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This paper restores many of the Ramsey tax/pricing lessons perceived as outdated in the optimal tax literature following the Atkinson and Stiglitz (J Public Econ 6:55–75, 1976) framework wherein differential commodity taxes are considered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.

1 Introduction

Frank Ramsey’s 1927 classic paper, Stiglitz (2015) writes, “can be thought of as launching the field of optimal taxation and revolutionizing public finance” (p. 1). And, while calling it “brilliant,” he goes on to conclude the Introduction of his own paper by this observation “... later analyses showed crucial qualifications, so that the policy relevance of Ramsey’s analysis may be limited”. This paper calls for a re-examination of these “qualifications”. They, we shall argue, stem from a Mirrleesian approach to optimal taxation that ignores public firms’ break-even constraints. Yet, in practice, regulation is almost always associated with a budget balancing requirement. A fact that has not escaped the attention of regulatory pricing literature forming a cornerstone of regulatory economics.¹

Prior to Mirrlees (1971), the Ramsey tax framework served as a cornerstone of the optimal tax theory. The central question in this literature was that of designing (linear) commodity taxes to collect a given tax revenue. Labor income went generally untaxed or subjected to a linear tax. The main point this literature made was that, except in very special cases, commodity taxes should not be uniform and that they should be set to balance efficiency and equity considerations. The efficiency aspects manifest themselves most clearly in the inverse-elasticity rules (derived when demand functions, Hicksian or Marshallian, are separable). These efficiency-driven tax rules entail a regressive bias in that goods with low price-elasticities are often necessities consumed proportionately more by poorer households. This bias is then mitigated by the equity terms in the tax rules (often appearing as covariance terms). These tend to increase the tax rate on goods that are consumed proportionately more by richer households.

The Mirrleesian approach posed a serious challenge to the Ramsey tax framework and the lessons drawn from it. Mirrlees (1971) argued that the existence or absence of tax instruments must be rationalized on the basis of the informational constraints in the economy. This approach turns the Ramsey tax framework on its head by making nonlinear income taxation the most powerful tax instrument at the disposal of the government. In turn, the reliance on the nonlinear income tax has a devastating implication for the usefulness of commodity taxes. In their classic contribution, Atkinson and Stiglitz (1976) show that under some conditions—identical and weakly separable preferences in labor supply and goods—an optimal nonlinear income tax is sufficient to implement any incentive compatible Pareto-efficient allocation. In other words, commodity taxes are not necessary for revenue raising or for redistribution.² The Ramsey results come about, it is thus argued, merely as an artifact of restricting the income tax to be linear (an inconsistent assumption given the assumed informational structure and also at odds with reality). The Atkinson and Stiglitz (AS) result has had a tremendous effect in shaping the views of public economists concerning the design of optimal tax systems: In particular that prices should not be used for redistribution (even in a second best setting), and that in-kind transfers are not useful.³

This strong view notwithstanding, the Ramsey-type rules have had a more or less independent second—or some might argue first—life as a model of regulatory pricing.⁴ In his pioneering paper that appeared prior to Mirrlees (1971), Boiteux (1956) studied linear pricing of a regulated multi-product monopoly that has to cover some “fixed cost” (for instance the infrastructure cost of the network). This is to be achieved through markups on the monopoly’s different products (equivalent to taxes).⁵ Formally, this problem is equivalent to a Ramsey tax model with the fixed cost playing the role of the government’s tax revenue requirement.

To sum, optimal tax and regulatory pricing literatures appear to have diverged in the way they view the practical relevance of the AS result (see Sect. 2.1 below). Whereas Ramsey pricing is considered totally passé in the optimal tax literature, Ramsey-type lessons permeate the field of regulatory economics. This is quite surprising considering the fact that the issues the two literatures address have an identical formal structure. In both cases, there is a public authority whose objective is to raise a fixed amount of revenue (where the revenue finances the government’s expenditures in the optimal tax literature and the firms’ fixed costs in the regulation literature).⁶ This paper is an attempt to reconcile these divergent views in order to bring them together. To this end, we incorporate the regulatory economics focus on the budget balance into a Mirrleesian optimal tax framework. As far as we know, a comprehensive analysis of this sort has not been attempted before.

The optimal tax setting we consider combines nonlinear income taxation with linear taxation/pricing of consumption goods. The informational structure that underlies this setting is now standard in the Mirrleesian optimal taxation literature. First, individuals’ earning abilities and labor supplies are not publicly observable, but their pre-tax incomes are. This rules out type-specific lump-sum taxes but allows for nonlinear income taxation. Second, individual consumption levels of goods, whether subject to regulation or not, are not observable so that nonlinear taxation and/or pricing of goods are not possible. On the other hand, anonymous transactions are observable making linear commodity taxation feasible.

To account for the regulatory economics balanced-budget concerns, we assume that a subset of the goods are produced by a public or regulated firm that has to cover a fixed cost through markups on the different commodities it sells.⁷ This latter constraint gives rise to a “break-even” constraint on the part of the firm. This comes on top of the overall government’s budget constraint. In other words, while our setting is that of AS, we depart from the existing optimal tax literature by formally incorporating a binding break-even constraint in our model. Break-even constraints have somehow been overlooked in the optimal tax literature; yet they are fundamental to the regulatory pricing literature. In Sect. 2.1 we draw on this literature to show that there exist good justifications for imposing such a constraint.

While we optimize over all tax instruments including income, we are not concerned with the properties of the income tax schedule. Our aim is to solely study the commodity taxation and pricing rules (for goods produced by public/regulated firms as well as those produced subject to no regulation). We derive these rules for general preferences but concentrate on the case of weakly-separable preferences between labor supply and goods that underlies the AS result. We shall refer to this environment as the AS setting/framework/model and contrast it with the Ramsey environment wherein all tax instruments are linear. The fundamental contribution of our paper is the finding that the AS setting with a break-even constraint restores many of the traditional Ramsey tax/pricing features which have been questioned by modern optimal tax theory.

