Optimal taxation with multiple incomes and types

1 Introduction

Studying optimal tax problems involving multiple income sources and dimensions of unobserved heterogeneity presents a significant challenge in the field of public finance. It is crucial to consider this dual-layered multidimensionality when exploring topics such as the optimal taxation of a couple's incomes, the optimal taxation of income from labor and capital, and the optimal means testing of benefits. However, the investigation of such issues introduces considerable theoretical challenges, complicating the formulation of policy recommendations.

Mirrlees (1976) pioneered a Mechanism Design (MD) approach to characterize the optimal incentive-compatible allocation of multiple taxable incomes. This approach results in a partial differential equation that proves challenging to interpret. Golosov, Tsyvinski, and Werquin (2014) develop a Tax Perturbation (TP) approach to characterize the optimal tax schedule. They, too, derive a partial differential equation describing the optimal tax function. Their formulation, however, offers the benefit of being expressed in terms of observable sufficient statistics.

The extent to which the MD approach by Mirrlees (1976) and the TP approach by Golosov, Tsyvinski, and Werquin (2014) are equivalent has not been examined yet. At first glance, it might appear that both approaches must be equivalent, since the taxation principle (Hammond (1979)) proves that selecting a tax function is equivalent to choosing an incentive-compatible allocation. However, this principle holds true only when there are no additional constraints on the tax schedule or the allocation of taxable incomes. Both Mirrlees' MD approach and Golosov, Tsyvinski, and Werquin's TP approach introduce additional smoothness assumptions—the former on the allocation and the latter on the tax schedule—to facilitate the application of variational calculus. These smoothness assumptions address the possibility that distinct types may be allocated the same income bundles, a situation referred to as “bunching,” or instances where the mapping between types and income is discontinuous, a situation referred to as “jumping.” In this paper, we derive optimal-tax formulas using both approaches. We study a model where taxpayers differ in multiple unobservable characteristics and in multiple incomes, and assume that individuals respond along the intensive margins. We assume that multidimensional versions of the single crossing conditions hold and make standard assumptions on the smoothness of the optimal-tax function and the optimal allocation. We show that under these assumptions the optimal-tax formulas derived through both approaches are equivalent when the number of unobservable characteristics equals the number of taxable incomes. The assumptions required to apply the TP approach are slightly more demanding than those required to apply the MD approach. With the TP approach, we need to assume the tax-function is thrice differentiable, whereas the MD approach only requires twice-differentiability of the tax function.

We also investigate the cases where the numbers of taxable incomes and characteristics are not equal. The TP approach solves the optimal-tax problem in the income space, whereas the MD approach solves the same problem in the type space. Therefore, if the number of characteristics exceeds the number of incomes, solving the optimal-tax problem with the TP approach reduces the dimensionality of the problem, and thus its complexity. Conversely, the MD approach reduces the dimensionality when the number of incomes exceeds the number of characteristics.

Determining the optimal tax schedule involves solving a partial differential equation, a task that is significantly more complex than solving the ordinary differential equation implied by the optimal tax formula for a single tax base. To illustrate the complexity, consider that in a one-dimensional scenario, one can examine the effects of perturbing the marginal tax rate at one income level. The optimal marginal tax rate at that income level is then determined by the ratio of mechanical and income effects at all incomes above, to compensated effects at the income level under consideration. However, in a multidimensional scenario, it is not possible to examine the effects of a change in the tax gradient at one combination of incomes without inducing additional changes in the tax gradients at other combinations of incomes.

We develop a numerical algorithm that tackles this geometric complexity and can solve the optimal multidimensional tax problem in its general form. We apply our algorithm to the taxation of couples. In our application, we adopt simplifying assumptions akin to those made by Kleven, Kreiner, and Saez (2007). We presume quasilinear and additively separable household preferences. Furthermore, consistent with the empirical literature, we assume that the labor supply of wives is more elastic (0.43) than that of husbands (0.11) (Bargain and Peichl (2016)). Lastly, we calibrate the joint distribution of skills nonparametrically, starting from the joint distribution of incomes in the Current Population Survey (CPS) of the US census.

