Disutility costs

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

$$\begin{array}{rl}{U}_{in}& =V_{in}+{\epsilon }_{in}={\beta }^{\prime }{x}_{in}+{\epsilon }_{in}\\ P(i|C_n)& =P(U_{in}=\underset{j\in C_n}{max}{U}_{jn})=\frac{\exp \mu V_{in}}{\sum _{j\in C_n}\exp \mu V_{jn}}\end{array}$$

wherein:

Symbol	Description
$C_n$	Choice set of alternatives for agent $n$
$U_{in}$	Agent $n$’s utility of alternative $i$
${\epsilon }_{in}$	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 ${\epsilon }_{in}$ 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:

  $$\begin{array}{rl}{V}_{\text{bus},n}& =a\cdot {\text{speed}}_n+b\cdot {\text{ticket price}}_n\\ V_{\text{walk},n}& =a\cdot {\text{speed}}_n+c\cdot {\text{temp}}_n\end{array}$$

  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:

  $$\begin{array}{rl}{V}_{in}& ={\beta }^{\prime }{x}_{in}\\ \beta & =\left[\begin{array}{ccc}a& b& c\end{array}\right]\\ x_{in}& =\left[\begin{array}{ccc}{\text{speed}}_{in}& {\text{ticket price}}_{in}& {\text{temp}}_{i}n\end{array}\right]\end{array}$$

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

  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.

$$\begin{array}{rl}{x}_{in}^{A\prime }& ={\left[\begin{array}{c}\text{purchase price}\\ \text{variable price}\\ \text{fuel/energy price}\\ \text{charging station availability}\\ \text{technology novelty}\\ \text{peer social influence}\end{array}\right]}_{in}\\ {\beta }^A& =\left[\begin{array}{cccccc}{b}_1& b_2& b_3& b_4& b_5& b_6\end{array}\right]\end{array}$$

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 $x_{in}^B$ and parameters ${\beta }^B$, but yields the same choice probabilities.

$$\begin{array}{rl}{\beta }^{A\prime }{x}_{in}^A& ={\beta }^{B\prime }{x}_{in}^B\\ x_{in}^{B\prime }& ={\left[\begin{array}{c}\text{purchase price}\\ \text{variable price}\\ \text{fuel/energy price}\\ \text{disutility cost}\end{array}\right]}_{in}\\ {\beta }^B& =\left[\begin{array}{cccc}{b}_1& b_2& b_3& -1\end{array}\right]\\ {\text{disutility cost}}_{in}& =\left[\begin{array}{c}{b}_4\\ b_5\\ b_6\end{array}\right]\cdot {\left[\begin{array}{c}\text{charging station availability}\\ \text{technology novelty}\\ \text{peer social influence}\end{array}\right]}_{in}^{\prime }\end{array}$$

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

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 $V_{in}^B$.