Restoring Ramsey tax lessons to Mirrleesian tax settings: Atkinson–Stiglitz and Ramsey reconciled

Abstract

This paper restores many of the Ramsey tax/pricing lessons perceived as outdated in the optimal tax literature following the Atkinson and Stiglitz (J Public Econ 6:55–75, 1976) framework wherein differential commodity taxes are considered to be redundant. The key to our findings is the incorporation of a “break-even constraint” for public firms into the Atkinson and Stiglitz framework. Break-even constraints are fundamental to the regulatory pricing literature but have somehow been overlooked in the optimal tax literature. Incorporating them reconciles the optimal-tax and the regulatory-pricing views on Ramsey tax/pricing rules.

Appendix A

1.1 First-order characterization of the (constrained) Pareto-efficient allocations

Rearranging the terms in (11), and dropping the constants $\bar{R}$ and F,  one may usefully rewrite the Lagrangian expression as

$$\begin{aligned} {\mathcal {L}}= & {} \sum _{j}\left( \eta ^{j}+\sum _{k\ne j}\lambda ^{jk}\right) v^{j}+\mu \sum _{j}\pi ^{j}\left[ (I^{j}-c^{j})+\sum _{i=1}^{n}(p_{i}-1)x_{i}^{j}\right] \nonumber \\&+\left( \mu +\delta \right) \sum _{j}\pi ^{j}\left[ \sum _{s=1}^{m} (q_{s}-1)y_{s}^{j}\right] -\sum _{j}\sum _{k\ne j}\lambda ^{jk}v^{jk}. \end{aligned}$$

(A1)

The first-order conditions of this problem are, for $j,k=1,2,\ldots ,H,$

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial I^{j}}= & {} \left( \eta ^{j}+\sum _{k\ne j}\lambda ^{jk}\right) v_{I}^{j}+\mu \pi ^{j}\left[ 1+\sum _{i=1}^{n} (p_{i}-1)\frac{\partial x_{i}^{j}}{\partial I^{j}}\right] \nonumber \\&+\left( \mu +\delta \right) \pi ^{j}\left[ \sum _{s=1}^{m}(q_{s} -1)\frac{\partial y_{s}^{j}}{\partial I^{j}}\right] -\sum _{k\ne j} \lambda ^{kj}v_{I}^{kj}=0, \end{aligned}$$

(A2)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial c^{j}}= & {} \left( \eta ^{j}+\sum _{k\ne j}\lambda ^{jk}\right) v_{c}^{j}+\mu \pi ^{j}\left[ -1+\sum _{i=1}^{n} (p_{i}-1)\frac{\partial x_{i}^{j}}{\partial c^{j}}\right] \nonumber \\&+\left( \mu +\delta \right) \pi ^{j}\left[ \sum _{s=1}^{m}(q_{s} -1)\frac{\partial y_{s}^{j}}{\partial c^{j}}\right] -\sum _{k\ne j} \lambda ^{kj}v_{c}^{kj}=0, \end{aligned}$$

(A3)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial p_{i}}= & {} \sum _{j}\left( \eta ^{j} +\sum _{k\ne j}\lambda ^{jk}\right) v_{i}^{j}+\mu \sum _{j}\pi ^{j}\left[ \sum _{e=1}^{n}(p_{e}-1)\frac{\partial x_{e}^{j}}{\partial p_{i}}+x_{i}^{j}\right] \nonumber \\&+\left( \mu +\delta \right) \sum _{j}\pi ^{j}\left[ \sum _{f=1}^{m} (q_{f}-1)\frac{\partial y_{f}^{j}}{\partial p_{i}}\right] -\sum _{j} \sum _{k\ne j}\lambda ^{jk}v_{i}^{jk}=0, \nonumber \\&\qquad i=2,3,\ldots ,n, \end{aligned}$$

(A4)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial q_{s}}= & {} \sum _{j}\left( \eta ^{j} +\sum _{k\ne j}\lambda ^{jk}\right) v_{s}^{j}+\mu \sum _{j}\pi ^{j}\left[ \sum _{e=1}^{n}(p_{e}-1)\frac{\partial x_{e}^{j}}{\partial q_{s}}\right] \nonumber \\&+\left( \mu +\delta \right) \sum _{j}\pi ^{j}\left[ \sum _{f=1}^{m} (q_{f}-1)\frac{\partial y_{f}^{j}}{\partial q_{s}}+y_{s}^{j}\right] -\sum _{j}\sum _{k\ne j}\lambda ^{jk}v_{s}^{jk}=0, \nonumber \\&\qquad s=1,2,\ldots ,m, \end{aligned}$$

(A5)

where a subscript on $v^{j}$ denotes a partial derivative. Equations (A2)–(A5) characterize the Pareto-efficient allocations constrained both by the public firms’ break-even constraint, the resource constraint, the self-selection constraints, as well as the linearity of commodity tax rates.

