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Semi-parametric estimation of the EASI model: Welfare implications of taxes identifying clusters due to unobserved preference heterogeneity
\pubMonth\pubYear\pubVolume\pubIssue \Keywords{Demand systems, Dirichlet process, EASI model, Equivalent variation, Taxes, Welfare analysis.}
Ex ante welfare analysis of new taxes should be a basic economic principle. Thus, policy makers have a baseline framework to make decisions that potentially minimize welfare losses due to tax distortions as taxation can only be a second-best option baumol1970optimal.
This paper provides a novel inferential framework to analyze potential welfare implications due to a consumption tax based on the equivalent variation. Our inferential approach is done using an econometric framework based on a non-parametric specification of the stochastic perturbations in the exact affine Stone index incomplete demand system using Dirichlet processes to identify clusters of consumers due to unobserved preference heterogeneity taking censoring, simultaneous endogeneity and non-linearity into account.
Consumer responses to changes in prices is instrumental for any assessment of welfare consequences of taxation, and demand models play an essential role in assessing such consequences. Our specification is based on an extension of the exact affine Stone index (EASI) demand system Lewbel2009 as this satisfies the axioms of choice such that additivity, homogeneity, and symmetry restrictions can easily be imposed in a simple setting to perform estimation. In addition, the rank in the function space spanned by the Engel curves is flexible (it can be more than three), allows the Engel curves to take arbitrary shapes, and the stochastic errors can be interpreted as unobserved consumer heterogeneity. These are nice features that other well known demand systems do not take into account, for instance, the almost ideal demand system (AIDS, Deaton1980), and quadratic AIDS (QAIDS, Banks1997). These properties are particularly relevant when working with micro level data due to variation in expenditure being not smoothed out by aggregation, and much of the demand variation may not be explained by observable variables Blundeletal2007, zhen2014predicting.
To the best of our knowledge, we are the first to propose a non-parametric specification of the stochastic errors in the EASI demand system. We extend Ramirez2021's inferential approach to an incomplete demand system in a semi-parametric specification, and Conley2008 and Jensen2014's proposals to a high dimensional system of simultaneous equations with cross-equation restrictions. Our approach easily allows to check and impose symmetry, strict cost monotonicity and concavity, and perform inference of the equivalent variation as a by-product of the posterior chains.
Our application is based on utilities demand, particularly electricity, gas, water and sewerage. These goods are approximately 5% of households budget share in developed countries like Australia (4.6%), Germany (5.8%), Japan (5.7%), South Korea (3.7%), Spain (5.7%) and United States (4.2%). The average is equal to 5.9% in developing countries like Colombia (5.8%), Chile (5.5%), Georgia (4.8%), Jordan (4.7%), Maldives (7.7%), Mexico (4.8%), Peru (5.9%) and Serbia (6.5%). Electricity represents the highest share (3.1%), followed by gas (1.3%) and water (1.1%) in these countries. In particular, we analyze the welfare implications of an approximately 1% electricity consumption tax rate on middle to high income households using Colombian data. This electricity demand tax was implemented by the national Colombian government in July 2019 charging these households to subsidy recovery of the electricity system attending the population in the north-western Colombian region after bankruptcy of its former provider.\footnote{This tax was declined in December 2020 by order of the Colombian National Constitutional Court because \href{https://www.eltiempo.com/justicia/cortes/las-razones-de-la-corte-para-tumbar-sobretasa-de-energia-en-estratos-4-5-y-6-para-electricaribe-553358}{“this extra charge is against equity, equality, efficiency and progresivity”} (Authors translation).} This provider attended about one third of the total residential subscribers in Colombia, that is, 2.4 million users, most of them belonging to the poorest segment, about 84% of its total subscribers.
There are remarkable welfare analysis due to price changes using demand systems, for instance, Banks1997 and Lewbel2009. In particular, previous studies analyzing implications associated with electricity price changes are Schulte2017,Tovar2018,Pereira2019,Ramirez-Hassan2019. The latter two use univariate demand approaches, whereas the former two use demand systems. Schulte2017 use a quadratic expenditure system, whereas Tovar2018 use the EASI demand system.
We matched household level data from different sources: \href{http://microdatos.dane.gov.co/index.php/catalog/566/get_microdata}{the Colombian national household budget survey}, \href{http://www.sui.gov.co/web/}{the utilities information system}, and the regulation councils of \href{https://www.creg.gov.co/}{energy and gas}, and \href{https://www.cra.gov.co/seccion/inicio.html}{water and sewerage} to perform our application. Our results seem to be consistent with previous literature in utilities, that is, electricity, water, gas and sewerage are inelastic normal services. It seems that Engel curves of utilities in our application are non-linear. This feature seems to be particularly relevant in the tails of the income distribution, and in sewerage. In addition, predictive distributions, which take simultaneous endogeneity into account, seem to be consistent with observable data, and predict substitutions effects between electricity and other non-utility goods given electricity price changes. Predictive p-values indicate good model assessment for inferring price effects based on the Slutsky matrix. Moreover, we identify four clusters due to unobserved preference heterogeneity, but evidence suggests that observable variables describe in good way preferences under the EASI model in our application. There is heterogeneity regarding welfare implications of this new tax on electricity demand; high socioeconomic households face the highest losses on average. In particular, there is a 95% probability that the equivalent variation as percentage of income of the representative household is between 0.60% to 1.49% given an approximately 1% electricity tariff increase. This result highlights the remarkable welfare implications due to tariff increases of inelastic goods.
After this introduction, we briefly sum up the incomplete demand EASI model for facility in exposition in Section (ref). Particular emphasis is on equivalent variation in Section (ref). Section (ref) shows our econometric proposal. In particular, we set our specification in Section (ref), sections (ref) and (ref) show our inferential framework and construction of the predictive distribution, respectively. The hypothesis testing setting of microeconomic restrictions and model assessment are set in sections (ref) and (ref). Section (ref) shows data construction and descriptive statistics, and Section (ref) displays inferential and predictive results regarding price effects, Engel curves and welfare estimates. Section (ref) concludes and give guidelines for future research. We show algorithmic details and descriptive statistics by relevant clusters (observable and unobservable) in the Appendix (Section (ref)).
The EASI incomplete demand system is an extension of the EASI demand system Lewbel2009 proposed by zhen2014predicting. Incomplete demand systems give correct welfare change measures without invoking the weak separability assumption and the exogeneity of group expenditures zhen2014predicting. This is done by introducing in the demand system a composite numéraire good representing all other goods and services, and using household income rather than group expenditure.
