Bunching and Taxing Multidimensional Skills^††thanks: We thank Hector Chade, Paolo Martellini, Chris Moser, Emmanuel Saez, Florian Scheuer, Andrew Shephard, Chris Taber and especially Jean-Charles Rochet for detailed insightful comments.

The planning problem contains an assignment problem. The planner pairs worker and coworker inputs with firm projects to maximize output given a distribution of worker inputs and firm projects. We show the planner optimally chooses a self-sorted assignment, meaning that workers are paired with identical coworkers, and also show that the planner pairs better teams with more valuable projects.⁴⁴4This assignment problem falls into a class of multimarginal, multidimensional optimal transportation problems. Multidimensional skill and the dependence of worker output on coworkers are central to recent advances in sorting models that utilize optimal transport theory to characterize equilibrium (Chiappori, McCann, and Nesheim, 2010; Dupuy and Galichon, 2014; Lindenlaub, 2017; Chiappori, McCann, and Pass, 2017; Galichon and Salanié, 2022).

We now develop the intuition for Proposition 1. Given a firm project, and since the technology for each task in equation (4) is supermodular, the planner optimally wants to positively sort both cognitive and manual inputs to project $z$. In our economy with multidimensional worker skills, positive sorting within each task is attained by self-sorting. An optimal assignment thus features self-sorting of workers with coworkers within projects $z$. Given that workers are optimally self-sorted, the planner remains to sort self-sorted workers with effective skill $X$ to firms $z$. Since the effective production technology is supermodular in team quality $X$ and project value $z$, the optimal assignment features positive sorting between teams and project values.⁶⁶6Positive sorting of effective skill $X$ with project values $z$ follows the classical Beckerian analysis (Becker, 1973).

Given that the assignment features self-sorting, the output per worker produced by a team of two workers supplying task inputs $(x_{c},x_{m})$ is $\frac{1}{2}z(x^{2}_{c}+x^{2}_{m})$. Aggregate output is $\int y(x_{1},x_{2},z)\text{d}\gamma=\frac{1}{2}\int z(x^{2}_{c}+x^{2}_{m})\text{d}\Phi$, and the resource cost (6) can be written as:

We next transform the planner problem from choosing consumption and task allocations to choosing consumption utility and labor disutility allocations.

For each task $s\in S$, we define the skill parameter $p_{s}=\kappa\alpha_{s}^{-\rho}$ so the skill parameter is inversely related to the underlying skill $\alpha_{s}$. The implied distribution function for the skill parameter vector $p$ is denoted $\pi$, and the corresponding assignment is denoted $z(p)$. We use this skill transformation to define a worker’s utility from consumption as a function of their skill vector as $c(p):=u(c(\alpha))$. Following this definition, the resource cost of consumption utility is $\mathcal{C}(c(p))=u^{-1}(c(p))$. Since the utility from consumption is strictly increasing and concave in the consumption allocation, the resource cost is strictly increasing and convex in consumption utility. Similarly, we define labor disutility in each task $s\in\mathcal{S}$ as a function of the transformed skill parameter $p$ as $x_{s}(p):=x_{s}(\alpha)^{\rho}$. The resource cost of providing disutility $\mathcal{X}(x_{s}(p)):=-\frac{1}{2}x_{s}(p)^{\frac{2}{\rho}}$ is decreasing and strictly convex in labor disutility for $\rho>2$.

Given the introduction of the skill parameter $p$ and the transformation of the choice variables from allocations to utilities, the planner chooses $\{(c(p),x_{c}(p),x_{m}(p))\}$ to minimize the resource cost of providing welfare:

subject to linear incentive constraints:

for all workers $(p,q)$, and a linear promise keeping condition:

We show that the indirect utility for workers is convex and decreasing in type $p=(p_{c},p_{m})$. The indirect utility is defined as:

which implies that for any incentive compatible allocation $\nabla u(p)=-x(p)=-(x_{c}(p),x_{m}(p))^{T}$ almost everywhere. Using the indirect utility function, the incentive constraints (12) are $u(p)\geq u(q)-(p_{c}-q_{c})x_{c}(q)-(p_{m}-q_{m})x_{m}(q)$ or, equivalently in notation of scalar product $\langle\cdot,\cdot\rangle$:

for the incentive constraint where worker type $p$ does not want to report to be of type $q$.

