Optimal tax problems with multidimensional heterogeneity: a mechanism design approach

Abstract

We propose a new method, that we call an allocation perturbation, to derive the optimal nonlinear income tax schedules with multidimensional individual characteristics on which taxes cannot be conditioned. It is well established that, when individuals differ in terms of preferences on top of their skills, optimal marginal tax rates can be negative. In contrast, we show that with heterogeneous behavioral responses and skills, one has optimal positive marginal tax rates, under utilitarian preferences and maximin.

A Appendix

1.1 A.1 Proof of Lemma 4

Proof

The proof consists of two steps. In step (i), we show that there exists at most one incentive-compatible allocation $(w,\theta )\mapsto ({\underline{C}}(w,\theta ),{\underline{Y}}(w,\theta ))$ that verifies Assumption 3 and such that $({\underline{C}}(w,\theta _0),{\underline{Y}}(w,\theta _0))=(C(w,\theta _0),Y(w,\theta _0))$. In step (ii), we show that this allocation verifies the whole set of incentive constraints (6).

Step (i). To build up the entire incentive-compatible allocation $(w,\theta )\mapsto ({\underline{C}}(w,\theta ),{\underline{Y}}(w,\theta ))$, we must choose $({\underline{C}}(w,\theta _0),{\underline{Y}}(w,\theta _0))=(C(w,\theta _0),Y(w,\theta _0))$ at any skill level. For each group $\theta$, ${\underline{Y}}(\cdot ,\theta )$ verifies Assumption 3 if and only if its reciprocal ${\underline{Y}}^{-1}(\cdot ;\theta )$ is smoothly increasing. Let $y\in \mathbb {R}_+$ be an income level. As $Y(\cdot ,\theta _0)$ is smoothly increasing from Assumption 3, there exists a unique skill level w such that $y=Y(w,\theta _0)$. Then according to Lemma 3, among individuals of group $\theta$, only those of skill ${\underline{W}}(w,\theta )$ must be assigned to the income level $y=Y(w,\theta _0)$ to verify incentive-compatibility.¹⁰ Therefore, ${\underline{Y}}^{-1}(\cdot ,\theta )$ must be defined by:

$$\begin{aligned} {\underline{Y}}^{-1}(\cdot ,\theta ):\qquad y\overset{Y^{-1}(\cdot ,\theta _0)}{\longmapsto }w=Y^{-1}(y,\theta _0) \overset{{\underline{W}}(\cdot ,\theta )}{\longmapsto }Y^{-1}(y,\theta ). \end{aligned}$$

${\underline{Y}}^{-1}(\cdot ,\theta )$ is then smoothly increasing as a combination of two smoothly increasing functions. Moreover, since for each type $(\omega ,\theta )$, there exists a single skill level $\omega$ such that ${\underline{Y}}(\omega ,\theta )=\textit{Y}(w,\theta _0)$, incentive compatibility requires that ${\underline{C}}(\omega ,\theta )$ also needs to be equal to ${\underline{C}}(w,\theta _0)$. This ends the proof of step (i).

Step (ii). Note that the allocation $(w,\theta )\mapsto ({\underline{Y}}(w,\theta ),{\underline{C}}(w,\theta ))$ is built in such a way that one has ${\underline{Y}}(\omega ,\theta )=\textit{Y}(w,\theta _0)\text { and } {\underline{C}}(\omega ,\theta )=\textit{C}(w,\theta _0)$ if and only if $\omega ={\underline{W}}(w,\theta )$ and (10) holds. Differentiating in w both sides of these two equations and rearranging terms, we obtain

$$\begin{aligned} \frac{{\dot{C}}\left( w,\theta _0\right) }{{\dot{Y}}\left( w,\theta _0\right) }= \frac{\dot{{\underline{C}}}\left( {\underline{W}}(w,\theta ),\theta _0\right) }{\dot{{\underline{Y}}}\left( {\underline{W}}(w,\theta ),\theta _0\right) }. \end{aligned}$$

As $w\mapsto (C(w,\theta _0),Y(w,\theta _0))$ is assumed to verify the within-group incentive constraints in Eq. (8b), we know that the left-hand side of the above equation is equal to

$$\begin{aligned} \mathscr {M}(C(w,\theta _0),Y(w,\theta _0);w,\theta _0). \end{aligned}$$

