A welfare evaluation of the 1986 tax reform for married couples in the United States

Abstract

This paper evaluates the welfare effects of the 1986 Tax Reform Act (TRA86). We rely on different welfare metrics, which fully retain preference heterogeneity and are based on different ethical priors. We estimate utility functions with preference heterogeneity on the basis of structural models of family labor supply. Then, using these estimated preferences, we compute and compare different well-being rankings corresponding to different ways of measuring well-being. Finally, we identify the losers and the winners of TRA86, in absolute and relative terms, for each of the welfare metrics.

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Fig. 1

Source: authors’ calculations from TAXSIM

Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    As the EITC depends on family income, it is expected to stimulate the labor supply at the extensive margin but, in the phase-out region, it should reduce incentives at the intensive margins. Empirical studies find that the EITC expansions had heterogeneous effects on labor supply, depending on family composition. On the one hand, Dickert et al. (1995), Eissa et al. (1996), Ellwood (2000) and Meier and Rosenbaum (2000, 2001) show that the EITC reforms increased the workforce participation of single women with children, since it made work more attractive. On the other hand, Ellwood (2000) and Eissa and Hoynes (2004) find that the EITC expansion between 1984 and 1996 reduced the total family labor supply of married couples. Overall, the positive effect on labor force participation seems to be large enough to offset the negative impact at the intensive margin in working hours (Scholz 1996).

  2. 2.

    As an efficiency measure they use the excess burden of taxation, i.e., the sum of how much each individual would be willing to pay to get rid of all taxes and transfers.

  3. 3.

    In a similar fashion, Piketty and Saez (2007) analyze changes in the progressivity of the federal income tax over time but cannot disentangle policy changes from other factors.

  4. 4.

    Bargain et al. (2013), Carpantier and Sapata (2016) and Decoster and Haan (2015) also use well-being indexes that are similar to those which we employ here. They do it, however, with different purposes: while Bargain et al. (2013) focus on a cross-country comparison of well-being, the other two studies deal with the sensitivity of different well-being rankings (in the USA and Germany) to different normative principles.

  5. 5.

    If we run the same simulation for earnings larger than $200,000, we arrive at the same conclusion. The only difference is that the reduction in the taxes at federal and state level induced by TRA86 is increasing with earnings in absolute value.

  6. 6.

    A deeper analysis of the indexes presented here is in Fleurbaey (2008), Fleurbaey and Maniquet (2011) and Fleurbaey and Blanchet (2013).

  7. 7.

    A general normative characteristic of money-metric utilities is that they do not depend on the initial cardinalization \(U_i\) of individual preferences \(R_i\), as the same value \(m_i(.)\) would be obtained with any increasing transformation of \(U_i\) (Fleurbaey and Blanchet 2013). As a consequence, the comparison of individual situations does not depend on the initial cardinalization of \(U_i\).

  8. 8.

    As a matter of fact, relying on the social welfare functions proposed by Fleurbaey and Maniquet (2011), just knowing the properties of the tax function one could assess the goodness of a reform. Of course, as explained in Fleurbaey and Maniquet (2006), the assessment of a tax-benefit system depends on the value judgments pertaining to the treatment of preference heterogeneity and they specifically discuss the optimality of an EITC-type of scheme. Their result, however, crucially relies on the fact that they use a social welfare function of the maximin type. Hence, in order to assess a reform and to know which part of the tax function should be changed first is sufficient to know the earnings of the worst-off individual.

  9. 9.

    A similar sample selection can be found in the methodological contribution of van Soest et al. (2002), Eissa and Hoynes (2004), Bargain et al. (2013) and Decoster and Haan (2015).

  10. 10.

    See Dagsvik and Strøm (2006) for an evaluation of the performance of the Box–Cox functional form in analyzing labor supply behavior compared to less parametric (polynomial) specifications as studied by van Soest et al. (2002).

  11. 11.

    See van Soest et al. (2002) for a discussion on the advantages and limits of the discrete labor supply model with respect to the continuous one.

  12. 12.