To facilitate our exposition, we introduce the notion of an “isotax curve,” which refers to a set of income bundles that incur the same tax liability. Our findings indicate that the optimal isotax curves are nearly linear and parallel, with both spouses subjected to positive marginal tax rates. A joint income tax that discounts female income by approximately 53% closely approximates the fully optimized schedule in terms of social welfare. Additionally, we explore the concept of negative jointness, which stipulates that the optimal marginal tax rates of males should decrease with an increase in female income (and vice versa). Kleven, Kreiner, and Saez (2007) analytically demonstrate that negative jointness is desirable when the productivities of both spouses are assumed to be uncorrelated. However, our numerical findings suggest that this result does not hold up under a more realistic joint distribution of productivities.

In addition to our comparison between the MD approach and the TP approach, and our numerical algorithm, we make several theoretical contributions. First, we formulate a test to verify the Pareto efficiency of a given tax schedule. If the welfare weights revealed by the optimal tax formula are negative for certain income bundles, then reducing tax liabilities at these income bundles results in a self-financed Pareto improvement. This extends the revealed social preference approach of Werning (2007), Bourguignon and Spadaro (2012), Bargain, Dolls, Neumann, Peichl, and Siegloch (2014), Jacobs, Jongen, and Zoutman (2017), Scheuer and Werning (2017), Hendren (2020), and Bierbrauer, Boyer, and Hansen (2023) to a multidimensional context.

Second, we employ the MD approach to establish conditions under which the first-order conditions are sufficient to characterize the optimal allocation. This holds true when the government's Lagrangian is concave both with respect to the taxpayers' utilities, and with respect to the gradient of the mapping between the taxpayers' types and utilities. We analytically confirm that the specification used in our numerical exercise complies with these sufficiency conditions. Therefore, once we have obtained a numerical solution that satisfies the government's first-order conditions, we know that it is the unique solution. Consequently, there is no need to perform sensitivity analyses concerning the initial conditions of our algorithm.

Third, we address a concern in the TP approach, where it is assumed by both Saez (2001) and Golosov, Tsyvinski, and Werquin (2014) that incomes respond smoothly to tax perturbations. We contribute by explicitly outlining the assumptions about the tax schedule that ensure smooth responses of taxpayers to tax perturbations. Our assumptions rule out kinks in the tax schedule and the presence of multiple global optima, thereby ensuring that incremental tax perturbations do not lead to jumps in taxpayers' behavior.

Fourth, we introduce a new method to derive the optimal mechanism. When Mirrlees (1976) and Kleven, Kreiner, and Saez (2007) derive necessary conditions for the optimal incentive-compatible allocation, they use taxpayers' utilities and taxable incomes as controls. The challenge here is that there are numerous income allocations that satisfy their necessary conditions for the optimum. We show how for each such income allocation, the first-order incentive constraints imply the partial derivatives of the attained utilities with respect to the types. At this point, nothing guarantees that the obtained partial derivatives of the achieved utilities are mutually consistent, meaning they imply symmetric second-order partial derivatives. Mirrlees (1976, p. 343) and Kleven, Kreiner, and Saez (2007, p. 18) recognize this issue by stating that among the different solutions of the partial differential equation, only the one implying symmetric second-order cross- derivatives should be considered. We circumvent these challenges by dividing the government's problem into two stages. In the first stage, the government chooses the optimal type-to-utility mapping from the set of possible mappings. In the second stage, the taxable incomes are determined as functions of the utility profile and its partial derivatives.

Finally, we investigate the cases where the number of characteristics p differs from the number of incomes n. If the number of characteristics exceeds the number of incomes, opting for the TP method and using average sufficient statistics among taxpayers with identical income bundles reduces the complexity of the problem. This extends the findings of Saez (2001), Scheuer and Werning (2016) and Jacquet and Lehmann (2021) to situations where taxpayers have multiple income sources. Conversely, applying the MD approach in this setting is only feasible under strong restrictions on preferences (see, e.g., Choné and Laroque (2010), Rothschild and Scheuer (2013, 2014, 2016), Scheuer (2014), and Jacquet and Lehmann (2023) for the $$ case).

