Efficient bilateral taxation of externalities


In the context of the example of a factory whose smoke emissions affect a near-by laundry, Coase (J Law Econ 3:1–44, 1960) argued for taxing the laundry as well as the factory, while Baumol (Am Econ Rev 62:307–322, 1972) argued for taxing only the factory. Consistent application of marginal cost pricing shows that the efficient tax on laundries is positive when the number of laundries is finite and that the tax approaches zero in the limit as the number of laundries approaches infinity. The efficient tax on factories is bounded away from zero, regardless of the number of factories. Our framework is an application of the Vickrey–Clarke–Groves family of truth-telling mechanisms that require each agent to bear the full social cost of changing the outcome that would have prevailed had she not participated in the decision. Until now, the literature has not fully resolved the discrepancies between Coase’s and Baumol’s arguments, and even contemporary textbooks on environmental economics and public economics do not offer correct and complete analyses.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Vickery (1961) describes a corresponding mechanism that offers each seller the larger of her revealed marginal cost and a prospective buyer’s willingness to pay; doing so provides sellers with incentives to reveal their true marginal costs because they might otherwise forego advantageous trades.

  2. 2.

    Ng (2007) makes a similar argument, although he does not relate the tax on residents to Vickrey’s framework. Instead, Ng achieves truthful revelation of marginal damages by restricting the permissible heterogeneity of marginal damages across residents.

  3. 3.

    This type of diagram is presented in Buchanan and Stubblebine (1962) as well as in Ng (1971, 2007). We assume that the marginal benefit schedule DKC is concave rather than convex to make it easier to relate our graphical argument to Fig. 2 in Vickrey (1961). Our argument holds for convex as well as concave marginal benefit schedules.

  4. 4.

    Thus, curve DKC measures the marginal benefit of the single factory’s emissions in a one-factory world and the joint marginal benefit of the emissions of two factories in the two-factory world.

  5. 5.

    Consider a factory whose production process exhibits variable economies of scale, and whose marginal benefit of emissions increases initially and falls after crossing a threshold. While the integral of the marginal cost of emissions is less than the factory’s total benefit from emitting, the product of the cost of the marginal unit and the quantity of its emissions might exceed this total benefit. In such a case, the standard Pigouvian tax leads to an inefficient location decision.

  6. 6.

    We write “usually” because while the VCG mechanism requires that each agent bear the net cost of her presence, it does not require that she pay this amount. The VCG mechanism also can be implemented by offering each agent a subsidy, equal to the net social benefit from her not affecting the social choice as much as she might have. By revealing her true benefit schedule, the agent ensures that her net marginal benefit on the marginal unit of change that is caused by her presence equals the foregone subsidy that she would have received had she not caused this marginal unit of change. Financing the subsidies would then lead to a budget deficit as well.

  7. 7.

    Kunreuther and Kleindorfer (1986, p. 295) claim that VCG mechanisms cannot be applied to the regulation of public bads since such an application “requires the public commodity to have a positive value to each participant so that there is a net surplus after the commodity tax is levied.” Our Fig. 2 shows that the area 0DKMLA represents such a net surplus even after all taxes are paid. Kunreuther and Kleindorfer reference Tideman and Tullock (1976) in support of their claim, but we are not aware of any statement in Tideman and Tullock that would provide such support.

  8. 8.

    We thank Glen Weyl for pointing this characteristic of VCG mechanisms out to us.

  9. 9.

    The existence of the integral in Eq. 6 is a consequence of our assumption that total emissions are constant. If the emissions of one factory increase, then all other factories must emit less—that is, when an additional factory incurs marginal damages, we move to a world in which the marginal benefit from emitting is smaller than it was before because all factories have shrunk in size.

  10. 10.

    See, for example, Tresch (2002, p. 417).

  11. 11.

    Consider Coase’s original example of a factory whose emissions cause annual damages worth \(\varGamma = \$ 100\) to a nearby resident. The resident can relocate at an annual cost of \(\gamma_{L} = \$ 40\), while the factory can abate all emissions at an annual cost of \(\gamma_{F} = \$ 90\). Because Coase (1960, p. 41) assumed that the factory is taxed $100 per year if it emits, taxing the factory but not the resident leads to socially inefficient spending of $90 – $40 = $50. Baumol (1972, p. 472, fn. 1) argued that the true marginal damage and, hence, the appropriate Pigouvian tax is zero rather than $100, because it is socially optimal for the resident to move, which he will do on his own in the absence of the factory tax. However, Thompson and Batchelder (1974, p. 470) pointed out that the appropriate Pigouvian tax on the factory is \(\gamma_{L} = \$ 40\), so that each party bears the full social cost \(\gamma_{L}\).