    Leisure time enters the utility function through l, which is wife’s fraction of time not devoted to work. More precisely, \(l\equiv 1-h=1-H/2500\). Hence, l is equal to 1 if \(H=0\) and is equal to 0 if \(H=2500\).

  13. 13.

    It is worthy to remark that the parametric assumption of the random component \(\varvec{\varepsilon }\) is not restrictive. McFadden and Train (2000) showed indeed that under mild regularity conditions any discrete choice model derived from random utility maximization can be approximated by EV-I errors in combination with a nonparametric specification of unobserved individual characteristics determining preferences.

  14. 14.

    Note that \(v_y\) and \(v_l\) are independent if and only if \(p_1p_2=p_3p_4\) (van den Berg et al. 1994; van den Berg and Lindeboom 1998), making it easy to test for the independence of the unobserved components.

  15. 15.

    In the selection equation, we include also the number of children younger than 17, an indicator for the presence of children younger than 6 and the number of family members.

  16. 16.

    We used the Internet Taxsim (v9). See Feenberg and Coutts (1993) for an introduction to TAXSIM. Further information can be found in Internet at http://www.nber.org/taxsim/. TAXSIM has already been used in the empirical studies of Eissa et al. (2008), Bargain et al. (2014) and Bargain et al. (2013).

  17. 17.

    We considered social security benefits as independent of wives’ working hours choice. In each year of our longitudinal dataset, less than 1% of the families received social security benefits. From 1984 to 1990, only 2.4% of the families (61 out of 2529) received social security benefits in at least one year in which they were in the sample. We instead did not include in the budget constraint food stamps and the Aid to Families with Dependent Children declared in the PSID, since they depend on family income (and therefore on wives’ working hours choice) but, differently from social security benefits, they are not handled by TAXSIM.

  18. 18.

    The PSID is produced and distributed by the Institute for Social Research, Survey Research Center, University of Michigan, Ann Arbor, MI.

  19. 19.

    Further information on the size of the sample used for the estimation of households preferences is provided in Appendix A.1 together with descriptive statistics.

  20. 20.

    We describe the simulation algorithm to obtain the figures in Table 2 in Appendix A.4.

  21. 21.

    Further details and labor supply elasticities for different demographic groups are in Table 14, Appendix A.3.

  22. 22.

    See also Bargain et al. (2016) for a recent survey on estimated labor supply elasticities in the USA and Western Europe.

  23. 23.

    Appendix A.5 describes the simulation algorithm to obtain the welfare metrics under the actual 1986 fiscal law and the counterfactual scenario with the 1988 fiscal rules.

  24. 24.

    See Table 13 in Appendix A.4.

  25. 25.

    \(8.3\%=[\exp (0.080)-1]\times 100\).

  26. 26.

    More details are provided in Appendix A.1.

  27. 27.

    The net income is deflated to 1990 prices using the consumer price index by the US Bureau of Labor Statistics and available at ftp://ftp.bls.gov/pub/special.requests/cpi/cpiai.txt.

  28. 28.

    The notation in this section is inspired by the one in Wooldridge (2010, Sect. 19.9.2).

  29. 29.

    We do not report the estimation result of the model without unobserved heterogeneity for the sake of brevity. They are available from the authors upon request.

  30. 30.

    We also tried to include time dummies in the linear preference terms of leisure and net income. Since their coefficients were not jointly significant at the usual 5% statistical level and the goodness of fit did not show any relevant improvement, we removed the time dummies from the set of covariates to save in degrees in freedom.

  31. 31.

    Since the unobserved heterogeneity \({\mathbf {v}}_i\) is a random term at household level, at each replication of the simulation algorithm, each individual is assigned a new vector \((\widehat{v}_{yi},\widehat{v}_{li})\) of unobserved characteristics.

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Acknowledgements

The collection of data used in this study was partly supported by the National Institutes of Health under Grant Number R01 HD069609 and the National Science Foundation under award number 1157698. We are very grateful to François Maniquet for helpful discussions and comments. We would like also to thank Marc Fleurbaey, Dirk Neumann, two anonymous reviewers, and the participants at the CORE seminar (Université catholique de Louvain) and at the Social Choice and Welfare Meeting 2014 (Boston) for their remarks and suggestions. Any remaining errors are our own.