When the number of incomes n exceeds the number of unobservable characteristics p, it generally makes more sense to use the MD approach. Indeed, by working within the type space rather than the income space, one reduces the problem's dimensionality. In this case, the government's problem involves the two-step process described above. The second step then is a subprogram that determines the most efficient distribution of income choices to produce the type-to-utility mapping that was selected in the first step. The solution to this subprogram is independent of government preferences and solely depends on the resource costs of providing these utility levels. Our findings shed light on similar subprograms implicitly present in the works of, among others, Atkinson and Stiglitz (1976), Golosov, Kocherlakota, and Tsyvinski (2003), Gerritsen, Jacobs, Rusu, and Spiritus (2025), and Ferey, Lockwood, and Taubinsky (2022).

2 The model

3 The tax perturbation approach

In this section, we derive the optimal tax formula using the TP approach, which was previously derived by Golosov, Tsyvinski, and Werquin (2014). A necessary condition for a tax schedule to be optimal is that small perturbations of the schedule do not change social welfare. Golosov, Tsyvinski, and Werquin (2014) assume that individuals respond smoothly to such perturbations. We contribute by revealing underlying assumptions that ensure that the responses of the taxpayers to tax reforms are smooth. We also identify additional assumptions that allow the characterization of the optimum in a partial differential equation. Identifying these assumptions allows us to compare the TP approach to the MD approach.

Assumption 1(i) rules out kinks like those in piecewise linear tax schedules. Moreover, it ensures that the first-order conditions (8) are twice continuously differentiable in t, w, and x, provided that the direction $$ is thrice continuously differentiable. We require thrice differentiability to derive Proposition 1 and 3, as we explain in more detail below. Assumption 1(ii) ensures that the first-order conditions (8) are associated with a local maximum of the taxpayers' program (7). Parts (i) and (ii) of Assumption 1 together enable one to apply the implicit function theorem to determine how a local maximum of (7) is affected by a small tax perturbation or a small change in types. Assumption 1(iii) rules out the existence of multiple global maxima. This prevents an incremental tax perturbation from causing a “jump” in the taxpayers' choices from one maximum to another. At such jumps, the derivative of $$ with respect to the size t of the perturbation tends to infinity.³ If Assumption 1 is satisfied, then the function $$ that solves (7) is continuously differentiable for t close to 0, that is, the behavioral responses to tax reforms are smooth.

Geometrically, Assumption 1 implies that for each type w, the indifference set defined by $$ admits a single tangency point with the budget set defined by $$ and lies strictly above the budget set elsewhere. Given that we assume that the indifference sets defined by $$ are strictly convex, Assumption 1 is automatically verified if the tax schedule is linear (see Appendix A.1).

We characterize the optimal tax schedule under the presumption that Assumption 1 holds, which then needs to be verified ex post in applications. This approach is analogous with the standard first-order MD approach, which presumes that the second-order incentive constraints do not bind in the optimum, and verifies ex post that this is indeed the case (Mirrlees (1971, p. 188)).

At the optimum, there should not exist an infinitesimal perturbation of the tax schedule that would induce a first-order effect on the government's objective. Therefore, the right-hand side of (11) should be equal to zero for any direction $$. We derive an optimal tax formula from this requirement in Appendix A.4. To do so, we first rewrite (11) in the income space. For this purpose, let $$ denote the range of the type set $$ under the allocation $$. Let $$ denote the joint density of incomes x, which is defined over $$. Furthermore, for each combination of incomes $$, let $$, $$, and $$, respectively, denote the means of $$, $$ and $$ among taxpayers that earn the combination of incomes $$. Second, we use the divergence theorem to rewrite the second line of (11). For this purpose, we need income densities and compensated responses to be continuously differentiable. Furthermore, we can only apply the divergence theorem on the income space $$ if it is of dimension n. We thus make the following assumption.