Details of the EASI demand system, and its incomplete version, can be found in Lewbel2009 and zhen2014predicting. However, we provide here the basic framework for exposition convenience. In our application, we model simultaneously five implicit Marshallian budget shares including the composite numéraire good ($J=5$), $\bm\omega(\bm p_i, y_i, \bm z_i, \bm \epsilon_i) \equiv \bm w_i =
^{\top}$ for $i=1,2,\dots, N$ households,\footnote{We do not include interaction effects between socioeconomic characteristics and prices in our main specification following zhen2014predicting and Tovar2018. Main results are robust regarding these effects in our application.}
where the price index for the numéraire good is $p_{iJ}=\frac{\log(\text{CPI}_i)-\sum_{j=1}^{J-1} w_{ij}p_{ij}}{w_{iJ}}$, $\text{CPI}$ is the consumer price index. The implicit utility, $y$, is an exact affine transformation of the (log) Stone index, $x-\bm{p}^{\top}\bm{w}$, $x$ is the (log) nominal income and $\bm p$ is the (log) Fisher ideal price index vector based on the sample average zhen2014predicting. The space spanned by the price vectors can have any rank up to $4$ in our case. The system of equations (ref) can involve any polynomial degree in $y$, which gives flexibility for the Engel curves, and take into account socioeconomic controls $l=1,2,\dots,L$ ($\bm z$, $z_0=1$ excluded from $\bm z$). In addition, the stochastic error ($\bm{\epsilon}$) can be interpreted as unobserved preference heterogeneity. Notice that this system (equations (ref) and (ref)) is endogenous and non-linear in parameters ($\bm b_r, \bm C, \bm D, \bm A, \bm B$). In our application, the main endogeneity concern is due to simultaneity between budget shares and the measure of real total income. We do not consider price endogeneity due to using household level data and regulated natural monopoly markets taking prices directly from provider records. The former prevents demand-supply simultaneity, and the latter prevents searching provider strategies and price measurement errors.\footnote{zhen2014predicting use a weighted average price based on adjacent locations as the instrument of price. Observe that this approach is equivalent to ours as price in a region is the same for all households conditional on strata in our application.}
To achieve cost function regularity in the EASI model, the sets of coefficients in the system of equations (ref) have to satisfy $\bm{1}_J^{\top}\bm{b}_0=1$, $\bm{1}_J^{\top}\bm{b}_r=0$ for $r\neq 0$, $\bm{1}_J^{\top}\bm{A}=\bm{1}_J^{\top}\bm{B}=\bm{0}_J^{\top}$, $\bm{1}_J^{\top}\bm{C}=\bm{1}_J^{\top}\bm{D}=\bm{0}_L^{\top}$ and $\bm 1_J^{\top}\bm{\epsilon}=0$ for adding up constraints, $\bm{A}\bm{1}_J=\bm{B}\bm{1}_J=\bm{0}_J$ for cost function homogeneity, strict cost monotonicity requires $\bm p^{\top}\left[\sum_{r=0}^R \bm b_r r y^{r-1}+\bm D \bm z + \bm B \bm p / 2\right]>-1$, Slutsky symmetry is ensured by symmetry of $\bm A$ and $\bm B$, and a sufficient and necessary condition for concavity of the cost function is negative semidefiniteness of the normalized Slutsky matrix $\bm A+\bm B y+\bm w \bm w^{\top}- \bm W$, where $\bm W$ is the diagonal matrix composed by $\bm w$, and $\bm{1}_J$ is a $J$-dimensional vector of ones.
The compensated share price semi-elasticities are given by
whose value for the representative individual is $\bm A$ given that $y=0$ because baseline prices are equal to 1, and demeaned nominal income ($x=0$).
The real income share semi-elasticities are
Household characteristics share semi-elasticities are given by
where $\bm{c}_{l}$ and $\bm{d}_{l}$ are columns from $\bm C$ and $\bm D$, respectively.
The normalized Slutsky matrix is
The Marshallian demand functions in the EASI model are implicitly given by:
Solving for the Marshallian share semi-elasticities with respect to nominal income is
Marshallian share price semi-elasticities are recovered from Hicksian share price semi-elasticities (equation (ref)) and the above Marshallian share income semi-elasticities using the Slutsky matrix.
Observe that using the causal path diagram (Figure (ref)) we can decompose the causal effect of price on Marshallian shares into the direct effect ($\nabla_{\bm p}\bm \omega(\bm{p},y,\bm {z},\bm\epsilon)$), and the indirect (mediator) effect through the nominal income ($\bm B$).
In addition, the Hicksian demand price elasticities are
where $\mathbbm{1}(\cdot)$ is the indicator function and $\nabla_{\bm p}\bm\omega(\bm p, y, \bm z, \bm \epsilon)_{lj}$ is the $lj$-th element of the matrix $\nabla_{\bm p}\bm\omega(\bm p, y, \bm z, \bm \epsilon)$. The Marshallian demand income elasticities are
where $\nabla_{x}\bm{w}(\bm p, x, \bm{z}, \bm\epsilon)_j$ is the $j$-th element of the vector $\nabla_{x}\bm{w}(\bm p, x, \bm{z}, \bm\epsilon)$. Using again the Slutsky decomposition with equations (ref) and (ref), we obtain the Marshallian price elasticities,
The Marshallian Engel curves for household $i$ at the base prices $\bm p=\bm 0$ where $x_i=y_i$ are
These functions summarize the parameter estimates associated with income.
We use the equivalent variation (EV) which is the maximum amount a consumer would be willing to pay at income level $x$ to avoid the change from price vector $\bm{P}^{0}$ to $\bm{P}^{1}$ Mascollel1995, where the superscripts 0 and 1 denote prices before and after change. \citet*{Chipman1980} showed that the equivalent variation (EV) is generally the relevant measure for performing welfare analysis in a context in which different tariff policies are ordered due to using the actual price vector as reference. In the case of the EASI model, this is Tovar2018:
where $a_{lj}$ is the $lj$-th element of matrix $\bm A$, $P_j^1=(1+\Delta_{j})P_j^0$ and $w_j^1=w_j^0+\nabla_{\bm p}\bm w_{jl}\times \Delta_{l}$, $\nabla_{\bm p}\bm w_{jl}$ is the $jl$-th element of the Marshallian share price semi-elasticities (equation (ref)), and $\Delta_{j}$ is the new tax rate on good $j$.
In the case where there is a new tax in just one good, say good $l$, equation (ref) becomes
where the second equality follows assuming symmetry.\footnote{We predict income shares in the denominator of equation (ref) using Marshallian elasticities to avoid singularity issues arising from using the predictive distribution, and improve precision of estimates (see Section (ref) for more discussion). This approach seems plausible in our application given the small price change (0.8%).}
Observe that this expression incorporates unobserved heterogeneous, substitution and income effects through $w_j^0$, $\bm{A}$ and $\nabla_{\bm p}\bm w_{jl}$, respectively.
We propose a semi-parametric estimation of the EASI incomplete demand system extending Ramirez2021's parametric proposal. We base our approach on Conley2008's ideas, but in a high dimensional system of equations with cross-equation restrictions in latent variables. In particular, we model the stochastic errors using Dirichlet processes, this is particularly appealing in the EASI model as stochastic errors are interpreted as unobserved consumer heterogeneity, then we can potentially identify conditional clusters, and analyze welfare implications associated with particular consumer groups. Our inferential framework also allows to test and impose symmetry, strict cost monotonicity and concavity of the cost function, handle censored data, take parameter non-linearities, simultaneous endogeneity and obtain coherent predictive distributions including estimation errors in systems of simultaneous equations. Conley2008 show that a Bayesian semi-parametric specification is more efficient than parametric Bayesian or classical methods when the errors are non-normal.