Specifically, we demonstrate that, given a break-even constraint, not only is it desirable to tax the goods produced by the public/regulated firm but also other goods. Intuitively, taxation of privately-produced goods are generally needed to offset the distortions created by the public/regulated firm’s departure from marginal cost pricing. This result stands in sharp contrast to AS result on the redundancy of commodity taxes. We then illustrate and elaborate on our findings by studying a simple framework with one publicly-provided and two privately-provided goods.

The two special cases of independent Hicksian and independent Marshallian demand curves provide further insights into the nature of the tax/pricing rules in our model. In the separable Hicksian demand case, we find that private goods, which are not included in the break-even constraint, should go untaxed. On the other hand, public firms should follow pricing rules that are purely efficiency-driven and Ramsey type: Goods are taxed inversely to their compensated demand elasticity regardless of their distributional implications. Redistribution is taken care of by the income tax (allowing public firms’ prices to be adjusted for revenue raising as in the Ramsey model with identical individuals). This is to be contrasted with today’s prevailing view—based on the AS framework and ignoring break-even constraints—that commodity taxes are redundant. It also differs from earlier Ramsey pricing views that commodity taxes should follow inverse elasticity rules adjusted for redistributive concerns.

Results become less predictable in the case where Marshallian demands are independent. Here, allowing for a break-even constraint in the AS framework, resurrects a role for commodity taxes that go beyond the goods produced subject to the break-even constraint. Instead, it spills over to the taxation of other goods as well. One continues to get inverse elasticity rules as in the Ramsey model; however, their structure differs from the traditional expressions in the Ramsey model. On the one hand, they are more complicated than the pure efficiency rules. On the other hand, there is no covariance or similar term that captures redistributive considerations. Instead, they contain “tax revenue terms” that measure the social value of the extra tax revenues generated from demand variations that follow the compensating adjustments in disposable income. These terms lead to predictions that are similar to those coming from the many-household Ramsey model albeit without redistributive concerns; namely, that goods with higher demand elasticities should be taxed more heavily.

Finally, we study what is arguably the most celebrated general result obtained in the Ramsey model; namely, the (un)equal proportional reduction in compensated demands property. We show that, in contrast to the single-household Ramsey model, the reductions differ across goods. This in and of itself is not particularly surprising given the presence of heterogeneous households. More interestingly, compared to the many-household Ramsey model with the break-even constraint, we find that the redistributive considerations are once again replaced by tax revenue terms.

2 The model

There are H types of individuals, indexed $j=1,2\ldots ,H$, who differ in their wages, $w^{j}$, but have identical preferences over goods and leisure.⁸ Let $\pi ^{j}$ denote the percentage of j-type individuals in total population and normalize the population size at one so that $\sum \pi ^{j}=1$. All goods are produced at a constant marginal cost which we normalize to one. Some, ${\underline{x}}=(x_{1},x_{2},\ldots ,x_{n})$, are produced by the private sector; the rest, ${\underline{y}}=(y_{1},y_{2},\ldots ,y_{m})$, are produced by a public or regulated firm which incurs a fixed cost. The firm is constrained to break even by marking up its marginal costs.⁹ Let ${\underline{p}} =(p_{1},p_{2},\ldots ,p_{n})$ denote the consumer price of ${\underline{x}}$ and ${\underline{q}}=(q_{1},q_{2},\ldots ,q_{n})$ the consumer price of ${\underline{y}}$. Finally, denote the commodity tax rates on ${\underline{x}}$ by $\underline{t}=$ $(t_{1},t_{2},\ldots ,t_{n})$ and the public firms’ commodity-tax-cum-mark-ups by $\underline{\tau }=(\tau _{1},\tau _{2},\ldots ,\tau _{m}).$ We have $p_{i}=1+t_{i}$ ($i=1,2,\ldots ,n$) and $q_{s}=1+\tau _{s}$ ($s=1,2,\ldots ,m$).

Individual consumption levels are not publicly observable but anonymous transactions can be observed. Consequently, commodity taxes must be proportional and public sector prices linear. For the remaining variables, the information structure is the one typically considered in mixed taxation models; see e.g., Christiansen (1984) and Cremer and Gahvari (1997). In particular, an individual’s type, $w^{j}$, and labor input, $L^{j}$, are not publicly observable; his before-tax income, $I^{j}=w^{j}L^{j}$, on the other hand, is. Hence type-specific lump-sum taxation is ruled out but non-linear taxation of incomes is feasible.

To characterize the (constrained) Pareto-efficient allocations we derive an optimal revelation mechanism. For our purpose, a mechanism consists of a set of type-specific before-tax incomes, $I^{j}$s, aggregate expenditures on private sector and public sector goods, $c^{j}$s, and two vectors of consumer prices (same for everyone) ${\underline{p}}$ and ${\underline{q}}$ (for ${\underline{x}}$ and ${\underline{y}}$). To proceed further, it is necessary to consider the optimization problem of an individual for a given mechanism $({\underline{p}},{\underline{q}},c,I)$. Formally, given any vector $({\underline{p}},{\underline{q}},c,I)$, an individual of type j maximizes utility $u=u({\underline{x}},{\underline{y}},I/w^{j})$ subject to the budget constraint $\sum _{i=1}^{n}p_{i}x_{i}+\sum _{s=1}^{m}q_{s}y_{s}=c$. The resulting conditional demand functions for ${\underline{x}}$ and ${\underline{y}}$ are denoted by $x_{i}=x_{i}({\underline{p}},\underline{q},c,I/w^{j})$ and $y_{s}=y_{s}({\underline{p}},\underline{q},c,I/w^{j})$.¹⁰ Substituting in the utility function yields the conditional indirect utility function

$$\begin{aligned} v({\underline{p}},{\underline{q}},c,I/w^{j})\equiv u[ {\underline{x}}({\underline{p}},{\underline{q}},c,I/w^{j}),{\underline{y}}( {\underline{p}},{\underline{q}},c,I/w^{j}),I/w^{j}]. \end{aligned}$$