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Correspondence to Giacomo Valletta.

Appendix

Appendix

More information about the data and sample

Column (1) of Table 9 displays the number of household observations in the original PSID dataset, while column (2) reports the number of observations after applying our selection criteria. The sample in column (2) is the one used to predict wives’ hourly wages. After predicting wives’ hourly wages and simulating the net income which corresponds to the discrete hour point of the interval actually chosen, we removed from the sample households in the last percentile of the pooled net income distribution (9 households removed). We also deleted households with nil net income when the wife is supposed not to work (4 households removed). Finally, column 3 provides detailed information about the size of the sample used for the estimation of households preferences.

Table 9 Number of household observations from the original PSID dataset to the final selected sample
Table 10 Working hours distribution of married women
Fig. 5
figure5

Distribution of wives’ working hours, their wage rates, yearly gross labor income and family net income

Table 10 reports the distribution of wives’ working hours. About 21% of wives in our sample do not work. The remaining married women are almost equally spl it into those working full time (more than 1750 h/year, in average 35 h/week) and those working part-time (less than 1750 h/year). The top-left panel of Fig. 5 displays the working hours distribution. The top-right panel is instead the distribution of married women’s predicted hourly wages. The bottom panels of Fig. 5 focus on wives’ gross labor income and family yearly after tax income.Footnote 27 Table 11 reports the descriptive statistics of the variables used in the estimation of the utility function. Married women are on average 35.5 years old, 2.5 years younger than their husbands. There are 1.5 dependent children (17 years old or younger) in each married couple, and in 68% of the couples there is a child younger than 6. One quarter of the couples are non-white. Almost 9% of the married women declare to be in bad health, while the same figure for men is about 7.4%. Married men are more likely to have a diploma of tertiary diploma (24.7%) than their wives (18.7%). However, married women are less likely than married men to have a low educational attainment: while 17.9% of the husbands stopped at grade 11 or less, only 14.7% of married women stopped at grade 11 or less.

Table 11 Summary statistics

In comparing the distribution of these demographic characteristics between husbands and wives, it must be reminded that we selected families with the husband working at least 1500 h/year and the wife between 20 and 55 years of age. These two selection criteria are the most important ones in terms of number of observations removed from the original sample of married couples. In order to shed more light on the consequences on the representativeness of the selected sample of married couples in terms of family income, in Fig. 6 we plot the kernel density estimates of the family yearly net incomes of: (i) our final sample of married couples (in blue); (ii) the whole sample of married couples including also those with the husband working less than 1500 h/year and with the wife older than 55 or younger than 20 (in green); (iii) the whole sample of married couples but excluding those with the wife older than 55 or younger than 20 (in red). The vertical lines indicate the average of each distribution. Figure 6 shows that the selection of families with the husband working more than 1500 h/year generated the oversampling of relatively richer couples. The age restriction had instead no influence. (Color figure online)

Fig. 6
figure6

Kernel density estimates of the family yearly net incomes for our selected sample of married couples (blue), for the whole sample of married couples (green), for the whole sample excluding married couples with wife younger than 20 or older than 55 (red). Notes: the vertical dashed lines indicate the mean of the household earnings of each distribution. We used the Epanechnikov kernel in calculating the kernel density estimates (Color figure online)

Sample selection corrected estimation of log wage equation

In order to account for non-workers’ missing wages, log wage equations for the wives are estimated taking into account sample selection bias. We first tested for sample selection following Wooldridge (1995).Footnote 28 The log wage equation is

$$\begin{aligned} \ln w_{it} = {\mathbf {x}}_{it,w}\varvec{\beta }_w+c_i+u_{it}, \quad t=1,\ldots ,T, \end{aligned}$$
(A.1)