1.2 Derivation of (15) for optimal commodity taxes

Multiply Eq. (A3) by $x_{i}^{j}$, sum over j and add the resulting equation to (A4). Similarly, multiply (A3) by $y_{s}^{j}$, sum over j and add the resulting equation to (A5). Simplifying results in the following system of equations for $i=2,\ldots ,n$ and $s=1,2,\ldots ,m,$

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial p_{i}}+\sum _{j}x_{i}^{j}\frac{\partial {\mathcal {L}}}{\partial c^{j}}= & {} \sum _{j}\left( \eta ^{j}+\sum _{k\ne j}\lambda ^{jk}\right) \left( v_{i}^{j}+x_{i}^{j}v_{c}^{j}\right) \nonumber \\&+\,\mu \sum _{j}\pi ^{j}\left[ \sum _{e=1} ^{n}(p_{e}-1)\left( \frac{\partial x_{e}^{j}}{\partial p_{i}}+x_{i}^{j} \frac{\partial x_{e}^{j}}{\partial c^{j}}\right) \right] \nonumber \\&+\left( \mu +\delta \right) \sum _{j}\pi ^{j}\left[ \sum _{f=1}^{m} (q_{f}-1)\left( \frac{\partial y_{f}^{j}}{\partial p_{i}}+x_{i}^{j} \frac{\partial y_{f}^{j}}{\partial c^{j}}\right) \right] \nonumber \\&-\,\sum _{j} \sum _{k\ne j}\lambda ^{kj}\left( v_{i}^{kj}+x_{i}^{j}v_{c}^{kj}\right) =0, \end{aligned}$$

(A6)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial q_{s}}+\sum _{j}y_{s}^{j}\frac{\partial {\mathcal {L}}}{\partial c^{j}}= & {} \sum _{j}\left( \eta ^{j}+\sum _{k\ne j}\lambda ^{jk}\right) \left( v_{s}^{j}+y_{s}^{j}v_{c}^{j}\right) \nonumber \\&+\,\mu \sum _{j}\pi ^{j}\sum _{e=1}^{n} (p_{e}-1)\left( \frac{\partial x_{e}^{j}}{\partial q_{s}}+y_{s}^{j} \frac{\partial x_{e}^{j}}{\partial c^{j}}\right) \nonumber \\&+\left( \mu +\delta \right) \sum _{j}\pi ^{j}\left[ \sum _{f=1}^{m} (q_{f}-1)\left( \frac{\partial y_{f}^{j}}{\partial q_{s}}+y_{s}^{j} \frac{\partial y_{f}^{j}}{\partial c^{j}}\right) \right] \nonumber \\&-\,\sum _{j} \sum _{k\ne j}\lambda ^{kj}\left( v_{s}^{kj}+y_{s}^{j}v_{c}^{kj}\right) +\delta \sum _{j}\pi ^{j}y_{s}^{j}=0. \end{aligned}$$

(A7)

With $v_{i}^{j}+x_{i}^{j}v_{c}^{j}=0$ from Roy’s identity, the left-hand side of (A6) shows the impact on the Lagrangian expression ${\mathcal {L}}$ of a variation in $p_{i}$ when the disposable income of individuals is adjusted according to

$$\begin{aligned} dc_{j}=x_{i}^{j}dt_{i}, \end{aligned}$$

(A8)

to keep their utility levels constant. With $v_{s}^{j}+y_{s}^{j}v_{c}^{j}=0$, the left-hand side of (A7) shows the same compensated effect for a variation in $q_{s}$ where

$$\begin{aligned} dc_{j}=y_{s}^{j}d\tau _{s}. \end{aligned}$$

(A9)

These compensated derivatives, $(\partial {\mathcal {L}}/\partial p_{i})_{v^{j}=\overline{v}^{j}}$ and $(\partial {\mathcal {L}} /\partial q_{s})_{v^{j}=\overline{v}^{j}}$ vanish at the optimal solution.