Thus, a j-type individual who is assigned $c^{j},I^{j}$ will have demand functions and an indirect utility function given by

$$\begin{aligned} x_{i}^{j}= & {} x_{i}({\underline{p}},{\underline{q}},c^{j},I^{j}/w^{j} ), \end{aligned}$$

(1)

$$\begin{aligned} y_{s}^{j}= & {} y_{s}( {\underline{p}},{\underline{q}},c^{j},I^{j} /w^{j}), \end{aligned}$$

(2)

$$\begin{aligned} v^{j}= & {} v( {\underline{p}},{\underline{q}},c^{j},I^{j}/w^{j}). \end{aligned}$$

(3)

Similarly, the demand functions and the indirect utility function for a j-type who claims to be of type k,  the so-called mimicker, is given by

$$\begin{aligned} x_{i}^{jk}= & {} x_{i}({\underline{p}},{\underline{q}},c^{k},I^{k}/w^{j} ), \end{aligned}$$

(4)

$$\begin{aligned} y_{s}^{jk}= & {} y_{s}( {\underline{p}},{\underline{q}},c^{k},I^{k} /w^{j}) , \end{aligned}$$

(5)

$$\begin{aligned} v^{jk}= & {} v( {\underline{p}},{\underline{q}},c^{k},I^{k}/w^{j}). \end{aligned}$$

(6)

2.1 The break-even constraint

The information structure posited above describes only the informational asymmetries between the tax administration and the taxpayers typically assumed in the optimal tax literature. This does not rationalize a break-even constraint which is the missing link between the optimal tax and regulatory economics literatures. In settings where tax policy is restricted only by informational considerations of this type, break-even constraints could be undone by simple lump-sum transfers from the government to the operators. Yet, in practice, regulation is almost always associated with budget balancing requirements, which explains why they form a cornerstone of regulatory economics.

2.1.1 Break-even constraints in regulatory economics

Ramsey–Boiteux (RB) pricing continues to play an important role in the sectors still subject to some form of regulation even though, over the last few decades, the scope of regulation has declined. A prominent example is the postal sector in the US where Ramsey–Boiteux pricing remains an important benchmark in regulatory hearings; see Crew and Kleindorfer (2011, 2012). As a matter of fact, not only has RB pricing kept its position but it has even found new applications in settings of regulatory reform and market liberalization. For instance, while the original Boiteux model concerns a monopoly, Ware and Winter (1986) show that generalized RB rules prevail in imperfectly competitive markets. Furthermore, Laffont and Tirole (1990) argue that one should price the network access an incumbent operator has to provide to its upstream competitors on the basis of RB logic.

Another interesting result, shown by Vogelsang and Finsinger (1979), is that Ramsey prices can be decentralized through an iterative procedure based on a global price cap; see also Laffont and Tirole (2000, page 67). More generally, in the literature on incentive regulation, Ramsey prices are viewed as a kind of “ideal” solution: they represent a so-called “full information” benchmark. One should bear in mind, however, that the informational asymmetries the incentive literature focuses on are between the regulator (public authority) and the firms. This is in contrast with the informational asymmetries between the tax administration (public authority) and the workers (taxpayers) that are at the heart of the optimal tax literature. Finally, the idea that prices ought to be used for redistributive purposes is the rationale for a great deal of regulatory policies including social tariffs and more generally universal service requirements; see Cremer et al. (2001a) for a discussion of the theoretical foundations of these policies and their practical implementation.

Now regulatory economists typically take the break-even requirement as an exogenously given constraint; it is neither questioned nor justified in the relevant literature. For instance, Brown and Sibley (1986, p. 35) refer to the United States “tradition of regulation: utilities are expected to cover their costs”. Viscusi et al. (1998), who present an otherwise very comprehensive overview of regulatory and anti-trust policy, devote just a few lines to the break-even issue and take a somewhat cynical view by stating (p. 371): “Regulators do not see as their primary objective achieving economic efficiency. Rather, they appear to seek a set of prices that are not unduly discriminatory but that permit total revenue to cover total cost.”

2.1.2 Rationalizing break-even constraints

Going beyond this near axiomatic view of break-even constraints, there are both informational and non-informational grounds that can rationalize them. Political economy considerations provide an obvious example for the latter. Public authorities are typically reluctant to finance a structural deficit in a given sector through subsidies from the general budget. Voters consider it to be more “fair” to have the natural gas users pay for the transportation costs associated with the network of pipelines and pumping stations (as compared to the taxpayers). Legal issues provide another reason. In the EU, for instance, “State Aid” is illegal: Member states are not allowed to finance their operators’ deficits through subsidies—a policy that is being enforced increasingly more vehemently.¹¹

The information-based rationalizations are not explicitly addressed in our setting. These are summarized by Laffont and Tirole (1993, pp. 23–30). They give two basic arguments. One, which they ascribe to Coase (1945, 1946), applies to “a firm or a product whose existence is not a forgone conclusion”. Unless an activity at least breaks even, one cannot be sure that its production is beneficial for the society to warrant the government covering its fixed costs. Roughly speaking, Coase argues that, absent a budget constraint, the government is entrusted to make this decision without having the appropriate information. With a break-even constraint, this decision is effectively made by consumers who thereby reveal if their willingness to pay for the product is sufficiently high. The second argument is related to incentives associated with informational asymmetries between the regulator and the firm (which we do not formally model).¹² For instance, Allais (1947) has argued that the absence of a break-even constraint would create “inappropriate incentives for cost reduction”.