Proposition 1 provides necessary conditions for the government's optimum in the income space. It is consistent with Proposition 3 in Golosov, Tsyvinski, and Werquin (2014). The Euler–Lagrange equation (12a) provides a divergence equation that should hold for any income $$. Note that we use thrice differentiability of the tax schedule (Assumption 1(i) to derive the proposition. Equation (26b) in Appendix A.2 shows that the sufficient statistics $$ depend on the second-order partial derivatives of the tax schedule. Since the right-hand side of (12a) once again contains partial derivatives of these sufficient statistics, we require the tax system to be at least thrice differentiable. Equations (12b) are boundary conditions that should hold at any income $$ in the boundary of $$. Finally, Equation (12c) states that, starting from the optimum, a lump sum perturbation implies no first-order effect on the Lagrangian. Using (6), the latter condition determines the Lagrange multiplier λ. To provide more intuition, the working paper version of this article contains a heuristic proof of Proposition 1 based on a reform that uniformly changes tax liability within a closed convex subset of $$ and changes marginal tax rates around that subset (Section III.3 of Spiritus et al. (2022)).

We have shown that the optimality conditions in Proposition 1 and the condition for a Pareto improvement in Proposition 2 are valid if Assumptions 1 and 2 hold. As Assumption 2 may appear overly demanding, we now discuss a microfoundation to demonstrate its plausibility.

Together, Assumptions 1 and 2′ guarantee that the tax schedule effectively separates taxpayers by type, so no two types choose the same bundle of incomes. We thus obtain the following lemma, which we prove in Appendix A.6.

Lemma 1 allows us to rewrite the necessary conditions for the optimal tax schedule in the type space. This is important because the type space $$ is exogenous to the tax schedule whereas the income space $$ is not. In the numerical computations, this enables us to solve the Euler–Lagrange partial differential equation over a fixed space. Additionally, it is useful because we will also be able to retrieve this optimal tax formula in the type space using the MD approach, proving the consistency of the two approaches. We derive the following proposition in Appendix A.7.⁹

4 The mechanism design approach

In this section, we rederive the optimal tax system using the mechanism-design approach instead of the tax-perturbation approach. This exercise serves two purposes. First, it allows us to verify under what conditions the two approaches result in the same optimal-tax function. Second, we use the mechanism-design approach to verify under what conditions the solution to the government's first-order conditions uniquely describes the social maximum.

We restrict our attention to allocations that are continuously differentiable and satisfy the incentive constraint (16). This is formalized in Assumption 3.

Our approach differs from the traditional approach in Mirrlees (1976), Kleven, Kreiner, and Saez (2007), and Renes and Zoutman (2017), who directly maximize Lagrangian (18) subject to the incentive constraint (17) with respect to both the utility profile and the allocation. As noted by Mirrlees (1976, p. 343) and Kleven, Kreiner, and Saez (2007, p. 18), the traditional approach hides a conceptual problem in the multidimensional context. To see this, consider an example in which utility is additively separable as in (14). In that case, for any given candidate allocation $$, the first-order incentive constraints (17) form a system of partial differential equations in $$. If there is only one type, $$, the system simplifies to an ordinary differential equation, which can be integrated to provide the corresponding mapping $$, up to a constant. Conversely, when $$, the system of partial differential equations (17) for a given candidate mapping $$ yields a candidate for the gradient of $$ with components $$ for all $$. However, not every combination of mappings $$ can be the gradient of a mapping $$. The utility profile $$ must exhibit symmetric second-order cross-derivatives, that is, $$ for all j, k, and all w. Hence, only candidate mappings $$ that imply a utility profile that verifies $$ for all j, k, and for all w, are implementable. These additional implementability constraints are irrelevant in one-dimensional optimal tax problems but cannot be ignored in the multidimensional case. Our approach overcomes this challenge by explicitly choosing the utility profile $$ in the first stage, and choosing $$ and $$ from the incentive-compatible allocations that implement that utility profile in the second stage. Therefore, the solution automatically satisfies the implementability condition $$.