  12. 12.

    To provide laundries with an incentive to reveal their respective net benefits, each laundry i whose announcement \(\omega_{Li}\) causes the sign of \(\sum\nolimits_{j = 1}^{m} {\omega_{Lj} }\) to differ from the sign of \(\sum\nolimits_{j = 1,j \ne i}^{m} {\omega_{Lj} }\) (a “pivotal” laundry) must pay a Clarke tax equal to the absolute value of \(\sum\nolimits_{j = 1,j \ne i}^{m} {\omega_{Lj} }\). For example, if \(\sum\nolimits_{j = 1,j \ne i}^{m} {\omega_{Lj} < 0}\) and \(\sum\nolimits_{j = 1}^{m} {\omega_{Lj} > 0}\), then the joint air cleaning facility would not have been installed had laundry i announced a net benefit less than \(\left| {\sum\nolimits_{j = 1,j \ne i}^{m} {\omega_{Lj} } } \right|\), but the facility is installed because laundry i has announced \(\omega_{Li} > \left| {\sum\nolimits_{j = 1,j \ne i}^{m} {\omega_{Lj} } } \right|\). Laundry i therefore pays a Clarke tax equal to \(\left| {\sum\nolimits_{j = 1,j \ne i}^{m} {\omega_{Lj} } } \right|\), which is the margin by which those in favor of rejecting the facility would have won in laundry i’s absence. Thus, each laundry pays the cost that its announcement imposes on all other laundries. Plassmann and Tideman (2011) offer a related application of the VCG mechanism to the problem of land assembly.

  13. 13.

    Applying the VCG mechanism again among the remaining m − k laundries leads to one of two possible scenarios: in scenario 1, the air cleaning facility is installed because \(\omega_{Li} > 0\) \(\forall i = 1, \ldots ,m - k\) (possibly after several iterations during which additional laundries leave), although the laundries that relocated could have remained in the area at no cost. In scenario 2, the air cleaning facility is not installed because \(\sum\nolimits_{i = 1}^{m - k} {\omega_{Li} < 0}\), and all laundries leave the area, even though there was a different set of \(\gamma_{Li}^{e}\)’s \(\forall i = 1, \ldots ,m\) under which all laundries would have preferred to stay.

  14. 14.

    Even though the date on Coase’s publication precedes that of Vickrey’s paper, Vickrey might deserve credit for an earlier publication because the 1960 issue of the Journal of Law and Economics has a 1961 copyright date. The issue arrived at the library of the University of Virginia in April 1961.

  15. 15.

    This approach might be financially feasible if team members must pay fixed fees to be on a team and a person not on the team collects the fees and pays the rewards.

  16. 16.

    See also Schulze and D'Arge (1974, p. 766, fn 8).

  17. 17.

    In their spatial model, laundries and factories compete with each other for locations. Hence, a laundry will choose to locate next to a factory that pays the tax rather than abate only if it is efficient to do so. White and Wittman did not address the concern raised by Buchanan and Stubblebine (1962) that laundries might try to extort payments from factories prior to divulging their final location decisions.

  18. 18.

    See Mas-Colell et al. (1995, pp. 354–356), Tresch (2002, pp. 194–202, 212–228), Anderson (2003, pp. 103–108), Gruber (2007, pp. 134–146), Hyman (2008, pp. 104–110), Tresch (2008, pp. 106–119), Seidman (2009, pp. 36–43), Tietenberg and Lewis (2014, pp. 371–396), Callan and Thomas (2013, pp. 318–321) and Stiglitz and Rosengard (2015, pp. 139–140).

  19. 19.

    Rosen and Gayer (2009, pp. 84–85). The example was already included in the 5th edition, published in 1999 by Harvey Rosen alone.

  20. 20.

    Situations with multiple pool owners and multiple families with children lead to the collective action problem that we identified in Sect. 4. In such cases, it will generally be easiest to require every pool owner to secure her pool rather than resolve every individual situation efficiently.

  21. 21.

    For example, the German army punishes soldiers who leave their lockers unsecured, thereby inviting their fellow soldiers to steal (“Anleitung zum Kameradendiebstahl”).

  22. 22.

    That is, the solid curve KF and the solid curve LMG are equidistant at all points.

  23. 23.

    Had the second factory’s presence shifted the marginal benefit curve to the solid curve DKF instead of DKC, then the second factory would obtain benefits from emitting the amount BN (since K differs from M) but no benefits from any additional emissions. Because the tax NLMKB would not provide an incentive for the second factory to reduce emissions below BN, there would be no shape FKC and the second factory would bear no control cost.