More recently, Joskow (2007) has extensively discussed the break-even requirement. He too concentrates on its implication rather than examining its rationale. Nevertheless he effectively presents another argument for its existence which, quite paradoxically, is linked to the recent decline in public ownership observed in most countries. Many firms that previously were publicly administered have now turned into totally- or partly-privatized “ordinary” cooperatives. Still, a large part of their activities remains regulated. Joskow argues strongly that these entities can attract private capital only if the regulatory mechanism yields prices that are compatible with break-even constraints. Private investors, Joskow (2007) maintains, would never consider an entity that has to rely on recurrent financing of its deficits through subsidies from the general budget, as a credible outlet for investment, particularly when its fixed costs are sunk.¹³

2.2 Constrained Pareto-efficient allocations

Denote the government’s external revenue requirement by $\bar{R}$ and the fixed costs of public firms by F. Constrained Pareto-efficient allocations are described, indirectly, as follows.¹⁴ Maximize

$$\begin{aligned} \sum _{j=1}^{H}\eta ^{j}v({\underline{p}},{\underline{q}},c^{j},I^{j} /w^{j}) \end{aligned}$$

(7)

with respect to ${\underline{p}},{\underline{q}},c^{j}$ and $I^{j}$ where $\eta ^{j}$s are positive constants with the normalization $\sum _{j=1}^{H} \eta ^{j}=1$.¹⁵ The maximization is subject to the resource constraint

$$\begin{aligned} \sum _{j=1}^{H}\pi ^{j}\left[ (I^{j}-c^{j})+\sum _{i=1}^{n}(p_{i}-1) x_{i}^{j}+\sum _{s=1}^{m}(q_{s}-1)y_{s}^{j}\right] \ge \bar{R}+F, \end{aligned}$$

(8)

the break-even constraint

$$\begin{aligned} \sum _{j=1}^{H}\pi ^{j}\left[ \sum _{s=1}^{m}(q_{s}-1)y_{s}^{j}\right] \ge F, \end{aligned}$$

(9)

and the self-selection constraints

$$\begin{aligned} v^{j}\ge v^{jk}, \quad j,k=1,2,\ldots ,H. \end{aligned}$$

(10)

Denote the Lagrangian expression by ${\mathcal {L}}$, and the Lagrangian multipliers associated with the resource constraint (8) by $\mu ,$ the public firms’ break-even constraint (9) by $\delta$, and with the self-selection constraints (10) by $\lambda ^{jk}$. We have

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

(11)

The first-order conditions of this problem with respect to $I^{j},c^{j}$, for $j,k=1,2,\ldots ,H$, and $p_{i},q_{s}$, for $i=2,3,\ldots ,n$, and $s=1,2,\ldots ,m$, are derived in Appendix A.¹⁶ They 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 (see the paragraph below as to why the optimization does not extend to $i=1$).

The reason that we do not optimize over $p_{1}$ is well-known in the optimal tax literature. With ${\underline{x}}$ and ${\underline{y}}$ being homogeneous of degree zero in ${\underline{p}},{\underline{q}},$ and c,  consumer prices can be determined only up to a proportionality factor. Consequently, one of the consumer prices must be normalized. We choose $p_{1}$ and set its value at $p_{1}=1$.¹⁷ Having stated this, a caveat is in order. In the absence of the break-even constraint, the normalization of one of the consumer prices is without any loss of generality. In our setting, the fact that consumer prices can be determined only up to a proportionality factor implies, at first glance, that the break-even constraint may be rendered inconsequential. This is the idea that the government can always raise all prices proportionately to cover the break-even constraint of the public firm, However, imposing a binding break-even constraint to ensure that public firm’s revenues cover its fixed costs also rules out the possibility of an across the board uniform increase in all consumer prices (including those of the public firm). Indeed, such an across-the-board increase in all prices is in contradiction with the very idea of imposing a break-even constraint in the first place. This “policy” will never work when there are multiple firms with different cost structures.¹⁸

3 Atkinson and Stiglitz theorem and optimal commodity taxes

In the standard mixed taxation model without the break-even constraint, assuming preferences are weakly separable in goods and labor supply, the Atkinson and Stiglitz (1976) theorem on the redundancy of commodity taxes holds. The particular feature of separability that drives the AS result is the property that a j-type who pretends to be of type k will have the same demand as type k. That is,

$$\begin{aligned} x_{i}^{jk}= & {} x_{i}^{k}=x_{i}({\underline{p}},{\underline{q}},c^{k}), \end{aligned}$$

(12)

$$\begin{aligned} y_{s}^{jk}= & {} y_{s}^{k}=y_{s}({\underline{p}},{\underline{q}},c^{k}). \end{aligned}$$

(13)

This arises because with weak-separability, preferences take the following form $u=u(f({\underline{x}},{\underline{y}}),I/w^{j})$. Under this circumstance, the (conditional) demand functions for ${\underline{x}}$ and ${\underline{y}}$ specified in Eqs. (1)–(2) and (4)–(5) will be independent of $I/w^{j}$ so that $x_{i}=x_{i}({\underline{p}},{\underline{q}},c)$ and $y_{s}=y_{s}({\underline{p}},{\underline{q}},c)$. Moreover, the indirect utility function too will be weakly separable in $({\underline{p}},{\underline{q}},c)$ and $I/w^{j}$ and written as $v(\phi ({\underline{p}},{\underline{q}},c),I/w^{j})$.

The above property has far reaching implications for optimal commodity taxes in our setting too; both those produced by the public firm as well as privately. To derive these, introduce the compensated version of demand functions (1)–(2). Specifically, denote the compensated demand for a good by a “tilde” over the corresponding variable. Let $\Delta$ denote the $(n+m-1)\times (n+m-1)$ matrix derived from the Slutsky matrix, aggregated over all individuals, by deleting its first row and column,

(14)

We prove in Appendix A that in this case optimal commodity taxes satisfy the following equations,¹⁹

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

(15)

Equation (15) demonstrates quite clearly that, notwithstanding the Atkinson and Stiglitz (1976) theorem, $\underline{t}=\underline{0}$ and $\underline{\tau }=\underline{0}$ are not a solution to (15). Commodity taxes and departures from marginal cost pricing are necessary components of Pareto-efficient tax structures.