A differentiable function $u$ on a convex domain is convex if and only if $u(p)\geq u(q)+\langle p-q,\nabla u(q)\rangle$. This implies that an incentive compatible indirect utility function is necessarily convex. Since the gradient of the indirect utility function is the negative of a worker’s production disutility, and production disutility is positive, the indirect utility function decreases in $p$, or $\nabla u(p)=-x(p)\leq 0$.⁷⁷7The indirect utility function thus increases in skill $\alpha$. In Appendix A.2, we discuss differentiability of the indirect utility function in more detail, and we also establish which incentive compatibility constraints are redundant.

We refer to bunching as different workers being assigned the same labor allocation $x$, and therefore the same consumption allocation $c$. We label the set of bunching points by $\mathcal{B}$.⁸⁸8Alternatively, one could define a worker $p$ being bunched when there exists another worker $p^{\prime}$ in its neighborhood such that $x(p)=x(p^{\prime})$. Our definition of bunching is the closure of this set. While these definitions are economically equivalent, our definition facilitates the presentation of Proposition 4.

We now state the notions of convexity and strong convexity. Assume that the indirect utility is twice continuously differentiable in the neighborhood of a type $p$. An indirect utility function $u$ is convex if and only if the Hessian matrix $H(u)$ is positive semidefinite. The indirect utility function is strongly convex if $H(u)-\alpha_{I}I$ is positive semidefinite for some strictly positive $\alpha_{I}$, where $I$ is the identity matrix.

In order to describe optimal distortions, we define tax wedges for each task. The tax wedge captures the difference between a worker’s marginal rate of substitution between task $x_{s}$ and consumption $c$, $v^{\prime}\big{(}\frac{x_{s}(\alpha)}{\alpha_{s}}\big{)}\frac{1}{\alpha_{s}}\big{/}u^{\prime}(c(\alpha))$, and the marginal rate of transformation, $zx_{s}(\alpha)$. We define the tax wedge as:

where it follows from the inverse function theorem that $\mathcal{C}^{\prime}(c(p))=1/u^{\prime}(c(\alpha))$. A positive wedge plays a role of an implicit tax on marginal income on task $s$.⁹⁹9Using the definition for the tax wedge, we write $\frac{\tau_{s}}{1-\tau_{s}}=-\frac{z\mathcal{X}^{\prime}(x_{s}(p))+p_{s}\mathcal{C}^{\prime}(c(p))}{p_{s}\mathcal{C}^{\prime}(c(p))}$.

The planner chooses consumption utility and labor disutility $(c,x)$ to minimize the Lagrangian:

subject to the incentive constraints (12), where $\lambda$ denotes the multiplier on the promise keeping constraint (13).¹⁰¹⁰10We suppress the dependence on $p$ to streamline notation.

Proposition 2 combines two variational arguments. First, consider a small proportional change in consumption utility and labor disutility. This variation is feasible. Since this scaling is unrestricted, meaning that it can either increase or decrease the utility allocations, it implies that (18) holds with equality at the optimal allocation $(c,x)$. Second, consider a convex combination of an optimal allocation and any other feasible allocation with a small weight. The convex combination is equivalent to scaling down the optimal allocation and adding a small positive perturbation. By the previous argument, rescaling does not change the Lagrangian at the optimum allocation. The positive perturbation should not decrease the Lagrangian. Since this perturbation is positive it gives an inequality condition.

Proposition 2 presents an implementability constraint for an incentive constrained economy. The implementability conditions are more common in the Ramsey taxation literature where they represent the distortions to allocations introduced by pre-specified taxes. In our model, we do not impose direct restrictions on the permissible taxes and, instead, an information friction endogenously restricts the set of allocations. Importantly, our implementability condition holds with inequality which, as we show, is essential for characterizing the bunching regions.

