Demand for culture in Spain and the 2012 VAT rise

Abstract

This paper analyzes the effects that the 2012 VAT rise in Spain had on household’ demand for cultural goods and services. Household’ demands are modeled as a two-stage QUAIDS. After estimating price and expenditure elasticities, and the pass-through parameter associated with the reform, our results show that the individual welfare loss and the increment in the tax bill increase, but less than proportionately, with income. Consequently, the reform can be considered as regressive. Relating the effects of the VAT reform to households’ incomes also implies a low quantitative effect, because of the low proportion of total household expenditure that cultural expenditure represents. From a social perspective, the size of the induced welfare loss would positively depend on society’s inequality aversion. Regardless of the latter, it cannot be concluded whether the reform would have increased or reduced inequality in the distribution of cultural spending. Our results prove qualitatively robust to alternative values of the pass-through parameter.

Appendix

1.1 Individual and social welfare: theory

We follow the methods of Urzúa (2001) and first established in King (1983b). We first obtain the tax revenues of the hth household before and (expected) after the tax reform, $R_{h}^{0}$ and $R_{h}^{1},$, respectively. Thus, one trivially has that $R_{h}^{0} = \sum _{i=1}^{n}t_{i}^{0}p_{i,h}^{0}q_{i,h} ^{0}/(1+t_{i}^{0})$ and $R_{h}^{1}=\sum _{i=1}^{n}t_{i}^{1}p_{i,h}^{1}q_{i,h}^{1}/(1+t_{i}^{1})$, where $q_{i,h}^{0}$ and $q_{i,h}^{1}$ denote the hth household’s quantity demanded of the ith good at the old and the new price vector, respectively. We next obtain $q_{i,h}^{1}$. Assuming a Marshallian demand function $q_{i,h}=q_{i,h}(p_{1,h},p_{2,h},\ldots p_{n,h} ,m_{h})$, where $m_{h}$ denotes total expenditure on cultural goods and services, totally differentiating both sides, assuming further as usual that the $m_{h}$ were invariant to the VAT reform, approximating $dp_{j,h}\approx p_{j,h}^{1}-p_{j,h}^{0}$, using Eq. (22) and, finally, approximating $q_{i,h}^{1}\approx$ $q_{i,h}^{0}+dq_{i,h}$, it can be shown that

$$\begin{aligned} q_{i,h}^{1}=q_{i,h}^{0}\left[ 1+\sum _{j=1}^{n}E_{i,j}^{u}\frac{(t_{j} ^{1}-t_{j}^{0})\times \gamma _{j}}{1+t_{j}^{0}}\right] , \end{aligned}$$

(23)

for $i=1,2,\ldots ,n$, where $E_{i,j}^{u}$ denotes the uncompensated (Marshallian) demand cross-price elasticity of good i with respect to price j, which was obtained in Eq. (10). Note that $q_{i,h}^{1}$ depends on the pass-through parameter, $\gamma _{j}$, and so does the (expected) after-tax reform, $R_{h}^{1}$, above defined.

Once price effects of changing the tax rates have been computed, the individual welfare change arising from the tax reform for each household can be estimated in different ways following standard microeconomics. One possible way is the equivalent variation, EV$_{h}$, or the amount of money which would have to be given to the hth household when it faces the initial price vector, $p_{h}^{0}$, to make it as well off as it would be facing the new price vector, $p_{h}^{1}$, with its initial cultural expenditure, $m_{h}$ (Gravelle and Rees 2004). More formally, upon denoting this household’s indirect utility function by V, EV$_{h}$ is implicitly defined by $V(m_{h}+EV_{h},p_{h}^{0})$ $\equiv V(m_{h},p_{h}^{1})$, so that EV$_{h}<0$ for $p_{h}^{1}\ge p_{h}^{0}$, $p_{h}^{1}\ne p_{h}^{0}$. Thus, from Eqs. (1)–(4) one can explicitly solve for EV$_{h}$ as

$$\begin{aligned} {\hbox {EV}}_{h}=a({\mathbf {p}}_{h}^{0})\times \exp \left\{ \frac{b({\mathbf {p}}_{h} ^{0})\times \ln \left[ m_{h}/a({\mathbf {p}}_{h}^{1})\right] }{b({\mathbf {p}} _{h}^{1})+\left[ \lambda ({\mathbf {p}}_{h}^{1})-\lambda ({\mathbf {p}}_{h} ^{0})\right] \times \ln \left[ m_{h}/a({\mathbf {p}}_{h}^{1})\right] }\right\} -m_{h}. \end{aligned}$$

(24)

As a closely related concept, one could also define the final equivalent expenditure, EE$_{h}^{F}$, as

$$\begin{aligned} V({\hbox {EE}}_{h}^{F},{\mathbf {p}}_{h}^{0})\equiv V(m_{h},{\mathbf {p}}_{h}^{1} ), \end{aligned}$$

(25)

(i.e., the expenditure required at pre-reform prices to attain the same level of utility as with post-reform prices) so that EE$_{h}^{F}\equiv m_{h}+{\hbox {EV}}_{h}<m_{h}$ for $p_{h}^{1}\ge p_{h}^{0}$, $p_{h}^{1}\ne p_{h}^{0}$.

