Voting over selfishly optimal income tax schedules with tax-driven migrations

Abstract

We study majority voting over selfishly optimal nonlinear income tax schedules proposed by a continuum of workers who can migrate between two competing jurisdictions. Both skill level and migration cost are the private information of each worker who will propose an allocation schedule that maximizes the utility of her own type. We identify reasonable scenarios in which the first-order approach applies and hence the second-order sufficient condition for incentive compatibility is fulfilled; otherwise, we need to apply the ironing surgery developed by Brett and Weymark (Games Econ Behav 101:172–188, 2017). Under quasilinear-in-consumption preferences, we show that the tax schedule proposed by the median skill type is the Condorcet winner, and provide a complete characterization of this tax schedule. While this schedule features negative marginal tax rates for low-skilled workers, it features positive rates for high-skilled workers with small migration elasticities; the marginal tax rates at the bottom and top skill levels cannot be unambiguously signed. Moreover, we detail the conditions under which migration induces uniformly higher or lower equilibrium marginal tax rates facing both low- and high-skilled workers than their counterparts in autarky, which leads us to conclude that geographic mobility does not always limit the government’s ability to redistribute incomes via tax-transfer systems.

Appendices

Appendix: Proofs

Proof of Lemma 3.1

By using (6), we have

$$\begin{aligned} U(w)=U({\underline{w}})+\int _{{\underline{w}}}^{w}\frac{y(t)}{t^{2}}h'\left( \frac{y(t)}{t} \right) dt. \end{aligned}$$

(38)

Integrating over the ex post support of the skill distribution yields

$$\begin{aligned} \int _{{\underline{w}}}^{{\overline{w}}}U(w){\tilde{f}}(w)dw =U({\underline{w}})\int _{{\underline{w}}}^{{\overline{w}}}{\tilde{f}}(w)dw+ \int _{{\underline{w}}}^{{\overline{w}}}\left[ \int _{{\underline{w}}}^{w}\frac{y(t)}{t^{2}}h'\left( \frac{y(t)}{t} \right) dt \right] {\tilde{f}}(w)dw. \end{aligned}$$

(39)

Reversing the order of integration in (39) gives rise to

$$\begin{aligned} \int _{{\underline{w}}}^{{\overline{w}}}U(w){\tilde{f}}(w)dw =U({\underline{w}})\int _{{\underline{w}}}^{{\overline{w}}}{\tilde{f}}(w)dw+ \int _{{\underline{w}}}^{{\overline{w}}}\frac{y(t)}{t^{2}}h'\left( \frac{y(t)}{t} \right) \left[ \int _{t}^{{\overline{w}}}{\tilde{f}}(w) dw\right] dt. \end{aligned}$$

(40)

Also, it follows from (5) that

$$\begin{aligned} \int _{{\underline{w}}}^{{\overline{w}}}U(w){\tilde{f}}(w)dw =\int _{{\underline{w}}}^{{\overline{w}}}c(w){\tilde{f}}(w)dw-\int _{{\underline{w}}}^{{\overline{w}}}h\left( \frac{y(w)}{w} \right) {\tilde{f}}(w)dw. \end{aligned}$$

(41)

Applying the equality form of (11) to (41) shows that

$$\begin{aligned} \int _{{\underline{w}}}^{{\overline{w}}}U(w){\tilde{f}}(w)dw =\int _{{\underline{w}}}^{{\overline{w}}}y(w){\tilde{f}}(w)dw-\int _{{\underline{w}}}^{{\overline{w}}}h\left( \frac{y(w)}{w} \right) {\tilde{f}}(w)dw. \end{aligned}$$

(42)

Combining (40) and (42) leads us to

$$\begin{aligned} \begin{aligned} U({\underline{w}})\int _{{\underline{w}}}^{{\overline{w}}}{\tilde{f}}(w)dw=&\int _{{\underline{w}}}^{{\overline{w}}}y(w){\tilde{f}}(w)dw-\int _{{\underline{w}}}^{{\overline{w}}}h\left( \frac{y(w)}{w} \right) {\tilde{f}}(w)dw\\&-\int _{{\underline{w}}}^{{\overline{w}}}\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \left[ \int _{w}^{{\overline{w}}}{\tilde{f}}(t) dt\right] dw. \end{aligned} \end{aligned}$$

(43)

Applying (10), we can rewrite (43) as

$$\begin{aligned} U({\underline{w}})=\frac{1}{\Gamma ({\underline{w}},{\overline{w}})} \int _{{\underline{w}}}^{{\overline{w}}}\left\{ \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] {\tilde{f}}(w)-\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \Gamma (w,{\overline{w}})\right\} dw. \end{aligned}$$

(44)

Substituting (44) into (38) and setting $w=k$, the maximand in (14) is established. $\square$

Proof of Lemma 3.2

By setting $k={\underline{w}}$, the maximand of problem (14) is hence given by (44). It is straightforward that the corresponding maximization problem can be solved point-wise. By letting $\partial U({\underline{w}})/ \partial y(w)=0$ and rearranging the algebra, we obtain

$$\begin{aligned} \begin{aligned}&\left[ 1-\frac{1}{w}h'\left( \frac{y(w)}{w} \right) \right] {\tilde{f}}(w) +\left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{\partial {\tilde{f}}(w)}{\partial y(w)}-\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \frac{\partial \Gamma (w,{\overline{w}})}{\partial y(w)} \\&\quad = \ U({\underline{w}})\frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial y(w)} + \left[ \frac{1}{w^{2}}h'\left( \frac{y(w)}{w} \right) +\frac{y(w)}{w^{3}}h''\left( \frac{y(w)}{w} \right) \right] \Gamma (w,{\overline{w}}). \end{aligned}\end{aligned}$$

(45)

As it is obvious by (10) that

$$\begin{aligned} \frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial y(w)} = \frac{\partial \Gamma (w,{\overline{w}})}{\partial y(w)}=\frac{\partial {\tilde{f}}(w)}{\partial y(w)}, \end{aligned}$$

(46)

applying (46) to (45) and rearranging the algebra, the desired (15) is hence established. By setting $k=w$ in the maximand of problem (14), then, for $\forall w\in ({\underline{w}},{\overline{w}})$, (17) is immediate by evaluating

$$\begin{aligned} \frac{\partial U(w)}{\partial y(w)}= \frac{\partial U({\underline{w}})}{\partial y(w)} + \frac{\partial }{\partial y(w)}\int _{{\underline{w}}}^{w}\frac{y(t)}{t^{2}}h'\left( \frac{y(t)}{t} \right) dt =\frac{\partial }{\partial y(w)}\int _{{\underline{w}}}^{w}\frac{y(t)}{t^{2}}h'\left( \frac{y(t)}{t} \right) dt \end{aligned}$$

at the maxi-min income schedule. Also, by using (8), (16) is immediate. $\square$

Proof of Lemma 3.3

By setting $k={\overline{w}}$, the maximand of problem (14) can be written as

$$\begin{aligned} \begin{aligned} U({\overline{w}})&=\frac{1}{\Gamma ({\underline{w}},{\overline{w}})} \int _{{\underline{w}}}^{{\overline{w}}}\left\{ \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] {\tilde{f}}(w)-\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \Gamma (w,{\overline{w}})\right\} dw \\&\quad +\int _{{\underline{w}}}^{{\overline{w}}}\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) dw. \end{aligned} \end{aligned}$$

(47)

The maximization problem can be solved point-wise. Applying $\partial U({\overline{w}})/ \partial y(w)=0$ and (46) to (47) and rearranging the algebra, we obtain

$$\begin{aligned} \begin{aligned}&\left[ 1-\frac{1}{w}h'\left( \frac{y(w)}{w} \right) \right] {\tilde{f}}(w) +\left[ y(w)-h\left( \frac{y(w)}{w} \right) -\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) -U({\underline{w}}) \right] \frac{\partial {\tilde{f}}(w)}{\partial y(w)} \\&\quad = \left[ \frac{1}{w^{2}}h'\left( \frac{y(w)}{w} \right) +\frac{y(w)}{w^{3}}h''\left( \frac{y(w)}{w} \right) \right] [\Gamma (w,{\overline{w}})-\Gamma ({\underline{w}},{\overline{w}})]. \end{aligned} \end{aligned}$$

(48)

Noting from (38) that $\partial U(w)/ \partial y(w)=\partial U({\overline{w}})/ \partial y(w)=0$ for all $w\in ({\underline{w}},{\overline{w}})$ whenever evaluated at the maxi-max income schedule. Applying this to (16) reveals that $\partial {\tilde{f}}(w)/\partial y(w)=0$ for all $w\in ({\underline{w}},{\overline{w}})$, thus (18) follows immediately from (48). Moreover, it follows from (47) that the first order condition can be expressed as:

$$\begin{aligned} \frac{\partial U({\overline{w}})}{\partial y(w)}= \frac{\partial U({\underline{w}})}{\partial y(w)} + \frac{\partial }{\partial y(w)}\int _{{\underline{w}}}^{{\overline{w}}}\frac{y(t)}{t^{2}}h'\left( \frac{y(t)}{t} \right) dt=0, \end{aligned}$$

evaluating which at $y(w)=y({\underline{w}})$ immediately gives (21). Evaluating (16) at $w={\underline{w}}$ yields (20). Then, we obtain (19) by evaluating (48) at $w={\underline{w}}$. $\square$

Proof of Proposition 3.1

Using the maximization problem (14) stated in Lemma 3.1, it is easy to show that

$$\begin{aligned} \frac{\partial U(k)}{\partial y(w)} = \frac{\partial U({\overline{w}})}{\partial y(w)} \ \ \text {for} \ \forall w\in [{\underline{w}},k) \end{aligned}$$

and

$$\begin{aligned} \frac{\partial U(k)}{\partial y(w)} = \frac{\partial U({\underline{w}})}{\partial y(w)} \ \ \text {for} \ \forall w\in (k,{\overline{w}}], \end{aligned}$$

for $\forall k\in ({\underline{w}},{\overline{w}})$. Therefore, the desired income schedule (22) follows from a direct application of Lemmas 3.2 and 3.3 . $\square$

Proof of Proposition 3.2

First, (27) in part (i) is immediate by applying (20)-(21) to the tax formula of (24). Part (ii) is also immediate by (25). Using the chain rule of calculus, (8) and (9), we have

$$\begin{aligned} \frac{\partial {\tilde{f}}(w)}{\partial y(w)} \frac{1}{{\tilde{f}}(w)} =\frac{\partial {\tilde{f}}(w)}{\partial \Delta (w)} \frac{c(w)}{{\tilde{f}}(w)}\frac{\partial U(w)}{\partial y(w)} \frac{1}{c(w)} = {\tilde{\theta }}(w)\frac{\partial U(w)}{\partial y(w)} \frac{1}{c(w)}. \end{aligned}$$

(49)

Then we get from (49), (17), (26), (2), (5)–(6) and the condition

$$\begin{aligned}&y(w)-h\left( \frac{y(w)}{w} \right) - \frac{y(w)}{w^2}h'\left( \frac{y(w)}{w} \right) \nonumber \\&\quad =T^{R}(y(w))+U(w)-U'(w)>U({\underline{w}}) \end{aligned}$$

(50)

that

$$\begin{aligned} \tau ^{R}(w)=\underset{>0}{\underbrace{\frac{\partial U(w)}{\partial y(w)}}} \left\{ \underset{>0}{\underbrace{\frac{\Gamma (w,{\overline{w}})}{{\tilde{f}}(w)}}} \ - \ \underset{>0}{\underbrace{\frac{{\tilde{\theta }}(w)}{c(w)} \left[ y(w)-h\left( \frac{y(w)}{w} \right) - \frac{y(w)}{w^2}h'\left( \frac{y(w)}{w} \right) - U({\underline{w}})\right] }} \right\} , \end{aligned}$$

by which assertion (28) is immediate. It follows from (6) that $U({\overline{w}})>U({\underline{w}})$. Making use of (50) again shows the desired assertion (29). $\square$

Proof of Proposition 3.3

We complete the proof in three steps.

