD.

• Some observables (the last three) are not costs. They are measured with non-monetary units.

• If we have estimated this model, then we have the choice probabilities \(P(i | C_n)\) for every agent, and the values of \(\beta^A\).

Then we define disutility cost equivalents as follows. We specify a ‘reduced’ model that has fewer elements in the vectors of observables \(x^B_{in}\) and parameters \(\beta^B\), but yields the same choice probabilities.

\[\begin{split}\beta^{A\prime} x^A_{in} & = \beta^{B\prime} x^B_{in} \\ x^{B\prime}_{in} & = \begin{bmatrix} \text{purchase price} \\ \text{variable price} \\ \text{fuel/energy price} \\ \text{disutility cost} \\ \end{bmatrix}_{in} \\ \beta^B & = \begin{bmatrix} b_1 & b_2 & b_3 & -1\end{bmatrix} \\ \text{disutility cost}_{in} & = \begin{bmatrix} b_4 \\ b_5 \\ b_6 \\ \end{bmatrix} \cdot \begin{bmatrix} \text{charging station availability} \\ \text{technology novelty} \\ \text{peer social influence} \\ \end{bmatrix}^\prime_{in}\end{split}\]

Exercise: use the expression at the top of the page to show that \(V^B_{in} = \mu \beta^{B\prime} x^B_{in}\) yields the same choice probabilities \(P(i | C_n)\) as \(V^A_{in} = \mu \beta^{A\prime} x^A_{in}\).

Note that:

• Each observable, e.g. \(\text{charging station availability}\), may be 0 for all \(n\) for certain alternatives \(i\) (e.g. ICEV) where it is not relevant.

• The \(\text{disutility cost}\) is a positive number with monetary units (e.g. 1024 USD); we arbitrarily select a fixed value of \(-1 \left[\frac{1}{\text{USD}}\right]\) for its parameter in \(\beta^B\). This is for an intuitive and consistent interpretation: a greater disutility reduces the total systematic utility \(V^B_{in}\).