We assume specific equations in the “structural latent forms” (system (ref) and equation (ref)) and reduce form (“first stage system” (ref)), but stochastic errors distribute non-parametrically. In particular, we have the following system of simultaneous equations for $i=1,2,\dots, N$:
The structural form system (ref) has the set of unobserved latent budget shares $\tilde{\bm w}_i^*=\left[w_{i1}^* \ w_{i2}^* \dots w_{iJ-1}^*\right]^{\top}$, $\bm F_i=\left[\bm Y_i \ \bm W_i^{w}\right]$, $\bm Y_i=\left[\left[\bm I_{J-1}\otimes (y_i \ y_i^2 \dots y_i^R \ \bm z_{i}^{\top} y_i)\right] \ \left[(\bm I_{J-1}\otimes \tilde{\bm p}_i^{\top})y_i\bm D_{J-1}\right]\right]$ where there is the log of price ratios $\tilde{\bm p}_i=\left[\log(P_{i1}/P_{iJ}) \ \log(P_{i2}/P_{iJ}) \dots \log(P_{iJ-1}/P_{iJ})\right]^{\top}$, $\bm D_{J-1}$ is the $(J-1)^2\times J(J-1)/2$ dimensional duplication matrix ($\bm D_{J-1}vech(\bm H^{\top})=vec(\bm H^{\top})$, $\bm H$ is a symmetric matrix), and $\otimes$ is the Kronecker product. Observe that $\bm Y_i$ has the endogenous regressors from the system of equations (ref). Meanwhile, the set of exogenous regressors is $\bm W_i^{w}=\left[\left[\bm I_{J-1}\otimes \bm z_{i}^{\top}\right] \ \left[(\bm I_{J-1}\otimes \tilde{\bm p}_i^{\top})\bm D_{J-1}\right]\right]$. Observe that we omit one latent share due to the adding up restriction implies that the $J$ dimensional joint density function is degenerate.
In addition, $\bm{\phi}=\left[\bm{\beta}^{\top} \ \bm{\delta}^{\top}\right]^{\top}$, $\bm{\beta}=\left[b_{11}\dots b_{1R} \ D_{11} \dots D_{1L} \ b_{21} \dots D_{J-1L} \ vech(\bm B^{\top})\right]^{\top}$ and $\bm{\delta}=\left[C_{11} \dots C_{1L} \ C_{21} \dots C_{2L} \dots C_{J-1L} \ vech(\bm{A}^{\top})\right]^{\top}$ are the vectors of endogenous coefficients and exogenous coefficients, respectively.
The system of reduced form equations is given by (ref) where $\bm y_i^*=\left[y_i \ y_i^2 \dots y_i^R \ \bm z_{i}^{\top} y_i \ \tilde{\bm p}_i^{\top}y_i \right]^{\top}$ is a $q=R+L+J-1$ dimensional vector, and $\bm G_i=\left[\bm Z_i \ \bm W_i^y\right]$ where the matrix of exogenous regressors is $\bm W_i^y=\left[\left[\bm I_{q}\otimes \bm z_{i}^{\top}\right] \ \left[\bm I_{q}\otimes \tilde{\bm p}_i^{\top}\right]\right]$, and the matrix of instruments is $\bm Z_i=\left[\left[\bm I_{q}\otimes (\tilde{y}_i \ \tilde{y}_i^2 \dots \tilde{y}_i^R \ \bm z_{i}^{\top} \tilde{y}_i)\right] \ \left[(\bm I_{q}\otimes \tilde{\bm p}_i^{\top})\tilde{y}_i\right]\right]$ where $\tilde{y}_i=x_i-\bm{p}_i^{\top}\bar{\bm w}$ is proposed by Lewbel2009 as baseline instrument, $\bar{\bm w}$ is the sample average.
Observe that we can give a causal interpretation of our econometric specification of the EASI model that can be represented using the Wright-Burks-Pearl causal diagram shown in Figure (ref) (see Wright1921,burks1926inadequacy,pearl1995causal for origins and details).
By construction, there is simultaneous causality between the income budget shares and the implicit utility. We can see this simultaneous causality as unmeasurable factors which influence both household's income shares and implicit utility. Therefore, the instruments built using $\tilde{y}_i$ help to identify the causal effect of the implicit utility on income shares, and as a consequence, identify direct and indirect effects associated with prices, nominal income and socioeconomic controls.
Regarding these unmeasurable factors, which are interpreted as unobserved preference heterogeneity in the EASI model, we assume the following:
where $\bm\Sigma_i=
$, $\bm\mu_i, \bm\Sigma_i| G\stackrel{i.i.d}\sim G$, and $G|G_0,\alpha\sim DP(\alpha,G_0(\cdot|\bm \kappa))$, that is, the unknown distribution $G$ has a Dirichlet process mixture prior (DPM) with base distribution $G_0(\cdot|\bm \kappa)$ with unknown parameters $\bm \kappa$, and precision parameter $\alpha$ \citep{Ferguson1973}. In particular, $\mathbb{E}\left[G(A)\right]=G_0(A)$ and $\mathbb{V}ar\left[G(A)\right]=G_0(A)\left[1-G_0(A)\right]/(1+\alpha)$ for a given event $A$ \cite[p. ~8]{Muller2015}; observe that $\alpha \rightarrow \infty$ then $G$ concentrates at $G_0$. We assume conjugate prior specifications for computational convenience Conley2008, Jensen2014,
that is the base distribution $G_0$ is a normal-inverse Wishart distribution.
This is a very flexible way to model the stochastic errors in our specification with a nice economic interpretation. The DPM establishes that the unknown distribution of $\left[\bm\epsilon_i \ \bm v_i\right]$ is an infinite mixture Antoniak1974 $\int F_N(\bm \mu, \bm \Sigma) dG(\bm \theta)$ where $\bm\theta = \left\{\bm \mu, \bm \Sigma\right\}$, $dG(\bm \theta)=\sum_{m=1}^{\infty}\pi_m\delta_{\bm\theta_m}(\cdot)$ Sethuraman1994, $F_N$ is a Normal distribution with mean $\bm \mu$ and variance matrix $\bm \Sigma$ (equation (ref)), $\delta_{\bm\theta_m}(\cdot)$ denotes the measure given mass one to the atom $\bm\theta_m$, $\pi_1=v_1$, $\pi_m=v_m\prod_{h<m}(1-v_h)$, $v_m\sim Be(1, \alpha)$. Observe that non-zero mean stochastic errors allows greater flexibility in this distribution than fixing it at zero.\footnote{This implies that we do not require intercepts in the system ((ref) and (ref)).} In addition, the hierarchical representation of the Dirichlet process Antoniak1974 induces a probability model on clusters. This implies two advantages in our setting: first, it facilities posterior computation, and second, the clusters can be interpreted as groups of households with same conditional unobserved preferences (random utility parameters). However, we should acknowledge that DPM is not consistent for the number of clusters, but there is posterior asymptotic concentration in the other model's components such as structural parameters and predictive densities Miller2014.
The hierarchical representation implies that there are latent assignment variables $s_i=m$ such that when $\bm\theta_i$ is equal to $m$-th unique $\bm\theta_m^*$, that is, $\bm\theta_i=\bm\theta_m^*$ then $s_i=m$. Given a DP prior,
and
where $\pi_m=P(\bm\theta_i=\bm\theta_m^*)$.
Following Kasteridis2011, the link between the latent shares and the observed shares is
Wales1983 note that the transformation (ref) has the property that the resulting density function is independent of which set of $J-1$ latent shares are used in its derivation.
Setting $\bm{Data}=\bm w, \bm p, \bm z, \bm x$, the posterior distribution is proportional to
where $M$ is the number of clusters.
Equation (ref) does not have a closed analytical expression. Then, we group parameters in order to obtain standard conditional posterior distributions that facilitate posterior computations due to the Pólya urn characterization of the DPM Blackwell1973.
where $\bm u_i=
-
$.
These blocks allow a Gibbs sampling algorithm. See Appendix (ref) for expressions of the posterior conditional distributions and algorithmic details.