${\underline{\mathrm{Step 1.}}}$ It follows from (25) and (26) that

$$\begin{aligned} \begin{aligned} \tau ^{M}(w)-\tau ^{R}(w)= \&\frac{\partial {\tilde{f}}(w)}{\partial y(w)} \frac{1}{{\tilde{f}}(w)}\left[ y(w)-h\left( \frac{y(w)}{w} \right) - \frac{y(w)}{w^2}h'\left( \frac{y(w)}{w} \right) -U({\underline{w}})\right] \\&-\frac{\Gamma ({\underline{w}},{\overline{w}})}{w{\tilde{f}}(w)}\left[ \frac{1}{w}h'\left( \frac{y(w)}{w} \right) +\frac{y(w)}{w^{2}}h''\left( \frac{y(w)}{w} \right) \right] . \end{aligned} \end{aligned}$$

(51)

Applying (50) to (51), we immediately get $\tau ^{M}(w)<\tau ^{R}(w)$ for $T^{R}(y(w))\le U'(w)+U({\underline{w}})-U(w)$. Then we have either $\tau ^{M}(w)<0<\tau ^{R}(w)$ or $\tau ^{M}(w)<\tau ^{R}(w)\le 0$. If $\tau ^{M}(w)<0<\tau ^{R}(w)$, then under tax schedule $\tau ^{M}(\cdot )$ each type-w worker has her income distorted upward compared to the full-information solution, whereas her income is distorted downward compared to the full-information solution under tax schedule $\tau ^{R}(\cdot )$. If $\tau ^{M}(w)<\tau ^{R}(w)\le 0$, then each type-w worker has her income distorted upward compared to the full-information solution under both tax schedules, but the magnitude of distortion is greater under tax schedule $\tau ^{M}(\cdot )$. Thus, regardless of which case we consider, we see an upward discontinuity of the income schedule, as desired in part (i-a).

${\underline{\mathrm{Step 2.}}}$ If, however, $T^{R}(y(w))> U'(w)+U({\underline{w}})-U(w)$, then applying (49) and (50) to (51) yields

$$\begin{aligned}&\tau ^{M}(w)-\tau ^{R}(w)\\&\quad = \underset{>0}{\underbrace{\frac{\partial U(w)}{\partial y(w)}}} \left\{ \underset{>0}{\underbrace{\frac{{\tilde{\theta }}(w)}{c(w)} \left[ y(w)-h\left( \frac{y(w)}{w} \right) - \frac{y(w)}{w^2}h'\left( \frac{y(w)}{w} \right) - U({\underline{w}})\right] }} \ - \ \underset{>0}{\underbrace{\frac{\Gamma ({\underline{w}},{\overline{w}})}{{\tilde{f}}(w)}}} \right\} , \end{aligned}$$

by which we arrive at the following result:

$$\begin{aligned} \tau ^{M}(w) {\left\{ \begin{array}{ll}<\tau ^{R}(w) &{} \hbox { for}\ {\tilde{\theta }}(w)<{\tilde{\theta }}^{*}(w),\\ =\tau ^{R}(w) &{} \hbox { for}\ {\tilde{\theta }}(w)={\tilde{\theta }}^{*}(w),\\>\tau ^{R}(w) &{} \hbox { for}\ {\tilde{\theta }}(w)>{\tilde{\theta }}^{*}(w), \end{array}\right. } \end{aligned}$$

(52)

in which the critical value of migration elasticity is given by

$$\begin{aligned} {\tilde{\theta }}^{*}(w)=\frac{c(w)\Gamma ({\underline{w}},{\overline{w}})}{\left[ y(w)-h( y(w)/w )- (y(w)/w^2)h'( y(w)/w)- U({\underline{w}})\right] {\tilde{f}}(w)}. \end{aligned}$$

(53)

Using (28), (52)–(53) and ${\tilde{\theta }}^{*}(w) >{\tilde{\theta }}_{\star }(w)$, we have the following results: $0>\tau ^{M}(w)>\tau ^{R}(w)$ if ${\tilde{\theta }}(w)>{\tilde{\theta }}^{*}(w)$, $0>\tau ^{M}(w)=\tau ^{R}(w)$ if ${\tilde{\theta }}(w)={\tilde{\theta }}^{*}(w)$, $\tau ^{M}(w) <0 \le \tau ^{R}(w)$ if ${\tilde{\theta }}(w)\le {\tilde{\theta }}_{\star }(w)<{\tilde{\theta }}^{*}(w)$, and $\tau ^{M}(w)<\tau ^{R}(w)<0$ if ${\tilde{\theta }}_{\star }(w)<{\tilde{\theta }}(w)<{\tilde{\theta }}^{*}(w)$. By applying the same reasoning used to prove part (i-a), the desired assertions in parts (i-b) and (ii) follow.

${\underline{\mathrm{Step 3.}}}$ As w approaches ${\underline{w}}$ from above, we get from Eq. (25) and the continuity of $\tau ^{M}(w)$ over the interval $({\underline{w}},k)$ that $\tau ^{M}({\underline{w}})=0$. As a result, under the marginal tax rate $\tau ^{M}({\underline{w}})$ each type-${\underline{w}}$ worker has her income undistorted as in the full-information solution. Thus according to Eqs. (24), (20), (21), (2), (5) and (6) we have

$$\begin{aligned} {\underline{\tau }}^{M}({\underline{w}})-\tau ^{M}({\underline{w}})=\underset{>0}{\underbrace{-\frac{\partial {\tilde{f}}({\underline{w}})}{\partial y({\underline{w}})} \frac{1}{{\tilde{f}}({\underline{w}})}}}\cdot \left[ T^{M}(y({\underline{w}}))- U'({\underline{w}})\right] . \end{aligned}$$

If $T^{M}(y({\underline{w}}))> U'({\underline{w}})$, then we immediately have ${\underline{\tau }}^{M}({\underline{w}})>\tau ^{M}({\underline{w}})=0$, which means that under the marginal tax rate ${\underline{\tau }}^{M}({\underline{w}})$, each type-${\underline{w}}$ worker has her income distorted downward compared to the full-information solution. We thus see an upward discontinuity of the income schedule at the bottom skill level, as desired in claim (iv). If, by contrast, $T^{M}(y({\underline{w}}))< U'({\underline{w}})$, then we have ${\underline{\tau }}^{M}({\underline{w}})<\tau ^{M}({\underline{w}})=0$, yielding a downward discontinuity of the income schedule at the bottom skill level, as desired in claim (iii). $\square$

Proof of Theorem 3.2

We complete the proof in four steps.

${\underline{\mathrm{Step 1.}}}$ Under the conditions given in Theorem 3.2, Proposition 3.3 suggests that there will be two downward discontinuities in the income schedule. Hence we need to build two bridges such that the resulting income schedule satisfies the SOIC condition. Obviously, the left endpoint of the first bridge is just ${\underline{w}}$. Let us fix the other bridge endpoints $w_{\eta }, w_{\alpha }$ and $w_{\beta }$ that shall be endogenously determined, and let $y^{*}({\underline{w}}, w_{\eta })$ and $y^{*}(w_{\alpha }, w_{\beta })$ denote the optimal before-tax income levels on the bridges over skill intervals $[{\underline{w}}, w_{\eta }]$ and $[w_{\alpha }, w_{\beta }]$, respectively. Without a loss of generality, we assume that the bridge in the middle cannot begin in the interior of a bunching interval of the maxi-max schedule $y^{M*}(\cdot )$, nor can it end in the interior of a bunching interval of maxi-min schedule $y^{R*}(\cdot )$. In what follows, let ${\mathcal {B}}^{M}$ and ${\mathcal {B}}^{R}$ denote the types that are bunched with some other types in the complete solution to the maxi-max and maxi-min problems, respectively. Also, whenever w is bunched, we let interval $[w_{-}, w_{+}]$ denote the set of types bunched with w.

${\underline{\mathrm{Step 2.}}}$ We now equivalently rewrite the maximand of problem (14) as follows:

$$\begin{aligned} \begin{aligned} U(k) =&\int _{{\underline{w}}}^{k}\left\{ \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{{\tilde{f}}(w)}{\Gamma ({\underline{w}},{\overline{w}})}+\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \left[ 1-\frac{\Gamma (w,{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] \right\} dw \\&+\int _{k}^{{\overline{w}}}\left\{ \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{{\tilde{f}}(w)}{\Gamma ({\underline{w}},{\overline{w}})}-\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \frac{\Gamma (w,{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right\} dw. \end{aligned} \end{aligned}$$

(54)

Taking into account the bunching possibility, (54) should be modified as follows:

$$\begin{aligned} \begin{aligned} U^{*}(k) =&\int _{{\underline{w}}}^{k}{\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w|w\notin {\mathcal {B}}^{M} \right\} }dw \\&+\int _{{\underline{w}}}^{k} {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w_{-},w_{+}),\Gamma (w_{-},{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w|w\in {\mathcal {B}}^{M} \right\} } dw \\&+\int _{k}^{{\overline{w}}}{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w|w\notin {\mathcal {B}}^{R} \right\} }dw \\&+\int _{k}^{{\overline{w}}} {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (w_{-},w_{+}),\Gamma (w_{+},{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w|w\in {\mathcal {B}}^{R} \right\} } dw, \end{aligned} \end{aligned}$$

(55)

in which

$$\begin{aligned} \begin{aligned}&{\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}})) \\&\quad \equiv \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{{\tilde{f}}(w)}{\Gamma ({\underline{w}},{\overline{w}})}+\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \left[ 1-\frac{\Gamma (w,{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] ;\\&\quad {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w_{-},w_{+}),\Gamma (w_{-},{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\\&\quad \equiv \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{\Gamma (w_{-},w_{+})}{\Gamma ({\underline{w}},{\overline{w}})}+\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \left[ 1-\frac{\Gamma (w_{-},{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] \end{aligned} \end{aligned}$$

(56)

and

$$\begin{aligned} \begin{aligned}&{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\\&\quad \equiv \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{{\tilde{f}}(w)}{\Gamma ({\underline{w}},{\overline{w}})}-\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \frac{\Gamma (w,{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})};\\&\quad {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (w_{-},w_{+}),\Gamma (w_{+},{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\\&\quad \equiv \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{\Gamma (w_{-},w_{+})}{\Gamma ({\underline{w}},{\overline{w}})}-\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \frac{\Gamma (w_{+},{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \end{aligned} \end{aligned}$$

(57)

with ${\mathbb {I}}$ being a standard indicator function.

