Análisis de constitucionalidad del impuesto propuesto

A choice modelling study seeks an understanding of the preferences of a non-marketed good or service. In a choice modelling survey, a set of alternatives is presented to respondents, who are asked to choose the most-preferred alternative. Lancaster’s Theory of Demand (1966) provides the framework for the design of the alternatives in the survey questionnaire. A set of attribute describes the alternatives and different levels of the attributes characterise each alternative. The analyst observes the choices made by the respondent and estimates the underlying utility of the respondent.

A commonly used model to describe individual choice is the random utility model introduced in McFadden (1974) (Scarpa & Rose 2008). The remainder of this section first introduces the random utility model before discussing the methods to estimate individual utility from the choices made by the respondent.

4.1.1 Random utility model

As discussed in Section 2.2.2, if the preferences of an individual are complete, transitive, continuous and monotonous, then the individual’s preference ranking can be mapped to a real-valued utility function that assigns higher numerical values to preferred alternatives.

61

Using the notation from Section 2.2.2, suppose individual 𝑖 chooses from 𝐾 number of alternatives, each indexed 𝑘, then the individual’s utility for alternative 𝑘 is denoted 𝑢_𝑖𝑘.

When faced with a set of 𝐾 number of alternatives, individuals will make choices based on their underlying preference ranking. Hence, if the individual chooses a specific alternative, 𝑘_𝑎, when faced with 𝐾 number of alternatives, then the individual reveals that she prefers 𝑘_𝑎 to all other alternatives in the choice situation. Concomitantly, the utility associated with 𝑘_𝑎 is the highest among all other alternatives, implying that:

𝑢𝑖𝑘𝑎 ≥ 𝑢𝑖𝑘𝑏,𝑎 ≠ 𝑏 (4–1)

or, equivalently,

𝑢𝑖𝑘𝑎 = argmax

𝐾∈{𝑘1,𝑘2,…,𝑘𝐾}

𝑢𝑖𝑘 (4–2)

The random utility model provides the conceptual base for modelling the choices of individuals. An individual’s preference for each alternative could be influenced by many factors. The factors affecting individual preferences could be broadly categorised into factors which can be observed by the analyst24 _{and factors which cannot be observed by the analyst}25_.

As such, the individual’s utility, 𝑢_𝑖𝑘, is partitioned into an observed component, 𝑉_𝑖𝑘 and an unobserved component, 𝜀𝑖𝑘 (Hensher, Rose & Greene 2015). If the observed and unobserved components of utility is additively separable, the utility function can be written as26:

𝑢_𝑖𝑘 = 𝑉_𝑖𝑘+ 𝜀_𝑖𝑘 (4–3)

The observed component of utility can be modelled to be a function of the variables of the attributes which describe each alternative and the covariates describing the individual as well as a set of parameters to be estimated. Specifically, 𝑉_𝑖𝑘 can be defined to be:

𝑉_𝑖𝑘 = 𝑓(𝑍_𝑖𝕚𝑘, 𝛽) (4–4)

24 _{An example of an observable characteristic is the demography of the respondent (e.g., age and income of the}

respondent). The analyst could design the survey questionnaire to include questions on the demography of the respondent. Hence, these characteristics of the respondent would be observable to the analyst.

25 _{Examples of an unobservable characteristic are the background level of noise and the mood of the individual.}

The analyst may not have the equipment to record the background level of noise when the respondent is completing the survey and this factor would be an unobserved characteristic of the respondent. Similarly, the analyst may not be able to quantify the mood of respondents when they are answering the survey questionnaire.

26 _{Similar to all utility functions, the utility function described in Equation (4–3) is ordinal and any positive}

monotonic transformation of this function will result in an equivalent ranking of preferences. Consequently, scaling Equation (4–3) by a positive real number, 𝜆 ∈ ℝ+_{, yields 𝜆𝑢}

𝑖𝑘= 𝜆𝑉𝑖𝑘+ 𝜆𝜀𝑖𝑘. For the remainder of this

62

where 𝑍_𝑖𝕚𝑘 is a vector of 𝕚 attributes describing alternative 𝑘. 𝛽 is a set of parameters to be estimated by the analyst and 𝛽 assigns weights to each attribute in 𝑍_𝑖𝕚𝑘.

