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Second-Best Pigouvian Taxation: A Clarification

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Abstract

This paper argues that the search for a “purely environmental” component of a tax on goods or factors of production that impact the environment—separate from its redistributive and distortive effects—is fraught with difficulties. The quest is often impossible because of the interconnectedness between labor supply, consumption decisions and the environmental quality. The paper differentiates between two conceptualization for “the Pigouvian tax” that have been employed in the literature and argues that each has tried to isolate the environmental component in its own way. One conceptualization, due to Cremer et al. (J Public Econ 70:343–364, 1998) does so by ruling out direct feedback from changes in environmental quality on the incentive effect of the tax. In the second conceptualization, due to Bovenberg and Ploeg’s (J Public Econ 55:349–390, 1994), incentive effects are ruled out by making consumers’ valuation of environmental quality independent of the labor supply. This is achieved by assuming separability between labor supply and other goods (including environmental equality). To convey its message, the paper studies the properties of optimal polluting and non-polluting non-labor input taxes in a Mirrleesian model with endogenously determined wages.

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Notes

  1. See, among others, Bovenberg and Ploeg (1994), Bovenberg and Mooij (1994); Bovenberg and Mooij (1997), Fullerton (1997), Schöb (1997), Cremer, Gahvari and Ladoux (1998); Cremer,Gahvari and Ladoux (2001), Cremer and Gahvari (2001), Boadway and Tremblay (2008), Micheletto (2008), and Gahvari (2010, 2012). See also the survey by Bovenberg and Goulder (2002) and the references therein.

  2. Some tentative numerical estimates of, and comparisons between, the optimal emission tax versus the Pigouvian tax are presented in Cremer et al. (2003); Cremer et al. (2010).

  3. In particular, I do not make any claims to the originality of my second-best tax characterizations.

  4. One can also interpret \(\gamma ^{s}\) and \(\gamma ^{u}\) as welfare weights associated with \(u^{s}\) and \(u^{u}\) in a utilitarian social welfare maximization problem. The set of (constrained) Pareto-efficient allocations corresponds to the set of allocations defined by different values of \(\gamma ^{s}\) and \(\gamma ^{u}\) as the welfare weights change (\(\gamma ^{u}/\gamma ^{s}\) goes from zero to infinity). To see the equivalence, recall that the Pareto-efficient allocations are customarily found by maximizing the utility of one agent subject to keeping the utility of other agents fixed plus the economy’s resource constraint (informational constraints are also included when describing constrained Pareto-efficient allocations). Thus, in the stated problem of this paper, one maximizes \(u^{s}\) subject to \(u^{u}\ge \overline{u}\) plus constraints (1)–(5). This is summarized by the Lagrangian

    $$\begin{aligned} \mathcal {L}=&u^{s}+\eta \left( u^{u}-\overline{u}\right) +\lambda \left( u^{s}\!-\!u^{su}\right) \!+\!\mu \left[ \mathbf {O}\left( \cdot \right) -\pi ^{s}c^{s}-\pi ^{u}c^{u}-rK\right. \\&\left. -pE-\overline{R}\right] +\delta ^{s}\left[ w^{s}-\mathbf {O}_{L^{s}}\left( \cdot \right) \right] +\delta ^{u}\left[ w^{u}-\mathbf {O}_{L^{u}}\left( \cdot \right) \right] . \end{aligned}$$

    Comparing this expression with the Lagrangian expression (6) in the text reveals that the two are equivalent except for the normalization rule. Whereas in the problem leading to Lagrangian (6) \(\gamma ^{s}+\gamma ^{u}\) is normalized to one; here one implicitly normalizes \(\gamma ^{s}=1\) while denoting \(\gamma ^{u}/\gamma ^{s}\) by \(\eta \).

  5. Gaube (2005) shows that wage endogeneity may lead to second-best solutions other than the traditional “redistributive” and “regressive” cases. Given the note’s goals, I nevertheless concentrate on the \(w^{s}\ge w^{u}\) case. See Micheletto (2004) for the characterization of the optimal redistributive policy with endogenous wages.

  6. In the language of the optimal income tax problem, mimicking refers to the act of one agent choosing the bundle intended for another type, thus misrepresenting his true type in the tax design problem.

  7. The subscript on a variable indicates its partial derivative with respect to the argument the subscript denotes, and two subscripts indicate second partial derivatives with respect to the two arguments the subscripts denote.