Equally crucial is to realize that the underlying reason for this result is the existence of the break-even constraint. To see this, observe that if $F=0$, i.e. if there is no fixed cost, the break-even constraint becomes irrelevant and thus non-binding in our optimization problem. Under this circumstance, $\delta =0$ and the right-hand side of (15) is reduced to a vector of $n+m-1$ zeros. One then obtains $t_{i}=0$ and $\tau _{s}=0$ (marginal cost pricing) for all $i=1,2,\ldots ,n$ and $s=1,2,\ldots ,m$, and returns to the Atkinson and Stiglitz result that commodity taxes are redundant. With $F>0$, the break-even constraint is necessarily violated under marginal cost pricing so that $\delta >0$. In this case, the first n elements of the vector in the right-hand side of (15) continue to be zero, but the other m elements differ from zero. It then follows that the solution no longer implies all ts and all $\tau$s are zero. Nor will it be the case that the ts are necessarily zero. This point, that the existence of a break-even point requires not only the taxation of goods produced by the public firm but also the taxation of privately-provided goods, constitutes the major lesson of our study.

The second important lesson is that the tax/pricing rules for both types of goods, unlike their traditional Ramsey counterparts, are not affected by redistribution concerns.²⁰ It is important to note that this statement concerns tax rules as opposed to tax levels which will obviously be affected.

To gain further insight into the nature of commodity taxes in (15), we next resort to a simple special case with two private-sector and one public-sector good.

3.1 Two privately-produced goods, one public

Under this simple structure, and with $t_{1}=0,$ $t_{2}$ and $\tau _{1}$ are found from Eq. (15) to equal to

$$\begin{aligned} \left( \begin{array} [c]{c} t_{2}\\ \left( 1+\frac{\delta }{\mu }\right) \tau _{1} \end{array} \right)= & {} \frac{-\delta }{\mu \left[ \sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{2}^{j}}{\partial p_{2}}\sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{1}^{j} }{\partial q_{1}}-\left( \sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{2}^{j} }{\partial q_{1}}\right) ^{2}\right] } \\&\times \left( \begin{array} [c]{c} -{\sum }_{j}\pi ^{j}\frac{\partial \tilde{x}_{2}^{j}}{\partial q_{1}}{\sum }_{j} \pi ^{j}y_{1}^{j}\\ {\sum }_{j}\pi ^{j}\frac{\partial \tilde{x}_{2}^{j}}{\partial p_{2}}{\sum }_{j}\pi ^{j}y_{1}^{j} \end{array} \right) . \end{aligned}$$

It immediately follows from the above that

$$\begin{aligned} t_{2}=\frac{\sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{2}^{j}}{\partial q_{1}} }{-\sum _{j}\pi ^{j}\frac{\partial \tilde{x}_{2}^{j}}{\partial p_{2}}}\left( 1+\frac{\delta }{\mu }\right) \tau _{1}. \end{aligned}$$

With $\tau _{1}>0$ to cover the fixed costs, and $\partial \tilde{x}_{2}^{j}/\partial p_{2}<0,$ $t_{2}$ has the same sign as $\sum _{j}\pi ^{j}(\partial \tilde{x}_{2}^{j}/\partial q_{1})=\sum _{j}\pi ^{j}(\partial \tilde{y}_{1}^{j}/\partial p_{2}).$ Thus if $x_{2}$ and $y_{1}$ are Hicksian substitutes with $\partial \tilde{y}_{1}^{j}/\partial p_{2}>0$, one sets $t_{2}>0.$ This increases $p_{2}$, and with it, $\tilde{y}_{1}^{j}$ and thus mitigates the distortion created by $\tau _{1}>0$. On the other hand, if $\partial \tilde{y}_{1}^{j}/\partial p_{2}<0,$ one sets $t_{2}<0.$ This lowers $p_{2}$ and, as a result, increases $\tilde{y}_{1}^{j}.$ Either way then, one sets $t_{2}$ to increase $\tilde{y}_{1}.$ The break-even condition increases the price of $y_{1}$ above its marginal cost so that its consumption is less than optimal. One attempts to reverse this through $t_{2}$. As a general lesson, taxation of privately-produced goods are necessitated to offset the distortions created by having to depart from marginal-cost pricing on the part of the public firm. ²¹

To gain a better intuition into the nature of the tax/pricing rules in our model, we next consider the two well-known special cases for which the Ramsey setting yields simple results. These are the independent Hicksian and Marshallian demand curves cases whose solutions in the Ramsey model indicate the famous inverse elasticity rules. Following these cases, we examine the most celebrated general result of the Ramsey model; namely the (un)equal proportional reduction in compensated demands property. But first we summarize our main results thus far as,

Proposition 1

Consider an Atkinson and Stiglitz setting wherein some goods,  including the untaxed numeraire, are produced by the private sector and some by a public or regulated firm subject to a break-even constraint. Then,  contrary to the Atkinson and Stiglitz result, 

• (i) Commodity taxes are desirable with optimal commodity taxes being characterized by Eq. (15).

• (ii) Break-even constraints for public/regulated firms have spill overs to other goods :  They should be taxed if they are Hicksian substitutes and subsidized if they are Hicksian complements to the good produced by the public firm. These positive or negative taxes are necessitated to offset the distortions created by the public firm having to depart from marginal-cost pricing.

• (iii) The tax/pricing rules for both types of goods,  those that are produced privately and those that are provided through the public firm,  are not affected by redistribution concerns.