Using the definition of ${\underline{W}}(\cdot ,\theta )$, we have that $w\mapsto ({\underline{C}}(w,\theta ),{\underline{Y}}(w,\theta ))$ also verifies Eq. (8b). From Lemma 2, it thus verifies the within-group incentive constraints (7). We now check whether the inequality (6) is verified for any $(w,w',\theta ,\theta ')\in \mathbb {R}_+^2\times \Theta ^2$. We know there exists $\omega \in \mathbb {R}_+$ such that ${\underline{Y}}(\omega ,\theta )={\underline{Y}}(w',\theta ')\text { and } {\underline{C}}(\omega ,\theta )={\underline{C}}(w',\theta ')$. The incentive constraints in (6) are therefore equivalent to:

$$\begin{aligned} \mathscr {U}\left( C(w,\theta ),Y(w,\theta );w,\theta \right) \ge \mathscr {U}\left( C(\omega ,\theta ),Y(\omega ,\theta );w,\theta \right) . \end{aligned}$$

The latter inequality is verified as $w\mapsto ({\underline{C}}(w,\theta ),{\underline{Y}}(w,\theta ))$ satisfies Eq. (8b). $\square$

1.2 A.2 Derivation of Eq. (17)

Proof

To derive (17), we must compute the various Gâteaux derivatives at $t=0$. For $A=C,Y,U$ and a given $\delta$, the Gâteaux derivative of A in the direction $\Delta _Y(\cdot ,\cdot ;\delta )$ at $t=0$ is denoted $\hat{{\hat{A}}}(x,\theta ;\delta )$. Let us remind its definition:

$$\begin{aligned} \hat{{\hat{A}}}(x,\theta ;\delta )\overset{\text {def}}{\equiv }\underset{t\mapsto 0}{\lim } \frac{{\hat{A}}(x,\theta ;t,\delta )-A(w,\theta )}{t}. \end{aligned}$$

By definition we get: $\hat{{\hat{Y}}}(x,\theta _0;\delta )=\Delta _Y(x;\delta )$, and from (15b) we obtain:

$$\begin{aligned} \hat{{\hat{Y}}}(x,\theta ;\delta )=0\qquad \text {if}\qquad x \in \left[ 0,W(w-\delta ,\theta )\right] \cup \left[ W(w,\theta ),+\infty \right) . \end{aligned}$$

(24a)

Equation (15c) imply that the Gâteaux derivatives of utilities are nil for skill below $W(w-\delta ,\theta )$. For skills x between $W(w-\delta ,\theta )$ and $W(w,\theta )$, Eq. (15e) implies:

$$\begin{aligned} \hat{{\hat{U}}}(x,\theta ;\delta )= -\int _{W(w-\delta ,\theta )}^{x} \upsilon _{yw}\left( Y(\omega ,\theta _0);\omega ,\theta _0\right) \ \hat{{\hat{Y}}}(\omega ,\theta _0;\delta )\ d\omega . \end{aligned}$$

(24b)

For skill x above $W(w,\theta )$, according to (15f), we have:

$$\begin{aligned} \hat{{\hat{U}}}(x,\theta ;\delta )= -\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(\omega ,\theta _0);\omega ,\theta _0\right) \ \hat{{\hat{Y}}}(\omega ,\theta _0;\delta )\ d\omega . \end{aligned}$$

(24c)

Moreover, Eq. (15h) implies that the Gâteaux derivatives of income must verify:

$$\begin{aligned} \int _{w-\delta }^{w} \upsilon _{yw}\left( Y(\omega ,\theta _0);\omega ,\theta \right) \ \hat{{\hat{Y}}}(\omega ,\theta _0;\delta ) \ d\omega = \int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(\omega ,\theta );\omega ,\theta \right) \ \hat{{\hat{Y}}}(\omega ,\theta ;\delta ) \ d\omega . \end{aligned}$$

(24d)