where \(w_{it}\) is observed only if wife i is working, \({\mathbf {x}}_{it,w}\) is the set of exogenous regressors determining wages at time t, \(c_i\) is time-invariant unobserved characteristics and \(u_{it}\) is an idiosyncratic error term. Denote by \(s_{it}\) the binary indicator equal to 1 if woman i is working at time t and 0 otherwise. Assume that, for each t, \(s_{it}\) is determined by the following probit model

$$\begin{aligned} s_{it}={\mathbbm {1}}[{\mathbf {x}}_{it,s}\varvec{\beta }_s+\bar{{\mathbf {x}}}_{i,s}\varvec{\delta }_s+v_{it}],\quad v_{it}|{\mathbf {x}}_{it,s}\sim \text {Normal}(0,1), \end{aligned}$$
(A.2)

where \({\mathbf {x}}_{it,s}\) contains \({\mathbf {x}}_{it,w}\) and \(\bar{{\mathbf {x}}}_{i,s}\) is the average of \({\mathbf {x}}_{it,s}\) over t and represents the Mundlak (1978) approach to allow the unobservables to be correlated with some elements of the observed characteristics in a discrete choice panel data model.

The parameters of the model in Eq. (A.2) are estimated by pooled probit and reported in the first columns of Table 12. We can test for sample selection by computing the inverse Mills ratio obtained from the pooled probit estimation and by plugging it into the log wage equation as a further regressor. Under the null assumption of no sample selection, the inverse Mills ratio should not be significant when the log wage equation is estimated by fixed effects (FE). The bottom line of Table 12 displays a Wald test for the significance of the inverse Mills ratio after the FE estimation of the log wage equation. The test is made robust to heteroskedasticity and serial correlation. As the inverse Mills ratio is highly significant, a problem of sample selection is detected and should be corrected for.

Table 12 Estimated parameters of the employment selection equation and sample selection corrected log wage equation

In order to correct for sample selection, we run pooled ordinary least square regression on the log wage equation augmented by the inverse Mills ratio, its interaction with the time dummies and \(\bar{{\mathbf {x}}}_{i,s}\). The estimation results of the coefficients of the log wage equation are reported in the last three columns of Table 12. These estimated coefficients are used to predict wages for both working and non-working wives.

Estimated parameters, labor supply elasticities and goodness of fit

Table 13 reports the estimation results of the Box–Cox utility function in Eq. (3). We find that it is important to account for the presence of unobserved heterogeneityFootnote 29: the log-likelihood function improves from −22,291.6 without unobserved heterogeneity to −18,052.4 with unobserved heterogeneity specified as explained in Sect. 4.1. We also find that the unobserved components determining preferences in leisure and the unobserved ones determining preferences in income are positively and significantly correlated.

Table 13 Estimated parameters of the Box–Cox utility function with unobserved heterogeneity

Since we constrained \(\alpha _l\) and \(\alpha _y\) to be between 0 and 1, the marginal utilities are positive and the concavity conditions are satisfied if and only if \(\phi _y({\mathbf {x}}_y,v_y)>0\) and \(\phi _l({\mathbf {x}}_l,v_l)>0\). The utility function and its first and second derivatives depend, however, not only on observed characteristics but also on unobservables. Since we have estimated the distribution function of the unobservables, we can derive the mixed utility function and check whether the corresponding first and second derivatives satisfy the monotonicity and the concavity conditions. The estimated mixed utility function is

$$\begin{aligned} \sum _{m=1}^4 \widehat{p}_m u(y,l;{\mathbf {x}}, \widehat{{\mathbf {v}}}_m)=\sum _{m=1}^4 \widehat{p}_m \widehat{\phi }_y({\mathbf {x}}_y,\widehat{v}_{my})\frac{y^{\widehat{\alpha }_y}-1}{\widehat{\alpha }_y}+ \sum _{m=1}^4 \widehat{p}_m \widehat{\phi }_l({\mathbf {x}}_l,\widehat{v}_{ml})\frac{l^{\widehat{\alpha }_l}-1}{\widehat{\alpha }_l}. \end{aligned}$$
(A.3)

Hence, the monotonicity and concavity conditions are satisfied if, for each \(i=1,\ldots ,N\) and \(t=1,\ldots ,T\),

$$\begin{aligned} \sum _{m=1}^4 \widehat{p}_m \widehat{\phi }_y({\mathbf {x}}_{it,y},\widehat{v}_{my})>0 \quad \text {and} \quad \sum _{m=1}^4 \widehat{p}_m \widehat{\phi }_l({\mathbf {x}}_{it,l},\widehat{v}_l)>0. \end{aligned}$$

We find that the monotonicity and concavity conditions are satisfied for every \(i=1,\ldots ,N\) and \(t=1,\ldots ,T\).