  1. Anderson, J. (2003). Public finance. Boston: Houghton Mifflin.

    Google Scholar 

  2. Arrow, K. (1979). The property rights doctrine and demand revelation under incomplete information. In Economics and human welfare (pp. 17–33). New York: Academic Press.

  3. Baumol, W. (1972). On taxation and the control of externalities. American Economic Review, 62, 307–322.

    Google Scholar 

  4. Baumol, W. (1974). On taxation and the control of externalities: Reply. American Economic Review, 64, 472.

    Google Scholar 

  5. Baumol, W., & Oats, W. (1975). The theory of environmental policy. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  6. Becker, G., DeGroot, M., & Marschak, J. (1964). Measuring utility by a single-response sequential method. Behavioral Science, 9, 226–232.

  7. Buchanan, J., & Stubblebine, W. (1962). Externality. Economica, 29, 371–384.

    Article  Google Scholar 

  8. Callan, S., & Thomas, J. (2013). Environmental economics and management: Theory, policy, and applications (6th ed.). Cincinnati: South-Western College Publishing.

    Google Scholar 

  9. Clarke, E. (1971). Multipart pricing of public goods. Public Choice, 11, 17–33.

    Article  Google Scholar 

  10. Clarke, E. (1972). Multipart pricing of public goods: An example. In S. Mushkin (Ed.), Public prices for public products. Washington: The Urban Institute.

    Google Scholar 

  11. Coase, R. (1960). The problem of social cost. Journal of Law and Economics, 3, 1–44.

    Article  Google Scholar 

  12. D’Aspremont, C., & Gérard-Varet, L. (1979). Incentives and incomplete information. Journal of Public Economics, 11, 25–45.

    Article  Google Scholar 

  13. Dupuit, J. (1844). De la mesure de l’utilité des travaux publics, Annales des Ponts et Chaussées, s. II, 2nd semester, 332–375. English translation: Barback, R.H. (1952). On the measurement of the utility of public works. International Economic Papers, 83–110.

  14. Goeree, J. & Zhang, J. (2013). Electoral engineering: One man, one bid. http://www.econ.uzh.ch/esei/Events/prague/Zhang.pdf.

  15. Groves, T. (1970). The allocation of resources under uncertainty. In Doctoral dissertation. Berkeley: University of California.

  16. Groves, T., & Ledyard, J. (1977). Optimal allocation of public goods: A solution to the “free rider” problem. Econometrica, 45, 783–809.

    Article  Google Scholar 

  17. Groves, T., & Loeb, M. (1975). Incentives and public inputs. Journal of Public Economics, 4, 211–226.

    Article  Google Scholar 

  18. Gruber, J. (2007). Public finance and public policy (2nd ed.). New York: Worth Publishers.

    Google Scholar 

  19. Hylland, A., & Zeckhauser, R. (1979). A mechanism for selecting public goods when preferences must be elicited. In Working paper. Kennedy School of Government, Harvard University.

  20. Hyman, D. (2008). Public finance (9th ed.). Mason: Thomson-Southwestern.

    Google Scholar 

  21. Kraus, M., & Mohring, H. (1975). The role of pollutee taxes in externality problems. Economica, 42, 171–176.

    Article  Google Scholar 

  22. Kunreuther, H., & Kleindorfer, P. (1986). A sealed-bid auction mechanism for siting noxious facilities. American Economic Review, 76, 295–299.

    Google Scholar 

  23. Lalley, S. & Weyl, E. G. (2016). Quadratic voting. Available at SSRN: http://dx.doi.org/10.2139/ssrn.2003531.

  24. Mas-Colell, A., Whinston, M., & Green, J. (1995). Microeconomic theory. New York and Oxford: Oxford University Press.

    Google Scholar 

  25. Mohring, H., & Boyd, H. (1971). Analyzing “externalities”: “Direct interaction” vs “asset utilization” frameworks. Economica, 38, 347–361.

    Article  Google Scholar 

  26. Ng, Y. (1971). Recent developments in the theory of externality and the Pigovian solution. Economic Record, 47, 169–185.

    Article  Google Scholar 

  27. Ng, Y. (2007). Eternal Coase and external costs: A case for bilateral taxation and amenity rights. European Journal of Political Economy, 23, 641–659.

    Article  Google Scholar 

  28. Plassmann, F., & Tideman, T. N. (2011). Marginal cost pricing and eminent domain. Foundations and Trends in Microeconomics, 7(2), 99–209.