We use Proposition 2 to derive an optimality condition for our multidimensional taxation problem in terms of a stochastic dominance condition.

We first use the indirect utility (14) for a feasible allocation $(\hat{c},\hat{x})$ to write the implementability condition (18) as:

for any nonnegative, decreasing and convex indirect utility function $\hat{u}$. By Proposition 2 it follows that (19) holds with equality for an optimal indirect utility function. Integrating implementability condition (19) by parts we obtain:

for any nonnegative, decreasing and convex indirect utility function $\hat{u}$, where boundary conditions act on $\hat{u}$ as $\Xi(\hat{u})=\int^{\bar{p}_{m}}_{\underline{p}_{m}}\pi(p_{c}\mathcal{C}^{\prime}(c)+z\mathcal{X}^{\prime}(x_{c}))\hat{u}\big{|}^{\bar{p}_{c}}_{\underline{p}_{c}}\text{d}p_{m}+\int^{\bar{p}_{c}}_{\underline{p}_{c}}\pi(p_{m}\mathcal{C}^{\prime}(c)+z\mathcal{X}^{\prime}(x_{m}))\hat{u}\big{|}^{\bar{p}_{m}}_{\underline{p}_{m}}\text{d}p_{c}$.

We now define second-order stochastic dominance (Shaked and Shanthikumar, 2007):

Comparing the costs and the benefits of taxes is the key insight of the classic ABC formula and the analysis of Diamond (1998) and Saez (2001). In the classic unidimensional case, these costs and benefits are exactly equated for each of the skill levels. Our theorem shows that for the multidimensional tax case with bunching the logic of the ABC formula applies as the costs and the benefits of the taxes are compared. However, those are not necessarily equated at each skill level. Instead, our optimal tax condition (21) considers the benefits and the costs of the entire schedule of taxes and states that the entire schedule of benefits of taxes should second-order stochastically dominate the entire schedule of distortions $-$ showing the non-local nature of the problem with multidimensional skills in the regions with bunching. Our formula applies both to the regions with and without bunching and, in the latter case reduces to equating the costs and the benefits of distortions at each skill level $-$ thus making it a local problem for the regions without bunching.¹¹¹¹11In Appendix A.5, we develop the connection between our general optimal tax conditions and the classic ABC formula. Our optimal taxation condition as stochastic dominance is also related to the sweeping operator in Rochet and Choné (1998). More specifically, the existence of a version of the sweeping operator can be established by using a variation of the Strassen Theorem (see Shaked and Shanthikumar (2007), Theorem 4.A.5). The optimal taxation formula goes beyond existence of such an operator by further relating the entire schedule of costs to the entire schedule of benefits of optimal taxes.

We next provide an optimal tax formula for the multidimensional taxation problem as an equality. This representation also connects to the optimal tax formulas in unidimensional taxation problems which are derived as equality measuring the marginal costs and benefits of taxation (Mirrlees, 1971; Diamond, 1998; Saez, 2001). The main difference with these results is that the optimal tax formula in our multidimensional taxation problem explicitly accounts for global incentive constraints.

Second, the indirect utility function $u(p)$ being convex is equivalent to its Hessian being positive semidefinite, $H(u)\succeq 0$ for all worker types $p$, which in turn is equivalent to:

for all vectors $v\in\mathbb{R}^{2}$. These inequalities are an infinite series of constraints parameterized by the vectors $v$ for each worker type $p$. For each of these constraints, we introduce a multiplier $\lambda(v,p)\geq 0$ and include these constraints into the Lagrangian for the planning problem.