Using Eqs. (12), (24a) and (24c), the Gâteaux derivative of the Lagrangian (16) is:

$$\begin{aligned}&\frac{\partial \hat{{\mathscr {L}}}}{\partial t}(0;\delta )= \int _{\theta \in \Theta }\left\{ \int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( 1-\frac{\upsilon _y(Y(x,\theta );x,\theta )}{u^\prime (C(x,\theta ))}\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) f(x|\theta )dx\right. \nonumber \\&+ \left. \int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( \frac{\Phi _U[x,\theta ]}{\lambda }-\frac{1}{u^\prime [x,\theta ]}\right) \hat{{\hat{U}}}(x,\theta ;\delta ) f(x|\theta )dx\right. \nonumber \\&- \left( \int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx\right) \nonumber \\&\times \left. \left( \int _{W(w,\theta )}^{\infty } \left( \frac{\Phi _U[x,\theta ]}{\lambda }-\frac{1}{u^\prime [x,\theta ]}\right) f(x|\theta )dx \right) \right\} d\mu (\theta ). \end{aligned}$$

(25)

Dividing the first-order condition $\frac{\partial \hat{\mathscr {L}}}{\partial t}(0;\delta )=0$ by $\int _{w-\delta }^{w} \upsilon _{yw}\left( Y(x,\theta _0);x,\theta _0\right) \ \hat{{\hat{Y}}}(x,\theta _0;\delta )\ dx$ implies, using (24b) and (24d), that:

$$\begin{aligned}&\int _{\theta \in \Theta } \dfrac{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( 1-\dfrac{\upsilon _y(Y(x,\theta );x,\theta )}{u^\prime (C(x,\theta ))}\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) f(x|\theta )dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx} \ d\mu (\theta ) \nonumber \\&\quad = \int _{\theta \in \Theta }\left\{ \int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( \dfrac{\Phi _U[x,\theta ]}{\lambda }-\dfrac{1}{u^\prime [x,\theta ]}\right) \dfrac{\int _{W(w-\delta ,\theta )}^{x} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx } f(x|\theta )dx \right. \nonumber \\&\qquad + \left. \int _{W(w,\theta )}^{\infty } \underset{}{\left( \frac{\Phi _U[x,\theta ]}{\lambda }-\frac{1}{u^\prime [x,\theta ]}\right) } f(x|\theta )dx\right\} d\mu (\theta ). \end{aligned}$$

(26)

We finally take the limit of the latter equality when $\delta$ tends to 0. Let us consider the first term in the right-hand side of (26). Since

$$\begin{aligned} \dfrac{\int _{W(w-\delta ,\theta )}^{x} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx } \in [0,1] \end{aligned}$$

we get that:

$$\begin{aligned}&\left| \int _{\theta \in \Theta } \int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( \dfrac{\Phi _U[x,\theta ]}{\lambda }-\dfrac{1}{u^\prime [x,\theta ]}\right) \dfrac{\int _{W(w-\delta ,\theta )}^{x} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx } f(x|\theta )dx d\mu (\theta )\right| \\&\quad \le \left| \int _{\theta \in \Theta } \int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( \dfrac{\Phi _U[x,\theta ]}{\lambda }-\dfrac{1}{u^\prime [x,\theta ]}\right) f(x|\theta )dx d\mu (\theta )\right| . \end{aligned}$$

As the right hand-side of the latter inequality tends to 0 when $\delta$ tends to 0, the limit of (26) when $\delta$ tends to zero leads to:

$$\begin{aligned}&\underset{\delta \mapsto 0}{\lim }\quad \int _{\theta \in \Theta } \dfrac{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( 1-\dfrac{\upsilon _y(Y(x,\theta );x,\theta )}{u^\prime (C(x,\theta ))}\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) f(x|\theta )dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx} d\mu (\theta ) \nonumber \\&\quad = \iint _{\theta \in \Theta ,x\ge W(w,\theta )} \left( \frac{\Phi _U[x,\theta ]}{\lambda }-\frac{1}{u^\prime [x,\theta ]}\right) f(x|\theta )dx\ d\mu (\theta ). \end{aligned}$$

(27)