As ironing does not affect the solution outside a bunching region, no modifications to the integrands in (55) are needed for types that are not bunched. Departing from the first-order approach, if an extra unit of consumption is given to type-w workers, it must be given to all workers who are bunched with them, whose mass is $\Gamma (w_{-},w_{+})$. Also, if w is bunched, in the maxi-max case, some of this extra consumption can be reclaimed from workers of lower types than those bunched with w, whose mass is $\Gamma ({\underline{w}},{\overline{w}})-\Gamma (w_{-},{\overline{w}})$. The corresponding workers in the maxi-min case are those of higher types than those bunched with w, whose mass is $\Gamma (w_{+},{\overline{w}})$.

${\underline{\mathrm{Step 3.}}}$ Now, the selfishly optimal income schedule of proposer $k\in ({\underline{w}},{\overline{w}})$ is obtained by solving the following problem:

$$\begin{aligned} \begin{aligned}&\max _{y(\cdot )} \left\{ \int _{{\underline{w}}}^{w_{\eta }} {\tilde{\Phi }}^{M*}(w, y(w),\Gamma ({\underline{w}},w_{\eta }),\Gamma ({\underline{w}},{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\eta }> {\underline{w}} \right\} } dw \right. \\&\quad + \int _{w_{\eta }}^{w_{\alpha }}{\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\eta }\le w_{\alpha } \right\} }dw\\&\quad +\int _{w_{\alpha }}^{k} {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w_{\alpha },w_{\beta }),\Gamma (w_{\alpha },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} } dw \\&\quad +\int _{k}^{w_{\beta }} {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (w_{\alpha },w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} } dw\\&\quad +\left. \int _{w_{\beta }}^{{\overline{w}}}{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ {\overline{w}}\ge w_{\beta } \right\} }dw \right\} , \end{aligned} \end{aligned}$$

(58)

subject to

$$\begin{aligned} y(w)= {\left\{ \begin{array}{ll} y^{*}\left( {\underline{w}},w_{\eta }\right) &{} \hbox { for}\ w\in \left[ {\underline{w}},w_{\eta }\right] ,\\ y^{*}\left( w_{\alpha },w_{\beta }\right) &{} \hbox { for}\ w\in \left[ w_{\alpha },w_{\beta }\right] . \end{array}\right. } \end{aligned}$$

Thus, problem (58) can be simplified as the following unconstrained maximization problem:

$$\begin{aligned}&\max _{y(\cdot )} \left\{ \int _{w_{\eta }}^{w_{\alpha }}{\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\eta }\le w_{\alpha } \right\} }dw \right. \\&\quad +\left. \int _{w_{\beta }}^{{\overline{w}}}{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ {\overline{w}}\ge w_{\beta } \right\} }dw \right\} . \end{aligned}$$

This problem can obviously be solved point-wise, and the solutions are implicitly determined by these first-order conditions:

$$\begin{aligned} \begin{aligned}&\frac{\partial {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial y(w)}\\&\quad +\frac{\partial {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial {\tilde{f}}(w)}\frac{\partial {\tilde{f}}(w)}{\partial y(w)}\\&\quad +\frac{\partial {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma (w,{\overline{w}})}\frac{\partial \Gamma (w,{\overline{w}})}{\partial y(w)}\\&\quad +\frac{\partial {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma ({\underline{w}},{\overline{w}})}\frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial y(w)}=0 \ \ \text {for} \ \forall w\in (w_{\eta }, w_{\alpha }), \end{aligned} \end{aligned}$$

(59)

and

$$\begin{aligned} \begin{aligned}&\frac{\partial {\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial y(w)}\\&\quad +\frac{\partial {\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial {\tilde{f}}(w)}\frac{\partial {\tilde{f}}(w)}{\partial y(w)}\\&\quad +\frac{\partial {\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma (w,{\overline{w}})}\frac{\partial \Gamma (w,{\overline{w}})}{\partial y(w)}\\&\quad +\frac{\partial {\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma ({\underline{w}},{\overline{w}})}\frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial y(w)}=0 \ \ \text {for} \ \forall w\in (w_{\beta }, {\overline{w}}]. \end{aligned} \end{aligned}$$

(60)

As above, we denote the resulting solutions as $y^{M*}(\cdot )$ and $y^{R*}(\cdot )$, respectively.

${\underline{\mathrm{Step 4.}}}$ We now show that $y^{*}(w_{\alpha },w_{\beta })=y^{M*}(w_{\alpha })$ if $w_{\alpha }>{\underline{w}}$ and $y^{*}(w_{\alpha },w_{\beta })=y^{R*}(w_{\beta })$ if $w_{\beta }<{\overline{w}}$. Based on the above ironing procedure, we can apply the same reasoning used to prove Proposition 3 of Brett and Weymark (2017) to show that $y^{*}(\cdot )$ is continuous on $[{\underline{w}}, {\overline{w}}]$. Suppose that there exists a type $k'>k$ for which $y^{*}(k')$ is not the maxi-min income, formally $y^{*}(k')\ne y^{R*}(k')$. The SOIC condition (7) must bind at $k'$, which implies that the slope of $y^{*}(\cdot )$ is zero at $k'$. Since $y^{*}(\cdot )$ is continuous, we obtain that there exists a $w_{\beta }>k'$ such that $y^{*}(\cdot )$ is constant on $[k, w_{\beta }]$ and coincides with the maxi-min income schedule $y^{R*}(\cdot )$ on $[w_{\beta }, {\overline{w}}]$. Similarly, if there exists a type $k'<k$ for which $y^{*}(k')$ is not the maxi-max income, formally $y^{*}(k')\ne y^{M*}(k')$, we can use the same argument to show that there exists a $w_{\alpha }<k'$ such that $y^{*}(\cdot )$ is constant on $[w_{\alpha },k]$ and coincides with the maxi-max income schedule $y^{M*}(\cdot )$ on $[w_{\eta }, w_{\alpha }]$. By further setting $y^{*}({\underline{w}},w_{\eta })\equiv {\underline{y}}^{M*}({\underline{w}})$, the desired income schedule given by (30) is therefore established. $\square$

Proof of Lemma 3.4

We get from (55), (58), (20) and (21) that

$$\begin{aligned} \frac{\partial ^2 U^{*}(k)}{\partial y({\underline{w}}) \partial k} = \&\frac{\partial {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w_{\alpha },w_{\beta }),\Gamma (w_{\alpha },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma ({\underline{w}},{\overline{w}})} \frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial y({\underline{w}})} \cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ w=k \right\} }\\&- \frac{\partial {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (w_{\alpha },w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma ({\underline{w}},{\overline{w}})} \frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial y({\underline{w}})} \cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ w=k \right\} }\\ = \&\frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial y({\underline{w}})} \cdot \frac{y(k)}{k^{2}} h'\left( \frac{y(k)}{k} \right) \frac{1}{\Gamma ({\underline{w}},{\overline{w}})^{2}}\cdot \left[ \Gamma (w_{\alpha },{\overline{w}})-\Gamma (w_{\beta },{\overline{w}}) \right] \cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ w_{\beta }<{\overline{w}} \right\} }\\ = \&\underset{<0}{\underbrace{\frac{\partial {\tilde{f}}({\underline{w}})}{\partial y({\underline{w}})} }}\cdot \frac{y(k)}{k^{2}} h'\left( \frac{y(k)}{k} \right) \frac{1}{\Gamma ({\underline{w}},{\overline{w}})^{2}}\cdot \underset{>0}{\underbrace{\left[ \Gamma (w_{\alpha },{\overline{w}})-\Gamma (w_{\beta },{\overline{w}}) \right] }} \cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ w_{\beta }<{\overline{w}} \right\} }\\ < \&0, \end{aligned}$$

which implies that $U^{*}(k)$ is a submodular function, and an application of the Topkis Theorem (see Topkis 1978) implies that $y({\underline{w}})$ is decreasing in k. Since the endpoint $w_{\eta }(k)$ is completely determined by the value of $y({\underline{w}})$, we get that $w_{\eta }(k)$ is decreasing in k. $\square$

Proof of Lemma 3.5

We will complete the proof in five steps.

${\underline{\mathrm{Step 1.}}}$ Our proof employs the procedure developed by Brett and Weymark (2017). Suppose $w_{\beta }<{\overline{w}}$ holds. According to the continuity of income schedule $y^{*}(\cdot )$, we get from Theorem 3.2 that $y^{*}(w_{\beta })=y^{R*}(w_{\beta })$. Also, $y^{*}(w_{\beta })=y^{*}(w_\alpha )$ because income is a constant on the bridge. If we also have $w_{\alpha }>{\underline{w}}$, then by continuity again, $y^{*}(w_\alpha )=y^{M*}(w_\alpha )$. Define

$$\begin{aligned} \psi (w_{\beta })\equiv {\left\{ \begin{array}{ll} (y^{M*})^{-1}(y^{R*}(w_{\beta })) &{} \hbox { if}\ w_{\alpha }>{\underline{w}},\\ w_{\alpha } &{} \hbox { if}\ w_{\alpha }={\underline{w}}. \end{array}\right. } \end{aligned}$$

(61)

We can therefore write proposer k’s objective function of choosing $w_{\beta }$ as follows:

$$\begin{aligned} \begin{aligned}&\Xi (w_{\beta };k) \\&\quad \equiv \int _{w_{\eta }}^{\psi (w_{\beta })}{\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\eta }\le w_{\alpha } \right\} \cap \left\{ y(w)=y^{M*}(w) \right\} }dw\\&\qquad + \int _{\psi (w_{\beta })}^{k} {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (\psi (w_{\beta }),{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} } dw \\&\qquad +\int _{k}^{w_{\beta }} {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} } dw\\&\qquad +\int _{w_{\beta }}^{{\overline{w}}}{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ {\overline{w}}\ge w_{\beta } \right\} \cap \left\{ y(w)=y^{R*}(w) \right\} }dw. \end{aligned} \end{aligned}$$

(62)

Using (62), the first-order condition with respect to $w_{\beta }$ can be derived as

$$\begin{aligned} \Psi _{1}+\Psi _{2}(k)+\Psi _{3}(k)+\Psi _{4}=0, \end{aligned}$$

(63)

in which

$$\begin{aligned}&\begin{aligned}&\Psi _{1} = \\&\quad \frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }} \left\{ {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\eta }\le w_{\alpha } \right\} \cap \left\{ y(w)=y^{M*}(w) \right\} \cap \left\{ w=\psi (w_{\beta }) \right\} } \right. \\&\quad -\left. {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (\psi (w_{\beta }),{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} \cap \left\{ w=\psi (w_{\beta }) \right\} } \right\} , \end{aligned} \end{aligned}$$