4 Constrained Pareto efficient pricing rules

4.1 Zero cross-price compensated elasticities

Assume that Hicksian demands are independent so that the compensated demand of any produced good does not depend on the prices of the other produced goods. In this case, the reduced Slutsky matrix, where the row and column pertaining to leisure are deleted, is diagonal. Hence Eq. (15) simplifies to

$$\begin{aligned} \left( \begin{array}[c]{c} t_{2}\\ \vdots \\ t_{n}\\ \left( 1+\frac{\delta }{\mu }\right) \tau _{1}\\ \vdots \\ \left( 1+\frac{\delta }{\mu }\right) \tau _{m} \end{array} \right) =\frac{\delta }{\mu }\left( \begin{array}[c]{c} 0\\ \vdots \\ 0\\ \frac{\sum _{j}\pi ^{j}y_{1}^{j}}{-\sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{1}^{j}}{\partial q_{1}}}\\ \vdots \\ \frac{\sum _{j}\pi ^{j}y_{m}^{j}}{-\sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{m}^{j}}{\partial q_{m}}} \end{array} \right) . \end{aligned}$$

Consequently, for all $i=2,\ldots ,n$ and $s=1,2,\ldots ,m,$

$$\begin{aligned} t_{i}= & {} 0,\\ \tau _{s}= & {} \frac{\delta }{\mu +\delta }\frac{\sum _{j}\pi ^{j}y_{s}^{j}}{-\sum _{j}\pi ^{j}\frac{\partial \tilde{y}_{s}^{j}}{\partial q_{s}}} =\frac{\delta }{\mu +\delta }\frac{q_{s}\sum _{j}\pi ^{j}y_{s}^{j}}{\sum _{j}\pi ^{j}\tilde{y}_{s}^{j}\widetilde{\varepsilon }_{ss}^{j}}, \end{aligned}$$

where $\widetilde{\varepsilon }_{ss}^{j}$ is the absolute value of the j-type’s own-price elasticity of compensated demand for $y_{s}$. That is, for all $i=2,\ldots ,n$ and $s=1,2,\ldots ,m,$ $t_{i}=0$ and

$$\begin{aligned} \frac{\tau _{s}}{1+\tau _{s}}=\frac{\delta \sum _{j}\pi ^{j}y_{s}^{j}}{(\mu +\delta ) \sum _{j}\pi ^{j}y_{s}^{j}\widetilde{\varepsilon }_{ss}^{j}}, \end{aligned}$$

(16)

which is an inverse elasticity rule. Again, the inverse elasticity rule arises only because of the existence of the break-even constraint. In the absence of this constraint, $\delta =0$ so that $\tau _{s}$, for all $s=1,2,\ldots ,m$, will also be equal to zero.

The next question concerns the difference between our results of $\underline{t}= \underline{0}$ and $\underline{\tau }$ characterized by Eq. (16) with the inverse elasticity rules derived in the Ramsey framework as presented for instance by Diamond (1975). Comparing the two sets of results reveals two differences. One is that whereas in the traditional Ramsey model, all goods are subject to the inverse elasticity rules, in the Atkinson–Stiglitz framework only the goods that are produced by public firms are subject to the inverse elasticity rule. The goods produced by private firms should not be taxed. The second difference is that in the traditional Ramsey framework, the inverse elasticity rules are adjusted for redistributive concerns (through the covariance terms). No such terms appear in the Atkinson–Stiglitz framework. Specifically, redistributive concerns are taken care of by nonlinear income taxes. The only role for commodity taxes is “efficiency,” or more precisely, revenue raising considerations.

To sum, we find that in this special case, the private goods (not included in the break-even constraint) continue to go untaxed as in the Atkinson–Stiglitz setting with no break-even constraint. On the other hand, the pricing rules used by the public firm are purely efficiency-driven Ramsey rules. Goods are taxed inversely to their compensated demand elasticity regardless of their distributional implications. Redistribution is taken care of by the income tax allowing the public firm’s prices to be adjusted for revenue raising (as in the Ramsey model with identical individuals).

It will become clear below that the apparent simplicity of this rule is to some extent misleading. It obscures some effects which are present but happen to cancel out in this special case. We shall return to this issue in the next subsections.

4.2 Zero cross-price elasticities

We now turn to the case where Marshallian demand functions are independent so that the demand for any produced good does not depend on the prices of other produced goods.²² To simplify the pricing rules that obtain in this case, it is simpler to start from the intermediate expressions (A10)–(A11) given in Appendix A rather than from (15). Rearranging these expressions, making use of the weak-separability assumption, and setting all the cross-price derivatives equal to zero, we obtain for all $i=2,\ldots ,n$ and $s=1,2,\ldots ,m,$

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

(17)

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

(18)

Before simplifying these expressions any further, it is informative to delve into their interpretation. Recall that we are considering a compensated variation in the tax rates such that $dc_{j}=x_{i}^{j}dt_{i}$ for a variation in $t_{i}$ and $dc_{j}=y_{s}^{j}dq_{s}$ for a variation in $\tau _{s}$. In other words, individuals’ disposable incomes are adjusted to keep utility levels constant for every individual. With utility levels unchanged, the impact of the variation on social welfare entirely depends on the extra tax revenue (or profit) it generates. The left-hand sides of (17) and (18) measure the social value of this extra tax revenue (for a variation in $t_{i}$ or in $\tau _{s}$ respectively). Obviously, when the tax system is optimized, this social value must be equal to zero (otherwise welfare could be increased by changing the tax rates).