By continuity, the variations of $f(x|\theta )$, $\upsilon _y(Y(x,\theta );x,\theta )$, $\upsilon _{yw}(Y(x,\theta );x,\theta )$ and $u^\prime (c(x,\theta ))$ within the skill intervals $[W(w-\delta ,\theta ),W(w,\theta )]$ are of second-order when $\delta$ tends to 0. As $\Theta$ and intervals $[W(w-\delta ,\theta ),W(w,\theta )]$ are compact, for any small $e>0$, there always exists ${\tilde{\delta }}(e)$ such that for all $(x,\theta )\in [W(w-{\tilde{\delta }}(e),\theta ),W(w,\theta )]\times \Theta$, one has:

$$\begin{aligned}&\left( \frac{1-\upsilon _y[ W(w,\theta ) ,\theta ]}{u^\prime (C(W(w,\theta ),\theta )}f(W(w,\theta )|\theta )-e\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) \le \left( \frac{1-\upsilon _y[ W(x,\theta ) ,\theta ]}{u^\prime (C(W(x,\theta ),\theta )}f(x|\theta )\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) \\&\quad \le \left( \frac{1-\upsilon _y[W(w,\theta ) ,\theta ]}{u^\prime (C(W(w,\theta ),\theta )}f(W(w,\theta )|\theta )+e \right) \hat{{\hat{Y}}}(x,\theta ;\delta ) \end{aligned}$$

and

$$\begin{aligned}&\left( \upsilon _{yw} [W(w,\theta ) ,\theta ] - e\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) \le \upsilon _{yw} [W(x,\theta ) ,\theta ] \ \hat{{\hat{Y}}}(x,\theta ;\delta ) \le \left( \upsilon _{yw} [W(w,\theta ) ,\theta ] +e \right) \hat{{\hat{Y}}}(x,\theta ;\delta )<0 \end{aligned}$$

so that for all $\delta <{\tilde{\delta }}(e)$:

$$\begin{aligned}&\int _{\theta \in \Theta } \dfrac{\left( 1-\dfrac{\upsilon _y(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )}{u^\prime (C(W(w,\theta ),\theta )}\right) f(W(w,\theta )|\theta )+e}{\upsilon _{yw}(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )-e}\ \dfrac{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \hat{{\hat{Y}}}(x,\theta ;\delta ) dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx} d\mu (\theta ) \\&\quad \le \int _{\theta \in \Theta } \dfrac{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( 1-\dfrac{\upsilon _y(Y(x,\theta );x,\theta )}{u^\prime (C(x,\theta ))}\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) f(x|\theta )dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx} d\mu (\theta ) \\&\quad \le \int _{\theta \in \Theta } \dfrac{\left( 1-\dfrac{\upsilon _y(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )}{u^\prime (C(W(w,\theta ),\theta )}\right) f(W(w,\theta )|\theta )-e}{\upsilon _{yw}(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )+e}\ \dfrac{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \hat{{\hat{Y}}}(x,\theta ;\delta ) dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx} d\mu (\theta ). \end{aligned}$$

and therefore, for all $\delta <{\tilde{\delta }}(e)$:

$$\begin{aligned}&\int _{\theta \in \Theta } \dfrac{\left( 1-\dfrac{\upsilon _y(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )}{u^\prime (C(W(w,\theta ),\theta )}\right) f(W(w,\theta )|\theta )+e}{\upsilon _{yw}(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )-e} d\mu (\theta ) \\&\quad \le \int _{\theta \in \Theta } \dfrac{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \left( 1-\dfrac{\upsilon _y(Y(x,\theta );x,\theta )}{u^\prime (C(x,\theta ))}\right) \hat{{\hat{Y}}}(x,\theta ;\delta ) f(x|\theta )dx}{\int _{W(w-\delta ,\theta )}^{W(w,\theta )} \upsilon _{yw}\left( Y(x,\theta );x,\theta \right) \ \hat{{\hat{Y}}}(x,\theta ;\delta )\ dx} d\mu (\theta ) \\&\quad \le \int _{\theta \in \Theta } \dfrac{\left( 1-\dfrac{\upsilon _y(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )}{u^\prime (C(W(w,\theta ),\theta )}\right) f(W(w,\theta )|\theta )-e}{\upsilon _{yw}(Y(W(w,\theta ),\theta );W(w,\theta ),\theta )+e} d\mu (\theta ) \end{aligned}$$

Hence, the left-hand side of (27) is equal to the left-hand side of (17). $\square$