To apply methods from variational calculus to the government's problem (19)–(21), we make regularity assumptions about subprogram (21) in Assumption 4. First, we rule out the possibility that two allocations that yield the same utility profile also extract an identical amount of resources. Second, we make differentiability assumptions about the unique solution to subprogram (21). Together, these assumptions ensure the differentiability of the function L, defined in Equation (20), with respect to all of its arguments. We will provide a plausible microfoundation in Assumption 4′.

Note that subprogram (21) selects n incomes subject to p constraints. Assumption 4 thus implies that the MD approach is generally restricted to cases with at least as many incomes as types, $$, although some exceptions exist, as we discuss in Section 5. Assumptions 3 and 4 allow us to derive necessary conditions for the FOMD problem (19) by considering continuously differentiable perturbations in the utility profile $$, and deducing the resulting perturbed allocations $$ from subprogram (21). Assumption 4 ensures a unique perturbed allocation exists for every perturbed $$. Mirrlees (1976, p. 342) implicitly makes a similar assumption to ensure his system of equations (63) admits a single solution. This leads to the following proposition, which we prove in Appendix B.1.

In a setting where the number of incomes equals the number of characteristics, $$, there is usually only one incentive-compatible allocation that can implement the same utility profile, because the number of free variables in the system of equations (17) is equal to the number of equations. In a setting with more incomes than types, $$, the same utility profile can typically be offered through multiple incentive compatible allocations. In that case, through subprogram (21), the n first-order conditions (22a) can be decomposed into p conditions characterizing the optimal $$ profile and $$ supplementary conditions describing how to decentralize the mapping $$ at the lowest cost.

Several results in the literature that are derived in settings where $$ correspond to these supplementary conditions. A famous example is the Atkinson and Stiglitz (1976) theorem, which states for $$ that when preferences are weakly separable between leisure and consumption, commodity taxes should be uniform. This result remains valid regardless of social preferences for redistribution and can be seen as a way to realize a desired distribution of utilities with the least distortions (see also Jacobs and Boadway (2014)). In the same vein, Boadway and Keen (1993), Gauthier and Laroque (2009), and Jacobs and de Mooij (2015) retrieve first-best principles such as the Samuelson rule for the provision of public good or the Pigouvian tax rule in case of externalities in models with weakly separable preferences and one-dimensional unobserved heterogeneity. Another strand of literature considers capital income taxation in settings with endogenous labor supply and savings and one dimension of unobserved heterogeneity. Assuming that preferences, inherited wealth or returns to capital vary along the ability distribution, the Atkinson and Stiglitz (1976)'s theorem no longer applies and the optimal capital tax is nonzero.¹⁰ These authors show how to split the deadweight losses of redistribution between labor and capital income taxation, relying only on efficiency considerations without reference to social preferences for redistribution. Finally, the “new dynamic public finance” literature (Golosov, Kocherlakota, and Tsyvinski (2003)) considers models, where at each period, there is a new productivity drawn (a new dimension of unobserved heterogeneity) and agents make a labor supply and a saving decision at each period, such that $$ where p is equal to the number of periods. The inverse Euler equation then describes how the planner should allocate consumption between the present period and each state of nature of the following period at the lowest cost. The finding that such supplementary efficiency conditions arise when $$ is summarized in Corollary 1.

An additional advantage of our approach is that it becomes straightforward to provide conditions under which the government's necessary conditions 22 are also unique and sufficient. We do so in the following proposition.

This result is especially important for our numerical simulations. We will demonstrate the use of Proposition 5 in Section 6, where we prove that in our simulations the mapping $$ is concave. As we find in the simulations an allocation that verifies the necessary conditions, Proposition 5 then ensures that this allocation is the unique solution to the government's problem.

We now provide a microfoundation to show the plausibility of Assumption 4. This additional assumption allows us to retrieve the optimal tax formula in the type space provided by Proposition 3 using the FOMD approach, instead of the TP approach.