To understand this interpretation, assume one changes $c_{j}$ after $t_{i}$ or $\tau _{s}$ changes. Start with a variation in $t_{i}$. With the tax revenues being given by

$$\begin{aligned} \sum _{j}\pi ^{j}\left( \sum _{e=1}^{n}t_{e}x_{e}^{j}+\sum _{f=1}^{m}\tau _{f}y_{f}^{j}\right) , \end{aligned}$$

and the cross-price derivatives being equal to zero, the change in $t_{i}$ produces an extra tax revenue of

$$\begin{aligned} \sum _{j}\pi ^{j}x_{i}^{j}+t_{i}\sum _{j}\pi ^{j}\frac{\partial x_{i}^{j} }{\partial p_{i}}. \end{aligned}$$

Our compensation rule requires $\sum _{j}\pi ^{j}x_{i}^{j}$ of this to be rebated to individuals.²³ The net change in revenue, minus compensation, is equal to

$$\begin{aligned} t_{i}\sum _{j}\pi ^{j}\frac{\partial x_{i}^{j}}{\partial p_{i}}. \end{aligned}$$

This is the first expression on the left-hand side of (17). At the same time, the $\sum _{j}\pi ^{j} x_{i}^{j}$ compensation leads to an additional tax revenue of

$$\begin{aligned}&\sum _{e=1}^{n}t_{e}x_{e}^{j}\left( \sum _{j}\pi ^{j}x_{i}^{j}\right) +\sum _{f=1}^{m}\tau _{f}y_{f}^{j}\left( \sum _{j}\pi ^{j}x_{i}^{j}\right) \\&\quad =\left( \sum _{j}\pi ^{j}x_{i}^{j}\right) \left( \sum _{e=1}^{n}t_{e}x_{e}^{j}+\sum _{f=1}^{m}\tau _{f}y_{f}^{j}\right) . \end{aligned}$$

To convert these tax revenue changes into social welfare (measured in units of general revenues), one must multiply tax revenue variations emanating from ${\underline{y}}$-goods by a factor of $(1+\delta /\mu ).$ This is because the revenue from ${\underline{y}}$-goods enters both the global government budget constraint and the break-even constraint. This results in the second expression on the left-hand side of (17).

The left-hand side of (18) can be decomposed in a similar way, except for one extra complication; namely the additional term $(\delta /\mu )\sum _{j}\pi ^{j}y_{s}^{j}$. In this exercise, $\sum _{j}\pi ^{j}y_{s}^{j}$ represents the value of the refunds to individuals. When collected as a tax, this amount has a social value of $(1+\delta /\mu )\sum _{j}\pi ^{j}y_{s}^{j}$. On the other hand, the refund “costs” only $\sum _{j}\pi ^{j}y_{s}^{j}$ (it comes from the general budget and has no impact on the break even constraint).

To ease the comparison with traditional Ramsey expressions, one can rewrite (17) and (18) in a form that depicts inverse elasticity rules. Introduce

$$\begin{aligned} A_{i}= & {} \sum _{j=1}^{H}\pi ^{j}x_{i}^{j}\left[ \sum _{e=1}^{n}t_{e} \frac{\partial x_{e}^{j}}{\partial c^{j}}+\left( 1+\frac{\delta }{\mu }\right) \sum _{f=1}^{m}\tau _{f}\frac{\partial y_{f}^{j}}{\partial c^{j}}\right] , \end{aligned}$$

(19)

$$\begin{aligned} B_{s}= & {} \sum _{j=1}^{H}\pi ^{j}y_{s}^{j}\left[ \sum _{e=1}^{n}t_{e} \frac{\partial x_{e}^{j}}{\partial c^{j}}+\left( 1+\frac{\delta }{\mu }\right) \sum _{f=1}^{m}\tau _{f}\frac{\partial y_{f}^{j}}{\partial c^{j}}\right] +\frac{\delta }{\mu }\sum _{j}\pi ^{j}y_{s}^{j}, \end{aligned}$$

(20)

where $A_{i}$ and $B_{s}$ measure the social value of the extra tax revenues due to refunds, with $B_{s}$ also including $(\delta /\mu )\sum _{j}\pi ^{j} y_{s}^{j}.$ We have, for all $i=2,\ldots ,n$ and $s=1,2,\ldots ,m,$

$$\begin{aligned} \frac{t_{i}}{1+t_{i}}= & {} \frac{A_{i}}{\sum _{j} \pi ^{j}x_{i}^{j}\varphi _{ii}^{j}}, \end{aligned}$$

(21)

$$\begin{aligned} \frac{\tau _{s}}{1+\tau _{s}}= & {} \frac{B_{s}}{(1+\delta /\mu ) \sum _{j}\pi ^{j}y_{s}^{j}\varepsilon _{ss}^{j}}, \end{aligned}$$

(22)

where and $\varphi _{ii}^{j}$ and $\varepsilon _{ss}^{j}$ denote the absolute value of the j-type’s own-price elasticity of Marshallian demands for $x_{i}$ and for $y_{s}.$

Expressions (21) and (22) have a number of interesting implications, particularly when compared to their traditional counterparts. First, the effect of the break-even constraint is no longer confined to the goods which enter this constraint. Instead, it spills over to the other goods which no longer go untaxed (compare with the result obtained in Sect. 4.1). Second, we get inverse elasticity rules as in the Ramsey model; albeit without redistributive terms. This becomes clear below. The numerator of both expressions contain the “tax revenue” terms $A_{i}$ and $B_{s}$. Recall that these expressions measure the social value of the extra tax revenue generated from the demand variations that follow the (compensating) adjustments in disposable income.

One may wonder why these terms were absent in Sect. 4.1. The key to understanding this property is that when Hicksian demands are independent, the price-cum-income variations we consider have by definition no impact on the demand of any of the other goods. And the effect on the good under consideration is already captured in the (compensated) elasticity term. To sum up, Sect. 4.1 has given simple results, not because the different effects were absent but because they happen to cancel out exactly under the considered assumptions.

4.3 Proportional reduction in compensated demands

When there are cross-price effects, the Ramsey model no longer yields results that can be presented as simple inverse elasticity rules. One popular way to present the tax rules in this case is in terms of proportional reductions in compensated demands. This leads to the celebrated “equal proportional reduction” pure efficiency result in the one-consumer Ramsey problem (and adjusted for redistributive considerations in its many household extension). We show in Appendix A that the proportional reductions in compensated demands for each privately-produced and each publicly-produced good are given by

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

(23)

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

(24)

The left-hand sides of (23) and (24) represent the proportional reductions in compensated demand of $x_{i}$ and for$y_{i}$. The right-hand sides of (23) and (24) show that these are proportional to the compensated impacts of the considered good’s tax rate on the break-even constraint. And as such, it differs across different commodities. Consequently, a version of the inverse elasticity rule also holds in the Atkinson–Stiglitz framework with a break-even constraint. However, the reduction is adjusted for tax revenue considerations; or more precisely the revenue of the regulated firm.