1.3 A.3 Proof of Lemma 5

With one-dimensional heterogeneity, we only consider within-group incentive constraints. Adopting a first-order approach, only (8a) is considered when building up the Hamiltonian:

$$\begin{aligned} \left( Y(w,\theta )-\mathscr {C}\left( Y(w,\theta ),U(w,\theta );w,\theta \right) +\dfrac{\Phi \left( U(w,\theta );w,\theta \right) }{\lambda } \right) \cdot f(w|\theta )-q(w|\theta )\cdot v_w\left( Y(w,\theta );w,\theta \right) . \end{aligned}$$

where $Y(w,\theta )$ and $U(w,\theta )$ are the control and state variables respectively. Using (12), the necessary conditions are:

$$\begin{aligned} 0= & {} \left( 1-\dfrac{v_y\left[ w,\theta \right] }{u^\prime \left[ w,\theta \right] }\right) \cdot f(w|\theta )- q(w|\theta )\cdot v_{yw}\left[ w,\theta \right] \end{aligned}$$

(28a)

$$\begin{aligned} -{\dot{q}}\left( w|\theta \right)= & {} \left( \dfrac{\Phi _U\left[ w,\theta \right] }{\lambda }-\dfrac{1}{u^\prime \left[ w,\theta \right] }\right) \cdot f(w|\theta ) \end{aligned}$$

(28b)

$$\begin{aligned} 0= & {} q(0|\theta ) \end{aligned}$$

(28c)

$$\begin{aligned} 0= & {} \underset{w\mapsto \infty }{\lim }q(w|\theta ). \end{aligned}$$

(28d)

Combining (28b) with (28d) leads to

$$\begin{aligned} q(w|\theta )=\int _w^\infty \left( \dfrac{\Phi _U\left[ w,\theta \right] }{\lambda }-\dfrac{1}{u^\prime \left[ w,\theta \right] }\right) \cdot f(\omega |\theta )d\omega . \end{aligned}$$

(28e)

Combining (3), (2), (28a) and (28e) leads to (18a). Combining (28c) with (28e) leads to (18b).

1.4 A.4 Proof of Proposition 2

Define a reform of a tax schedule $y\mapsto T(y)$ with its direction, which is a differentiable function $y\mapsto R(y)$ defined on ${\mathbb {R}}_+$, and with its algebraic magnitude $m\in {\mathbb {R}}$. After a reform, the tax schedule becomes $y\mapsto T(y)-m \ R(y)$ and the utility of an individuals of type $(w,\theta )$ is:

$$\begin{aligned} U^R(m;w,\theta )\overset{\text {def}}{\equiv }\quad \underset{y}{\max }\quad u(y-T(y)+m\ R(y))-\upsilon (y;w,\theta ). \end{aligned}$$

(29)

We denote by $Y^R(m;w,\theta )$ the income of workers of types $(w,\theta )$ after the reform and her consumption becomes $C^R(m;w,\theta )=Y^R(m;w,\theta )-T(Y^R(m;w,\theta ))+m\ R(Y^R(m;w,\theta ))$. When $m=0$, we have $Y^R(0;w,\theta )=Y(w,\theta )$ and $C^R(0;w,\theta )=C(w,\theta )$. Applying the envelope theorem to (29), we get:

$$\begin{aligned} \frac{\partial U^R}{\partial m}(m;w,\theta )=u_c\left( C^R(m;w,\theta )\right) \ R(y). \end{aligned}$$

(30)

Using (3), the first-order condition associated to (29) equalizes to zero the following expression:

$$\begin{aligned} {\mathscr {Y}}^R(y,m;w,\theta ) \overset{\text {def}}{\equiv }1-T^\prime (y)+m\ R^\prime (y)-\mathscr {M}\left( y-T(y)+m\ R(y),y;w,\theta \right) . \end{aligned}$$

(31)

For simplicity, we drop the superscript R and write $\mathscr {Y}_y(Y(w,\theta );w,\theta )$ for $\mathscr {Y}_y^R(Y(w,\theta ),0;w,\theta )$.¹¹ We define behavioral responses to tax reforms of direction R by applying the implicit function theorem to $\mathscr {Y}^R(y,m;w,\theta )=0$ at $m=0$, which yields:

$$\begin{aligned} \frac{\partial Y^R}{\partial m}(0;w,\theta )=-\frac{\mathscr {Y}^R_{m}(Y(w,\theta ),0;w,\theta )}{\mathscr {Y}_{y}(Y(w,\theta ),0;w,\theta )} \end{aligned}$$