Third, we establish (Lemma 8) that one can represent the constraint that the indirect utility function has to be convex as a matrix condition by introducing a positive semidefinite Kuhn-Tucker matrix $M(p)$ for each worker $p\in P$. Instead of considering the infinite series of constraints for each worker $p$, a single positive semidefinite matrix $M(p)$ induces convexity of the indirect utility function. Upon integration by parts, this restriction appears as a modified social welfare weight $\omega(p)=1+\frac{\Delta M(p)}{\lambda\pi}$. In summary, the main contribution of the convexity constraint to the planning problem is modifying the social welfare weights through a convexity correction. Finally, we derive in Appendix A.6.2 and Appendix A.6.3 how this modified social welfare weight translates into the optimal tax condition (24).

The main difficulty in analyzing bunching in the multidimensional case is that the possible indirect utility perturbations $\hat{u}$ are required to be convex. The convexity of perturbations thus acts as an additional constraint on the entire tax schedule. Without bunching, the perturbation argument is straightforward to construct and leads to equating of cost and benefits of taxes at each skill level. Intuitively, if the underlying utility function is strongly convex, a small enough additive perturbation preserves convexity. As a result, the optimal tax condition (24) in Theorem Theorem applies with the convexity correction $\Delta M=0$ at the types where there is no bunching.

Finally, we provide a converse to Corollary 3 that allows to determine the regions of bunching.

In this section, we discuss the main technique that enables the numerical solution of our problem. Specifically, we transform our problem into a linear problem using Legendre transformations for convex functions that translates convex functions into the upper envelopes of their tangent lines. In order to explain the Legendre transform, and show its importance, we use the resource cost of providing consumption utility $\mathcal{C}$ as an example.

A convex function exceeds all tangent lines. For any consumption utility $c$, and for any point of tangency $a$:

where the equality follows by parameterizing the tangent lines with their slope $\varphi:=\mathcal{C}^{\prime}(a)$ and by letting $\mathcal{C}^{*}(\varphi):=-\mathcal{C}(a)+a\mathcal{C}^{\prime}(a)$ for $a={\mathcal{C}^{\prime}}^{-1}(\varphi)$. The function $\mathcal{C}^{*}$ is the Legendre transform for the resource cost of providing consumption utility $\mathcal{C}$. Since a convex function exceeds all its tangent lines, and since the function value equals the value of the tangent line at the point of tangency:

The Legendre transformation converts the convex resource cost of providing consumption utility on the left side of (29) into a family of linear constraints on the right. The family of linear constraints is parameterized by the slopes of the tangent lines of the cost function. Since the resource cost increases with consumption utility, the slopes of the tangent lines are positive, or $\varphi\geq 0$.

The previous steps apply for any convex function, allowing us to use the same argument to transform the resource cost of providing work disutility into a family of linear constraints:

for each skill $s\in\mathcal{S}$. An increase in production disutility increases production and therefore lowers resource costs. The resource cost of production disutility is decreasing, implying negative slopes of the tangent lines, or $\psi_{s}\leq 0$.

To summarize, the transformed planning problem is to minimize the resource cost of providing utilitarian welfare $\mathcal{U}$:

subject to incentive constraints (12) for all workers $(p,q)\in P\times P$, and the linear promise keeping condition (13).¹²¹²12In Section A.9, we show this problem is equivalent to maximizing utilitarian welfare subject to the resource constraint, and the incentive constraints. In Section A.10 we establish how to derive the stochastic dominance condition and the general optimal tax formula directly from the transformed problem.

First, we consider only a small set of incentive constraints by adding incentive constraints between two worker types only if the distance between them is small.¹³¹³13Oberman (2013) shows that the solution to the problem with only local constraints provides a reasonable initial guess. We then use an iterative procedure to update the set of incentive constraints. On each step, we add all violated incentive constraints to the problem.¹⁴¹⁴14After the final step, the candidate solution satisfies all constraints to the strictly convex optimization problem and hence is the unique solution. In practice, we always obtain the same solution for different initial conditions. With 40 thousand types, this procedure allows to reduce the number of incentive constraints to about 4 million constraints instead of 1.6 billion. Second, an important step that helps us reduce the number of incentive constraints is that we do not need to consider reducible incentive constraints (see Section A.2). This observation additionally reduces the number of constraints by a factor of two. In Appendix A.11 we further prove the accuracy of the approximate planner problem and describe the algorithm that we use to characterize the numerical solution. We finally note that without introducing Legendre transforms the objective is nonlinear. Currently, even the state-of-the-art nonlinear solvers cannot handle the characterization of the solution even for small numbers of types.