As an alternative to EV$_{h}$, one can also consider the compensating variation, CV$_{h}$, or the amount of money which must be taken from the hth household’s cultural expenditure, $m_{h}$, when facing the new price vector, $p_{h}^{1}$, in order to make it as well off as it was when it faced the old price vector, $p_{h}^{0}$ (Gravelle and Rees 2004). In other words, CV$_{h}$ is implicitly defined as $V(m_{h}-{\hbox {CV}}_{h},p_{h}^{1})$ $\equiv V(m_{h},p_{h}^{0} )$, so that CV$_{h}<0$ for $p_{h}^{1}\ge p_{h}^{0}$, $p_{h}^{1}\ne p_{h}^{0}$. From Eqs. (1)–(4) one has that CV$_{h}$ is explicitly solved for as

$$\begin{aligned} {\hbox {CV}}_{h}=m_{h}-a({\mathbf {p}}_{h}^{1})\times \exp \left\{ \frac{b({\mathbf {p}} _{h}^{1})\times \ln \left[ m_{h}/a({\mathbf {p}}_{h}^{0})\right] }{b({\mathbf {p}} _{h}^{0})+\left[ \lambda ({\mathbf {p}}_{h}^{0})-\lambda ({\mathbf {p}}_{h} ^{1})\right] \times \ln \left[ m_{h}/a({\mathbf {p}}_{h}^{0})\right] }\right\} . \end{aligned}$$

(26)

As was the case with the equivalent variation, the compensating variation allows one to define the initial equivalent expenditure, EE$_{h}^{I}$, as

$$\begin{aligned} V({\hbox {EE}}_{h}^{I},{\mathbf {p}}_{h}^{1})\equiv V(m_{h},{\mathbf {p}}_{h}^{0} ), \end{aligned}$$

(27)

(i.e., the expenditure required at post-reform prices to attain the same level of utility as with pre-reform prices) so that EE$_{h}^{I}\equiv m_{h}-{\hbox {CV}}_{h}>m_{h}$ for $p_{h}^{1}\ge p_{h}^{0}$, $p_{h}^{1}\ne p_{h}^{0}$. Both EE$_{h}^{F}$ and EE$_{h}^{I}$ are, therefore, monetary measures of the hth household’s welfare after the tax reform which can be easily computed after Eqs. (24) and (26), respectively. A welfare-enhancing reform (i.e., EV$_{h}$, CV$_{h}>0$) will imply that EE$_{h}^{I}<m_{h}<{\hbox {EE}}_{h}^{F}$. And, similarly, a reducing welfare reform (i.e., EV$_{h}$, CV$_{h}<0$) will imply that EE$_{h}^{I}> m_{h}>{\hbox {EE}}_{h}^{F}$. Note that, by construction, EE$_{h}^{F}-{\hbox {EE}}_{h}^{I}\equiv {\hbox {EV}}_{h}+{\hbox {CV}}_{h}$, which, in case of a welfare loss (gain) as a result of the tax policy, will be negative (positive).

As a complement to these individual welfare change measures, we also consider the welfare effects from a social point of view, i.e., the social value of the reform. Borrowing from the tradition in the related literature, we assume the existence of an indirect social welfare function, W, defined in terms of the vector of equivalent expenditures $\widehat{\hbox {EE}}=[{\hbox {EE}}_{1},{\hbox {EE}}_{2},\ldots ,{\hbox {EE}}_{H}]$ given by

$$\begin{aligned} W(\cdot )=\left\{ \begin{array}{ll} \frac{1}{H}\sum _{h=1}^{H}\frac{{\hbox {EE}}_{h}^{1-\varepsilon }}{1-\varepsilon },&{}\quad {\text { for} } \quad \varepsilon \ne 1\\ \\ \frac{1}{H}\sum _{h=1}^{H}\ln {\hbox {EE}}_{h},&{}\quad {\text { for} } \quad \varepsilon =1 \end{array} \right. , \end{aligned}$$