We used non-informative hyperparameters, that is, hyperparameters that are consistent with sample information in the set of prior distributions (expression (ref)). In particular, $\bm\phi_0=\bm 0_{\dim\left\{\bm\phi\right\}}$, $\bm\psi_0=\bm 0_{\dim\left\{\bm\psi\right\}}$, $\bm\Phi_0=100\bm I_{\dim\left\{\bm\phi\right\}}$, $\bm\Psi_0=100\bm I_{\dim\left\{\bm\psi\right\}}$, $r_0=q+J+1$, $\bm R_0=\bm I_{q+J-1}$, $\bm\mu_0=\bm 0_{q+J-1}$, $\tau_0 = 0.01$, and $\alpha_0=\beta_0=0.1$. In general, structural parameters and predictive densities are not sensitive to hyperparameters Escobar1995,Conley2008,Jensen2014. However, higher values of the smoothing hyperparameter ($\tau_0$) affects the number of modes in the stochastic errors, although, from a practical point of view, the differences are small Escobar1995. We decided to choose a small value to avoid this issue. The number of clusters is affected by $\alpha_0$ and $\beta_0$ Escobar1995,Conley2008,Jensen2014, but in general there is not posterior consistency regarding this not matter the values of the hyperparameters are Miller2014.
Observe that we based our inferential framework on conditional statements of each household's income shares to sample the latent shares. This is not possible in the predictive analysis due to not observing budget shares for the potentially unobserved household $0$. We follow ideas of Jensen2014 extending their approach to a high dimensional system of contemporaneous simultaneous equations with cross-equation restrictions.
Taking into account that $
\perp
\Bigg|\Lambda$, where $\Lambda=\left\{\bm\phi, \bm\psi,\bm s,\left\{\bm\Sigma_i\right\},\left\{\bm\mu_i\right\},\alpha\right\}$, the marginal predictive density of $\tilde{\bm w}_{0}^*$ at the potentially unobserved household $0$ is given by\footnote{Observe that we can get the posterior predictive distribution based on an observed household. We perform this in our application based on the representative household to perform model assessment based on predictive p-values (see section (ref)).}
where for each random draw from the posterior distribution of $\bm\Lambda^{(s^1)}, s^1=1,2,\dots,S^1$, there are $S^2$ random draws from the posterior predictive distribution of $y_0^{(s^2)}, s^2=1,2,\dots,S^2$.
The conditional predictive density of $\tilde{\bm w}_{0}^*$ is
where $\bm R_{11,0}$ is the upper-right $J-1 \times J-1$ block matrix of $\bm R_{0}$, $\bm\mu_{k,\tilde{\bm w}_{0}^*}$ is the first $J-1$ elements of $\bm\mu_k$, $k=\left\{m,0\right\}$, $N_m$ is the number of observations such that $s_i=m$, $v=r_0+1-(q+J-1)$, and $F_0^{(s)}$ has the same structure than $\bm F_i$, but using values for the household 0 regarding prices and socioeconomic characteristics, and the predictive distribution of $y_0^{(s)}$ is
where $\mu_{m,y}$ is the $J$-th element of $\bm\mu_k$, $\bm G_{1,0}$ is the first row $\bm G_0$, which takes same structure as $\bm G_i$, but using values associated with household $0$, $\sigma_{1,vv,m}$ is the $11$-th element of the matrix $\bm\Sigma_{vv,m}$, and $r_{JJ,0}$ is the $JJ$-th element of the matrix $\bm R_0$.
In addition, $w_{0J}^*$ has a degenerate density with probability one at $1-\sum_{j=1}^{J-1}w_{0j}^*$. Observe that the latent budget shares will not necessarily satisfy the requirement of being between zero and one, although they sum to one.
We obtain the predictive distribution of $\bm w_0$ taking into account that
and using the transformation Wales1983
Letting $\bm w_0 = \left[\tilde{\bm w}^{\top}_0 \ w_{0J}\right]^{\top} = \left[\tilde{\bm w}^{+\top}_0 \ \tilde{\bm w}^{-\top}_0 \ w_{0J}\right]^{\top}$, where $\tilde{\bm w}^{+}_0$ is a $K$ dimensional vector of positive shares ($0\leq K \leq J-1$), $\tilde{\bm w}^{-}_0$ is a $J-1-K$ dimensional vector of 0's, $\tilde{\bm w}^{+}_0\perp \tilde{\bm w}^{-}_0|\bm w_0^*$, $w_{0J}\perp \tilde{\bm w}^*_0|\tilde{\bm w}_0$, using the change of variable theorem, and taken into account that given $K$ positive shares there are $(J-1)!/(K!(J-1-K)!)$ cases to consider,
where $(-\bm\infty, \bm 0]_{J-1}$ is the negative orthant in $\mathbb{R}^{J-1}$, $P_{{\bm w}_0^{*}}(\cdot$) and $f_{{\bm w}_0^{*}}(\cdot)$ are the probability and density function induced by the distribution of $\tilde{\bm w}_{0}^*|\bm\Lambda^{(s)}$.
Equations (ref) to (ref) help us to obtain the predictive density function for $\bm w_0$. Observe that this predictive density is coherent by construction, that is, the marginal predictive distributions ($f(w_{0j}|\bm{Data}), j=1,2,\dots,J$) are in the probability simplex set,
We test the microeconomic restrictions following the formal Bayesian framework, model posterior odds, which reduces to Bayes factors given a priori equal model probabilities, that is, $PO_{01}=BF_{01}=\frac{m(\bm w|\mathcal{M}_0)}{m(\bm w|\mathcal{M}_1)}$ where $m({\bm w}|\mathcal{M}_r)$ is the marginal likelihood, $r=0,1$, $\mathcal{M}_0$ is the restricted model and $\mathcal{M}_1$ is the unrestricted (encompassing) model.
Evaluation of the marginal likelihood in DPMs is challenging due to requiring integration over the infinite dimensional space of $G$ to obtain the likelihood function Basu2003. Fortunately, we can take advantage of the nested structure of the microeconomic restrictions, and use the Savage-Dickey density ratio (or variations) to calculate the Bayes factors of $H_0$ against $H_1$ given that we use the same configuration of priors in both models dickey1971weighted. Then, we can estimate posterior model probabilities, $P(\mathcal{M}_0|\bm{Data})=\frac{BF_{01}}{1+BF_{01}}$.
Our point of departure is the unrestricted model. This specification follows exactly same stages as in the restricted model, it is just that in the specification of the system (ref) we set $\bm D_{J-1}=\bm I_{(J-1)^2}$. Slutsky symmetry is ensured by symmetry of $\bm A$ and $\bm B$. The null hypothesis of symmetry restrictions can be written as $H_0. \ \bm R\bm\phi=\bm 0_{(J-1)(J-2)}$ versus the alternative $H_1. \ \bm R\bm\phi\neq\bm 0_{(J-1)(J-2)}$, where $\bm\phi$ in this case is the vector of unrestricted coefficients, $\bm R$ is the $(J-1)(J-2)\times (J-1)[R+L+L+(J-1)+(J-1)]$ dimensional restriction matrix, and $(J-1)(J-2)$ is the number of symmetry restrictions in our specification.
Following the Savage-Dickey density ratio dickey1971weighted,verdinelli1995computing,
We show in Appendix (ref) that $\bm\phi|\bm\psi,\left\{\bm \Sigma_i\right\},\left\{\bm \mu_i\right\},\bm s,\tilde{\bm w}^*,\bm y^* \sim MN(\bar{\bm\phi},\bar{\bm\Phi})$, where
and
Then, $\bm R\bm\phi|\bm\psi,\left\{\bm \Sigma_i\right\},\left\{\bm \mu_i\right\},\bm s,\tilde{\bm w}^*,\bm y^* \sim MN(\bm R\bar{\bm\phi},\bm R\bar{\bm\Phi}\bm R^{\top})$.