(32)

where:

$$\begin{aligned} {\mathscr {Y}}_y^R(y,m;w,\theta )= & {} -T^{\prime \prime }(y)-\mathscr {M}_y(y-T(y)+m\ R(y),y;w,\theta ) \nonumber \\&-{\mathscr {M}}(y-T(y)+m\ R(y),y;w,\theta )\ {\mathscr {M}}_c(y-T(y)+m\ R(y),y;w,\theta ), \end{aligned}$$

(33a)

$$\begin{aligned} \mathscr {Y}_m^R(y,m;w,\theta )= & {} R^\prime (y)- R(y)\ {\mathscr {M}}_c(y-T(y)+m\ R(y),y;w,\theta ). \end{aligned}$$

(33b)

Using (2) and plugging $R(Y(w,\theta ))=0$ and $R^\prime (Y(w,\theta ))=0$ into (33b), the compensated elasticity of earnings (19a) can be rewritten as:

$$\begin{aligned} \varepsilon (w,\theta )=\frac{\mathscr {M}(C(w,\theta ),Y(w,\theta );w,\theta )}{-Y(w,\theta )\ \mathscr {Y}_y(Y(w,\theta );w,\theta ) }>0 \end{aligned}$$

(34a)

which is positive since $\mathscr {Y}_y\left( Y(w,\theta );w,\theta \right) <0$. Plugging $R(Y(w,\theta ))=1$ and $R^\prime (Y(w,\theta ))=0$ into (33b), the income effect (19b) can be rewritten as:

$$\begin{aligned} \eta (w,\theta )= \frac{\mathscr {M}_c(C(w,\theta ),Y(w,\theta );w,\theta )}{\mathscr {Y}_y(Y(w,\theta );w,\theta )} \end{aligned}$$

(34b)

which is negative if leisure is a normal good, since then $\mathscr {M}_c>0$. The elasticity $\alpha (w;\theta )$ of earnings with respect to the skill level can be expressed as:

$$\begin{aligned} \alpha (w,\theta )= \frac{w\ \mathscr {M}_w(C(w,\theta ),Y(w,\theta );w,\theta )}{Y(w,\theta )\ \mathscr {Y}_y(Y(w,\theta );w,\theta )} >0. \end{aligned}$$

(34c)

Dividing (34a) by (34c) we get:

$$\begin{aligned} \frac{\varepsilon (w,\theta )}{\alpha (w,\theta )}=-\frac{v_{y}\left[ w,\theta \right] }{w\cdot v_{yw}\left[ w,\theta \right] }. \end{aligned}$$

(35)

Plugging (34a) into (34b) leads to:

$$\begin{aligned} \eta (w,\theta )= Y(w,\theta ) \cdot \frac{u^{\prime \prime }\left[ w,\theta \right] }{u^{\prime }\left[ w,\theta \right] } \cdot \varepsilon (w,\theta ). \end{aligned}$$

It is then straightforward to obtain:

$$\begin{aligned} {\hat{\eta }}(Y(w,\theta _0))=Y(w,\theta _0)\cdot \frac{u^{\prime \prime }\left[ w,\theta _0\right] }{u^{\prime }\left[ w,\theta _0\right] }\cdot {\hat{\varepsilon }}(Y(w,\theta _0)). \end{aligned}$$

(36)

Let $y\in \mathbb {R}_+$. Since $\mathscr {Y}_y\left( Y(w,\theta );w,\theta \right) <0$, there exists a single skill level w such that $y=Y(w,\theta _0)$. From (2), we know that:

$$\begin{aligned} 1-T^\prime \left[ w,\theta \right] =\frac{v_y\left[ w,\theta \right] }{u^\prime \left[ w,\theta \right] }. \end{aligned}$$

(37)