When the utility function is of the additively separable form described in (14), Assumption 4′ is equivalent to $$. Hence, Assumption 4′ is a way to extend the single crossing condition to a multidimensional context. Recall that in subprogram (21), for given type w, utility level u and utility gradient z, the government chooses the incentive compatible allocation x that maximizes the government's revenues. Because of the incentive constraints of subprogram (21), Assumption 4′ implies that for each type w, the value of $$ uniquely determines the allocation x. The subprogram thus admits a single solution and Assumption 4 is verified. When the number of incomes is equal to the number of unobserved characteristics ($$), the incentive constraints of subprogram (21) imply that Assumption 4 also leads to Assumption 4′, so both assumptions are equivalent.

The fact that Assumptions 4 and 4′ are equivalent when $$ enables us to show that the formulation of the optimality conditions in Proposition 3 can also be derived using the MD approach. We prove the following proposition in Appendix B.2.

We have thus shown that with the right assumptions, the same optimal-tax equations can be derived in the type space using either the FOMD approach or the TP approach. We elaborate further on the correspondence between the two approaches in the next section.

5 Comparing the TP and MD approaches

We now compare TP and MD approaches. In Section 5.1, we focus on the case when the numbers of incomes and characteristics are equal ($$), before turning our attention to the settings where incomes outnumber characteristics ($$, Section 5.2) and where characteristics outnumber incomes ($$, Section 5.3).

6 Numerical simulations

If both the type space and the income space are multidimensional, the optimal-tax formulas do not take the form of ordinary differential equations as in Mirrlees (1971), Diamond (1998), and Saez (2001), but they take the form of a second-order partial differential equation, as in Mirrlees (1976) and Golosov, Tsyvinski, and Werquin (2014). This significantly complicates the process of solving the optimal tax equations. To understand this, it helps to consider the effects of a tax perturbation from a geometric perspective. In the one-dimensional case, the change in the marginal tax rate at a given income level is directly connected to changes in tax liabilities at all higher incomes. In the multidimensional case, the relation is more complicated. To change the gradient of the tax function at a given point, one must change the tax liabilities near that point, causing changes in the tax gradient elsewhere; see, for instance, the graphical proof in the working paper version of this article, Section III.3 of Spiritus et al. (2022). To deal with this complexity, we rely on numerical simulations.

We calibrate the skill density $$ using the Current Population Survey (CPS) of the US census of March 2016. We focus on married, mixed-gender couples that live together. We only consider income from labor. We drop couples in which either partner earns less than $1000 per year or in which either of the partners' incomes is top-coded. We drop same-sex couples because in our simulations we attach labor elasticities based on gender in each couple. From each observed couple, we recover their type $$ from their labor earnings $$ by inverting the first-order conditions (3). For this purpose, we use a rough approximation of the current tax schedule in the US by assuming a constant marginal tax rate of 37%, a figure which is consistent with Barro and Redlick (2011, Table 1). Next, we estimate the type density through a bi-dimensional kernel. We specify the social welfare function to be CARA with $$, where $$ stands for the degree of inequality aversion. For our baseline simulation, we select β such that the assumed 37% tax rate coincides with the optimal linear tax rate. This leads to $$. Throughout the simulations, we assume that the government's revenue requirement equals 15% of GDP, which is close to the observed share of public spending in GDP for the US.

We first give an overview of the simulation algorithm, in Section 6.1. Next, in Section 6.2, we report the results of the simulations for the baseline calibration.

6.2 Results

Figure 1 displays the solution to the optimal tax problem using our baseline calibration. The optimal tax schedule is represented by the isotax curves, which are the loci of incomes for which the tax liability is constant at a given value. Male income is shown on the horizontal axis, while female income is shown on the vertical axis. The left panel displays the whole domain of the simulations running up to $$, while the right panel zooms in at incomes below $$, where we find most taxpayers, roughly 97% of males and 99% of females.