    Google Scholar 

  29. Rosen, H., & Gayer, T. (2009). Public finance (9th ed.). New York: McGraw-Hill Irwin.

    Google Scholar 

  30. Schulz, W., & D’Arge, R. (1974). The Coase proposition, information constraints, and long-run equilibrium. American Economic Review, 64, 763–772.

    Google Scholar 

  31. Seidman, L. (2009). Public finance. New York: McGraw-Hill Irwin.

    Google Scholar 

  32. Stiglitz, J., & Rosengard, J. (2015). Economics of the public sector (4th ed.). New York and London: W.W. Norton & Company.

    Google Scholar 

  33. Thompson, E., & Batchelder, R. (1974). On taxation and the control of externalities: Comment. American Economic Review, 64, 467–471.

    Google Scholar 

  34. Tideman, T. N., & Tullock, G. (1976). A new and superior process for making social decisions. Journal of Political Economy, 84, 1145–1159.

  35. Tideman, T. N., & Plassmann, F. (2017). Efficient collective decision-making, marginal cost pricing, and quadratic voting. Public Choice. doi:10.1007/s11127-017-0411-6.

    Google Scholar 

  36. Tietenberg, T., & Lewis, L. (2014). Environmental & natural resource economics (9th ed.). Upper Saddle River: Prentice Hall.

    Google Scholar 

  37. Tresch, R. (2002). Public finance (2nd ed.). London and San Diego: Academic Press.

    Google Scholar 

  38. Tresch, R. (2008). Public sector economics. New York: Palgrave Macmillan.

    Google Scholar 

  39. Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16, 8–37.

    Article  Google Scholar 

  40. Vickrey, W. (1968). Automobile accidents, tort law, externalities, and insurance: An economist’s critique. Law and Contemporary Problems, 33, 464–487.

    Article  Google Scholar 

  41. Weyl, E. G. (2012). Quadratic vote buying. In Working paper. Economics Department, University of Chicago.

  42. White, M., & Wittman, D. (1982). Pollution taxes and optimal spatial location. Economica, 49, 297–311.

    Article  Google Scholar 

  43. Zeckhauser, R. (1968). Studies in interdependence. In Doctoral dissertation. Cambridge: Harvard University.

Download references


We thank Glen Weyl for very helpful suggestions and comments on an earlier draft.

Author information



Corresponding author

Correspondence to Florenz Plassmann.



This appendix confirms that, in Fig. 2, the sum of the areas of the shapes NLMS and SMKB (= NLMKB) equals the area of GMKF by construction. Figure 6 describes the same situation as Fig. 2 without emphasizing the shaded areas. Start with rectangle NLKB with height NL and width NB, which consists of the shapes NLMS, SMKB, and LKM. Shift the base of rectangle NLKB to the right, thus turning it into a parallelogram whose lower left vertex coincides with point G. The lower right vertex of the parallelogram defines point F so that the distances NG and BF are identical. Because the area of a parallelogram is the product of its base and its height, the areas of parallelogram GLKF and rectangle NLKB are identical.

Fig. 6

The equivalence of the areas NLMKB and GMKF

Now account for the non-linearity of the marginal benefit curve DC. Consider the grey shape that is defined by the line LG and the arc formed by the part of the marginal benefit curve DG that starts at L, goes through M and ends at G. Subtract this grey shape from parallelogram GLKF. Add an identical shape at the right edge KF of parallelogram GLKF, thereby defining the solid curve that connects points K and F.Footnote 22 It follows that the area of the shape defined by the four solid lines GLKF is the same as the area of parallelogram GLKF and thus the same as the area of rectangle NLKB.

Finally, note that shape LKM is part of the rectangle NLKB as well as of the shape defined by the four solid lines GLKF. Subtracting LKM from both shapes leaves the shapes NLMKB and GMKF. Thus, the areas of NLMKB and GMKF are identical.

The area of AKB represents the cost of the emissions by the two factories that the resident bears, and the area of BKC represents the control cost that the two factories bear. The area of BKF, defined by the solid line KF, that measures the control cost of the first factory when the second factory operates is the sum of the first factory’s control cost SMG before the second factory started operating and the area NLMS. Hence the area of FKC that is defined by the solid lines is the second factory’s control cost.Footnote 23

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tideman, N., Plassmann, F. Efficient bilateral taxation of externalities. Public Choice 173, 109–130 (2017). https://doi.org/10.1007/s11127-017-0466-4

Download citation


  • Non-rival bads
  • Truth-telling mechanisms
  • Marginal cost pricing
  • Pigouvian tax

JEL Classifications

  • H21
  • H23