(64)

$$\begin{aligned}&\begin{aligned}&\Psi _{2}(k) = \\&\quad \int _{\psi (w_{\beta })}^{k} \frac{\partial {\tilde{\Phi }}^{M*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (\psi (w_{\beta }),{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial y^{R*}(w_{\beta })} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} } dw \\&\qquad +\int _{\psi (w_{\beta })}^{k} \frac{\partial {\tilde{\Phi }}^{M*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (\psi (w_{\beta }),{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma (\psi (w_{\beta }),w_{\beta })} \frac{\partial \Gamma (\psi (w_{\beta }),w_{\beta })}{\partial w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} } dw\\&\qquad +\int _{\psi (w_{\beta })}^{k} \frac{\partial {\tilde{\Phi }}^{M*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (\psi (w_{\beta }),{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma (\psi (w_{\beta }),{\overline{w}})} \frac{\partial \Gamma (\psi (w_{\beta }),{\overline{w}})}{\partial w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} } dw\\&\qquad +\int _{\psi (w_{\beta })}^{k} \frac{\partial {\tilde{\Phi }}^{M*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (\psi (w_{\beta }),{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma ({\underline{w}},{\overline{w}})} \frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} } dw\\&\quad \equiv \int _{\psi (w_{\beta })}^{k} \left[ \Psi _{21}(w)+\Psi _{22}(w)+\Psi _{23}(w)+\Psi _{24}(w) \right] \cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} } dw, \end{aligned} \end{aligned}$$

(65)

$$\begin{aligned}&\begin{aligned}&\Psi _{3}(k) = \\&\quad \int _{k}^{w_{\beta }} \frac{\partial {\tilde{\Phi }}^{R*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial y^{R*}(w_{\beta })}\frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} } dw \\&\qquad +\int _{k}^{w_{\beta }} \frac{\partial {\tilde{\Phi }}^{R*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma (\psi (w_{\beta }),w_{\beta })}\frac{\partial \Gamma (\psi (w_{\beta }),w_{\beta })}{\partial w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} } dw\\&\qquad +\int _{k}^{w_{\beta }} \frac{\partial {\tilde{\Phi }}^{R*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma (w_{\beta },{\overline{w}})}\frac{\partial \Gamma (w_{\beta },{\overline{w}})}{\partial w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} } dw\\&\qquad +\int _{k}^{w_{\beta }} \frac{\partial {\tilde{\Phi }}^{R*}(w, y^{R*}(w_{\beta }),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))}{\partial \Gamma ({\underline{w}},{\overline{w}})}\frac{\partial \Gamma ({\underline{w}},{\overline{w}})}{\partial w_{\beta }}\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} } dw\\&\quad \equiv \int _{k}^{w_{\beta }} \left[ \Psi _{31}(w)+\Psi _{32}(w)+\Psi _{33}(w)+\Psi _{34}(w) \right] \cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} } dw, \end{aligned} \end{aligned}$$

(66)

and

$$\begin{aligned} \begin{aligned} \Psi _{4}= \&{\tilde{\Phi }}^{R*}(w, y(w),\Gamma (\psi (w_{\beta }),w_{\beta }),\Gamma (w_{\beta },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} \cap \left\{ w=w_{\beta } \right\} }\\&-{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ {\overline{w}} \ge w_{\beta } \right\} \cap \left\{ y(w)=y^{R*}(w) \right\} \cap \left\{ w=w_{\beta } \right\} }. \end{aligned} \end{aligned}$$

(67)

${\underline{\mathrm{Step 2.}}}$ Using (56) and (57) gives us

$$\begin{aligned} \begin{aligned}&{\tilde{\Phi }}^{M*}(w,y(w),\Gamma (w_{\alpha },w_{\beta }), \Gamma (w_{\alpha },{\overline{w}}), \Gamma ({\underline{w}},{\overline{w}}))\\&\quad \equiv \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{\Gamma (w_{\alpha },w_{\beta })}{\Gamma ({\underline{w}},{\overline{w}})}+\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \left[ 1-\frac{\Gamma (w_{\alpha },{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] ,\\&\quad {\tilde{\Phi }}^{R*}(w,y(w),\Gamma (w_{\alpha },w_{\beta }), \Gamma (w_{\beta },{\overline{w}}), \Gamma ({\underline{w}},{\overline{w}}))\\&\quad \equiv \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{\Gamma (w_{\alpha },w_{\beta })}{\Gamma ({\underline{w}},{\overline{w}})}-\frac{y(w)}{w^{2}}h'\left( \frac{y(w)}{w} \right) \frac{\Gamma (w_{\beta },{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})}, \end{aligned} \end{aligned}$$

(68)

for $\forall w \in [w_{\alpha },w_{\beta }]$.

By using (61), (68), (56) and (57), we can rewrite (64) as

$$\begin{aligned} \begin{aligned} \Psi _{1} =&\frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }} \left\{ {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\eta } \le w_{\alpha } \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} \cap \left\{ w=\psi (w_{\beta }) \right\} } \right. \\&-\left. {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w,w_{\beta }),\Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} \cap \left\{ w=\psi (w_{\beta }) \right\} } \right\} \\ =&\frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }} \left\{ {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\right. \\&-\left. {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w,w_{\beta }),\Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}})) \right\} \cdot {\mathbb {I}}_{\left\{ w_{\eta } \le w_{\alpha } \right\} \cap \left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} \cap \left\{ w=\psi (w_{\beta }) \right\} }\\ =&\frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }} \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{{\tilde{f}}(w)-\Gamma (w,w_{\beta })}{\Gamma ({\underline{w}},{\overline{w}})}\cdot {\mathbb {I}}_{ \left\{ w_{\alpha }\ge w_{\eta } \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} \cap \left\{ w=\psi (w_{\beta }) \right\} }\\ =&\frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }} \left[ y^{R*}(w_{\beta })-h\left( \frac{y^{R*}(w_{\beta })}{\psi (w_{\beta })} \right) \right] \frac{{\tilde{f}}(\psi (w_{\beta }))-\Gamma (\psi (w_{\beta }),w_{\beta })}{\Gamma ({\underline{w}},{\overline{w}})}\cdot {\mathbb {I}}_{\left\{ w_{\alpha }\ge w_{\eta } \right\} }. \end{aligned} \end{aligned}$$

(69)

Using (65) and (68), we have

$$\begin{aligned}&\begin{aligned} \Psi _{21}(w) =&\left\{ \left[ 1-\frac{1}{w}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \frac{\Gamma (\psi \left( w_{\beta } \right) ,w_{\beta })}{\Gamma ({\underline{w}},{\overline{w}})} \right. \\&+ \left. \left[ \frac{1}{w^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) + \frac{y^{R*}(w_{\beta })}{w^{3}}h''\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \left[ 1-\frac{\Gamma (\psi \left( w_{\beta } \right) ,{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] \right\} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}, \end{aligned} \end{aligned}$$

(70)

$$\begin{aligned}&\begin{aligned} \Psi _{22}(w) =&\left[ y^{R*}(w_{\beta })-h\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \frac{1}{\Gamma ({\underline{w}},{\overline{w}})} \\&\times \left\{ {\tilde{f}}(w_{\beta })-{\tilde{f}}\left( \psi (w_{\beta }) \right) \frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }}+\left[ \int _{\psi (w_{\beta })}^{w_{\beta }}\frac{\partial {\tilde{f}}(w)}{\partial y^{R*}(w_{\beta })} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}dw \right] \right\} , \end{aligned} \end{aligned}$$

(71)

$$\begin{aligned}&\begin{aligned} \Psi _{23}(w) = \&\frac{y^{R*}(w_{\beta })}{w^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) \frac{1}{\Gamma ({\underline{w}},{\overline{w}})} \\&\times \left[ {\tilde{f}}\left( \psi (w_{\beta }) \right) \frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }}-\int _{\psi (w_{\beta })}^{w_{\beta }}\frac{\partial {\tilde{f}}(w)}{\partial y^{R*}(w_{\beta })} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}dw \right] , \end{aligned} \end{aligned}$$

(72)

and

$$\begin{aligned} \begin{aligned} \Psi _{24}(w) = \&\left\{ \frac{y^{R*}(w_{\beta })}{w^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) \Gamma (\psi (w_{\beta }),{\overline{w}}) - \left[ y^{R*}(w_{\beta })-h\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \Gamma (\psi (w_{\beta }),w_{\beta }) \right\} \\&\times \frac{1}{\Gamma ({\underline{w}},{\overline{w}})^{2}} \int _{\psi (w_{\beta })}^{w_{\beta }}\frac{\partial {\tilde{f}}(w)}{\partial y^{R*}(w_{\beta })} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}dw. \end{aligned} \end{aligned}$$

(73)

Similarly, using (66) and (68) gives us

$$\begin{aligned}&\begin{aligned} \Psi _{31}(w) =&\left\{ \left[ 1-\frac{1}{w}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \frac{\Gamma (\psi \left( w_{\beta } \right) ,w_{\beta })}{\Gamma ({\underline{w}},{\overline{w}})} \right. \\&- \left. \left[ \frac{1}{w^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) + \frac{y^{R*}(w_{\beta })}{w^{3}}h''\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \frac{\Gamma (w_{\beta },{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right\} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}, \end{aligned} \end{aligned}$$

(74)

$$\begin{aligned}&\begin{aligned} \Psi _{32}(w) =&\left[ y^{R*}(w_{\beta })-h\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \frac{1}{\Gamma ({\underline{w}},{\overline{w}})} \\&\times \left\{ {\tilde{f}}(w_{\beta })-{\tilde{f}}\left( \psi (w_{\beta }) \right) \frac{\mathrm {d} \psi (w_{\beta })}{\mathrm {d} w_{\beta }}+\left[ \int _{\psi (w_{\beta })}^{w_{\beta }}\frac{\partial {\tilde{f}}(w)}{\partial y^{R*}(w_{\beta })}\frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }} dw \right] \right\} , \end{aligned} \end{aligned}$$

(75)

$$\begin{aligned}&\Psi _{33}(w) = \frac{y^{R*}(w_{\beta })}{w^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) \frac{{\tilde{f}}\left( w_{\beta } \right) }{\Gamma ({\underline{w}},{\overline{w}})}, \end{aligned}$$

(76)

and

$$\begin{aligned} \begin{aligned} \Psi _{34}(w) = \&\left\{ \frac{y^{R*}(w_{\beta })}{w^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{w} \right) \Gamma (w_{\beta },{\overline{w}}) - \left[ y^{R*}(w_{\beta })-h\left( \frac{y^{R*}(w_{\beta })}{w} \right) \right] \Gamma (\psi (w_{\beta }),w_{\beta }) \right\} \\&\times \frac{1}{\Gamma ({\underline{w}},{\overline{w}})^{2}} \int _{\psi (w_{\beta })}^{w_{\beta }}\frac{\partial {\tilde{f}}(w)}{\partial y^{R*}(w_{\beta })} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}dw. \end{aligned} \end{aligned}$$

(77)