(28)

where parameter $\varepsilon$ captures the degree of aversion to social inequality (see Atkinson 1970). From Eq. (28) one can derive a measure of social value, the proportional increment in initial equivalent expenditure $\widehat{\hbox {EE}}^{I}=[{\hbox {EE}}_{1}^{I}, {\hbox {EE}}_{2}^{I} ,$ $\ldots ,{\hbox {EE}}_{H}^{I}]$, which we denote by $\lambda$, and which is defined as follows: the proportional increase in initial equivalent expenditure that would make it possible to match the social welfare created by the reform $\widehat{\hbox {EE}}^{F}=[{\hbox {EE}}_{1}^{F},{\hbox {EE}}_{2}^{F},\ldots ,{\hbox {EE}}_{H}^{F}]$. Or, more formally,

$$\begin{aligned}&W(\lambda \times {\hbox {EE}}_{1}^{I},\lambda \times {\hbox {EE}}_{2}^{I},\ldots ,\lambda \times {\hbox {EE}}_{H}^{I})\nonumber \\&\quad \quad =\,W({\hbox {EE}}_{1}^{F},{\hbox {EE}}_{2}^{F},\ldots ,{\hbox {EE}}_{H}^{F}), \end{aligned}$$

(29)

so that, given that EE$_{h}^{I}>{\hbox {EE}}_{h}^{F}$ as a result of $p_{h}^{1}\ge p_{h}^{0}$ and $p_{h}^{1}\ne p_{h}^{0}$, a value $\lambda <1$ denotes a social welfare loss induced by the reform.

Along the same lines, the equivalent expenditure function can also be used to construct inequality indices for the distribution of equivalent expenditure. Borrowing from Prieto-Rodriguez et al. (2005) who, in turn, follow Atkinson (1970) and Sen (1973), we define the equally distributed equivalent expenditure, G, as the equivalent expenditure level that, distributed equally among all households, would provide the same level of social welfare as the actual distribution of equivalent expenditure. We can define two alternative expressions for G, depending on whether we consider the initial equivalent expenditure, $G^{I}$, or the final equivalent expenditure, $G^{F}$, and whose precise definitions are given by

$$\begin{aligned} W(G^{I},G^{I},\ldots ,G^{I})\equiv W({\hbox {EE}}_{1}^{I},{\hbox {EE}}_{2}^{I},\ldots ,{\hbox {EE}}_{H} ^{I}) \end{aligned}$$

(30)

and

$$\begin{aligned} W(G^{F},G^{F},\ldots ,G^{F})\equiv W({\hbox {EE}}_{1}^{F},{\hbox {EE}}_{2}^{F},\ldots ,{\hbox {EE}}_{H} ^{F}). \end{aligned}$$

(31)

Inequality indices can be easily computed as follows. Denote the average initial and final equivalent expenditures as $\overline{{\hbox {EE}}^{I}} \equiv H^{-1}\sum _{h=1}^{H}{\hbox {EE}}_{h}^{I}$ and $\overline{\hbox {EE}^{F}}\equiv H^{-1}\sum _{h=1}^{H}{\hbox {EE}}_{h}^{F},$ respectively. If W is concave (therefore denoting inequality aversion), then $G^{I}\le \overline{\hbox {EE}^{I}}$ and $G^{F}\le \overline{\hbox {EE}^{F}}$. Under the assumption that $W(\cdot )$ is symmetrical and concave, previous definitions provide two inequality indices (one for the initial equivalent expenditure, the other for the final equivalent expenditure):

$$\begin{aligned} A^{I}\equiv 1-\frac{G^{I}}{\overline{\hbox {EE}^{I}}}, \quad {\text {and } }\quad A^{F} \equiv 1-\frac{G^{F}}{\overline{\hbox {EE}^{F}}}, \end{aligned}$$

(32)

where $A^{I},A^{F}\in [0,1]$.

Given Eq. (28) is conveniently solved to yield $G^{I}$ and $G^{F}$ as

$$\begin{aligned} G^{I}(\varepsilon )=\left[ \frac{1}{H}\sum _{h=1}^{H}\left( {\hbox {EE}}_{h}^{I}\right) ^{1-\varepsilon }\right] ^{\frac{1}{1-\varepsilon }}, \quad {\text {and } }\quad G^{F}(\varepsilon )=\left[ \frac{1}{H}\sum _{h=1}^{H}\left( {\hbox {EE}}_{h}^{F}\right) ^{1-\varepsilon }\right] ^{\frac{1}{1-\varepsilon }}, \end{aligned}$$

for $\varepsilon \ne 1$, and

$$\begin{aligned} G^{I}(\varepsilon )=\exp \left\{ \frac{1}{H}\sum _{h=1}^{H}\ln {\hbox {EE}}_{h} ^{I}\right\}, \quad {\text {and } }\quad G^{F}(\varepsilon )=\exp \left\{ \frac{1}{H}\sum _{h=1}^{H}\ln {\hbox {EE}}_{h}^{F}\right\} , \end{aligned}$$

for $\varepsilon =1$. Four remarks follow. First, note that from Eqs. (28)–(31) the welfare change can be easily computed as

$$\begin{aligned} \lambda (\varepsilon )=\frac{G^{F}(\varepsilon )}{G^{I}(\varepsilon )}, \end{aligned}$$