Observe that the denominator in Equation (ref) is easy to calculate given our prior assumptions (see expression (ref)). We take into account the following facts to calculate the numerator:
where $\bm{\Lambda}_{(-\bm{\phi})}=\bm\psi,\left\{\bm \Sigma_i\right\},\left\{\bm \mu_i\right\},\bm s,\tilde{\bm w}^*,\bm y^*$ are draws from the posterior distribution.
We follow common practice in empirical demand systems to estimate the model without imposing inequality restrictions, and then check the inequalities associated with utility function regularity using the Slutsky matrix estimates. In particular, we use the encompassing approach to check inequality restrictions proposed by klugkist2007bayes to test negative semidefiniteness of the Slutsky matrix, $\bm A + \bm By + \bm w\bm w^{\top}-\bm W$. Observe that the this condition is equivalent to $\lambda_{J-1}\leq \lambda_{J-2}\leq \dots\leq \lambda_{1}\leq 0$, where $\lambda_{\cdot}$ are the eigenvalues of the normalized symmetric Slutsky matrix.\footnote{See wetzels2010encompassing for a nice proof of the relationship between the encompassing approach for inequality and about restrictions, and the Savage-Dickey density ratio.}
Given the encompassing model $\mathcal{M}_1$, where there is no requirement of inequality restrictions, and the restricted model $\mathcal{M}_0$ subject to the inequality restrictions, the Bayes factor is
where $1/d_0$ and $1/c_0$ are the proportions of the encompassing posterior and encompassing prior that are in agreement with the constraints of model $\mathcal{M}_0$, respectively, and $\mathbbm{1}(\mathcal{M}_0)$ is the indicator function regarding model $\mathcal{M}_0$.
Observe that we can use the predictive density to generate pseudo-shares from our model ($\bm w^{(s)}, s=1,2,\dots,S$), and calculate a discrepancy function $D(\bm w^{(s)}, \bm\Lambda^{(s)})$ to estimate $p_D(\bm w)=P[D(\bm w^{(s)}, \bm\Lambda)\geq D(\bm w, \bm\Lambda)]$ using the proportion of the $S$ pairs for which $D(\bm w^{(s)}, \bm\Lambda^{(s)})\geq D(\bm w, \bm\Lambda^{(s)})$, where $\bm w$ are the observed shares. We also use the average discrepancy statistic $D(\bm w)=\mathbb{E}[D(\bm w, \bm\Lambda)]=\int D(\bm w, \bm\Lambda) \pi(\bm\Lambda|\bm{Data})d\bm\Lambda\approx \frac{1}{S}\sum_{s=1}^S D(\bm w, \bm\Lambda^{(s)})$ to estimate $p_{\mathbb{E}[D]}(\bm w)=P[D(\bm w^{(s)}, \bm\Lambda)\geq D(\bm w)]$ based on the posterior distribution of $D(\bm w^{(s)},\bm\lambda^{(s)})$ (see Appendix (ref) for algorithm details). Those are posterior predictive p-values gelman1996posterior which are measures of realized discrepancy assessments between our model and the data. Extreme tail probabilities ($p_D(\bm w) \leq 0.05 \ \text{or} \ p_D(\bm w) \geq 0.95$) suggest potential discrepancies. In particular, our assessments are based on the observed income shares and the normalized Slutsky matrix (Equation (ref)).
Observed that we used data augmentation strategy Tanner1987 to handle censoring at zero. The endogeneity is tackled using different ways to write the likelihood function
and accordingly deduce particular conditional posterior distributions. In addition, cost homogeneity is imposed by using $\tilde{\bm p}$ rather than ${\bm p}$, and adding up constraints are used to get coefficients associated with the excluded good ($w^*_{J}$). Slutsky symmetry is imposed using the duplication matrix. The non-linearity issue is tackled iteratively updating $\bm y$ using equation (ref) and draws of $\bm A$ and $\bm B$ from $\bm\phi$. Unobserved preference heterogeneity is introduced using the Dirichlet process mixture, and the coherent predictive distribution of the shares is done through the predictive distribution of the latent shares using equations (ref), (ref) and (ref). Slutsky symmetry and negative semidefiniteness are tested based on the Savage-Dickey density ratio dickey1971weighted, and the encompassing approach with inequality restrictions klugkist2007bayes. Finally, model assessment is based on predictive p-values gelman1996posterior.
We merge different data sets involving utility providers of electricity, water, gas and sewerage, and households in strata 4, 5, and 6, which are the households subject to this new electricity tax.\footnote{Colombian households are classified by the government into socioeconomic strata; the aim is to implement cross subsidies and social programs. There are 6 categories, from low-low to high-high, stratum 1 signals the most vulnerable socioeconomic households, whereas stratum 6 households are supposed to be the wealthiest.}
We used the cross sectional Encuesta nacional de presupuesto de los hogares (ENPH) - 2016, which is a representative national survey based on a probabilistic multistage, stratified and cluster sample, inquiring about household budget carried out by the Colombian national institute of statistics (DANE) between July 2016 - June 2017.\footnote{This survey defines three stages. The fist stage has inclusion probability equal to one for 32 provinces with their metropolitan areas, and 6 important municipalities. The inclusion probability of other municipalities is stratified according to population size, urbanization, urban/rural population proportion, unsatisfied basic needs index, and strata. The second stage defines groups of contiguous blocks (urban area) and sections (rural area) as sampling units, each group has on average 10 blocks or sections that are proportionally drawn according to the population defined in the first stage. In the third stage an average group of 10 contiguous households (clusters) is randomly drawn.} We processed 5,780 households in strata 4, 5 and 6 getting information about monthly expenditures on each utility (electricity, water, gas and sewerage), total household monthly income (\$COP), municipality (location), gender and age of household head, highest level of education attained by any household member, number of people living in the household and strata.
Equation (ref) shows expenditure ($E_{ismt}^u$) on utility $u=\left\{electricity, water, gas, sewerage\right\}$ at household $i$ in stratum $s$ with provider $m$ at time $t$.
This expenditure is equal to payment for variable consumption ($V$) plus fixed charge ($F$). The former is equal to price ($P$) at consumption range $c$ times consumption in this range ($Q$) due to tariff depending on consumption level in some utilities by regulation.
Electricity does not have fixed charge for any strata, whereas gas does have for strata 4, 5 and 6. Tariffs depend on stratum and providers, households in strata 5 and 6 have to pay a contribution to subsidize households in strata 1, 2 and 3, whereas households in stratum 4 pay the reference cost.\footnote{Tariffs for households in strata 1, 2 and 3 depend on consumption levels of electricity and gas.} We used household municipality location to identify its providers, and therefore to get the electricity tariff (\$COP/kWh) and gas tariff (\$COP/m3) including contributions if apply. This information is available at Sistema Único de Información (\href{www.sui.gov.co}{SUI}) and Comisión de Regulación de Energía y Gas (\href{www.creg.gov.co}{CREG}). The former is a public available information repository where utility providers have to upload business information, and the latter is the electricity and gas regulatory counsil.
Water and sewerage also have fixed charges which depend on providers and strata. This information is available at CREG for gas, and SUI for water and sewerage. Therefore, we obtain variable expenditure on these utilities subtracting fixed charges from total expenditures ($V_{ismt}^u=E_{ismt}^u-F_{smt}^u, u=\left\{water, gas, sewerage, \right\}$).