Make use of Roy’s identity to set,

$$\begin{aligned} v_{i}^{j}+x_{i}^{j}v_{c}^{j}= & {} 0,\\ v_{i}^{kj}+x_{i}^{kj}v_{c}^{kj}= & {} 0,\\ v_{s}^{j}+y_{s}^{j}v_{c}^{j}= & {} 0,\\ v_{s}^{kj}+y_{s}^{kj}v_{c}^{kj}= & {} 0. \end{aligned}$$

Substitute these values in Eqs. (A6) and (A7), set $p_{i}-1=t_{i}$ and $q_{s}-1=\tau _{s}$, and divide by $\mu .$ Upon changing the order of summation and further simplification one arrives at, for all $i=2,\ldots ,n,$ and $s=1,2,\ldots ,m,$

$$\begin{aligned}&\sum _{e=1}^{n}t_{e}\left[ \sum _{j}\pi ^{j}\left( \frac{\partial x_{e}^{j} }{\partial p_{i}}+x_{i}^{j}\frac{\partial x_{e}^{j}}{\partial c^{j}}\right) \right] +\left( 1+\frac{\delta }{\mu }\right) \sum _{f=1}^{m}\tau _{f}\left[ \sum _{j}\pi ^{j}\left( \frac{\partial y_{f}^{j}}{\partial p_{i}}+x_{i} ^{j}\frac{\partial y_{f}^{j}}{\partial c^{j}}\right) \right] \nonumber \\&\quad -\frac{1}{\mu }\sum _{j}\sum _{k\ne j}\lambda ^{kj}\left( x_{i}^{j} -x_{i}^{kj}\right) v_{c}^{kj}=0, \end{aligned}$$

(A10)

$$\begin{aligned}&\sum _{e=1}^{n}t_{e}\left[ \sum _{j}\pi ^{j}\left( \frac{\partial x_{e}^{j} }{\partial q_{s}}+y_{s}^{j}\frac{\partial x_{e}^{j}}{\partial c^{j}}\right) \right] +\left( 1+\frac{\delta }{\mu }\right) \sum _{f=1}^{m}\tau _{f}\left[ \sum _{j}\pi ^{j}\left( \frac{\partial y_{f}^{j}}{\partial q_{s}}+y_{s}^{j}\frac{\partial y_{f}^{j}}{\partial c^{j}}\right) \right] \nonumber \\&\quad -\frac{1}{\mu }\sum _{j}\sum _{k\ne j}\lambda ^{kj}\left( y_{s}^{j} -y_{s}^{kj}\right) v_{c}^{kj}+\frac{\delta }{\mu }\sum _{j}\pi ^{j}y_{s}^{j}=0. \end{aligned}$$

(A11)

Next, using the Slutsky equations,

$$\begin{aligned} \frac{\partial x_{e}^{j}}{\partial p_{i}}= & {} \frac{\partial \tilde{x}_{e}^{j}}{\partial p_{i}}-x_{i}^{j}\frac{\partial x_{e}^{j}}{\partial c^{j}},\\ \frac{\partial y_{f}^{j}}{\partial p_{i}}= & {} \frac{\partial \tilde{y}_{f}^{j}}{\partial p_{i}}-x_{i}^{j}\frac{\partial y_{f}^{j}}{\partial c^{j}},\\ \frac{\partial x_{e}^{j}}{\partial q_{s}}= & {} \frac{\partial \tilde{x}_{e}^{j}}{\partial q_{s}}-y_{s}^{j}\frac{\partial x_{e}^{j}}{\partial c^{j}},\\ \frac{\partial y_{f}^{j}}{\partial q_{s}}= & {} \frac{\partial \tilde{y}_{f}^{j}}{\partial q_{s}}-y_{s}^{j}\frac{\partial y_{f}^{j}}{\partial c^{j}}, \end{aligned}$$

while making use of the symmetry of the Slutsky matrix, one can further simplify (A10) and (A11) to

$$\begin{aligned}&\sum _{e=1}^{n}t_{e}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{i}^{j} }{\partial p_{e}}\right) +\left( 1+\frac{\delta }{\mu }\right) \sum _{f=1}^{m}\tau _{f}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{i}^{j}}{\partial q_{f}}\right) \nonumber \\&\quad =\frac{1}{\mu }\sum _{j}\sum _{k\ne j}\lambda ^{kj}\left( x_{i}^{j}-x_{i}^{kj}\right) v_{c}^{kj}, \end{aligned}$$

(A12)

$$\begin{aligned}&\sum _{e=1}^{n}t_{e}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{s}^{j} }{\partial p_{e}}\right) +\left( 1+\frac{\delta }{\mu }\right) \sum _{f=1}^{m}\tau _{f}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{s}^{j}}{\partial q_{f}}\right) \nonumber \\&\quad =\frac{1}{\mu }\sum _{j}\sum _{k\ne j}\lambda ^{kj}\left( y_{s}^{j}-y_{s}^{kj}\right) v_{c}^{kj}-\frac{\delta }{\mu }\sum _{j}\pi ^{j} y_{s}^{j}, \end{aligned}$$