Equations (23) and (24) differ from their traditional Ramsey counterpart in that they contain no redistributive (covariance) terms. Consequently, as in Sect. 4.2, nonlinear income taxation fully takes care of redistributive concerns and obviates the need to adjust the inverse elasticity rules for redistribution. We summarize the main results of this section as,

Proposition 2

In the Atkinson and Stiglitz setting of Proposition 1:

• (i) Assume compensated demands are independent. Then,  (a) the goods that are produced by non-regulated firms should not be taxed but public/regulated firms’ goods should be. (b) Taxation of public/regulated firms’ goods follow the Ramsey inverse elasticity rule as characterized by Eq. (16). (c) The taxation/pricing rules are purely efficiency-driven. Redistribution is taken care of by the income tax allowing the public firm’s prices to be adjusted for revenue raising (as in the Ramsey model with identical individuals).

• (ii) Assume Marshallian demands are independent. Then,  (a) all goods are taxed with the optimal taxes characterized by expressions (21)–(22). (b) Tax rates follows inverse elasticity rules as in the Ramsey model;  albeit without redistributive terms. (c) The tax rules also include “tax revenue” terms characterized by (19)–(20). (d) These terms measure the social value of the extra tax revenue generated from the demand variations that follow the (compensating) adjustments in disposable income.

• (iii) A version of the proportional reduction in compensated demands apply for all goods as characterized by (23) and (24). The reductions are proportional to the compensated impacts of the considered good’s tax rate on the break-even constraint and differs across different goods.

5 Conclusion

This paper has examined if the optimal tax and regulatory pricing approaches to Ramsey pricing can be reconciled. It has incorporated the government’s objective of revenue raising for financing its expenditures (including redistributive transfers), and a regulated firm’s objective to cover its fixed cost, into a single framework. Tax instruments were restricted by informational considerations as posited in the optimal tax literature and by the firm’s break-even constraint as stipulated in the regulatory economics. This is because in practice regulation is almost always associated with balanced budget requirements. As such, the paper has argued, ignoring break-even constraints may be considered a shortcoming of the modern optimal tax theory. This theory is based solely on informational asymmetries between the tax administration and taxpayers—an information structure that precludes break-even constraints. There are of course a variety reasons, with or without informational underpinnings, that necessitate break-even constraints. The paper has not aimed to formally and explicitly model a particular source for them. Instead, it has discussed in detail the many reasons that rationalize them while also showing that their neglect has led to an unwarranted view of the role of commodity taxes in the optimal tax literature.

By incorporating a break-even constraint into the Atkinson and Stiglitz framework, the paper has challenged the modern optimal tax view that considers commodity taxes redundant. It has restored many of the earlier Ramsey tax/pricing lessons within the Atkinson and Stiglitz framework. In particular, it has shown that while nonlinear income taxes do take care of all redistributive concerns, this does not mean that commodity taxes are redundant. Break-even constraints create a role for commodity taxes. Interestingly too, this role goes beyond the taxation of goods produced by the public/regulated firm and to the taxation of other goods as well. Put differently, break-even constraints for “regulated goods” have spill overs to “non-regulated goods”.

Throughout the paper, we have assumed that individual consumption levels are observable neither to the firm nor to the tax authorities; this rules out nonlinear prices and taxes. To be sure, nonlinear pricing would make it “easier” for the regulated firm to finance its fixed costs and this would typically—but not always—involve less distortions as compared to linear prices. However, nonlinear pricing does not undermine the basic message of the paper, namely, that there are distortions in the public sector which spill over to privately-produced goods.²⁴ Observe also that nonlinear pricing is done by firms. This requires that the firms observe individual consumption levels for their products. Nonlinear taxes, on the other hand, would require that the tax administration observes individual consumption levels for all goods—a far more demanding assumption.

We conclude by pointing out a number of possible extensions to our study. First, it would be interesting to compare the spill-over effects on the prices of non-regulated goods to the markups imposed on the goods subject to break-even constraints. Our various expressions suggest that this depends mainly on the size of the (compensated) cross-price effects. However, the complex way they operate does not allow one to draw clear-cut conclusions at this level of generality. Numerical examples could provide some illustrative indications, while an empirical study may lead to more satisfactory answers.

Second, we have not formally modeled the informational asymmetries between public authorities and regulated firms. These have been at the heart of the regulatory economics literature during the last two decades.²⁵ Such asymmetries of information would introduce an additional layer of complexity. For the regulator, the design of an incentive scheme comes on top of, and is intertwined with, the traditional pricing problem; see e.g., Laffont and Tirole (1993, chapter 3). However, the use of sophisticated incentive regulation does not in itself solve the problem of breaking even. In particular, this literature uses Ramsey pricing as a benchmark obtained under full information. Under asymmetric information pertaining to a public/regulated firm’s cost, pricing rules are more complex and often incorporate incentive corrections. As far as our results are concerned, this complexity can only reinforce our main findings. The use of Ramsey-type prices in the Atkinson–Stiglitz setting, and the spillover to the taxation of goods produced in the private sector, are expected to be robust. What may not be robust, though, are the simple results we obtained for separable demands. However, even these results may well continue to hold—at least in some circumstances. In particular, Laffont and Tirole (1993, p. 173) show that when the cost function satisfies some separability conditions, one obtains the so-called “incentive-pricing dichotomy”. Then, the incentive design leaves the pricing rules unaffected and we return to traditional Ramsey pricing.

Last but not least, regulation often pursues specific and likely non-welfarist redistributive objectives as in “universal access”. It would be interesting to study how these objectives interact with the general objectives of tax policy.

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).