Water and sewerage tariffs (\$COP/m3) depend on municipality location (meters above sea level, m.a.s.l), providers, household consumption and strata. There are three ranges defined by the regulatory entity (CRA): basic, complementary and luxury, which depend on m.a.s.l. \href{https://www.cra.gov.co/documents/Resolucion_CRA_750_de_2016-Edicion_y_copia.pdf}{(CRA 750 de 2016)}. The first range is for municipalities with an average altitude below 1,000 m.a.s.l., the second for municipalities with an average altitude between 1,000 and 2,000 m.a.s.l., and finally the last range for municipalities with average altitude above 2,000 meters. A municipality in a lower range can consume more cubic meters (m3) at lower consumption prices due to weather conditions; so, prices associated with basic water consumption is up to 16 m3, 13 m3 and 11 m3 for municipalities in first, second and third levels, respectively. Prices at complementary level is between upper basic range and 32 m3, 26 m3, and 22 m3, respectively, and luxury prices apply to additional consumption.
We can deduce average water and sewerage tariff per m3 using variable expenditure on these utilities, municipality location and strata from ENPH survey, consumption ranges from CRA, and provider marginal prices from SUI in conjunction with equation (ref). Observe that average prices are weighted averages of marginal prices, the former being the price signal that households perceive.
We use the utility shares based on variable expenditures because these are directly under household's control, whereas fixed charges are not.
Due to households were surveyed in different months between 2016 and 2017, prices will be expressed at June 2017 prices. Then, all prices, expenditures, and fixed charges are in dollars at the exchange rate of June 30, 2017, the month in which the survey ended.
Table (ref) shows descriptive statistics of household income shares in utilities. It can be seen that in this group, on average, electricity share is the greatest (3%), followed by water (2%). On average, the variable expenditure in utilities represents 6% of the income.\footnote{We omit households with shares equal to 0 to calculate sample means and standard deviations to avoid distortions.} We also report proportion of zeros in the sample, where electricity and water have the lowest figures (5%), while sewerage has the highest (51%). This highlights the importance of taking into account the censoring issue in the econometric framework.
Electricity average tariff is 0.16 USD/kWh with a range from 0.13 USD/kWh to 0.27 USD/kWh. This high variability in electricity prices is also present in the other utilities: water prices range from 0.16 USD/m3 to 1.46 USD/m3, sewerage from 0.16 USD/m3 to 1.31 USD/m3, and gas from 0.09 USD/m3 to 0.91 USD/m3. This heterogeneity is due to different socioeconomic conditions, municipalities location, regulatory legislation and utility providers efficiency. We performed mean differences statistical tests comparing households at different strata, we reject the null hypothesis of equal share and price means. As expected by regulatory means, utility prices are increasing with strata, and there are also statistical differences regarding budget shares (see tables (ref) and (ref) in the Appendix (ref)).
Table (ref) presents descriptive statistics of household characteristics, which were selected according to data availability and literature review Deaton1980,Banks1997,Lewbel2009,Tovar2018. We see from this table that the representative (modal) household has 3 members, classified in stratum 4, a 55 years-old man as household head, the highest education level is undergraduate, and it is located at an altitude more than 1,000 m.a.s.l. There is a high level of variability regarding income that shows high socioeconomic heterogeneity (Table (ref) in the Appendix (ref) shows this information by strata, and tables (ref) and (ref) show mean and factor difference tests).
Figure (ref) shows unconditional Engel curves kernel estimations (left panels, solid black line), their derivatives (right panels, solid black line), and their predictive 95% intervals (gray shaded area). It seems that there are polynomial complexities in these Engel curves that cannot be recovered using linear Deaton1980 or quadratic Banks1997 Engel demand systems.
We normalize prices using the representative household ($i= 4,334$) such that the baseline log price vector is equal to zero, and centered all observable household characteristics using this household. Then, all socioeconomic controls are zero for the representative household. We also normalize income using this household to mitigate computational problems due to using high order polynomial in $y_i$. This transformation does not change model fit, and helps to perform predictive exercises for the representative household at baseline prices due to $y=x=0$, then $f(\tilde{\bm w}_{0}^*|y_0,\bm\Lambda)=f(\tilde{\bm w}_{0}^*|\bm\Lambda)$.
We run 1,600 iterations with a burn-in equal to 600 using parallel processing in a server composed by two processors each with 12 cores in an Intel(R) Xeon(R) CPU E5-2670 v3 @ 2.30GHz RAM 397 GB architecture x86_64 CPU 64-bit. Total computing time 7 days 2.4 hours approx.
The parametric part of our structural equation (ref) would have $(J-1)\left[2(L+J-1)+R\right]$ parameters without imposing symmetry on $\bm A$ and $\bm B$. This would be an exactly identified model. We set $R=5$ as Lewbel2009,zhen2014predicting, this seems to give enough flexibility to handle Engel curves like those displayed in Figure (ref).\footnote{We perform robustness analysis regarding polynomial degree up to 5. We have computational issues using higher polynomial degrees.} In addition, we have $L=10$ and $J=5$; therefore, we would have 132 unrestricted parameters (33 for each equation). Imposing symmetry implies $(J-1)(J-2) = 12$ overidentifying restrictions. On the other hand, the reduced form equation has $q[2(L+J-1)+R]$ parameters, where $q=R+L+J-1$, this means 627 parameters. Therefore, our system of equations involves $(J-1+q)$ equations, that is, 23 equations, with 747 location parameters imposing symmetry restrictions, without taking unobserved heterogeneity parameters into account.
We compute several diagnostics to assess the convergence and stationarity of the posterior chains. In general, it seems that the posterior chains of the structural parameters in equation (ref) are stable. In particular, 100% of the parameters have dependence factors less than 5 using the Raftery1992's diagnostic with a 95% probability of obtaining an estimate in the interval $5\% \pm 2.5\%$, just 2 out of 120 have dependence factors higher than 2. Regarding the Heidelberger1983's and Geweke1992's tests at 5% significance level, 116 and 99 structural coefficients pass these tests, respectively. The former uses the Cramer-von-Mises statistic to test the null hypothesis that the sampled values come from a stationary distribution, and the latter tests for equality of the posterior means using the first 10% and the last 50% of the Markov chains. Results available upon authors request.
We check symmetry and negative semidefiniteness of the Slutsky matrix for the representative household using equations (ref) and (ref). We find that $2\log(BF_{01})$ are equal to 157.1 for the former, and 11.4 for the latter. These values suggest very strong evidence in favor of these hypothesis kass1995bayes. This implies that the posterior model probabilities imposing each of these restrictions for the representative household are approximately equal to 1.\footnote{We use non-informative priors for testing Slutsky symmetry. Observe that the Savage-Dickey density ratio compares equal dimensional densities; thus the Bayes factor does not necessarily give overwhelming support to parsimonious models in this setting. In addition, the way that we standardized observations means that the Bayes factor does not depend upon units of measurement. We perform some simulation exercises to test these assessments.}
Figure (ref) shows posterior estimates for the representative household. In particular, Hicksian and Marshallian price share semi-elasticities (top-left and top-right panels, equations (ref) and (ref)), and Marshallian price quantity and income elasticities (bottom-left and bottom-right panels, equations (ref) and (ref)).