We note that solving for the equilibrium assignment in the positive economy follows the same steps as solving for the planner assignment (9) in Section 3.1. It follows from Proposition 1 that the equilibrium features self-sorting between workers and coworkers, and positive sorting between team quality and firm project values.

In order to characterize wages and firm values, we solve the dual transport problem. Since the surplus is negative for any triplet in equilibrium, $S(x_{1},x_{2},z)\leq 0$, and since the aggregate resource constraint (35), the government budget constraint and the household budget constraints hold in equilibrium, the surplus equals zero almost everywhere with respect to the equilibrium assignment, so $w(x_{1})+w(x_{2})+\Omega(z)=y(x_{1},x_{2},z)$. Output is distributed to the owner and to the workers. We use this condition to establish further properties of the firm value function and the wage schedule in Section A.14.

In Section A.14, we first note that wages are only a function of effective worker skills $X=x^{2}_{c}+x^{2}_{m}$, and we define $h(X)$, the firm’s total wage bill, as $h(X)=2w(x)$. By applying standard arguments from optimal transport, wages are convex in effective worker skill $X$. In other words, small differences in effective worker skill translate into increasingly large differences in earnings.¹⁵¹⁵15The hedonic pricing condition $z=h^{\prime}(X)$ delivers superstar effects in our model as well as a number of other assignment models (see, for example, Rosen (1981), Gabaix and Landier (2008), Tervio (2008), Scheuer and Werning (2017), and Boerma, Tsyvinski, and Zimin (2025)). Moreover, the firm value function is the Legendre transform of the wage bill, $\Omega=h^{*}$. As a result, $h(X)+h^{*}(z)=zX$.

We apply this logic to the parametric continuously differentiable function $h(X)=X^{\eta}+2\zeta$ where $\eta\geq 1$ governs the convexity of wages and $\zeta$ captures the lowest wage per worker. Using the derived fact that $z=h^{\prime}(X)$, we can relate the distribution of firm projects $z$ to the convexity parameter $\eta$ of the wage bill. If $\eta=1$, there is no dispersion in firm productivity. We formalize this in Lemma 3.

We use data from the American Community Survey (ACS). We consider individuals between 25 and 60 years of age. The final sample from the ACS includes almost 16 million individuals between 2000 and 2019. For all our results, we use sample weights provided by the survey. Our measure of labor income is wage and salary income before taxes over the past 12 months.¹⁶¹⁶16This measure includes wages, salaries, commissions, cash bonuses, tips, and other money income received from an employer. We drop individuals with earnings below a threshold to focus on workers who are attached to the labor market. This minimum is one-half of the federal minimum wage times 13 weeks at 40 hours per week (as in Guvenen, Ozkan, and Song (2014)).

The ACS contains occupational information for every worker. We combine a worker with the task intensity for their occupation using O*NET task measures from Acemoglu and Autor (2011). Our cognitive measure is the average of their cognitive measures, and our manual measure is the average of their manual measures. Our resulting scores are approximately normally distributed across occupations.

For identification, we first construct a measure of relative task intensity by occupation. To obtain aggregated task production levels we use a Cobb-Douglas technology to map worker subtasks into final task production similar to Kremer (1993), Acemoglu and Autor (2011) and Deming (2017):

Letting $\log q_{s\nu}$ be the $Z$-score by subtask $\nu$, we obtain cognitive and manual task production levels. Since our aggregated cognitive and manual measure are approximately normally distributed, task production levels are approximately lognormal. We now make an identification assumption that the relative task input is equal to the relative task production level, $x_{m}/x_{c}=q_{m}/q_{c}$, which is hence also approximately lognormally distributed across occupations.