(33)

where $G^{I}(\varepsilon )$ and $G^{F}(\varepsilon )$ have just been obtained immediately above, and both $G^{I}$ and $G^{F}$ are expressed as explicitly dependent on $\varepsilon$, the parameter reflecting the degree of aversion to social inequality for the social welfare function W in Eq. (28). Second, it is the case that $\overline{\hbox {EE}^{I}}=G^{I}(\varepsilon )$ and $\overline{\hbox {EE}^{F}}=G^{F}(\varepsilon )$ if and only if $\varepsilon =0$; that is to say, equally distributed equivalent expenditures equal average equivalent expenditures if and only if there is no inequality aversion. Third, $\overline{\hbox {EE}^{I}}>G^{I}(\varepsilon )$ and $\overline{\hbox {EE}^{F}}>G^{F}(\varepsilon )$ if and only if $\varepsilon >0$. And, fourth, taking into account Eq. (32), the social welfare change associated with the reform in Eq. (33) can be rewritten as

$$\begin{aligned} \lambda (\varepsilon )=\frac{\left[ 1-A^{F}(\varepsilon )\right] \times \overline{\hbox {EE}^{F}}}{\left[ 1-A^{I}(\varepsilon )\right] \times \overline{\hbox {EE}^{I}}}, \end{aligned}$$

(34)

where the inequality indices in Eq. (32) have been explicitly expressed as functions of $\varepsilon$. This means that the proportional social gain equals the increment in the mean equivalent expenditure $\overline{\hbox {EE}^{F}}\times \overline{\hbox {EE}^{I}}^{-1}$ times the change in the (equality) indices $\left[ 1-A^{F}(\varepsilon )\right] \times \left[ 1-A^{I}(\varepsilon )\right] ^{-1}$.

1.2 Robustness check exercise

We show below the results corresponding to one alternative value for the pass-through parameter estimation: the extreme case of $\hat{\gamma }=100.00\%$ (i.e., VAT tax rate changes are completely shifted to consumers, so that producers’ prices stay constant; as noted above, the usual assumption in this kind work) (see Tables  12,  13,  14).

Consider the complete shift case whose results are shown in Tables  12 and  13 (in absolute and in relative terms, respectively) and in Table 14. The pattern is clear. First, post-reform tax revenues (and the corresponding increments) are higher for higher pass-through parameter values, which is simply a natural consequence of the price-inelastic demands for cultural goods and services [see own-price elasticities in Table 8]. Second, a higher pass-through parameter value implies a higher welfare loss from the consumers’ stand point as, of course, consumers’ prices must rise more. For instance, the value that we obtain for the median equivalent variation is closer to those obtained by Prieto-Rodriguez et al. (2005) for the hypothetical reforms that they consider and, also, under the complete shift assumption. The reader can compare the values in the last row of Table 12 with the values in Table 8, last row, in Prieto-Rodriguez et al. (2005). Third, here we have also computed two alternative measures of the excess burden referred to above (one based on the equivalent variation, the other on the compensating variation). As expected both measures show a negative sign. Higher levels of income are associated with higher levels of excess burden (see Table 12, columns 8 and 9). Fourth, and most important, when considering the effects relative to households’ incomes, the regressive nature of the VAT policy reform is clearly confirmed on average (see Table 13).

Table 12 Distributive analysis of welfare ($\hat{\gamma }=100.00\%$)

Table 13 Distributive analysis of welfare relative to annual income ($\hat{\gamma }=100.00\%$)

Table 14 shows the counterpart to Table 11: social welfare effects of the VAT reform, but this time under the assumption that the change in tax rates is completely shifted to consumers. Comparing columns 5 in Tables 11 and 14 the observed result is natural: everyone would expect that if the shift of the increments in tax rates are higher, for any inequality aversion parameter, the welfare loss will be higher (the values of King’s $\lambda$ in Table 14 fall below those in Table 11). The ordering in the Atkinson’s inequality indices remains the same, i.e., $A^{I}(\varepsilon )<A^{0} (\varepsilon )<A^{F}(\varepsilon )$ for all the $\varepsilon$’s considered, so that one cannot determine whether the VAT reform would have reduced or increased the inequality of the distribution of cultural expenditure among Spanish households.

Table 14 King’s proportional increase in initial equivalent income ($\hat{\gamma }=100\%$)