Table 14 reports simulated labor supply elasticities both at the intensive and the extensive margins for the whole sample, but also by selected characteristics. The simulation algorithm with regard to labor supply elasticities is described in Appendix A.4. Panel a) of Table 14 focuses on the results from the benchmark model, i.e., the one including the unobserved heterogeneity \({\mathbf {v}}\). Panel b) reports instead labor supply elasticities after the estimation of the model without unobserved heterogeneity. Panel a) shows that intensive elasticities, which measure the percent change in working hours when the net income increases by 1%, are large. Across the whole sample, a 1% increase in net income implies an overall increase in working hours of 0.53%, corresponding to an increase of 6.65 yearly working hours on average, and an increase in the employment rate by 0.19% (0.13% points on average). Similar to the marginal rates of substitution of income with leisure, the labor supply elasticities are quite homogeneous across different demographic groups. The labor supply seems to be more rigid for women with tertiary education and for women without children, suggesting that they are more career oriented. When we estimate the model neglecting the unobserved component \({\mathbf {v}}\), we get a labor supply elasticity which is marginally smaller on average, as displayed in the first line of panel b). Also when the unobserved heterogeneity is neglected we get quite homogeneous labor supply elasticities across different demographic groups, apart from the labor supply elasticity of women in bad health.

Table 14 Simulated labor supply elasticities

In the evaluation of the effects of TRA86 on well-being, the estimated parameters of the utility function are used to simulate the choice of net income–leisure bundles of families under a counterfactual scenario in order to isolate the effect of TRA86. As the reliability of the simulations depends on the capability of our discrete choice structural labor supply model to predict the realized household choices between net income and leisure, we first report goodness-of-fit checks of the estimated model. The goodness-of-fit statistics are constructed on the basis of simulations of 9999 bundle choices for each family in the sample and for each year the family remains in the sample. Since we replicate the simulations 9999 times, we can construct 95% confidence intervals of the predicted frequencies of working hours and check whether the predicted frequencies are close enough to the empirical ones. The simulation algorithm with regard the goodness of fit is in “Appendix A.4”.

The first panel of Table 15 contrasts the empirical frequencies of the yearly working hours discrete categories with the simulated ones across the whole time window 1984–1990. The bottom panel of Table 15 reports the model fit for two particular years, the first one (1984) and the last one (1990) of our time window. The aim is to check whether the estimated model is able to replicate the actual data over time although time enters the model specification in a very limited way (only through the wage equation and time-varying regressors).Footnote 30 The goodness of fit of the frequencies in 1984 and 1990 is in line with the model fit on the full sample.

Table 15 Goodness of fit: empirical and predicted frequencies, 1984–1990

Simulation algorithms with regard to marginal rates of substitution, labor supply elasticities and the goodness of fit

We start by describing the simulation algorithm for the goodness of fit, since it is the starting point of the algorithms for the estimation of the marginal rates of substitution and labor supply elasticities.

Simulation algorithm with regard to the goodness of fit

The simulation algorithm with regard to the goodness of fit proceeds according to the following steps:

  1. 1.

    Draw a vector of parameter estimates \(\widehat{\varTheta }\) assuming normality around the point estimates with a variance-covariance matrix equal to the estimated one. This ensures that the Monte Carlo confidence intervals encompass the parameter estimation precision.

  2. 2.

    Assign to each family the observed explanatory variables, observed family net income for each wife’s leisure choice and a vector of unobserved characteristics drawn with probabilities given by Eq. (8).Footnote 31

  3. 3.

