Hidrogeología y vulnerabilidad de los acuíferos

Due to the flat rate of tax imposed on lottery tickets in the UK, analysis of the tax incidence over the income distribution is equivalent to estimating the income elasticity of demand. This follows simply from the fact that the tax contributed by any given consumer is directly proportional to their expenditure. By far the most common approach in previous studies of the regressivity of lottery products using micro-level data is to estimate Engel curves in which the number of tickets purchased by an individual or household is linear in the level income or expenditure. Only minor deviations from this specification have been considered

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when using micro-data – for example, Farrell and Walker (1999) include a squared income term – and aggregate-level studies only deviate by taking logs of the regressand. For comparative purposes, Section 4.6.1 briefly replicates this approach by estimating the functional form:

𝐸₍ = 𝑋₍+_{𝛽 + 𝛾𝑌}

(+ 𝜀( (4.1)

where 𝐸₍ is expenditure by individual 𝑖 on lottery tickets, 𝑌₍ is some measure of income, and

𝑋( are controls for various demographic characteristics. Though clearly convenient, this naïve

specification is deficient in both theoretical and empirical aspects.

Although linear Engel curves as described in equation (4.1) will satisfy the additivity criterion – that income elasticities sum to 1 – when estimated as a set of all commodity groups, such functional forms are unnecessarily restrictive and only result from specific, and unlikely, utility functions (Pollak, 1971). Moreover, Leser (1963) notes that such a specification often produces negative expenditure values within the observed range of incomes and that changes in elasticity across the distribution are not consistent with economic theory (e.g. that the implied income elasticity of inferior goods increases with income). As a result, these models are typically ‘rejected’ by the data since they produce poor fit statistics and large estimated standard errors.

The preferred model of this chapter is one in which the budget share of lottery tickets for a given individual, 𝑤( = 𝐸(/𝑌(, is linear in log total expenditure, ln 𝑌(, and estimates are

presented in Section 4.6.2. This model was proposed by Working (1943) and later discussed by Leser (1963). The Working-Leser model is formally described as,

𝑤₍ = 𝑋₍+_{𝛽 + 𝛾 ln 𝑌}

( + 𝜀( (4.2)

where 𝑋₍+ _{are again controls for various demographic characteristics}70_{, 𝛽 is a vector of}

parameters to be estimated, and 𝛾 is a parameter to be estimated describing the effect of log income, ln 𝑌₍. Since 𝑤₍ is simply lottery expenditure divided by income, eliciting income elasticity of lottery tickets is trivial and given by 𝜂 = 1 +_æå. This model underpins the Almost

70_{Experiments with the specification to include controls for the number of people in the household made}

very little difference to the resultant estimates of income elasticity. Estimates obtained using Heckman selection and double hurdle routines with household size controls are therefore relegated to Appendix Table A4.19. Accounting for household size with equivalised income resulted in smaller income elasticities than those already reported, some unbelievably so.

Ideal Demand System (AIDS) developed by Deaton and Muellbauer (1980) which is often used with macro-level data to determine the income elasticities of broad commodity groups. The AIDS model is popular in the wider economics literature due, in part, to its derivation from consumer theory and the simple functional form.

The model used here deviates from the AIDS specification only by omitting price information for three reasons. First, the inclusion of price is only indirectly relevant for investigating the relationship between expenditure on lottery tickets and income. Second, due the repeated cross-sectional nature of the data any draw-by-draw price variation, such as that arising from rollovers, for all observations in each 2-week diary window is the same. Thus, variation in observed expenditure resulting from a change in price can be easily captured using year-month fixed effects, which are employed throughout the analysis in Section 4.6. Third, this strategy also circumvents issues posed by the anonymization processes of the data which results in only the month of the diary being included. That is, even if there were some advantage to be gained from including draw-by-draw price as an explanatory variable in the model, doing so is impossible since the data are censored to protect the identity of respondents in such a way that it is not clear which are the relevant draws in the diary window. Using fixed effects for each month within the sample should sufficiently capture variation in expenditure caused by prices, and is unlikely to be improved upon by, say, including average price of lotto within each month which would hardly vary at all. By adopting the Working-Leser specification here, the literature on the income elasticity of lotto and the regressivity of its taxation is brought up to date with the broader economic studies of other goods such as carbon taxes (Brannlund and Nordstrom, 2004), alcohol (Gil and Molina, 2009), and comprehensive studies over several commodity groups (e.g. Banks et al 1997).

The use of total expenditure, rather than income, on the right-hand side of equation (4.2) is common throughout the literature and respects a plausible two-stage budgeting process in which utility-maximising households first allocate disposable income between consumption and saving, then decide on the quantities of specific goods – in this case lottery tickets – to be consumed. Total expenditure then reflects the disposable income available to households in the current period. Moreover, reported gross or net incomes – particularly for those who are self- employed – can vary significantly over the short and medium term. Total expenditure, meanwhile, can be seen as a measure of average (or expected) levels of income, thus providing a better measure of income in the long-run.

In Section 4.6.4 the assumption that budget share of lottery tickets is linear in log income is relaxed by estimating the following equation using semi-parametric estimation as described in Robinson (1988),

𝑤₍ = 𝑋₍+_{𝛽 + 𝑓(ln 𝑌}

() + 𝜀( (4.3)

in which 𝑓 is some unknown function. This agnostic methodology allows full flexibility in the possible shape of 𝑓. Moreover, following Hardle and Mammen (1993) it is possible to test whether the function 𝑓 can be sufficiently approximated using polynomial expansions of ln 𝑌₍. This essentially becomes a test of whether the functional form imposed by the Working-Leser specification in equation (4.2) is rejected by the data.