Figure 1 shows the distribution of manual and cognitive task production levels across occupations in logs together with the relative distribution of manual and cognitive task intensity. The first two panels show that the distribution of manual task production levels and the distribution of cognitive task production levels can be described by a lognormal distribution. The right panel shows that the same holds for the relative manual task intensity.

Figure 2 displays the relation between relative task intensity and average earnings across occupations. Earnings are low for occupations with high manual task intensity, such as gardeners and truck drivers, while earnings are high for occupations with high cognitive task intensity such as software developers and actuaries. Moving from the 25th percentile to the 75th percentile in relative manual task intensity decreases earnings from 62 to 35 thousand dollars.

We now calibrate the positive model. We parameterize fiscal policy and preferences, and infer the underlying multidimensional skill distribution.

The government taxes labor earnings to finance expenditures $G$. If pre-tax earnings are $w$, then taxes are given by $T(w)=\tau w$. After-tax earnings are thus $(1-\tau)w$, we set $\tau=0.3$.

Firm heterogeneity governs the convexity of the wage schedule (see Lemma 3). We set the curvature parameter for the wage schedule $\eta$ to align the added variation in log wages due to firm heterogeneity with evidence from the literature on variation in log wages due to firm effects. Using the wage equation (36), the variation in firm projects multiplies the underlying variation across workers by $\eta^{2}$. We set $\eta=1.1$ to attribute 17 percent of the added variation in wages to firm effects. Our target of 17 percent is in line with estimates from the literature.¹⁷¹⁷17For example, Abowd, Lengermann, and McKinney (2003) find that firm variation makes up 17 percent of the variance in wages while Song, Price, Guvenen, Bloom, and Von Wachter (2019) instead report that firm variation makes up between 8 percent and 12 percent.

We next discuss the calibration of worker preferences. We use linear preferences with respect to consumption goods, $u(c)=c$, and estimate the parameter governing the curvature of the disutility function to efforts in each task $\rho$. We set $\rho$ such that a regression of log market hours on hourly wages, holding constant the marginal value of wealth, yields a coefficient of 0.55. This target value comes from the meta-analysis of estimates of the intensive margin Frisch elasticity from Chetty, Guren, Manoli, and Weber (2012).

To use estimates for the Frisch elasticity for total hours with respect to hourly productivity to calibrate the curvature of the utility function with respect to effort, we derive this expression within our model. Given the specification for the disutility from work (2), the linear utility from consumption, and the worker technology (3), the worker’s problem (34) is:

The optimality condition to the worker’s problem for each task $s\in\mathcal{S}$ is:

where $w(x)=\frac{1}{2}(x^{2}_{c}+x^{2}_{m})^{\eta}+\zeta$ by wage equation (36) with $\zeta$ representing minimum earnings in our data. That is, the marginal consumption utility from supplying extra tasks equals the marginal cost of effort. Taking the ratio of these optimality conditions, this implies that the skill, effort and task intensity ratio are related by:

where the second equality follows from the worker task technology (3). The marginal rate of substitution between activities, $\big{(}\frac{\ell_{c}}{\ell_{m}}\big{)}^{\rho-1}$, is equal to the ratio of marginal benefits between activities, $\big{(}\frac{\alpha_{c}}{\alpha_{m}}\big{)}^{2}\frac{\ell_{c}}{\ell_{m}}$. Relative efforts are determined by relative skills $\frac{\alpha_{c}}{\alpha_{m}}$. Workers spend more effort on tasks in which they are more talented.

Using the first-order conditions for effort, and observing that the share of total efforts on each task is constant by (40), we can express the Frisch elasticity of total hours $\ell_{c}+\ell_{m}$ as:¹⁸¹⁸18See Section A.16.

where $\lambda$ is the marginal value of wealth, and $z(x):=w(x)/(\ell_{c}+\ell_{m})$ is productivity per hour. We set $\rho=2.8$ so that the Frisch elasticity $\varepsilon$ is indeed $0.55$. Finally, we normalize $\kappa=\frac{1}{2\rho}$.

