Denoting, respectively, by
\(V^{i,p}\left( t^{p},G^{p},s\right) \) and
\( V^{i,np}\left( t^{np},G^{np}\right) \) the indirect utility of parents and non-parents of ability type
i, and denoting by
\(\alpha ^{ij}\) the welfare weight used by the government for agents of type
ij, the design problem solved by the government can be written as:
$$\begin{aligned} \underset{t^{p},G^{p},t^{np},G^{np}}{\max }\qquad \underset{i=1}{ \overset{N}{\sum }}\alpha ^{i,p}V^{i,p}\left( t^{p},G^{p},s\right) +\underset{i=1}{\overset{N}{\sum }}\alpha ^{i,np}V^{i,np}\left( t^{np},G^{np}\right) \end{aligned}$$
subject to:
$$\begin{aligned} \sum \limits _{j=p,np}t^{j}\underset{i=1}{\overset{N}{\sum }}\pi ^{ij}Y^{i,j}\ge \sum \limits _{j=p,np}\underset{i=1}{\overset{N}{\sum }}\pi ^{ij}G^{j}+sq\underset{i=1}{\overset{N}{\sum }}\pi ^{ip}\frac{Y^{i,p}}{ w^{i,p}},\qquad \left( \mu \right) \end{aligned}$$
where
\(\mu \) is the Lagrange multiplier associated with the government’s budget constraint.
Denote
\(\pi ^{i,p}/\underset{k=1}{\overset{N}{\sum }}\pi ^{k,p}\) by
\( {\overline{\pi }}^{ip}\) and
\(\pi ^{i,np}/\underset{k=1}{\overset{N}{\sum }} \pi ^{k,np}\) by
\({\overline{\pi }}^{i,np}\). The first-order condition with respect to
\(G^{p}\) and
\(G^{np}\) are, respectively, given by:
$$\begin{aligned}&\displaystyle \frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,p}}\underset{i=1}{\overset{N}{\sum }}\frac{\alpha ^{i,p}}{\mu }\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{\partial G^{p}}+\underset{i=1}{\overset{N}{\sum }} \left( t^{p}w^{ip}-sq\right) \overline{\pi }^{i,p}\frac{\partial h^{i,p}}{ \partial G^{p}}=1, \end{aligned}$$
(9)
$$\begin{aligned}&\displaystyle \frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,np}}\underset{i=1}{ \overset{N}{\sum }}\frac{\alpha ^{i,np}}{\mu }\frac{\partial V^{i,np}\left( t^{np},G^{np}\right) }{\partial G^{np}}+\underset{i=1}{\overset{N}{\sum }} t^{np}\overline{\pi }^{i,np}w^{i,np}\frac{\partial h^{i,np}}{\partial G^{np}} =1. \end{aligned}$$
(10)
Define the net social marginal valuation of a lump-sum transfer to a parent of type
i and to a non-parent of type
i as, respectively:
$$\begin{aligned} b^{i,p}\equiv & {} \frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,p}}\left[ \frac{1}{\overline{\pi }^{i,p}}\frac{\alpha ^{i,p}}{\mu }\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{\partial G^{p}}+\left( t^{p}w^{i,p}-sq\right) \frac{\partial h^{i,p}}{\partial G^{p}}\right] , \\ b^{i,np}\equiv & {} \frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,np}} \left[ \frac{1}{\overline{\pi }^{i,np}}\frac{\alpha ^{i,np}}{\mu }\frac{ \partial V^{i,np}\left( t^{np},G^{np}\right) }{\partial G^{np}}+\left( t^{np}w^{i,np}\right) \frac{\partial h^{i,np}}{\partial G^{np}}\right] . \end{aligned}$$
Having defined
\(b^{i,p}\) and
\(b^{i,np}\) we can easily see that condition (
9), (
10) boil down to requiring
\(E\left( b^{p}\right) =E\left( b^{np}\right) =1\), where
\(E\left( \cdot \right) \) denotes the expectation operator.
39 In other words, it prescribes that at an optimum the lump-sum component should be adjusted such that
\(b^{j}\), the government’s net social marginal valuation of a transfer of 1 currency unit (measured in terms of government’s revenue) to agents of group
j (with
\(j=p,np\)) should on average be equal to its marginal cost.