Finally, by using (68), we can rewrite (67) as

$$\begin{aligned} \begin{aligned} \Psi _{4}= \&{\tilde{\Phi }}^{R*}(w, y(w),\Gamma (\psi (w),w),\Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w_{\beta }) \right\} \cap \left\{ w=w_{\beta } \right\} }\\&-{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ {\overline{w}}\ge w_{\beta } \right\} \cap \left\{ y(w)=y^{R*}(w) \right\} \cap \left\{ w=w_{\beta } \right\} }\\ =\&\left[ {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (\psi (w),w),\Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))- {\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}})) \right] \\&\times {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w) \right\} \cap \left\{ w=w_{\beta } \right\} }\\ =\&\left\{ \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{\Gamma \left( \psi (w),w \right) -{\tilde{f}}(w)}{\Gamma \left( {\underline{w}},{\overline{w}} \right) } \right\} {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{R*}(w) \right\} \cap \left\{ w=w_{\beta } \right\} }. \end{aligned} \end{aligned}$$

(78)

${\underline{\mathrm{Step 3.}}}$ Suppose $w_{\alpha }>{\underline{w}}$ holds. By continuity of income schedule $y^{*}(\cdot )$, we get from Theorem 3.2 that $y^{*}(w_\alpha )=y^{M*}(w_\alpha )$. Also, $y^{*}(w_{\beta })=y^{*}(w_\alpha )$ because income is a constant on the bridge. If we also have $w_{\beta }<{\overline{w}}$, then by continuity again, $y^{*}(w_\beta )=y^{R*}(w_\beta )$. Define

$$\begin{aligned} \varphi (w_{\alpha })\equiv {\left\{ \begin{array}{ll} (y^{R*})^{-1}(y^{M*}(w_{\alpha })) &{} \hbox { if}\ w_{\beta }<{\overline{w}},\\ w_{\beta } &{} \hbox { if}\ w_{\beta }={\overline{w}}. \end{array}\right. } \end{aligned}$$

(79)

We can therefore write proposer k’s objective function of choosing $w_{\alpha }$ as follows:

$$\begin{aligned} \begin{aligned}&\Xi (w_{\alpha };k) \\&\quad \equiv \int _{w_{\eta }}^{w_{\alpha }}{\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\eta }\le w_{\alpha } \right\} \cap \left\{ y(w)=y^{M*}(w) \right\} }dw\\&\qquad + \int _{w_{\alpha }}^{k} {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w_{\alpha },\varphi (w_{\alpha })),\Gamma (w_{\alpha },{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ y(w)=y^{M*}(w_{\alpha }) \right\} } dw \\&\qquad +\int _{k}^{\varphi (w_{\alpha })} {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (w_{\alpha },\varphi (w_{\alpha })),\Gamma (\varphi (w_{\alpha }),{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{M*}(w_{\alpha }) \right\} } dw\\&\qquad +\int _{\varphi (w_{\alpha })}^{{\overline{w}}}{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ {\overline{w}} \ge w_{\beta } \right\} \cap \left\{ y(w)=y^{R*}(w) \right\} }dw. \end{aligned} \end{aligned}$$

(80)

Using (80), the first-order condition with respect to $w_{\alpha }$ can be derived as

$$\begin{aligned} \Lambda _{1}+\Lambda _{2}(k)+\Lambda _{3}(k)+\Lambda _{4}=0, \end{aligned}$$

(81)

in which

$$\begin{aligned}&\begin{aligned} \Lambda _{1}= \&\left[ {\tilde{\Phi }}^{M*}(w, y(w),{\tilde{f}}(w),\Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))- {\tilde{\Phi }}^{M*}(w, y(w),\Gamma (w,\varphi (w)), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}})) \right] \\&\times {\mathbb {I}}_{\left\{ w_{\eta } \le w_{\alpha } \right\} \cap \left\{ w_{\alpha }>{\underline{w}} \right\} \cap \left\{ y(w)=y^{M*}(w) \right\} \cap \left\{ w=w_{\alpha } \right\} }\\ =\&\left\{ \left[ y(w)-h\left( \frac{y(w)}{w} \right) \right] \frac{{\tilde{f}}(w)-\Gamma (w,\varphi (w))}{\Gamma \left( {\underline{w}},{\overline{w}} \right) } \right\} {\mathbb {I}}_{\left\{ w_{\alpha }\ge w_{\eta } \right\} \cap \left\{ y(w)=y^{M*}(w) \right\} \cap \left\{ w=w_{\alpha } \right\} }, \end{aligned} \end{aligned}$$

(82)

$$\begin{aligned}&\Lambda _{2}(k)=\int _{w_{\alpha }}^{k}\left[ \Lambda _{21}(w)+\Lambda _{22}(w)+\Lambda _{23}(w)+\Lambda _{24}(w) \right] \cdot {\mathbb {I}}_{\left\{ w_{\alpha }>{\underline{w}} \right\} } dw, \end{aligned}$$

(83)

with

$$\begin{aligned}&\begin{aligned} \Lambda _{21}(w) =&\left\{ \left[ 1-\frac{1}{w}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \frac{\Gamma (w_{\alpha },\varphi (w_{\alpha }))}{\Gamma ({\underline{w}},{\overline{w}})} \right. \\&+ \left. \left[ \frac{1}{w^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) + \frac{y^{M*}(w_{\alpha })}{w^{3}}h''\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \left[ 1-\frac{\Gamma (w_{\alpha },{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] \right\} \frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}, \end{aligned} \end{aligned}$$

(84)

$$\begin{aligned}&\begin{aligned} \Lambda _{22}(w) =&\left[ y^{M*}(w_{\alpha })-h\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \frac{1}{\Gamma ({\underline{w}},{\overline{w}})} \\&\times \left\{ {\tilde{f}}\left( \varphi (w_{\alpha }) \right) \frac{\mathrm {d} \varphi (w_{\alpha })}{\mathrm {d} w_{\alpha }}-{\tilde{f}}(w_{\alpha })+\left[ \int _{w_{\alpha }}^{\varphi (w_{\alpha })}\frac{\partial {\tilde{f}}(w)}{\partial y^{M*}(w_{\alpha })}\frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}dw \right] \right\} , \end{aligned} \end{aligned}$$

(85)

$$\begin{aligned}&\begin{aligned} \Lambda _{23}(w) = &\frac{y^{M*}(w_{\alpha })}{w^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \frac{1}{\Gamma ({\underline{w}},{\overline{w}})}\\&\times \left\{ {\tilde{f}}(w_{\alpha }) - \left[ \int _{w_{\alpha }}^{\varphi (w_{\alpha })}\frac{\partial {\tilde{f}}(w)}{\partial y^{M*}(w_{\alpha })}\frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}dw \right] \right\} , \end{aligned} \end{aligned}$$

(86)

and

$$\begin{aligned}&\begin{aligned}&\Lambda _{24}(w) \\&\quad = \left\{ \frac{y^{M*}(w_{\alpha })}{w^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \Gamma (w_{\alpha },{\overline{w}}) - \left[ y^{M*}(w_{\alpha })-h\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \Gamma (w_{\alpha },\varphi (w_{\alpha })) \right\} \\&\qquad \times \frac{1}{\Gamma ({\underline{w}},{\overline{w}})^{2}} \int _{w_{\alpha }}^{\varphi (w_{\alpha })}\frac{\partial {\tilde{f}}(w)}{\partial y^{M*}(w_{\alpha })} \frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}dw; \end{aligned} \end{aligned}$$

(87)

$$\begin{aligned}&\Lambda _{3}(k)=\int _{k}^{\varphi (w_{\alpha })}\left[ \Lambda _{31}(w)+\Lambda _{32}(w)+\Lambda _{33}(w)+\Lambda _{34}(w) \right] \cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} } dw, \end{aligned}$$

(88)

with

$$\begin{aligned}&\begin{aligned} \Lambda _{31}(w) =&\left\{ \left[ 1-\frac{1}{w}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \frac{\Gamma (w_{\alpha },\varphi (w_{\alpha }))}{\Gamma ({\underline{w}},{\overline{w}})} \right. \\&- \left. \left[ \frac{1}{w^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) + \frac{y^{M*}(w_{\alpha })}{w^{3}}h''\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \frac{\Gamma (\varphi (w_{\alpha }),{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right\} \frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}, \end{aligned} \end{aligned}$$

(89)

$$\begin{aligned}&\begin{aligned} \Lambda _{32}(w) =&\left[ y^{M*}(w_{\alpha })-h\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \frac{1}{\Gamma ({\underline{w}},{\overline{w}})} \\&\times \left\{ {\tilde{f}}\left( \varphi (w_{\alpha }) \right) \frac{\mathrm {d} \varphi (w_{\alpha })}{\mathrm {d} w_{\alpha }}-{\tilde{f}}(w_{\alpha })+\left[ \int _{w_{\alpha }}^{\varphi (w_{\alpha })}\frac{\partial {\tilde{f}}(w)}{\partial y^{M*}(w_{\alpha })}\frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}dw \right] \right\} , \end{aligned} \end{aligned}$$

(90)

$$\begin{aligned}&\Lambda _{33}(w) = \frac{y^{M*}(w_{\alpha })}{w^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \frac{{\tilde{f}}\left( \varphi (w_{\alpha })\right) }{\Gamma ({\underline{w}},{\overline{w}})}\frac{\mathrm {d} \varphi (w_{\alpha })}{\mathrm {d} w_{\alpha }}, \end{aligned}$$

(91)

and

$$\begin{aligned} \begin{aligned}&\Lambda _{34}(w) \\&\quad = \left\{ \frac{y^{M*}(w_{\alpha })}{w^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \Gamma (\varphi (w_{\alpha }),{\overline{w}}) - \left[ y^{M*}(w_{\alpha })-h\left( \frac{y^{M*}(w_{\alpha })}{w} \right) \right] \Gamma (w_{\alpha }, \varphi (w_{\alpha })) \right\} \\&\qquad \times \frac{1}{\Gamma ({\underline{w}},{\overline{w}})^{2}} \int _{w_{\alpha }}^{\varphi (w_{\alpha })}\frac{\partial {\tilde{f}}(w)}{\partial y^{M*}(w_{\alpha })} \frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}dw; \end{aligned} \end{aligned}$$

(92)

and finally

$$\begin{aligned} \begin{aligned} \Lambda _{4} =&\frac{\mathrm {d} \varphi (w_{\alpha })}{\mathrm {d} w_{\alpha }} \\&\quad \times [{\tilde{\Phi }}^{R*}(w, y(w),\Gamma (w_{\alpha },w),\Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{M*}(w_{\alpha }) \right\} \cap \left\{ w=\varphi (w_{\alpha }) \right\} } \\&-{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))\cdot {\mathbb {I}}_{\left\{ {\overline{w}}\ge w_{\beta } \right\} \cap \left\{ y(w)=y^{M*}(w_{\alpha }) \right\} \cap \left\{ w=\varphi (w_{\alpha }) \right\} }]\\ =&\left[ {\tilde{\Phi }}^{R*}(w, y(w),\Gamma (w_{\alpha },w),\Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}}))-{\tilde{\Phi }}^{R*}(w, y(w),{\tilde{f}}(w), \Gamma (w,{\overline{w}}),\Gamma ({\underline{w}},{\overline{w}})) \right] \\&\times {\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} \cap \left\{ y(w)=y^{M*}(w_{\alpha }) \right\} \cap \left\{ w=\varphi (w_{\alpha }) \right\} } \cdot \frac{\mathrm {d} \varphi (w_{\alpha })}{\mathrm {d} w_{\alpha }}\\ =&\left\{ \left[ y^{M*}(w_{\alpha })-h\left( \frac{y^{M*}(w_{\alpha })}{\varphi (w_{\alpha })} \right) \right] \frac{\Gamma \left( w_{\alpha },\varphi (w_{\alpha }) \right) -{\tilde{f}}(\varphi (w_{\alpha }))}{\Gamma \left( {\underline{w}},{\overline{w}} \right) } \right\} \frac{\mathrm {d} \varphi (w_{\alpha })}{\mathrm {d} w_{\alpha }}{\mathbb {I}}_{\left\{ w_{\beta }<{\overline{w}} \right\} }. \end{aligned} \end{aligned}$$