Using the O*NET task measures, we have information on the relative task intensity for each occupation $\frac{x_{m}}{x_{c}}$ and, hence, we identify the relative skills $\frac{\alpha_{m}}{\alpha_{c}}$ by equation (40). In order to determine the level of tasks, we use the wage equation (36):

Given the skill ratio for an individual’s occupation, $\frac{x_{m}}{x_{c}}$, and an individual’s earnings $w(x)$, this equation uniquely determines the level of cognitive tasks $x_{c}$, and hence the level of manual tasks $x_{m}$. By the optimality condition (39), we identify both cognitive skills $\alpha_{c}$ and manual skills $\alpha_{m}$ for each worker.

Suppose a worker’s occupational relative task intensity is equal to one, $\frac{q_{m}}{q_{c}}=\frac{x_{m}}{x_{c}}=1$, and their earnings equal mean earnings, which we normalize to one. By equation (42), the worker’s cognitive task intensity and the worker’s manual task intensity are equal to $1$. Using the optimality condition for task inputs (39), $\alpha_{s}^{\rho}=\frac{1}{2}$, implying the worker is equally skilled in both tasks. This worker is presented in the first row of Table 1.

Inferred manual skill increases with manual task intensity. Consider some worker with relative manual task intensity equal to three, $\frac{x_{m}}{x_{c}}=3$, and average earnings. By equation (42), the cognitive task intensity is $x_{c}=\frac{1}{\sqrt{5}}<1$ and hence the worker’s manual task intensity is greater with $x_{m}=\frac{3}{\sqrt{5}}>1$. Since $\alpha^{\rho}_{s}=\frac{1}{2}x_{s}^{\rho-2}$, it follows that the worker’s inferred manual skill increases with relative manual task intensity, while the worker’s cognitive skills decreases, as shown in the second row of Table 1.

Inferred skill levels increase with earnings. For a worker with a relative task intensity of one, but a high level of earnings, the relative skill intensity is one but the level of each task is greater. Consider a worker earning four times average earnings. By equation (42), we identify the worker’s cognitive task intensity, and therefore the worker’s manual task intensity, to be equal to $2$. Using the worker’s optimality condition for task inputs (39), $\alpha_{s}^{\rho}=\frac{1}{2}2^{\rho-2}$, implying that the worker is equally skilled in both tasks, and almost 1.75 times as skilled as a worker in the same occupation earning average earnings. This worker is presented in the third row of Table 1.

The presence of taxes does not affect inferred task intensities $x$ but does increase the inferred skill levels $\alpha$. Since the identification of the task intensity is based on pretax earnings (42), inferred task intensities do not vary with taxes. For $\eta=1$, since the task intensity does not change with taxes, we obtain $\alpha^{\rho}_{s}=\frac{1}{2(1-\tau)}x_{s}^{\rho-2}$. When workers are taxed, the marginal benefit from completing tasks is reduced. In order to rationalize the same levels of cognitive and manual task intensity supplied by a worker, it must be less costly for the worker to complete tasks due to increased levels of skills, as shown in the fourth row of Table 1.

Finally, increased dispersion in firm project values decreases wage dispersion that is attributed to dispersion in task intensity. Consider the dispersion in firm projects with $\eta>1$. Reorganizing the wage equation (42), $x_{c}=\left(2w(x)\right)^{\frac{1}{2\eta}}\Big{/}\sqrt{1+\left(\frac{x_{m}}{x_{c}}\right)^{2}}$, shows that higher values of $\eta$ compress the dispersion in task intensity. Further, by combining the first-order condition (39) with wage equation (42), we obtain $\alpha^{\rho}_{s}\propto w(x)^{\frac{\rho}{2\eta}-1}$. An increase in $\eta$ decreases the effective dispersion in skills. Dispersion in firm projects magnifies underlying differences in task intensity due to the positive sorting between workers and projects. Equivalently, small differences in effective worker skills generate large earnings differences.