We can see in Figure (ref) panel a that there are 8 out of 15 Hicksian price share semi-elasticities whose 95% credible interval does not cross zero, 3 of them are own-price share semi-elasticities (electricity, water and sewerage). Electricity is the most sensitive income share utility, an electricity tariff increase of 10 percent would imply a share 0.17 percentage points higher when income is raised to equate utility with that in the initial situation. We also see that 5 of the cross-price semi-elasticities are “statistically significant”. For instance, the water share compensated cross-price electricity semi-elasticity is -0.016, implying that a 10 percent water tariff increase would generate a statistically significant 0.16 percent points decrease in the income share for electricity, even when income is raised to hold utility constant.
Figure (ref) shows in panel b that there are 16 out of 25 Marshallian price share semi-elasticities whose 95% credible interval does not cross zero. Observe that the numeraire “good" is the most sensitive, a 10 percent price increase would imply approximately 0.4 percentage points real income share decrease when there is not compensation to keep utility level at the initial state. Uncompensated utility own-price share semi-elasticities are very similar to compensated ones. Observe that several uncompensated cross-price semi-elasticities are statistically significant, numeraire price increase causes the most relevant income share sensitivity. For instance, 10 percent price index numeraire increase implies 0.28, 0.15, 0.05 and -0.05 percent points change in the real income share for electricity, water, gas and sewerage, respectively.
Figure (ref) shows in panel c the Marshallian quantity demand elasticities. All 95% credible intervals for own-price elasticities are negative. In particular, 95% symmetric credible intervals for electricity, water and sewerage indicate inelastic goods, whereas gas own-price 95% posterior credible interval is $(-1.2, -0.7)$. All posterior mean utility own-price estimates suggest inelastic goods. Observe that posterior mean own-price elasticity estimates of water and sewerage are very similar (-0.75). However, the uncertainty regarding the latter is higher due to its higher level of censoring. On the other hand, the numeraire own-price elasticity is close to -1, although its 95% credible interval does not embrace this value. We see in this figure that there are relevant substitution and complementary price effects.
We can see in Figure (ref) panel d the Marshallian income elasticities. All utilities seem to be normal goods with 95% credible intervals less than 1, but greater than 0, except marginally for electricity. According to the posterior income elasticity mean estimates, sewerage is the most sensitive utility, followed by gas and water. The numeraire seems to be a luxury good, its very precise 95% credible interval is higher than 1, although very close to this value.
Inelastic and normal results for utilities agree with previous literature of demand systems, most of them QAIDS. In particular, renzetti1999municipal,Blundell1999,Gundimeda2008, Schulte2017,moshiri2018welfare,Tovar2018,diaz2021price found similar results for electricity, di2011consumers,Galves2016,suarez2020modeling for water, and renzetti1999municipal,Blundell1999,Gundimeda2008,diaz2021price for gas.\footnote{We did not find any references for sewerage. It should be because data limitations, but we suspect that sewerage elasticities should be similar to water elasticities as we got in our application.}
We also perform price and income effects analysis for representative households at strata level. We identify these households based on modal values of socioeconomic characteristics conditional on strata (see Table (ref) in the Appendix (ref)). These representative households by stratum are the observations 4135, 4924 and 3611 for strata 4, 5 and 6. This is relevant in the Colombian economy as cross subsidies on utilities depend on this classification. We can see in Figure (ref) in the Appendix (ref) these results. In general, we observe same patter in Hicksian and Marshallian share semielasticities compared with the sample representative household; although uncertainty level increases with strata. This is expected as socioeconomic characteristics differences between strata increase. Regarding Marshallian quantity demand elasticities, it seems that there are some differences between strata 6 and the other two strata; particularly regarding the effect of the numeraire price on utilities. It seems that the representative household of strata 6 is more elastic. This higher sensitivity is also in evidence in the income elasticity, where we also observe more differences between representative households. To sum up, it seems that quantity and income Marshallian elasticities are more sensitive to socioeconomic differences than semi-elasticities.
Figure (ref) shows the posterior mean and the 90%, 95% and 99% credible intervals of the conditional Engel curves for the representative household taking into account its unobserved preference heterogeneity cluster (see section (ref) for cluster details). They suggest statistical significant Engel curves in most of the range of the centered log real income, except for sewerage (see Panel d). This result is intuitive as the income share for sewerage is very low, and there is a high level of uncertainty due to censoring issues. Most of the Engel curves look quadratic, except income share in sewerage. Higher order polynomials seem not to be economically significant in the middle range of income. However, if we increase the range of variability of income, we observe that polynomial order higher than 2 are meaningful, but uncertainty also increases. It seems that in our application higher polynomial orders are relevant for low income share goods, like sewerage, or in the tails of the income distribution. This may explain why previous literature found demand system ranks less or equal than three Lewbel2009. More research should be done in this matter requiring very detailed data sets to confirm these hypothesis. We also estimate the Engel curves for representative households of each strata (see Figure (ref) in the Appendix (ref)). We observe same shapes as expected, but with different scales.
In general, we observe that income share in utilities would be higher for poor households, and there is an asymptote around 1% income share for the wealthiest households given that these households have observable characteristics equal to the representative household.
We estimate the posterior predictive distribution for the representative household using the framework in section (ref). We find that the modal predictive values of the joint distribution are 4.18%, 1.38%, 0.54%, 0.18% and 93.71% for electricity, water, gas, sewerage and numeraire using the real income at its modal value and baseline prices. We also estimate the predictive distribution assuming a 0.8% electricity tariff increase, which is equivalent to a new tax of 0.12 USD cents/kWh on electricity consumption taking into account an average tariff 0.16 USD/kWh for the representative household.\footnote{The Colombian government imposed a tax of 4 COP/kWh in July/2019, which is equal to 0.12 USD cents/kWh using an exchange rate equal to 3,038.26 COP/USD.} The modal predictive values assuming a 0.8% (19%) electricity tariff increase are 4.23% (4.50%), 1.39% (1.25%), 0.55% (0.57%), 0.19% (0.26%) and 93.63% (93.34%), respectively.\footnote{We also perform a predictive exercise assuming a 19% tariff increase due to the failed fiscal tax reform proposed by the Colombian government in the first semester of 2021 suggesting this percentage.}
Figure (ref) shows the empirical cumulative distribution functions associated with the marginal predictive distributions. The predictive mean values without electricity tariff increase are 5.2%, 2.2%, 1.5%, 1.3% and 89.6% that suggest good predictive performance as observed income shares for electricity, water, gas, sewerage and numeraire for the representative household are 4%, 2%, 1.2%, 1% and 91.8%, respectively.
In Figure (ref) we have that higher ordinate values conditional on electricity tariff change, that is, $P(W\leq w|\Delta\bm{p}\neq\bm{0})>P(W\leq w|\Delta\bm{p}=\bm{0})$, means that there is a higher probability of spending less income on a particular good given an electricity tariff increase. Panel a in Figure (ref) suggests that after a 4% income share in electricity the representative household facing the 0.8% electricity tariff variation increases the probability of spending less in electricity. We found that 63% of time the predictive distribution for electricity with tariff increase is greater than the predictive distribution without tariff increase. This figure is equal to 35%, 31%, 0% and 29% for water, gas, sewerage and the numeraire. However, results from the predictive distribution given a small tariff change can be difficult to identify. Then, we also analyze a 19% tariff increase in electricity demand. We have that there is potentially stochastic dominance in the marginal predictive distribution of electricity, and as a consequence, the numeraire. Panel a implies that there is a higher probability of spending more on electricity given a 19% tariff increase over all the support compared to the baseline situation. This meant substitution effects on other non-utility goods (numeraire), and small spillover effects on gas and sewerage.