(93)

$\underline{\mathrm{Step 4.}}$ We first consider the case with $w_{\beta }<{\overline{w}}$. It follows from (62) and (63) that

$$\begin{aligned} \frac{\partial ^2 \Xi (w_{\beta };k)}{\partial w_{\beta } \partial k}=\frac{\mathrm {d} \Psi _{2}(k) }{\mathrm {d} k}+\frac{\mathrm {d} \Psi _{3}(k) }{\mathrm {d} k}. \end{aligned}$$

(94)

Using equations (70)-(76), (94) can be explicitly expressed as

$$\begin{aligned} \begin{aligned}&\frac{\mathrm {d} \Psi _{2}(k) }{\mathrm {d} k}+\frac{\mathrm {d} \Psi _{3}(k) }{\mathrm {d} k}\\&\quad = \left[ \Psi _{21}(k)+\Psi _{22}(k)+\Psi _{23}(k) +\Psi _{24}(k) \right] -\left[ \Psi _{31}(k)+\Psi _{32}(k)+\Psi _{33}(k)+\Psi _{34}(k) \right] \\&\quad = \left[ \Psi _{21}(k)-\Psi _{31}(k) \right] + \underset{=0}{\underbrace{\left[ \Psi _{22}(k)-\Psi _{32}(k) \right] }} + \left[ \Psi _{23}(k)-\Psi _{33}(k) \right] + \left[ \Psi _{24}(k)-\Psi _{34}(k) \right] \\&\quad = \left[ \frac{1}{k^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{k} \right) + \frac{y^{R*}(w_{\beta })}{k^{3}}h''\left( \frac{y^{R*}(w_{\beta })}{k} \right) \right] \left[ 1-\frac{\Gamma (\psi (w_{\beta }),{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} +\frac{\Gamma (w_{\beta },{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}\\&\qquad + \frac{y^{R*}(w_{\beta })}{k^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{k} \right) \frac{1}{\Gamma ({\underline{w}},{\overline{w}})}\left[ -\frac{\mathrm {d} \Gamma (\psi (w_{\beta }),w_{\beta })}{\mathrm {d} w_{\beta }}\right] \\&\qquad + \frac{y^{R*}(w_{\beta })}{k^{2}}h'\left( \frac{y^{R*}(w_{\beta })}{k} \right) \frac{\Gamma (\psi (w_{\beta }),{\overline{w}})-\Gamma (w_{\beta },{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})^{2}}\int _{\psi (w_{\beta })}^{w_{\beta }}\frac{\partial {\tilde{f}}(w)}{\partial y^{R*}(w_{\beta })} \frac{\mathrm {d} y^{R*}(w_{\beta })}{\mathrm {d} w_{\beta }}dw. \end{aligned} \end{aligned}$$

(95)

Since $\mathrm {d} y^{R*}(w_{\beta })/\mathrm {d} w_{\beta }>0$ by assumption and $\Gamma (\psi (w_{\beta }),{\overline{w}})>\Gamma (w_{\beta },{\overline{w}})$, we have

$$\begin{aligned} \frac{\partial ^2 \Xi (w_{\beta };k)}{\partial w_{\beta } \partial k}=\frac{\mathrm {d} \Psi _{2}(k) }{\mathrm {d} k}+\frac{\mathrm {d} \Psi _{3}(k) }{\mathrm {d} k}>0 \end{aligned}$$

(96)

whenever

$$\begin{aligned} \frac{\mathrm {d} \Gamma (\psi (w_{\beta }),w_{\beta })}{\mathrm {d} w_{\beta }}\le 0, \end{aligned}$$

as desired. In particular, $\mathrm {d} \psi (w_{\beta })/\mathrm {d} w_{\beta }>0$ based on the construction of $\psi (\cdot )$ as well as the monotonicity of the income schedule. In the case of (96), $\Xi (w_{\beta };k)$ is a supermodular function, and an application of Topkis Theorem (see Theorem 6.1 of Topkis (1978)) implies that $w_{\beta }(k)$ is nondecreasing in k.

${\underline{\mathrm{Step 5.}}}$ We now consider the case with $w_{\alpha }>{\underline{w}}$. It follows from (80) and (81) that

$$\begin{aligned} \frac{\partial ^2 \Xi (w_{\alpha };k)}{\partial w_{\alpha } \partial k}=\frac{\mathrm {d} \Lambda _{2}(k) }{\mathrm {d} k}+\frac{\mathrm {d} \Lambda _{3}(k) }{\mathrm {d} k}. \end{aligned}$$

(97)

Using equations (83)-(91), (97) can be explicitly expressed as

$$\begin{aligned} \begin{aligned}&\frac{\mathrm {d} \Lambda _{2}(k) }{\mathrm {d} k}+\frac{\mathrm {d} \Lambda _{3}(k) }{\mathrm {d} k}\\&\quad = \left[ \Lambda _{21}(k)+\Lambda _{22}(k)+\Lambda _{23}(k) +\Lambda _{24}(k) \right] -\left[ \Lambda _{31}(k)+\Lambda _{32}(k)+\Lambda _{33}(k) +\Lambda _{34}(k)\right] \\&\quad = \left[ \Lambda _{21}(k)-\Lambda _{31}(k) \right] + \underset{=0}{\underbrace{\left[ \Lambda _{22}(k)-\Lambda _{32}(k) \right] }} + \left[ \Lambda _{23}(k)-\Lambda _{33}(k) \right] + \left[ \Lambda _{24}(k)-\Lambda _{34}(k) \right] \\&\quad = \left[ \frac{1}{k^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{k} \right) + \frac{y^{M*}(w_{\alpha })}{k^{3}}h''\left( \frac{y^{M*}(w_{\alpha })}{k} \right) \right] \left[ 1-\frac{\Gamma ( w_{\alpha },{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} +\frac{\Gamma (\varphi (w_{\alpha }),{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})} \right] \frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}\\&\qquad + \frac{y^{M*}(w_{\alpha })}{k^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{k} \right) \frac{1}{\Gamma ({\underline{w}},{\overline{w}})}\left[ -\frac{\mathrm {d} \Gamma (w_{\alpha }, \varphi (w_{\alpha }))}{\mathrm {d} w_{\alpha }} \right] \\&\qquad + \frac{y^{M*}(w_{\alpha })}{k^{2}}h'\left( \frac{y^{M*}(w_{\alpha })}{k} \right) \frac{\Gamma ( w_{\alpha },{\overline{w}})- \Gamma (\varphi (w_{\alpha }),{\overline{w}})}{\Gamma ({\underline{w}},{\overline{w}})^{2}} \int _{w_{\alpha }}^{\varphi (w_{\alpha })}\frac{\partial {\tilde{f}}(w)}{\partial y^{M*}(w_{\alpha })} \frac{\mathrm {d} y^{M*}(w_{\alpha })}{\mathrm {d} w_{\alpha }}dw. \end{aligned} \end{aligned}$$

(98)

Since $\mathrm {d} y^{M*}(w_{\alpha })/\mathrm {d} w_{\alpha }>0$ by assumption, we know according to (98) that

$$\begin{aligned} \frac{\partial ^2 \Xi (w_{\alpha };k)}{\partial w_{\alpha } \partial k}=\frac{\mathrm {d} \Lambda _{2}(k) }{\mathrm {d} k}+\frac{\mathrm {d} \Lambda _{3}(k) }{\mathrm {d} k}>0 \end{aligned}$$

(99)

whenever

$$\begin{aligned} \frac{\mathrm {d} \Gamma (w_{\alpha }, \varphi (w_{\alpha }))}{\mathrm {d} w_{\alpha }} \le 0, \end{aligned}$$

as desired. In the case of (99), $\Xi (w_{\alpha };k)$ is a supermodular function, and an application of Topkis Theorem (see Theorem 6.1 of Topkis (1978)) again implies that $w_{\alpha }(k)$ is nondecreasing in k. $\square$

Proof of Theorem 4.1

We complete the proof in four steps.

${\underline{\mathrm{Step 1.}}}$ Based on Proposition 3.3, we first consider the case with a downward discontinuity at the type of proposer, namely the income schedules with the SOIC condition being violated. Let us consider two alternative proposers of types $k_{1}$ and $k_{2}$, for $k_{1}<k_{2}$. Since the income schedules they proposed coincide with the maxi-max schedule for types below their type and coincide with the maxi-min schedule for types above their type, and the maxi-max income schedule lies everywhere above the maxi-min income schedule, the higher the type of proposer, the more workers who are below the proposer and the more who are allocated with the maxi-max incomes. Precisely, if the proposer changes from type $k_{1}$ to type $k_{2}$, all workers of types belonging to set $[k_{1}, k_{2}]$ are strictly better off in terms of pre-tax income, while all other workers of the remaining types are neutral to this change. We hence have that $y(w,k_{1})\le y(w,k_{2})$ for $\forall w, k_{1}, k_{2}\in [{\underline{w}}, {\overline{w}}]$, and $y(w,k_{1})< y(w,k_{2})$ for $\forall w\in [k_{1}, k_{2}]$ (see Fig. 7). In addition, as all proposers face the same government budget and incentive constraints, each proposer must weakly prefer what she obtains with her own schedule to what any other worker proposed for her. Formally,

$$\begin{aligned} U(w,w)\ \ge \ U(w,k) \ \ \ \text {for} \ \forall w,k\in \left[ {\underline{w}}, {\overline{w}}\right] . \end{aligned}$$

(100)

Fig. 7

The type of proposer changes from $k_{1}$ to $k_{2}$ when SOIC is violated

We next show that a worker of any type w has a weakly single-peaked preference on the set of types. To this end, we need to consider two cases with the proof procedure being directly borrowed from Brett and Weymark (2017).