The term in the left-hand side integral of (14a) can be rewritten as:

$$\begin{aligned} \frac{v_y\left[ W(w,\theta ),\theta \right] }{- W(w,\theta )\ v_{yw}\left[ W(w,\theta ),\theta \right] }\ W(w,\theta ) \ f(W(w,\theta )|\theta )= & {} \frac{\varepsilon \left( W(w,\theta ),\theta \right) }{\alpha \left( W(w,\theta ),\theta \right) }\cdot W(w,\theta ) \ f(W(w,\theta )|\theta )\\= & {} \varepsilon \left( W(w,\theta ),\theta \right) \ Y(w,\theta _0)\ h(Y(w,\theta _0)|\theta ). \end{aligned}$$

The first equality is obtained using Eq. (35). The second equality uses (21). It implies with (22b) that Eq. (14a) can be rewritten as:

$$\begin{aligned} \frac{T^\prime \left[ w,\theta _0\right] }{1-T^\prime \left[ w,\theta _0\right] }\cdot {\hat{\varepsilon }}\left( Y(w,\theta _0)\right) \cdot Y(w,\theta _0)\cdot \hat{h}(Y(w,\theta _0)) = J(w) \end{aligned}$$

(38)

where J(w) is defined by the right-hand side of (14a). $J(\cdot )$ admits for derivative ${\dot{J}}(w)$ where:

$$\begin{aligned}&{\dot{J}}(w)={\dot{C}}(w,\theta _0) \frac{u^{\prime \prime }\left[ w,\theta _0\right] }{ u^{\prime }\left[ w,\theta _0\right] } J(w)\\&\qquad + \int \limits _{\theta \in \Theta }\left\{ \frac{ \Phi _U\left[ W(w,\theta ),\theta \right] \ u^{\prime }\left[ W(w,\theta ),\theta \right] }{\lambda } -1\right\} {\dot{W}}(w,\theta )\ f\left( W(w,\theta )|\theta \right) d\mu (\theta ) \\&\quad = \int _{\theta \in \Theta }\left\{ g\left( W(w,\theta ),\theta \right) -1\right\} \cdot {\dot{W}}(w,\theta )\cdot f\left( W(w,\theta ; \theta _0)|\theta \right) \cdot d\mu (\theta ) +{\dot{C}}(w,\theta _0) \cdot \frac{ u^{\prime \prime }\left[ w,\theta _0\right] }{ u^{\prime }\left[ w,\theta _0\right] } \cdot J(w) \end{aligned}$$

where (20) has been used. Deriving with respect to the skill w both sides of (9) and of $C(w,\theta _0)=Y(w,\theta _0)-T\left( Y(w,\theta _0)\right)$, we obtain:

$$\begin{aligned} {\dot{W}}(w,\theta )=\frac{{\dot{Y}}\left( w,\theta _0\right) }{{\dot{Y}}\left( W(w,\theta ),\theta \right) } \qquad \text {and}\qquad {\dot{C}}(w,\theta _0) =\left( 1-T^\prime \left( Y(w,\theta _0)\right) \right) \ {\dot{Y}}(w,\theta _0). \end{aligned}$$

We thus obtain:

$$\begin{aligned} {\dot{J}}(w)=\left( \int \limits _{\theta \in \Theta }\left\{ g\left( W(w,\theta ),\theta \right) -1\right\} \frac{f\left( W(w,\theta )|\theta \right) }{{\dot{Y}}(W(w,\theta ),\theta )} d\mu (\theta ) + \left( 1-T^\prime \left[ w,\theta _0\right] \right) \frac{u^{\prime \prime }\left[ w,\theta _0\right] }{u^{\prime }\left[ w,\theta _0\right] } J(w) \right) {\dot{Y}}(w,\theta _0). \end{aligned}$$

Using (21) and (38), ${\dot{J}}(w)$ can be rewritten as:

$$\begin{aligned} {\dot{J}}(w)= & {} \left( \int \limits _{\theta \in \Theta }\left\{ g\left( W(w,\theta ),\theta \right) -1\right\} h\left( Y(w,\theta _0)|\theta \right) d\mu (\theta ) \right. \\&+ \left. T^\prime \left( Y(w,\theta _0)\right) Y(w,\theta _0) \frac{u^{\prime \prime }\left( C(w,\theta _0)\right) }{ u^{\prime }\left( C(w,\theta _0)\right) } {\hat{\varepsilon }}(Y(w,\theta _0)) \hat{h}(Y(w,\theta _0)) \right) {\dot{Y}}(w,\theta _0). \end{aligned}$$