We analyze welfare implications of the 0.12 USD cents/kWh on electricity tariff increase imposed by the Colombian government. We use the equivalent variation as percentage of the income (equation (ref)) and posterior estimates to perform inference of this change on the welfare of the representative household. Figure (ref) suggests that there is a 95% probability that the equivalent variation is between 0.60% and 1.49% with a mean equal to 1.02%, that is, on average the representative household faces a utility loss equivalent to 1.02% of its income, evaluated at the baseline prices, due to the 0.8% increase on the electricity tariff.
Actually, the percentage of the electricity tariff tax depends on socioeconomic strata because this is a fixed charge (0.12 US cents/Kwh) that does not discriminate strata, which in turn have different electricity tariffs depending on specific market conditions and regulation. We have that the average electricity tariff rates are 0.08%, 0.07% and 0.075% for strata 4, 5 and 6, respectively.
We estimate the posterior distribution of the equivalent variation for representative households in each stratum. Observe that welfare implications vary with household as consequence of different perceptual variations in electricity tariff, income shares, prices and socioeconomic characteristics. We observe that there are higher welfare losses as socioeconomic strata increase, that is, average equivalent variations equal to 1.0%, 1.4% and 2.0% for strata 4, 5 and 6. All 95% credible intervals are positive, except for stratum 6 where there is a higher level of uncertainty.
We also estimate the sampling distribution of the equivalent variation by strata taking estimation error into account, that is, we estimate the posterior expected value of the equivalent variation for each household with electricity service, and plot the sampling histogram by stratum. Figure (ref) shows that the modal value for all strata is approximately 1%. However, the sampling trimmed average for strata 4, 5 and 6 are 1.2%, 1.6% and 2.1%.\footnote{We calculated the sampling trimmed mean eliminating 2.5% of the estimates to avoid asymmetry outliers influence.} This confirms the previous pattern that was found by representative households.
These results highlight the potential remarkable welfare implications of taxes on inelastic goods.
Our proposal identifies 4 clusters due to unobserved preference heterogeneity. The clusters have 233, 5517, 29 and 1 households, respectively. We can see in Figure (ref) that approximately 95% of the households belong to cluster 2. This suggests that there is no a lot of heterogeneity due to unobserved preferences, and potentially observable variables describe in a good way preferences in our setting. In general, there is stability regarding cluster membership for all households.
Tables (ref) and (ref) in the Appendix show descriptive statistics of shares, prices, and household characteristics by cluster. As expected, representative (modal) household in cluster 2 is very similar to the representative (modal) household in our sample. Clusters 1 and 2 are also similar regarding observable characteristics, except that on average cluster 2 has the lowest proportions of zero utility shares (excluding the singleton cluster 4), and more educated household heads with a comparative low income living in a municipality located 1,000 m.a.s.l. On the other hand, cluster 3 has the highest proportion of zero shares, except in sewerage, and the most proportion of less educated, young and women as household heads, but the highest average income. On the other hand, cluster 4 is composed by the household 1973, which has a very low income facing low utility tariffs and expending just 4% of its income in utilities.
Figure (ref) shows box plots from posterior draws of the predictive distributions conditional on cluster membership evaluated at observed values of the representative household. We can see in Figure (ref) that the mean values of the representative household (red vertical lines) are very close to the mean values of the predictive distributions (green dots) of cluster 2, which is its cluster.
Observe that this is a kind of counterfactual exercise, that is, what would potentially be the posterior distribution of shares of the representative household, if this would belong to a different unobserved preference heterogeneity cluster. Using as a point predictive value the mean, we see that a hypothetical household whose observable variables would have been equal to the representative household, but with unobserved preference heterogeneity equal to cluster 1, 2, 3 and 4, would potentially have had electricity shares equal to 10.2%, 5.0%, 15.7% and 33.3%, respectively (see panel a in Figure (ref)). As expected, the level of uncertainty of these predictive values increases as the number of households in each cluster decreases. Therefore, variability of cluster 2 is the lowest due to most of the households belonging to this cluster, followed by cluster 1, 3 and 4. Similar analysis can be done using the other panels in Figure (ref).
We also perform welfare analysis by cluster, the 2.5% sampling trimmed mean of the equivalent variation is 1.8%, 1.3%, -11.7% and 12.0% for clusters 1, 2, 3 and 4, respectively. In addition, the 2.5% and 97.5% quantiles of the expected values of the equivalent variation are (-29.7%, 23.0%), (-2.2%, 6.6%) and (-124.8%, 38.1%) for clusters 1, 2 and 3. This shows that we may have unreasonable values, particularly for clusters 1, 3 and 4. These results suggest a limitation of our econometric framework; we take unobserved preference heterogeneity into account imposing the restriction of equality in location parameters. Therefore, price and income effects are heterogeneous due to observable variables, but not due to unobserved sensitivity (coefficients). We focus our attention on cluster 2 as most of households are in this group, and results seem sensible.
We perform model assessment conditional on membership to cluster 2 as 95% of our sample seems to belong to it. We use predictive p-values, where our “discrepancy functions" are the observed income shares and the normalized Slutsky matrix for the representative household that belongs to this cluster.
Figure (ref) shows the histograms associated with the posterior predictive distributions for the representative household conditional on cluster membership due to unobserved preference heterogeneity. Predictive p-values for income shares in electricity, water, gas, sewerage and numeraire are 0.63, 0.48, 0.37, 0.29 and 0.43, respectively. These values suggest good model assessment regarding prediction for the representative household.
We also calculate predictive p-values for the Slutsky matrix. Figure (ref) shows scatter plots of predictive versus realized discrepancies of the main diagonal elements of the Slutsky matrix for the representative household. The predictive p-value is estimated by the proportion of points above the 45$^{\circ}$ line. These are 0.32, 0.56, 0.63, 0.66 and 0.43 for electricity, water, gas, sewerage and numeraire, respectively. We have similar results using the predictive p-value based on the average discrepancy statistic. This evidence suggests good model assessment for this microeconomic concept.
We can see in Figure (ref) that mean own-price Slutsky terms are all negative, and their symmetric 95% credible intervals are also negative. This suggests that substitution effects are statistically significant.
Unobserved preference heterogeneity analysis suggests that there is not a lot of heterogeneity in this aspect, and as a consequence, observable variables seem to describe in a good way preferences in our setting. In addition, it seems that there is good model assessment.
We propose a semi-parametric inferential framework for the EASI incomplete demand system using a Dirichlet process mixture to identify clusters due to unobserved preference heterogeneity. Our application suggests that there are four clusters; although 95% of the households belong to one of them. This suggests that observable variables describe preferences in a good way. In addition, we find that utilities seem to be inelastic normal goods in the Colombian economy, substitution effects are relevant, particularly between utilities and other goods, and Engel curves are non-linear.
Equivalent variation analysis suggests that there is potential remarkable welfare implications due taxation on inelastic services. In particular, the 0.8% electricity tariff increase caused an average welfare loss equal to 1.02% on the representative household. In general, predictive p-values show good model assessment regarding prediction and Slutsky matrix. However, welfare analysis based on unobserved preference heterogeneity gives non sensible results for households that have high observable discrepancies compared to the representative household. This reveals some limitations in our econometric framework.
Future research should consider a full non-parametric specification that would allow to observe price and income heterogeneous effects associated with unobserved preferences. Observe that there is variability in price and income effects just due to observable variables, and Engel curves from different unobserved preference clusters have exactly the same shape, there is just a scale change. In addition, a hypothesis testing framework under more general conditions (non-nested models) should be formulated. For instance, comparing parametric versus non-parametric models.