(A13)

which hold for all $i=2,\ldots ,n,$ and $s=1,2,\ldots ,m.$ Finally, collect the terms involving $\delta /\mu$ and use the definition of $\Delta$ in (14) to write out Eqs. (A12) and (A13) in matrix notation:

$$\begin{aligned} \Delta \left( \begin{array}[c]{c} t_{2}\\ \vdots \\ t_{n}\\ \left( 1+\frac{\delta }{\mu }\right) \tau _{1}\\ \vdots \\ \left( 1+\frac{\delta }{\mu }\right) \tau _{m} \end{array} \right)= & {} \frac{1}{\mu }\left( \begin{array}[c]{c} {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( x_{2}^{j}-x_{2}^{kj}\right) v_{c}^{kj}\\ \vdots \\ {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( x_{n}^{j}-x_{n}^{kj}\right) v_{c}^{kj}\\ {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( y_{1}^{j}-y_{1}^{kj}\right) v_{c}^{kj}\\ \vdots \\ {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( y_{m}^{j}-y_{m}^{kj}\right) v_{c}^{kj} \end{array} \right) -\frac{\delta }{\mu }\left( \begin{array}[c]{c} 0\\ \vdots \\ 0\\ {\sum }_{j}\pi ^{j}y_{1}^{j}\\ \vdots \\ {\sum }_{j}\pi ^{j}y_{m}^{j} \end{array} \right) .\nonumber \\ \end{aligned}$$

(A14)

Premultiplying through by $\Delta ^{-1}$ yields

$$\begin{aligned} \left( \begin{array}[c]{c} t_{2}\\ \vdots \\ t_{n}\\ \left( 1+\frac{\delta }{\mu }\right) \tau _{1}\\ \vdots \\ \left( 1+\frac{\delta }{\mu }\right) \tau _{m} \end{array} \right)= & {} \frac{1}{\mu }\Delta ^{-1}\left( \begin{array}[c]{c} {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( x_{2}^{j}-x_{2}^{kj}\right) v_{c}^{kj}\\ \vdots \\ {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( x_{n}^{j}-x_{n}^{kj}\right) v_{c}^{kj}\\ {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( y_{1}^{j}-y_{1}^{kj}\right) v_{c}^{kj}\\ \vdots \\ {\sum }_{j}{\sum }_{k\ne j}\lambda ^{kj}\left( y_{m}^{j}-y_{m}^{kj}\right) v_{c}^{kj} \end{array} \right) \nonumber \\&-\,\frac{\delta }{\mu }\Delta ^{-1}\left( \begin{array}[c]{c} 0\\ \vdots \\ 0\\ {\sum }_{j}\pi ^{j}y_{1}^{j}\\ \vdots \\ {\sum }_{j}\pi ^{j}y_{m}^{j} \end{array} \right) . \end{aligned}$$

(A15)

With weak-separability of preferences, Eqs. (12) and (13) hold so that $x_{i}^{jk}=x_{i}^{k}$ and $y_{s}^{jk}=y_{s}^{k}$. It then follows immediately that the first vector on the right-hand side of (A15) vanishes, reducing it to (15).

1.3 Derivation of (23)–(24)

With weakly-separable preferences, one can rearrange Eqs. (A12) and (A13) as

$$\begin{aligned} -\mu \sum _{e=1}^{n}t_{e}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{i}^{j} }{\partial p_{e}}\right) -\mu \sum _{f=1}^{m}\tau _{f}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{i}^{j}}{\partial q_{f}}\right)= & {} \delta \sum _{f=1}^{m}\tau _{f}\left( \sum _{j}\pi ^{j}\frac{\partial \widetilde{y}_{f}^{j}}{\partial p_{i}}\right) ,\\ -\mu \sum _{e=1}^{n}t_{e}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{s}^{j} }{\partial p_{e}}\right) -\mu \sum _{f=1}^{m}\tau _{f}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{s}^{j}}{\partial q_{f}}\right)= & {} \delta \sum _{f=1}^{m}\tau _{f}\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{f}^{j} }{\partial q_{s}}\right) \\&+\,\delta \sum _{j}\pi ^{j}y_{s}^{j}. \end{aligned}$$

Then divide the first set of equations by $\mu \sum _{j}\pi ^{j}x_{i}^{j}$ and the second set of equations by $\mu \sum _{j}\pi ^{j}y_{s}^{j}$. They will then be rewritten as (23) and (24).