    For each bundle \(j=1,\ldots ,6\) and each \(i,\ldots ,N\), compute the predicted utility as the sum of the deterministic part of the utility function and a random draw \(\varepsilon _{ij}\) from the type I extreme value distribution, i.e., \(U_{ij}=u(y_{ij},l_{j}| {\mathbf {x}}_i,\widehat{{\mathbf {v}}}_i;\widehat{\varTheta })+\varepsilon _{ij}\).

  4. 4.

    Family i is predicted to choose the bundle j of net income and leisure when \(U_{ij}>U_{ik}, \ \forall \ k\ne j\).

  5. 5.

    Repeat steps 1 to 4 \(R=9,999\) times to get R independent realizations and build Monte Carlo confidence intervals.

Simulation algorithm with regard to the marginal rates of substitution

The simulation algorithm with regard to the marginal rates of substitution is very similar to the one for the goodness of fit. The difference is that after step 4, i.e., after simulating the bundle choice of family i, we computed the variation in the level of income needed to reach the same level of predicted optimal utility if leisure would increase by 50 h/year (decrease of 50 h of work). Finally, we averaged across the sample to get the average marginal rate of substitution and we repeated the procedure 999 times to build Monte Carlo confidence intervals.

The average partial effects in Table 2 of selected characteristics on the marginal rate of substitution are computed by taking the variation in the simulated marginal rate of substitution when we let one particular covariate vary. If the covariate is a dummy variable, we measure the variation in the marginal rate of substitution when the covariate changes from 0 to 1. When the covariate has a continuous support (wife’s age, husband’s age, or the number of children), we look at the variation when it increases by the amount indicated in Table 2.

Simulation algorithm with regard to the labor supply elasticity

The labor supply elasticity predicted by the model is computed by looking at the change in the predicted discrete working hour points generated by a 1% increase in the net income corresponding to each discrete working hour point. We modified the simulation algorithm for the goodness of fit and replaced steps 4 and 5 with the following steps:

  • \(4^\prime \) For each bundle \(j=1,\ldots ,6\) and each \(i,\ldots ,N\), compute also the counterfactual predicted utility with a 1% increase in net income \(\widetilde{U}_{ij}=u(1.01\times y_{ij},l_{j}| {\mathbf {x}}_i,\widehat{{\mathbf {v}}}_i;\widehat{\varTheta })+\varepsilon _{ij}\).

  • \(5^\prime \) Family i is predicted to choose the bundle j of net income and leisure when \(U_{ij}>U_{ik}, \ \forall \ k\ne j\). In the counterfactual scenario in which the disposable income is 1% larger, family i is predicted to choose the bundle j of net income and leisure when \(\widetilde{U}_{ij}>\widetilde{U}_{ik}, \ \forall \ k\ne j\). On the basis of the predicted working hours without and with the 1% increase in net income, we calculate different measures of labor supply elasticity both at the extensive and the intensive margins by averaging across the sample (or across subgroups conditional on selected observed characteristics).

  • \(6^\prime \) Repeat steps 1, 2, 3, \(4^\prime \) and \(5^\prime \) 999 times to get 999 independent realizations of the statistics of interest and build Monte Carlo confidence intervals.

Simulation algorithms to calculate welfare metrics

In order to understand the effect of TRA86 on households’ choices and, thereby, on the attained well-being level and on the position in the well-being ranking, we predicted twice the choices of the households in our sample in 1986 using the first 4 steps of the simulation algorithm used for the goodness of fit in Sect. A.4: first, using the actual 1986 tax rule to go from gross to net incomes, getting the optimal bundle \((l^*,y^*)\); secondly, using the 1988 tax rule to transform 1986 gross incomes into net incomes, getting the optimal bundle \((l^\prime ,y^\prime )\). By substituting these bundles into the estimated utility functions, we get the indirect utilities attained in the actual and counterfactual scenarios. Since the welfare metrics used in our empirical analysis are invariant to an additive rescale of the utility function, it does not matter whether we compute them on the basis of the indirect utility \(U(y^*,l^*;{\mathbf {x}},{\mathbf {v}},\varvec{\varepsilon })=u(y^*,l^*;{\mathbf {x}},{\mathbf {v}})+\varepsilon _l\) or on the basis of its deterministic component \(u(y^*,l^*;{\mathbf {x}},{\mathbf {v}})\).