The first-order condition with respect to
\(t^{np}\) is the following:
$$\begin{aligned}&\frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,np}}\underset{i=1}{ \overset{N}{\sum }}\alpha ^{i,np}\frac{\partial V^{i,np}\left( t^{np},G^{np}\right) }{\partial t^{np}}\\&\quad +\,\mu \left[ \underset{i=1}{\overset{N}{\sum }}\overline{\pi }^{i,np}w^{i,np}h^{i,np}+\underset{i=1}{\overset{N}{ \sum }}t^{np}\overline{\pi }^{i,np}w^{i,np}\frac{\partial h^{i,np}}{\partial t^{np}}\right] = 0, \end{aligned}$$
or, equivalently, applying the Slutsky equation and denoting by a tilde symbol a compensated variable:
$$\begin{aligned}&\frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,np}}\underset{i=1}{ \overset{N}{\sum }}\alpha ^{i,np}\frac{\partial V^{i,np}\left( t^{np},G^{np}\right) }{\partial t^{np}}\\&\quad + \mu \left[ \underset{i=1}{\overset{N}{\sum }}\overline{\pi } ^{i,np}w^{i,np}h^{i,np}+\underset{i=1}{\overset{N}{\sum }}t^{np}\overline{ \pi }^{i,np}w^{i,np}\left( \frac{\partial {\widetilde{h}}^{i,np}}{\partial t^{np}}-w^{i,np}h^{i,np}\frac{\partial h^{i,np}}{\partial G^{np}}\right) \right] = 0. \end{aligned}$$
Noticing that
\(\frac{\partial V^{i,np}\left( t^{np},G^{np}\right) }{\partial t^{np}}=-\frac{\partial V^{i,np}\left( t^{np},G^{np}\right) }{\partial G^{np} }w^{i,np}h^{i,np}\) (by applying Roy’s identity) and
\(\frac{\partial {\widetilde{h}}^{i,np}}{\partial t^{np}}=-w^{i,np}\frac{\partial {\widetilde{h}} ^{i,np}}{\partial w^{i,np}(1-t^{np})}\), and using (
10), we can derive the following implicit expression for the optimal
\(t^{np}\):
$$\begin{aligned} \frac{t^{np}}{1-t^{np}}=-\frac{cov\left( b^{np},Y^{np}\right) }{\underset{i=1 }{\overset{N}{\sum }}\overline{\pi }^{i,np}Y^{i,np}\frac{w^{i,np}\left( 1-t^{np}\right) }{h^{i,np}}\frac{\partial {\widetilde{h}}^{i,np}}{\partial w^{i,np}(1-t^{np})}}, \end{aligned}$$
or, equivalently, denoting by
\({\widetilde{\eta }} _{h^{i,np},w^{i,np}(1-t^{np})}\) the compensated elasticity of labor supply with respect to the net wage rate for a non-parent of skill type
i:
$$\begin{aligned} \frac{t^{np}}{1-t^{np}}=-\frac{cov\left( b^{np},Y^{np}\right) }{\underset{i=1 }{\overset{N}{\sum }}\overline{\pi }^{i,np}Y^{i,np}{\widetilde{\eta }} _{h^{i,np},w^{i,np}(1-t^{np})}}. \end{aligned}$$
The first-order conditions with respect to
\(t^{p}\) and
s are, respectively, given by:
$$\begin{aligned}&\frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,p}}\underset{i=1}{\overset{N}{\sum }}\alpha ^{i,p}\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{ \partial t^{p}}\nonumber \\&\quad +\,\mu \left[ \underset{i=1}{\overset{N}{\sum }}\overline{\pi } ^{i,p}w^{i,p}h^{i,p}+\underset{i=1}{\overset{N}{\sum }}\left( t^{p}w^{i,p}-sq\right) \overline{\pi }^{i,p}\frac{\partial h^{i,p}}{\partial t^{p}}\right] =0, \end{aligned}$$
(11)
$$\begin{aligned}&\frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,p}}\underset{i=1}{\overset{N}{\sum }}\alpha ^{i,p}\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{ \partial s}\nonumber \\&\quad +\,\mu \left[ -q\underset{i=1}{\overset{N}{\sum }}\overline{\pi } ^{i,p}h^{i,p}+\underset{i=1}{\overset{N}{\sum }}\left( t^{p}w^{i,p}-sq\right) \overline{\pi }^{i,p}\frac{\partial h^{i,p}}{\partial s}\right] =0. \end{aligned}$$
(12)
Using the Slutsky equation and denoting by a tilde symbol a compensated variable, we can rewrite Eqs. (
11), (
12) respectively as:
$$\begin{aligned}&\frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,p}}\underset{i=1}{\overset{N}{\sum }}\alpha ^{i,p}\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{ \partial t^{p}}\nonumber \\&+\,\mu \left[ \underset{i=1}{\overset{N}{\sum }}\overline{\pi } ^{i,p}w^{i,p}h^{i,p}+\underset{i=1}{\overset{N}{\sum }}\left( t^{p}w^{i,p}-sq\right) \overline{\pi }^{i,p}\left( \frac{\partial {\widetilde{h}}^{i,p}}{\partial t^{p}}-w^{i,p}h^{i,p}\frac{\partial h^{i,p}}{\partial G^{p}}\right) \right] =0, \qquad \end{aligned}$$
(13)