To model the unobserved component of utility it is necessary to make assumptions about how these preferences are formed. A common assumption is that for each alternative, the unobserved component of utility, 𝜀_𝑖𝑘, is randomly distributed with some density over individuals 𝑛 and choice situation 𝑠 (Hensher, Rose & Greene 2015). This probability density function is denoted 𝑓(𝜀𝑖𝑘) and assumptions about the functional form of the probability density function leads to different econometric models.

Suppose that the probability density function, 𝑓(𝜀𝑖𝑘), exists, then it is possible to reframe Equation (4–1) and Equation (4–2) in probabilistic terms. Define 𝑃_𝑖𝑘𝑎 to be the probability that alternative 𝑘_𝑎 yields the highest level of utility among 𝐾 number of alternatives, i.e., the probability that alternative 𝑘_𝑎 is chosen among 𝐾 alternatives, then,

𝑃_𝑖𝑘𝑎 = Pr(𝑢_𝑖𝑘𝑎 > 𝑢_𝑖𝑘𝑏) ,𝑎 ≠ 𝑏 (4–5)

substituting Equation (4–3) yields:

𝑃_𝑖𝑘𝑎 = Pr(𝑉_𝑖𝑘𝑎+ 𝜀_𝑖𝑘𝑎 > 𝑉_𝑖𝑘𝑏+ 𝜀_𝑖𝑘𝑏) ,𝑎 ≠ 𝑏 (4–6)

Equation (4–5) and (4–6) implies that the utilities for each choice situation, 𝑢_𝑖𝑘𝑎, are linked to other utilities by the probability function (Hensher, Rose & Greene 2015). Since 𝑃_𝑖𝑘𝑎 is a probability, for all alternatives 𝑃_𝑖𝑘𝑎 is bounded by 0 and 1. In addition, choosing 𝑘_𝑎 precludes the choice of 𝑘_𝑏, 𝑏 ≠ 𝑎. Assuming further that the set of alternatives, 𝐾, is exhaustive, then all the probabilities must sum to one, i.e., ∑𝐾 𝑃_𝑖𝑘

𝑘=1 = 1. Consequently, ceteris

paribus, an increase in 𝑢_𝑖𝑘𝑎 implies that 𝑃_𝑖𝑘𝑎 will increase, i.e., it is more likely that the respondent will choose 𝑘_𝑎. Furthermore, 𝑃_𝑖𝑘𝑏, 𝑏 ≠ 𝑎 will fall, i.e., it is less likely that the respondent will choose 𝑘𝑏, 𝑏 ≠ 𝑎.

4.1.2 Conditional logit model

While different probability density functions can be used to describe 𝑓(𝜀_𝑖𝑘)27_{, a} commonly used function is the extreme value type one (EV1) distribution, i.e., a Gumbel

27 _{Another candidate distribution function is the normal probability distribution function. However, there is no}

closed form solution to estimate the probabilities that a particular alternative will be chosen. Consequently, estimating the probabilities is computationally intensive, which has led to the popularity of the use of the Gumbel distribution (Hensher, Rose & Greene 2015).

63

distribution (Train 2003). The probability distribution function of a standard Gumbel distribution is (Gumbel 1954):

𝑓(𝜀_𝑖𝑘) = 𝑒−(𝜀𝑖𝑘+𝑒−𝜀𝑖𝑘) (4–7)

with a corresponding cumulative distribution of:

𝐹(𝜀_𝑖𝑘) = 𝑒−𝑒−𝜀𝑖𝑘 _(4–8)

McFadden (1974) showed that if 𝑓(𝜀_𝑖𝑘) is described by the standard Gumbel distribution, where 𝜀𝑛𝑠𝑗 is independently and identically distributed, with a variance of

𝜋2

6, then the probability that 𝑗𝑎 is chosen can be described by the logistic function:

𝑃_𝑖𝑘𝑎 = 𝑒

𝑉𝑖𝑘

∑𝐾 𝑒𝑉𝑖𝑘

𝑘=1

,𝑘_𝑎 ∈ 𝐾 (4–9)

Recall from Equation (4–4) that the observed component of utility, 𝑉_𝑖𝑘, is a function of 𝕚 attributes describing alternative 𝑘, 𝑍𝑖𝕚𝑘 and a set of parameters, 𝛽. If 𝑉𝑖𝑘 is described by a linear combination of 𝑍_𝑖𝕚𝑘 and 𝛽, then 𝑉_𝑖𝑘 can be defined as:

𝑉𝑖𝑘 = ∑ 𝛽𝕚𝑍𝑖𝕚𝑘 𝕀

𝕚=1

(4–10)