Calculation of the money-metric utility with \(\widetilde{w}=0\)

Once we get the indirect utilities from the first three steps of the simulation algorithm for the goodness of fit, the computation of the money-metric utility with \(\widetilde{w}=0\) for individual i boils down to the calculation of the vertical intercept in the space (hy) of the indifference curve reached by individual i (see Fig. 3), i.e.,

$$\begin{aligned} m_i^*(\widetilde{w}=0,y^*_i,l^*_i)= & {} \left[ 1+\frac{\widehat{\alpha _l}}{\widehat{\phi }_y}\widehat{u}(y^*,l^*)\right] ^{1/\widehat{\alpha _l}} \ \text {for the actual tax scenario};\end{aligned}$$
(A.4)
$$\begin{aligned} m_i^\prime (\widetilde{w}=0,y^\prime _i,l^\prime _i)= & {} \left[ 1+\frac{\widehat{\alpha _l}}{\widehat{\phi }_y}\widehat{u}(y^\prime ,l^\prime )\right] ^{1/\widehat{\alpha _l}} \ \text {for the counterfactual tax scenario.}\nonumber \\ \end{aligned}$$
(A.5)

Calculation of the money-metric utility with \(\widetilde{w}>0\)

To compute the money-metric utility with \(\widetilde{w}>0\), we must first identify the bundle \((l_0,y_0)\) located along the indifference curve attained by individual i with optimal choice \((l^*,y^*)\) (or \((l^\prime ,y^\prime )\) in the counterfactual scenario), where the indifference curve has slope equal to \(\widetilde{w}\). In other words, we have to solve a system of two equations for \(y_0\) and \(l_0\): the marginal rate of substitution between l and y equal to \(\widetilde{w}\) and the indirect utility at the optimal bundle equal to the usual formula in Eq. (3) for \((l_0,y_0)\):

figurea

Solving Eq. (A.6) for \(y_0\) and substituting into Eq. (A.7) yield

$$\begin{aligned} \frac{\widehat{\phi }_y}{\widehat{\alpha }_y}\left[ \left( \frac{\widehat{\phi }_l}{\widehat{\phi }_y} \frac{l_0^{\widehat{\alpha }_l-1}}{\widetilde{w}} \right) ^{\frac{\widehat{\alpha }_y}{\widehat{\alpha }_y-1}} -1\right] + \widehat{\phi }_l \frac{l_0^{\widehat{\alpha }_l}-1}{\widehat{\alpha }_l} -\widehat{u}(y^*,l^*)=0, \end{aligned}$$
(A.8)

which cannot be solved analytically for \(l_0\). Hence, we numerically minimize the absolute value of Eq. (A.8) with respect to \(l_0\) and, by substituting the solution into Eq. (A.6), we get the solution for \(y_0\). If \(l_0<0\) or \(l_0>1\), we are in the presence of a corner solution and \(l_0\) is replaced by \(l_0=0\) or \(l_0=1\), respectively.

Once we know the location of \((l_0,y_0)\), the money-metric utility with \(\widetilde{w}>0\) is the vertical intercept of the budget set passing through \((l_0,y_0)\) with slope \(\widetilde{w}\). Hence, \(m_i^*(\widetilde{w},y^*_i,l^*_i)=y_0-\widetilde{w}(1-l_0)\).

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Picchio, M., Valletta, G. A welfare evaluation of the 1986 tax reform for married couples in the United States. Int Tax Public Finance 25, 757–807 (2018). https://doi.org/10.1007/s10797-017-9472-1

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Keywords

  • Welfare measures
  • Tax reform
  • Preference heterogeneity
  • Discrete model
  • Labor supply

JEL Classification

  • D63
  • H31
  • J22