Strikingly, isotax curves are almost linear and parallel, except close to the boundaries. There, isotax curves are curved to satisfy boundary constraints (12b). This curvature pattern is most notable at high income levels where there are very few taxpayers. For lower incomes, the curvature only affects isotax curves very close to the lower bound.

Compared with the current economy, which is approximated by a linear tax rate of 37%, the optimal tax schedule leads to an improvement of the social objective equivalent to 0.82% of GDP in monetary terms. To understand which forces drive this gain, we decompose the welfare gain in different steps. Going from our approximation of the current economy (where we assume linear tax rates) to the optimal joint tax $$ captures the welfare gain of allowing the joint income tax schedule to be nonlinear. We find this welfare gain to be only 0.03%. If we now maintain the requirement that the isotax curves are linear and parallel but remove the requirement that both marginal tax rates are equal, so $$ where α is optimized, we obtain a welfare gain from the current economy equal to 0.81%. The optimal value of α is 2.13, which implies that female income is discounted by 53%. Hence, while the gain of optimizing the slope of the isotax curves (optimizing α) is economically significant, the welfare gain of relaxing the constraint that isotax curves must be linear and parallel appears to be small.

Kleven, Kreiner, and Saez (2007) show that under our individual and social preferences, when the abilities of both spouses are not correlated, the optimal marginal tax rates of each partner decrease in the income of the other partner. This is the so-called negative jointness of the optimal tax system. In a separate simulation with a population that replicates the moments of male and female incomes, but removes any correlation between the two, we confirm the optimality of the negative jointness of the tax system. In reality, however, the assumption that the skills of both partners are not correlated does not hold. We show in Figure 2 that the optimality of negative jointness is not robust to using more realistic type densities with positive assortative matching. Figure 2a (resp., Figure 2b) displays the marginal tax rate for females (males) as a function of their own income. Each curve graphs this marginal income while fixing male (female) income at the $$, $$, and $$ percentile of the male (female) income distribution. In case of negative jointness, the curve corresponding to male (female) income at the $$ percentile should be everywhere above the curve corresponding to male (female) income at $$ and $$ percentiles of the distribution. Figures 2a and 2b contradict this prediction, thereby rejecting the idea that negative jointness holds at the optimum. We rather find that, except at the very bottom of the income distribution, marginal tax rates exhibit minimal jointness, since in Figures 2a and 2b the three lines are close.¹³

7 Conclusion

We study the optimal tax problem with multiple incomes and multiple dimensions of unobserved heterogeneity. We identify assumptions on the smoothness of the allocation and of the tax schedule, and multidimensional extensions of the single crossing assumptions, that enable the use of variational calculus to characterize the optimum. When comparing the MD approach to the TP approach, we demonstrate that when the numbers of types and of incomes are equal, the latter implies slightly more demanding restrictions on the smoothness of the tax schedule. When there are more unobserved characteristics than incomes, the TP approach is more suitable than the MD approach. Conversely, when there are more incomes than unobserved characteristics, the TP approach as we apply it cannot be used to solve the problem. We show that in terms of rigor, the TP method is on par with the MD method.

We propose a numerical algorithm that addresses the difficulties inherent to the multidimensional tax problem. We apply this algorithm to the optimal taxation of couples. Our findings indicate that the optimal isotax curves are nearly linear and parallel. We show that the optimal negative jointness of the tax schedules when skills are uncorrelated does not hold up when a more realistic distribution is introduced.

In addition to our primary findings, we obtain several theoretical results. First, we identify a necessary condition for the tax schedule to be Pareto efficient. If this condition is not met, we describe a Pareto-improving tax reform. Second, we identify conditions that ensure the necessary conditions of the optimal tax problem are unique and sufficient. Third, we contribute to the TP approach by proposing conditions under which income bundles respond smoothly to small tax reforms. Fourth, we introduce a MD approach that encapsulates not only incentive constraints, but also the implementability constraints embedded in the multidimensional optimal tax problem. Lastly, we examine the cases where the number of incomes differs from the number of characteristics.