Substituting Equation (4–10) into Equation (4–9) yields: 𝑃𝑛𝑠𝑗𝑎 = 𝑒∑𝕀𝕚=1𝛽𝕚𝑍𝑖𝕚𝑘 ∑𝐾 𝑒∑𝕀𝕚=1𝛽𝕚𝑍𝑖𝕚𝑘 𝑘=1 ,𝑗𝑎 ∈ 𝐽 (4–11)

Equation (4–11) also reframes the interpretation of the respondent’s observed component of utility as an underlying latent variable and the logistic function as a link function. The observed component of utility is a continuous variable which can take the value of any real number, which is mapped to the probability space (bounded by 0 and 1) by the logistic function. Hence, the analyst is able to estimate the utility function of the respondent by observing which alternatives are chosen by the respondent when the respondent encounters a given choice situation.

4.1.3 Limitations of the conditional logit model and alternative models

While the conditional logit model provides a closed-form solution to understanding the utility function of the respondent, the assumption that the unobserved component of utility is independently and identically distributed is restrictive (Train 2003). This assumption implies that the unobserved component of utility is not related to the unobserved component of another

64

alternative. Consequently, alternatives to the conditional logit model have been developed to relax this assumption.

One possible specification which relaxes the restrictive assumptions that the unobserved component of utility is independently and identically distributed is the random parameter logit model (Hensher, Rose & Greene 2015). Similar to the conditional logit model, the probability that an alternative, 𝑗_𝑎, is chosen when individual 𝑛 is faced with choice situation 𝑠, is defined by Equation (4–10). However, unlike the conditional logit model, the random parameter logit model assumes that at least some of the parameters in the model are randomly distributed over the population such that:

𝑉_𝑖𝑘 = ∑ 𝛽_𝑖𝕚𝑍_𝑖𝕚𝑘 𝕀

𝕚=1

(4–12)

where 𝛽_𝑖𝕚 is described by a constant, 𝛽̅_𝕚, a vector of observable characteristics of respondent, 𝑧𝑖, and a vector of unobservable characteristics, 𝑣𝑖𝕚. If these terms are linearly separable, then, 𝛽𝑖𝕚 is defined as:

𝛽_𝑖𝕚 = 𝛽̅ + ∆𝑧_{𝕚 𝑖}+ Γ𝑣_𝑖𝕚 (4–13)

where ∆ and Γ are the parameters to be estimated.

Equation (4–13) indicates that the mixed logit model is a generalisation of the conditional logit model. Specifically, if ∆= 0 and Γ = 0, then 𝛽_𝑖𝕚 = 𝛽̅_𝕚 and Equation (4–12) is equivalent to Equation (4–10). This also implies that the analyst could model the observed characteristics of the respondents in order to control for potential correlation between the unobserved components of utility.

4.1.4 Welfare estimates

Recall from Section 2.2.2 that the willingness-to-pay for publicly-provided noise abatement corresponds to the compensating variation. Taking the total derivative of the utility function and setting the change in utility to be zero yields the marginal benefits associated with a change in attribute, 𝑀𝐵𝑖(𝑍𝕚𝑖𝑘), measured in dollar terms. This marginal benefits function is:

𝑀𝐵𝑖(𝑍𝕚𝑖𝑘) = − 𝜕𝑢_𝑖𝑘 𝜕𝑍𝕚𝑖𝑘 ⁄ 𝜕𝑢_𝑖𝑘 𝜕𝑝𝑟𝑖𝑐𝑒𝑖𝑘 ⁄ (4–14)

Since the observable component of utility described in Equation (4–12) is linear in the attributes, the partial derivative of the utility function with respect to changes in the levels of

65

each attribute is the regression coefficient, 𝛽_𝕚. As such, the marginal benefits associated with the change in each attribute can be rewritten as:

𝑀𝐵𝑖(𝑍𝕚𝑖𝑘) = −

𝛽𝕚 𝛽𝑐𝑜𝑠𝑡

(4–15) Since the marginal benefits described in Equation (4–15) is a function of the regression parameters, each of which is distributed around the mean described by some variance, the willingness-to-pay is also distributed around the point estimate. In order to estimate the variance of the marginal benefits, a variety of methods have been developed. Examples include the delta method, the Fieller method, and the Krinsky & Robb bootstrapping procedure (Hole 2007a; Krinsky & Robb 1986). Unlike the delta and Fieller methods, both of which assumes a normal distribution, the Krinsky & Robb method does not assume that the marginal benefits follow any distribution.