$$\begin{aligned}&\frac{1}{\underset{k=1}{\overset{N}{\sum }}\pi ^{k,p}}\underset{i=1}{\overset{N}{\sum }}\alpha ^{i,p}\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{ \partial s} \nonumber \\&\quad +\,\mu \left[ -q\underset{i=1}{\overset{N}{\sum }}\overline{\pi }^{i,p}h^{i,p}+ \underset{i=1}{\overset{N}{\sum }}\left( t^{p}w^{i,p}-sq\right) \overline{ \pi }^{i,p}\left( \frac{\partial {\widetilde{h}}^{i,p}}{\partial s}+qh^{i,p} \frac{\partial h^{i,p}}{\partial G^{p}}\right) \right] =0. \end{aligned}$$
(14)
Notice that, by applying Roy’s identity we can write
\(\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{\partial t^{p}}=-\frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{\partial G^{p}}w^{i,p}h^{i,p}\) and
\( \frac{\partial V^{i,p}\left( t^{p},G^{p},s\right) }{\partial s}=\frac{ \partial V^{i,p}\left( t^{p},G^{p},s\right) }{\partial G^{p}}qh^{i,p}\). Moreover, we have that
\(\frac{\partial {\widetilde{h}}^{i,p}}{\partial t^{p}} =-w^{i,p}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }\) and
\(\frac{\partial {\widetilde{h}}^{i,p}}{\partial s}=q\frac{\partial {\widetilde{h}}^{i,p}}{ \partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }\). Thus, using (
9), we can rewrite (
13), (
14) in matrix form as:
$$\begin{aligned}&\begin{bmatrix} \underset{i=1}{\overset{N}{\sum }}\overline{\pi }^{i,p}\left( w^{i,p}\right) ^{2}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})- \left( 1-s\right) q\right] }&-\underset{i=1}{\overset{N}{\sum }}\overline{ \pi }^{i,p}w^{i,p}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] } \\ \underset{i=1}{\overset{N}{\sum }}\overline{\pi }^{i,p}w^{i,p}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q \right] }&-\underset{i=1}{\overset{N}{\sum }}\overline{\pi }^{i,p}\frac{ \partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] } \end{bmatrix} \begin{bmatrix} t^{p} \\ qs \end{bmatrix}\nonumber \\&\quad = \begin{bmatrix} -\mathrm{cov}(b^{p},Y^{p}) \\ -\mathrm{cov}(b^{p},h^{p}) \end{bmatrix} \end{aligned}$$
(15)
Denoting by
\(\Delta \) the determinant of the 2x2 matrix on the left hand side of (
15), we have:
$$\begin{aligned} \Delta= & {} -\left( \underset{i=1}{\overset{N}{\sum }}\overline{\pi } ^{i,p}\left( w^{i,p}\right) ^{2}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }\right) \\&\times \left( \underset{ i=1}{\overset{N}{\sum }}\overline{\pi }^{i,p}\frac{\partial {\widetilde{h}} ^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }\right) \\&+\left( \underset{i=1}{\overset{N}{\sum }}\overline{\pi }^{i,p}w^{i,p} \frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }\right) \\&\times \left( \underset{i=1}{\overset{N}{\sum }} \overline{\pi }^{i,p}w^{i,p}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }\right) \\= & {} -\underset{i=1}{\overset{N-1}{\sum }}\left[ \overline{\pi }^{i,p}\frac{ \partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }\right. \\&\times \left. \left( \underset{j>i}{\overset{}{\sum }}\left( w^{i,p}-w^{j,p}\right) ^{2}\overline{\pi }^{j,p}\frac{\partial {\widetilde{h}} ^{j,p}}{\partial \left[ w^{j,p}(1-t^{p})-\left( 1-s\right) q\right] }\right) \right] \\< & {} 0. \end{aligned}$$
Thus, we have:
$$\begin{aligned} t^{p}= & {} \frac{\mathrm{cov}(b^{p},Y^{p})\underset{i=1}{\overset{N}{\sum }}\overline{ \pi }^{i,p}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }-\mathrm{cov}(b^{p},h^{p})\underset{i=1}{ \overset{N}{\sum }}\overline{\pi }^{i,p}w^{i,p}\frac{\partial {\widetilde{h}} ^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }}{ \Delta }, \\ s= & {} \frac{\mathrm{cov}(b^{p},Y^{p})\underset{i=1}{\overset{N}{\sum }}\overline{\pi } ^{i,p}w^{i,p}\frac{\partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }-\mathrm{cov}(b^{p},h^{p})\underset{i=1}{ \overset{N}{\sum }}\overline{\pi }^{i,p}\left( w^{i,p}\right) ^{2}\frac{ \partial {\widetilde{h}}^{i,p}}{\partial \left[ w^{i,p}(1-t^{p})-\left( 1-s\right) q\right] }}{q\Delta }. \end{aligned}$$