Using (36) and (22d), we get:

$$\begin{aligned} -{\dot{J}}(w)= & {} \left\{ 1-\hat{g}(Y(w,\theta _0))- {\hat{\eta }}(Y(w,\theta _0))\cdot T^\prime \left( Y(w,\theta _0)\right) \right\} \cdot \hat{h}\left( Y(w,\theta )\right) \cdot {\dot{Y}}(w,\theta _0). \end{aligned}$$

As $J(w)=\int _{x\ge w} (-{\dot{J}}(x))dx$, we get

$$\begin{aligned} J(w)= & {} \int _{x \ge w} \left\{ 1-\hat{g}(Y(x,\theta _0))- {\hat{\eta }}(Y(x,\theta _0))\cdot T^\prime \left( Y(x,\theta _0)\right) \right\} \cdot \hat{h}\left( Y(x,\theta )\right) \cdot {\dot{Y}}(x,\theta _0)\cdot dx. \end{aligned}$$

Changing variables by posing $z=Y(x,\theta _0)$, we get

$$\begin{aligned} J(w)=\int _{z \ge Y(w,\theta _0)} \left\{ 1-\hat{g}(z)- {\hat{\eta }}(z)\cdot T^\prime \left( Y(z)\right) \right\} \cdot \hat{h}\left( Y(x,\theta )\right) \cdot dz. \end{aligned}$$

(39)

Plugging (39) into (38) gives (23a). Combining (14b) and (39) leads to (23b).

1.5 A.5 Proof of Proposition 3

Let us denote

$$\begin{aligned} K(w) \overset{\text {def}}{\equiv }\iint \limits _{\theta \in \Theta ,x\ge W(w,\theta )} \left( \frac{1}{u^{\prime }(C(x,\theta ))}\ -\frac{\Phi _{U}(U(x,\theta );x,\theta )}{\lambda }\right) f(x|\theta )dx\ d\mu (\theta ) \end{aligned}$$

(40)

the ratio of the right-hand side of (14a) at the skill level w divided by $u^\prime \left( Y(w,\theta _0)-T(Y(w,\theta _0))\right)$. According to Proposition 1, Eq. (14a) and $\upsilon _y>0>\upsilon _{yw}$, the sign of $T^\prime (Y(w,\theta _0))$ is the sign of K(w).

Under utilitarian preferences, $\Phi _{u}=1$. Changing variable in (40) from x to t such that $x=W(t,\theta )$, (i.e. $Y(x,\theta )\equiv Y(t,\theta _0)$ and $C(x,\theta )\equiv C(t,\theta _0)$ according to (9)), we get:

$$\begin{aligned} K(w) = \int _{t\ge w}\left( \frac{1}{u^{\prime }(C(t,\theta _0))}-\frac{1}{\lambda }\right) \ \left( \int _{\theta \in \Theta } {\dot{W}}(t,\theta )\ f\left( W(t,\theta )\vert \theta \right) d\mu \left( \theta \right) \right) \ dt \end{aligned}$$

The derivative of K(w) has the sign of $1/\lambda -1/u^{\prime }(C(w,\theta _0))$, which is decreasing in w because of the concavity of $u(\cdot )$. Moreover, $\underset{w\mapsto \infty }{\lim }K(w)=0$ and Eq. (14b) implies that $K(0)=0$. Therefore, $K(\cdot )$ first increases and then decreases. It is thus positive for all (interior) skill levels. So, optimal marginal tax rates are positive.

Under maximin, one has $U(x,\theta )>U(0,\theta )$ for all $x>0$ from (8a). Therefore, within each group, the most deserving individuals are those whose skill $w=0$. The maximin objective implies $\Phi _{U}\left[ x,\theta \right] =0$ for all $x>0$. Hence, Eq. (40) simplifies to:

$$\begin{aligned} K(w) = \iint \limits _{\theta \in \Theta ,x\ge W(w,\theta )} \frac{1}{u^{\prime }(C(x,\theta ))}\ f(x|\theta )dx\ d\mu (\theta ) \end{aligned}$$

for all $w>0$, which is always positive, thereby leading to positive optimal marginal tax rates.