VOLUMEN DE SUMINISTRO

We used Bayesian procedures to estimate the individual-level utility parameters within a mixed logit model (described by Train 2003). The mixed logit approach introduces preference heterogeneity by ‘individualizing’ preferences; each respondent has a possibly unique set of preference parameters. As it is not practical to estimate the parameter vector governing the behaviour of individual respondents, preference parameters are instead defined as random draws from a joint distribution and mixed logit models estimate a distribution of these parameters from the full sample (see Train 2003). Mixed logit models also eliminate the assumption of independence from irrelevant alternatives50 _{which is a} restrictive assumption of the standard logit model. Bayesian procedures estimate the distribution of coefficients in the population and combine such information with the individual respondents’ choices to derive posterior or conditional estimates of the individual respondents’ tastes. Bayesian estimation outperforms the classical maximum likelihood simulation in three ways. First Bayesian procedures do not require maximisation of the simulated maximum likelihood function and therefore imposes less computational burden. In addition to problems of convergence, there is often no guarantee that the global maximum has been reached with the chosen starting value. Second, Bayesian estimation procedures are more consistent since they do not rely on the number of draws used in simulation approaches (Train 2003). Third and last, they can easily produce individual-level parameters and hence the calculation of a random parameter difference between the first and revised individual-level utility parameters.

All parameters are specified as random and given a normal distribution, except the price parameter which is fixed. While responses to the price parameter are very likely to vary across respondents

50 _{This independence means that the unobserved portion of utility for one alternative is unrelated to the}

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according to unobservable factors (i.e. factors independent of observable socioeconomic variables), fixing the price parameter greatly facilitates the estimation of WTA estimates51 _{and hence avoids} implausibly large WTA estimates which is often the case when the price parameter is set as random (Scarpa et al. 2008).

The utility function of an individual i facing a choice between two experimentally created alternatives and a reference level alternative in choice task t can be described as:

𝑉(𝐴𝑆𝐶, 𝑋𝑖𝑗𝑘, 𝛽𝑖) + 𝜀𝑖𝑗𝑡 if j=reference level alternative, otherwise,

𝑉(𝑋𝑖𝑗𝑘, 𝛽𝑖) + 𝜀𝑖𝑗𝑡

Where 𝑈𝑖𝑗 is the utility function for individual i, for alternative j, 𝛽𝑖 is a vector of preference parameters

that vary over households in the population (rather than being fixed) and are hence treated as random variables, 𝑋𝑖𝑗𝑘 is a vector of observed variables that relate to the alternatives j and respondents i, and

𝜀_𝑖𝑗𝑡 is a Gumbel distributed error term, ASC is an alternative specific constant (ASC) for the reference level.

On day 1, we specify the utility function (𝑈𝑖𝑗) of an individual i of the alternative j as:

𝑈

= 𝛽

𝑋

+ 𝜀

𝑈

= (𝛽

− 𝛽

)𝑋𝑖𝑗𝑡 + 𝜀

(

𝑈

= 𝑈

+ 𝑈

Bayesian procedures are often called hierarchical Bayes since they involve a hierarchy of parameters. 𝛽_𝑖 is a vector of individual-level parameters for individual i which describe the tastes of that individual. In the absence of any prior knowledge the parameters 𝛽𝑖 are assumed to be normally distributed in

the population with mean vector 𝛽̃ and covariance matrix Σ. From Bayes rule, the joint posterior distribution of the parameters is proportional to the individual likelihood times the prior distribution of the model parameters and can be expressed as:

51 _{The analytical expression for WTA calculation involves a ratio where the denominator is the price coefficient.}

When the price parameter is fixed, the distribution of WTA estimates may be inferred directly from the distribution of the non-price coefficient

(4.8)

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where α means “is proportional to”, i refers to the ith _{household out of N, Li(𝛽}

𝑖) is the likelihood of

household i's data conditional on 𝛽𝑖 and N(𝛽𝑖|𝛽̃, Σ) is the random effect distribution indexed by the

parameters, 𝛽̃ and Σ; p(𝛽̃,Σ) is a prior distribution placed on 𝛽̃ and Σ.

Draws from this joint posterior distribution are obtained through Gibbs sampling. That is, a draw is taken from the conditional posterior of each parameter, given the previous draw of other parameters (see Casella and George 1992). Under unrestrictive conditions, the mean of the Bayesian posterior of a parameter is asymptotically equivalent to the maximum likelihood estimator of the parameter. Similarly, the variance of the posterior distribution is the asymptotic variance of this estimator. Hence, the results obtained by Bayesian procedures can be interpreted from a purely classical perspective. We present the results the same way as that of classical estimation by giving the mean estimates and the 95% confidence intervals for each parameter. The model was estimated in R 3.2.1 using the RStan package (Stan Development Team 2015). We also accounted for the effect of socio-demographic variables (table 4.2) by interacting them with the random parameter differences (equation 4.7). We were particularly interested in whether literacy significantly affects the effect of more time to deliberate.

The derivation of the marginal rate of substitution is straightforward and leads to WTA estimates. They are defined as follows:

𝑊𝑇𝐴 = 𝛽𝑖

𝛽𝑝𝑟𝑖𝑐𝑒 (4.9)

Where 𝛽𝑖 are the attribute coefficients and 𝛽𝑝𝑟𝑖𝑐𝑒 are the price coefficients.

The standard errors and the 95% confidence intervals for these estimates of mean marginal willingness to pay are obtained by using the Delta method (Hensher et al. 2005). Hierarchical procedures produce a unique set of coefficients for each respondent. We can calculate average WTA for any number of different subgroups (e.g. by literacy level) by linking these coefficients with each respondent’s socioeconomic data. To compare the WTA estimates between day 1 and day 2, we cannot rely on a direct comparison of 95% confidence intervals for the mean WTA estimates using paired t-tests as non-overlapping confidence intervals may be biased indicators of the significance of differences in estimated means (Poe et al. 2005). To test for WTA differences between day 1 and day 2, we use the complete combinatorial method suggested by Poe et al. (2005) to develop precise confidence bounds for the difference between the means. This is a non-parametric test that involves empirically estimating the confidence interval around the difference of the mean WTA values.

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4.4 Results