${\underline{\mathrm{Step 2.}}}$ First, we consider the right-hand side of w. We arbitrarily pick three types $w, k_{1}, k_{2}$ satisfying $w<k_{1}<k_{2}$. Using (6) and (100), we have

$$\begin{aligned} \begin{aligned} U\left( w,k_{1}\right) \&= \ U(k_{1},k_{1})-\int _{w}^{k_{1}}h'\left( \frac{y(t,k_{1})}{t} \right) \frac{y(t,k_{1})}{t^{2}}dt\\&\ge \ U\left( k_{1},k_{2}\right) -\int _{w}^{k_{1}}h'\left( \frac{y(t,k_{1})}{t} \right) \frac{y(t,k_{1})}{t^{2}}dt. \end{aligned} \end{aligned}$$

(101)

Similarly, according to (6),

$$\begin{aligned} U(w,k_{2})\ = \ U(k_{1},k_{2})-\int _{w}^{k_{1}}h'\left( \frac{y(t,k_{2})}{t} \right) \frac{y(t,k_{2})}{t^{2}}dt. \end{aligned}$$

(102)

Solving for $U(k_{1},k_{2})$ from (102) and inserting it into (101) produces

$$\begin{aligned} U(w,k_{1})-U(w,k_{2})\ \ge \ \int _{w}^{k_{1}}\left[ h'\left( \frac{y(t,k_{2})}{t} \right) \frac{y(t,k_{2})}{t^{2}}-h'\left( \frac{y(t,k_{1})}{t} \right) \frac{y(t,k_{1})}{t^{2}} \right] dt. \end{aligned}$$

(103)

Since h is strictly increasing and convex, we hence have by using (103) that $U(w,k_{1})\ge U(w,k_{2})$, which combined with (100) reveals that

$$\begin{aligned} U(w,w)\ \ge \ U(w,k_{1})\ \ge \ U(w,k_{2}), \ \ \ \forall w<k_{1}<k_{2}. \end{aligned}$$

(104)

${\underline{\mathrm{Step 3.}}}$ Second, for the case with $w>k_{1}>k_{2}$, using (6) and (100) yields

$$\begin{aligned} \begin{aligned} U(w,k_{1})\&= \ U(k_{1},k_{1})+\int _{k_{1}}^{w}h'\left( \frac{y(t,k_{1})}{t} \right) \frac{y(t,k_{1})}{t^{2}}dt\\&\ge \ U(k_{1},k_{2})+\int _{k_{1}}^{w}h'\left( \frac{y(t,k_{1})}{t} \right) \frac{y(t,k_{1})}{t^{2}}dt. \end{aligned} \end{aligned}$$

(105)

Similarly,

$$\begin{aligned} U(w,k_{2})\ = \ U(k_{1},k_{2})+\int _{k_{1}}^{w}h'\left( \frac{y(t,k_{2})}{t} \right) \frac{y(t,k_{2})}{t^{2}}dt. \end{aligned}$$

(106)

Making use of (105) and (106) gives rise to

$$\begin{aligned} U(w,k_{1})-U(w,k_{2})\ \ge \ \int _{k_{1}}^{w}\left[ h'\left( \frac{y(t,k_{1})}{t} \right) \frac{y(t,k_{1})}{t^{2}}-h'\left( \frac{y(t,k_{2})}{t} \right) \frac{y(t,k_{2})}{t^{2}} \right] dt. \end{aligned}$$

(107)

Applying the same reasoning used in step 2 to (107), we arrive at

$$\begin{aligned} U(w,w)\ \ge \ U(w,k_{1})\ \ge \ U(w,k_{2}), \ \ \ \forall w>k_{1}>k_{2}. \end{aligned}$$

(108)

Accordingly, (104) combined with (108) reveals that the preference of any given type of worker is indeed (weakly) single-peaked on the set of types. Applying the Black’s Median Voter Theorem (see Black, 1948), the desired assertion hence follows. After ironing the downward discontinuity by building a bridge, we establish the income schedule that meets the SOIC condition in Theorem 3.2. Using Lemma 3.5, it is immediately apparent that the reasoning used above applies as well, and hence the desired assertion holds true for the complete solution of the tax design problem.

${\underline{\mathrm{Step 4.}}}$ We now move to the case in which the SOIC condition is met under the first-order approach. As shown in Fig. 8, if the proposer changes from type $k_{1}$ to type $k_{2}$, all workers of types belonging to set $[k_{1}, k_{2}]$ are strictly worse off in terms of pre-tax income, while all other workers of the remaining types are neutral to this change. We hence have that $y(w,k_{1})\ge y(w,k_{2})$ for $\forall w, k_{1}, k_{2}\in [{\underline{w}}, {\overline{w}}]$, and $y(w,k_{1})> y(w,k_{2})$ for $\forall w\in [k_{1}, k_{2}]$. In addition, condition (100) still applies here. It is easy to show that we still have (103) for $w<k_{1}<k_{2}$ and (107) for $w>k_{1}>k_{2}$. Since the right-hand side of inequalities (103) and (107) is equal to zero, the assertion established above holds true in the current case. $\square$

Fig. 8

The type of proposer changes from $k_{1}$ to $k_{2}$ when SOIC is met

Proof of Lemma 5.1

In the case of autarky considered by Brett and Weymark (2017), the marginal tax rates are obtained as follows:

$$\begin{aligned} \begin{aligned} {\hat{\tau }}^{M}(w)&\ = \ -\frac{F(w)}{wf(w)}\left[ \frac{1}{w}h'\left( \frac{y(w)}{w} \right) +\frac{y(w)}{w^{2}}h''\left( \frac{y(w)}{w} \right) \right] ,\\ {\hat{\tau }}^{R}(w)&\ = \ \frac{1-F(w)}{wf(w)}\left[ \frac{1}{w}h'\left( \frac{y(w)}{w} \right) +\frac{y(w)}{w^{2}}h''\left( \frac{y(w)}{w} \right) \right] . \end{aligned} \end{aligned}$$

(109)

Using (25) and (109), we obtain

$$\begin{aligned} {\hat{\tau }}^{M}(w) -\tau ^{M}(w)=- \left[ \frac{1}{w^2}h'\left( \frac{y(w)}{w} \right) +\frac{y(w)}{w^{3}}h''\left( \frac{y(w)}{w} \right) \right] \left[ \frac{F(w)}{f(w)}- \frac{\Gamma ({\underline{w}},w)}{{\tilde{f}}(w)} \right] , \end{aligned}$$

from which assertion (i) immediately follows.

Using (26), (109) and (17), we obtain

$$\begin{aligned} {\hat{\tau }}^{R}(w) -\tau ^{R}(w)= \&\frac{\partial U(w)}{\partial y(w)}\left[ \frac{1-F(w)}{f(w)} - \frac{\Gamma ({\underline{w}},{\overline{w}})-\Gamma ({\underline{w}},w)}{{\tilde{f}}(w)} \right] \\&+ \frac{\partial U(w)}{\partial y(w)}\frac{{\tilde{\theta }}(w)}{c(w)} \left[ T^{R}(y(w))+U(w)-U'(w)-U({\underline{w}}) \right] , \end{aligned}$$

from which assertion (ii) follows.

Applying (109) and (26) reveals that

$$\begin{aligned}&{\hat{\tau }}^{M}(w) -\tau ^{R}(w)\\&\quad =-\frac{\partial U(w)}{\partial y(w)} \left\{ \frac{\Gamma ({\underline{w}},{\overline{w}})-\Gamma ({\underline{w}},w)}{{\tilde{f}}(w)}+\frac{F(w)}{f(w)} - \frac{{\tilde{\theta }}(w)}{c(w)} \left[ T^{R}(y(w))+U(w)-U'(w)-U({\underline{w}}) \right] \right\} , \end{aligned}$$

by which we can establish assertion (iii).

Applying (25) and (109) reveals that

$$\begin{aligned} {\hat{\tau }}^{R}(w) -\tau ^{M}(w) = \left[ \frac{1}{w^2}h'\left( \frac{y(w)}{w} \right) +\frac{y(w)}{w^{3}}h''\left( \frac{y(w)}{w} \right) \right] \cdot \left[ \frac{\Gamma ({\underline{w}},w)}{{\tilde{f}}(w)}+\frac{1-F(w)}{f(w)}\right] , \end{aligned}$$

by which assertion (iv) is immediate. $\square$

Proof of Proposition 5.1

First, it is easy to see from Propositions 3.2 and 3.3 , (35) and (37) that $\Theta ^{R}(w)<{\tilde{\theta }}_{\star }(w)<\min \{\Theta ^{MR}(w), {\tilde{\theta }}^{*}(w)\}$. The relative magnitude of $\Theta ^{MR}(w)$ and ${\tilde{\theta }}^{*}(w)$ is generally ambiguous. Using claim (i) of Lemma 5.1 and Proposition 3.3(iii), we get ${\hat{\tau }}^{M}(w)>\tau ^{M}(w)$ whenever $T^{M}(y({\underline{w}}))< U'({\underline{w}})$ and $F(w)/f(w)<\Gamma ({\underline{w}},w)/{\tilde{f}}(w)$ for all $w\in ({\underline{w}},w_{m}]$, as desired in all three conditions. Furthermore, using (109), we have $0>{\hat{\tau }}^{M}(w)>\tau ^{M}(w)$. Using (34), the remaining requirements of condition (a) lead to ${\hat{\tau }}^{R}(w)>\tau ^{R}(w)$. Using claim (ii) of Proposition 3.2, the requirement of $T^{R}(y(w))< U'(w)+U({\underline{w}})-U(w)$ in condition (a) leads to $\tau ^{R}(w)>0$, and hence condition (a) implies that ${\hat{\tau }}^{R}(w)>\tau ^{R}(w)>0>{\hat{\tau }}^{M}(w)>\tau ^{M}(w)$, as desired in claim (i).

Using claim (ii) of Lemma 5.1, the requirements of $T^{R}(y(w))> U'(w)+U({\underline{w}})-U(w)$ and $[1-F(w)]/f(w)\ge [\Gamma ({\underline{w}},{\overline{w}})-\Gamma ({\underline{w}},w)]/{\tilde{f}}(w)$ in condition (b) imply that ${\hat{\tau }}^{R}(w)>\tau ^{R}(w)$. If ${\tilde{\theta }}(w) \le {\tilde{\theta }}_{\star }(w)$, then we get from claim (ii) of Proposition 3.2 that $\tau ^{R}(w)>0$, and hence condition (b) also implies that ${\hat{\tau }}^{R}(w)>\tau ^{R}(w)>0>{\hat{\tau }}^{M}(w)>\tau ^{M}(w)$, as desired in claim (i). Departing from condition (a), the migration elasticity is allowed to be greater than the threshold ${\tilde{\theta }}_{\star }(w)$, which means that it is possible that $\tau ^{R}(w)<0$ based on claim (ii) of Proposition 3.2. However, using claim (iii) of Lemma 5.1, it is still guaranteed that $0>\tau ^{R}(w)>{\hat{\tau }}^{M}(w)$ for ${\tilde{\theta }}_{\star }(w)<{\tilde{\theta }}(w)<\min \{\Theta ^{MR}(w), {\tilde{\theta }}^{*}(w)\}$. Using (109), we have in this case that ${\hat{\tau }}^{R}(w)>0>\tau ^{R}(w)>{\hat{\tau }}^{M}(w)>\tau ^{M}(w)$. Once again, claim (i) follows.