  8. This term is represented by \(\left[ \delta ^{s}\mathbf {O}_{L^{s}K}\left( \cdot \right) +\delta ^{u}\mathbf {O}_{L^{u}K}\left( \cdot \right) \right] /\mu \) in (7), and \(\left[ \delta ^{s}\mathbf {O}_{L^{s}E}\left( \cdot \right) +\delta ^{u}\mathbf {O}_{L^{u}E}\left( \cdot \right) \right] /\mu \) in (8) and (9). Obviously, while \(\mathbf {O}_{L^{s}K}\left( \cdot \right) \) has the same format as \(\mathbf {O} _{L^{s}E}\left( \cdot \right) \), they are not identical. The same observation applies to \(\mathbf {O}_{L^{u}K}\left( \cdot \right) \) and \(\mathbf {O}_{L^{u}E}\left( \cdot \right) \).

  9. Indeed, with \(\mathbf {O}_{L^{s}K}\left( \cdot \right) =\mathbf {O} _{L^{u}K}\left( \cdot \right) =\mathbf {O}_{L^{s}E}\left( \cdot \right) = \mathbf {O}_{L^{u}E}=0,\) equation (8) collapses to equation (17), given (16), in Cremer and Gahvari (2001).

  10. Thus, with exogenous wages, \(\mathbf {O}_{K}\left( \cdot \right) =r\) and there should be no taxes on non-polluting inputs. With endogenous wages, on the other hand, this is no longer the case because \(\mathbf {O}_{L^{s}K}\left( \cdot \right) \ne 0\) and \(\mathbf {O}_{L^{u}K}\left( \cdot \right) \ne 0\). In this note, I am interested only in the difference between non-polluting and polluting inputs. I will thus not discuss the properties of the common element between them which constitutes the tax on the non-polluting factor.

  11. The resource constraint (1)

    $$\begin{aligned} \mathbf {O}\left( \cdot \right) \ge \pi ^{s}c^{s}+\pi ^{u}c^{u}+rK+pE+ \overline{R}, \end{aligned}$$

    can also be written as

    $$\begin{aligned} \left[ \mathbf {O}\left( \cdot \right) -\left( rK+pE\right) \right] -\left( \pi ^{s}c^{s}+\pi ^{u}c^{u}\right) \ge \overline{R}. \end{aligned}$$

    Observe that \(\mathbf {O}\left( \cdot \right) -\left( rK+pE\right) \) is aggregate domestic expenditures (incomes) and \(\pi ^{s}c^{s}+\pi ^{u}c^{u}\) is the aggregate domestic private sector consumption. Put differently, the left-hand side of the latter inequality shows tax revenues so that this constraint can be considered as the government’s budget constraint. This is why one can refer to \(\mu \) as both the shadow price for the resource constraint as well as the shadow cost of public funds.

  12. By referring to the last expression on the right-hand side of (16) as the “tax difference,” I do not mean to imply that the first expressions on the right-hand sides of (7) and (16) take the same values; only that they have identical formulas.

  13. If there are other taxes in the system, these changes also affect the extent of the welfare cost associated with those taxes.

  14. Based on this latter type of separability, Kaplow (1996) argues that environmental taxes should be Pigouvian even in a tax reform exercise wherein the tax rates are suboptimal. His definition of Pigouvian is the same as Bovenberg and Ploeg (1994). The argument assumes that the government is able to adjust its income policy whenever it introduces or changes an environmental tax. This raises the following question: If the government is indeed able to adjust its income policy at will, why not set it optimally so that the question becomes one of tax design as in Bovenberg and Ploeg (1994) rather than one of tax reform (which is by definition constrained)? Kaplow (1996) finding has also been interpreted to mean that the “marginal cost of public funds” is equal to one. This interpretation is unwarranted; see Gahvari (2006). More recently Jacobs (2010) also argues that the marginal cost of public funds is always equal to one. However, Jacobs faults the literature for using a “wrong” concept of the marginal cost of public funds and advocates using a different definition. In his case, the marginal cost of public funds is indeed always equal to one but definitionally. It also requires that the government tax tools include an optimized uniform cash rebate (or lump-sum tax).

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Correspondence to Firouz Gahvari.

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I thank the Editor and two anonymous referees for helpful comments.

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Gahvari, F. Second-Best Pigouvian Taxation: A Clarification. Environ Resource Econ 59, 525–535 (2014). https://doi.org/10.1007/s10640-013-9747-7

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