Using claim (ii) of Lemma 5.1, the requirements of $T^{R}(y(w))>U'(w)+U({\underline{w}})-U(w)$, $[1-F(w)]/f(w)< [\Gamma ({\underline{w}},{\overline{w}})-\Gamma ({\underline{w}},w)]/{\tilde{f}}(w)$ and $\Theta ^{R}(w)<{\tilde{\theta }}(w)$ for all $w\in (w_{m},{\overline{w}}]$ in condition (c) imply that ${\hat{\tau }}^{R}(w)>\tau ^{R}(w)$. If $\Theta ^{R}(w)<{\tilde{\theta }}(w) \le {\tilde{\theta }}_{\star }(w)$, then we get from claim (ii) of Proposition 3.2 that $\tau ^{R}(w)>0$, and hence condition (c) also implies that ${\hat{\tau }}^{R}(w)>\tau ^{R}(w)>0>{\hat{\tau }}^{M}(w)>\tau ^{M}(w)$, as desired in claim (i). Similar to condition (b), here the migration elasticity is allowed to be greater than the threshold ${\tilde{\theta }}_{\star }(w)$, which leads to the possible prediction of $\tau ^{R}(w)<0$ based on claim (ii) of Proposition 3.2. However, using claim (iii) of Lemma 5.1, it is still guaranteed that $0>\tau ^{R}(w)>{\hat{\tau }}^{M}(w)$ for ${\tilde{\theta }}_{\star }(w)<{\tilde{\theta }}(w)<\min \{\Theta ^{MR}(w), {\tilde{\theta }}^{*}(w)\}$. Using (109), we have in this case that ${\hat{\tau }}^{R}(w)>0>\tau ^{R}(w)>{\hat{\tau }}^{M}(w)>\tau ^{M}(w)$. Once again, claim (i) follows. Finally, applying claim (i) and Fig. 3, claims (ii)-(iii) of this proposition are immediate. $\square$

Proof of Proposition 5.2

The proof is quite similar to that of the previous proposition and hence the detail is omitted to save space. Basically, using Lemma 5.1, Proposition 3.2 and (109), we get $\tau ^{R}(w)>{\hat{\tau }}^{R}(w)>0>\tau ^{M}(w)>{\hat{\tau }}^{M}(w)$ under one of the four conditions, then claim (i) holds true. Using claim (i) and Figure 4, claims (ii)-(iii) are immediate. $\square$

Proof of Corollary 5.1

According to the proof of Theorem 3.2 and Proposition 5.1, the relationship between the income schedule under migration and that in autarky is illustrated by Fig. 5, in which the thick black curve represents the income schedule derived by Brett and Weymark (2017) after ironing the downward discontinuity at the ex ante median skill level, with the bridge endpoints denoted by ${\hat{w}}_{\alpha }$ and ${\hat{w}}_{\beta }$. Also, $w_{\alpha }, w_{\beta }$ denote the bridge endpoints endogenously determined in the proof of Lemma 3.5. Immediately, result (i) follows from the left one of Fig. 5, and result (ii) follows from the right one. $\square$

Proof of Proposition 5.3

Given that the maxi-max income schedule is above the maxi-min income schedule in the autarky equilibrium but not in the migration equilibrium, claim (i) follows from applying part (i) of Lemma 5.1 and part (i) of Proposition 3.2. Thus, in order for all skill types to face higher tax rates under migration than in autarky, the maxi-min tax rates in the migration equilibrium must be larger than those in the autarky equilibrium, which cannot be satisfied by using part (ii) of Lemma 5.1 under the assumption of $T^{R}(y(w))> U'(w)+U({\underline{w}})-U(w)$ and ${\tilde{\theta }}(w)\ge {\tilde{\theta }}^{*}(w)$ for all $w \in ({\tilde{w}}_{m},{\overline{w}}]$. As such, we must have higher maxi-min tax rates in the autarky equilibrium than those in the migration equilibrium. Again applying the fact given at the beginning, claim (iii) is immediate by using part (i) of Lemma 5.1 and part (i) of Proposition 3.2. $\square$

Appendix B: The relative magnitude of ex ante and ex post median skill levels

After combining the migration decisions, the ex post measure of workers is given by

$$\begin{aligned} \Gamma ({\underline{w}},{\overline{w}})=\int _{{\underline{w}}}^{{\overline{w}}}{\tilde{f}}(w)dw \ = \ \int _{{\underline{w}}}^{w_{m}}{\tilde{f}}(w)dw + \int _{w_{m}}^{{\overline{w}}}{\tilde{f}}(w)dw. \end{aligned}$$

(110)

By using (8), we get the right-hand terms of (110) as²⁰

$$\begin{aligned} \int _{{\underline{w}}}^{w_{m}}{\tilde{f}}(w)dw = \frac{1}{2} \ + \ \underset{L^{NI}([{\underline{w}},w_{m}])\ =\ \text {net labor inflow}}{\underbrace{L^{I}([{\underline{w}},w_{m}])-L^{O}([{\underline{w}},w_{m}])}} \end{aligned}$$

(111)

and

$$\begin{aligned} \int _{w_{m}}^{{\overline{w}}}{\tilde{f}}(w)dw = \frac{1}{2} \ + \ \underset{L^{NI}([w_{m},{\overline{w}}])\ =\ \text {net labor inflow}}{\underbrace{L^{I}([w_{m},{\overline{w}}])-L^{O}([w_{m},{\overline{w}}])}}, \end{aligned}$$

(112)

in which the measures of labor inflows are defined as

$$\begin{aligned} \begin{aligned} L^{I}([{\underline{w}},w_{m}])&\equiv \int _{\left\{ w\in [{\underline{w}},w_{m}]| \Delta (w)\ge 0 \right\} }G_{-}(\Delta (w)|w)f_{-}(w)n_{-}dw, \\ L^{I}([w_{m},{\overline{w}}])&\equiv \int _{\left\{ w\in [w_{m},{\overline{w}}]| \Delta (w)\ge 0 \right\} }G_{-}(\Delta (w)|w)f_{-}(w)n_{-}dw, \end{aligned} \end{aligned}$$

(113)

and the measures of labor outflows are defined as

$$\begin{aligned} \begin{aligned} L^{O}([{\underline{w}},w_{m}])&\equiv \int _{\left\{ w\in [{\underline{w}},w_{m}]| \Delta (w)\le 0 \right\} }G(-\Delta (w)|w)f(w)dw, \\ L^{O}([w_{m},{\overline{w}}])&\equiv \int _{\left\{ w\in [w_{m},{\overline{w}}]| \Delta (w)\le 0 \right\} }G(-\Delta (w)|w)f(w)dw. \end{aligned} \end{aligned}$$

(114)

Using (110)–(114), we can identify the relationship of the ex post median skill level ${\tilde{w}}_{m}$ with the ex ante median skill level $w_{m}$ and summarize the results as three propositions.

Proposition 6.1

Suppose $\Gamma ({\underline{w}},{\overline{w}})=1$. We have: (a) If $L^{NI}([{\underline{w}},w_{m}])=L^{NI}([w_{m},{\overline{w}}])=0$, then ${\tilde{w}}_{m}=w_{m}$; (b) If $L^{NI}([{\underline{w}},w_{m}])>0$ and $L^{NI}([w_{m},{\overline{w}}])<0$, then ${\tilde{w}}_{m}<w_{m}$; (c) If $L^{NI}([{\underline{w}},w_{m}])<0$ and $L^{NI}([w_{m},{\overline{w}}])>0$, then ${\tilde{w}}_{m}>w_{m}$.

Proposition 6.2

Suppose $\Gamma ({\underline{w}},{\overline{w}})>1$. We have: (a) If $L^{NI}([{\underline{w}},w_{m}])=0$ and $L^{NI}([w_{m},{\overline{w}}])$ $>0$, then ${\tilde{w}}_{m}>w_{m}$; (b) If $L^{NI}([{\underline{w}},w_{m}])>0$ and $L^{NI}([w_{m},{\overline{w}}])=0$, then ${\tilde{w}}_{m}<w_{m}$; (c) If $L^{NI}([{\underline{w}},w_{m}])>0$ and $L^{NI}([w_{m},{\overline{w}}])>0$, then ${\tilde{w}}_{m}<w_{m}$ for $L^{NI}([{\underline{w}},w_{m}])>L^{NI}([w_{m},{\overline{w}}])$, ${\tilde{w}}_{m}=w_{m}$ for $L^{NI}([{\underline{w}},w_{m}])=L^{NI}([w_{m},{\overline{w}}])$, and ${\tilde{w}}_{m}>w_{m}$ for $L^{NI}([{\underline{w}},w_{m}])<L^{NI}([w_{m},{\overline{w}}])$; (d) If $L^{NI}([{\underline{w}},w_{m}])>0$ and $L^{NI}([w_{m},{\overline{w}}])<0$, then ${\tilde{w}}_{m}<w_{m}$; (e) If $L^{NI}([{\underline{w}},w_{m}])<0$ and $L^{NI}([w_{m},{\overline{w}}])>0$, then ${\tilde{w}}_{m}>w_{m}$.

Proposition 6.3

Suppose $\Gamma ({\underline{w}},{\overline{w}})<1$. We have: (a) If $L^{NI}([{\underline{w}},w_{m}])=0$ and $L^{NI}([w_{m},{\overline{w}}])$ $<0$, then ${\tilde{w}}_{m}<w_{m}$; (b) If $L^{NI}([{\underline{w}},w_{m}])<0$ and $L^{NI}([w_{m},{\overline{w}}])=0$, then ${\tilde{w}}_{m}>w_{m}$; (c) If $L^{NI}([{\underline{w}},w_{m}])<0$ and $L^{NI}([w_{m},{\overline{w}}])<0$, then ${\tilde{w}}_{m}<w_{m}$ for $L^{NI}([{\underline{w}},w_{m}])>L^{NI}([w_{m},{\overline{w}}])$, ${\tilde{w}}_{m}=w_{m}$ for $L^{NI}([{\underline{w}},w_{m}])=L^{NI}([w_{m},{\overline{w}}])$, and ${\tilde{w}}_{m}>w_{m}$ for $L^{NI}([{\underline{w}},w_{m}])<L^{NI}([w_{m},{\overline{w}}])$; (d) If $L^{NI}([{\underline{w}},w_{m}])>0$ and $L^{NI}([w_{m},{\overline{w}}])<0$, then ${\tilde{w}}_{m}<w_{m}$; (e) If $L^{NI}([{\underline{w}},w_{m}])<0$ and $L^{NI}([w_{m},{\overline{w}}])>0$, then ${\tilde{w}}_{m}>w_{m}$.

To identity the relationship between ex ante and ex post median skill levels, we divide the ex post population of workers into two groups: the first group has skill levels lower than the ex ante median skill level and the second group has skill levels higher than the ex ante median level. Propositions 6.1–6.3 consider three possible cases corresponding to three possible ex post measures of workers of all skill levels.

Proposition 6.1 considers the case that migrations do not change the total measure of workers. This proposition yields three possible subcases. Subcase (a) shows that labor inflow and labor outflow cancel each other out for both groups, and hence the median skill level should be the same under the same total measure. Subcase (b) demonstrates that the first group faces a positive net labor inflow while the second group faces a positive net labor outflow, hence the position of ex post median skill level should move towards the left under the same total measure, leading to a smaller median skill level than the ex ante one. Subcase (c) illustrates that the first group faces a positive net labor outflow while the second group faces a positive net labor inflow, hence the position of ex post median skill level should move towards the right under the same total measure, leading to a larger median skill level than the ex ante one. We can analyze Propositions 6.2–6.3 in the same way.