Abstract
This paper, which builds on Chipman (The economist’s vision. Essays in modern economic perspectives, 131–162, 1998), analyzes a simple model formulated by Hurwicz (Japan World Economics 7:49–74, 1995) of two agents—a polluter and a pollutee—and two commodities: “money” (standing for an exchangeable private good desired by both agents) and “pollution” (a public commodity desired by the polluter but undesired by the pollutee). There is also a government which issues legal rights to the two agents to emit a certain amount of pollution, which can be bought and sold with money. It is assumed that both agents act as price-takers in the market for pollution rights, so that competitive equilibrium is possible. The “Coase theorem” (so-called by Stigler (The theory of price, 1966) asserts that the equilibrium amount of pollution is independent of the allocation of pollution rights. A sufficient condition for this was (in another context) obtained by Edgeworth (Giorn Economics 2:233–245, 1891), namely that preferences of the two agents be “parallel” in the money commodity, whose marginal utility is constant. Hurwicz (Japan World Economics 7:49–74, 1995) argued that this parallelism is also necessary. This paper, which provides an exposition of the problem, raises some questions about this result, and provides an alternative necessary and sufficient condition.
This paper is dedicated to the memory of our esteemed respective former colleague and former thesis advisor Leonid Hurwicz. We greatly regret not having been able to discuss the final section with him before his death. Thanks are due to Augustine Mok for his help with the diagrams.
Originally published in Economic Theory, Volume 49, Number 2, February 2012 DOI 10.1007/s00199-010-0573-7.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This formulation differs from that of Hurwicz (1995) (which it otherwise follows closely), who considers pollution rights \(z_1 \equiv y_1\) and \(z_2 \equiv \eta - y_2\), which must satisfy \(z_1 = z_2\), analogously to the theory of public goods. This corresponds to the second equation of (1.6), since \(z_1 = y_1 = \eta - y_2 = z_2\). Thus the difference is largely one of notation. The present formulation, which provides a notation for individual i’s final holdings of money and of pollution rights \((x_i,y_i)\), makes it somewhat easier to interpret the diagrams in Figs. 1 and 2 below.
- 2.
This shows that the validity of the analysis in this paper is limited to the case of a single pollutee. The introduction of a second pollutee at once introduces a “free-rider” problem.
- 3.
Note that the first-order necessary (Lindahl-Samuelson) condition for the Pareto optimality for public-goods economies is given by \(\frac{\partial U_1}{\partial s}/{\frac{\partial U_1}{\partial x_1}} + {\frac{\partial U_2}{\partial s}}/{\frac{\partial U_2}{\partial x_2}}= g'(s)\), where g(s) is the x-input requirement for producing s units of the public good s. But in the pollution model there is no x-input requirement to produce s, so \(g(s) = 0\) identically. Hence the two marginal rates of substitution add up to zero.
- 4.
A parallel preference ordering is one that is representable by a quasi-linear utility function \(U(x,y) = \nu x + \phi (y)\) for \(\nu >0\) and \(\phi '(y)>0\) (cf. Hurwicz 1995, p. 55n). The term “parallel” was introduced by Boulding (1945) and followed by Samuelson (1964), though the concept goes back to Auspitz and Lieben (1889, Appendix II, Sect. 2, pp. 470–483), Edgeworth (1891, p. 237n; 1925, p. 317n), and Berry (1891, p. 550). The concept was also analyzed by Pareto (1892) and Samuelson (1942), and by Katzner (1970, pp. 23–26) who describes such preferences as “quasi-linear”. See also Chipman and Moore (1976, pp. 86–91, 108–110; 1980, pp. 940–946); Chipman (2006, p. 109).
- 5.
The swastika is an ancient Buddhist and Hindu symbol found on temples in central Asia. Hitler adopted it (after rotating it clockwise 45\(^\circ \)) as the symbol of his Nazi party. As L. Hurwicz reminded the first author in a seminar presentation, the German word for swastika is Hakenkreuz (hook-cross); so the set of Pareto optima in the Edgeworth box with parallel preferences is one of these hooks.
- 6.
The symbols x and y need to be interchanged to reconcile the present notation with Edgeworth’s.
- 7.
Edgeworth’s interest in this problem stemmed from his inquiry into the conditions under which the bargaining process introduced by Marshall in his Note on Berry (1891, pp. 395–397, 755–756; 1961, I, pp. 844–845; II, pp. 791–798), involving a succession of partial contracts at independently reached prices, with recontracting ruled out, would lead to a competitive equilibrium with in fact a uniform price, and thus a “determinate” solution.
- 8.
- 9.
Hurwicz mentioned (p. 68) that he had found an example (unfortunately not displayed) of cubic utility functions generating horizontal contract curves, but he dismissed this on the ground that “it must be possible to choose the two utility functions independently, while in the cubic case ‘the choice of \(u^2\) is limited by the choice of \(u^1\).”’ But formula (4.8) above, as well as the mutual tangency (4.3), shows that assuming the horizontality of the contract curve to be true, the two terms in brackets cannot be entirely unrelated. This does not imply any psychological dependence between the utility functions, but simply that there must be some kind of relationship, e.g. as in Edgeworth’s formula (4.5), in order for the horizontality of the contract curve to be true.
- 10.
Since Eq. (1.7) defines a continuous one-to-one mapping between the individuals’ pollution rights \(y_i \) with \(y_1+y_2=\eta \) and allocation of pollution s, the two constrained optimization problems are equivalent. Note that through such a monotonic transformation, one may transform an original problem into a concave optimization problem, in which the object function is (quasi)concave and the constraint sets are convex. Since this technique has been widely used in the literature such as in the moral hazard model in the Principal-Agent Theory (cf. Laffont and Martimort (2002, pp. 158–159).
- 11.
It can be also seen from http://eqworld.ipmnet.ru/en/solutions/fpde/fpde1104.pdf.
References
Asheim, G. B., Mitra, T., & Tungodden, B. (2016). Sustainable recursive social welfare functions. Economic Theory. reprinted in this volume
Auspitz, R., & Lieben, R. (1889). Untersuchungen über die Theorie des Preises. Leipzig: Verlag von Duncker & Humblot.
Berry, A. (1891). Alcune breve parole sulla teoria del baratto di A. Marshall. Giorn Economics, Series 2, 2, 549–553.
Boulding, K. E. (1945). The concept of economic surplus. American Economic Review, 35, 851–869.
Burniaux, J. -M., & Martins, J. O. (2016). Carbon leakage: a general equilibrium view. Economic Theory. reprinted in this volume.
Chichilnisky, G. (2016). Sustainable markets with short sales. Economic Theory. reprinted in this volume.
Chichilnisky, G., & Heal, G. (Eds.). (2000a). Environmental markets. Equity and efficiency. New York: Columbia University Press.
Chichilnisky, G., & Heal, G. (2000b). Markets for tradable carbon dioxide emission quotas: Principles and practice, 13–45.
Chichilnisky, G., Heal, G., & Starrett, D. (2000). Equity and efficiency in environmental markets: Global trade in carbon dioxide emissions, 46–67.
Chipman, J. S. (1974). Homothetic preferences and aggregation. Journal of Economic Theory, 8, 26–38.
Chipman, J. S. (1998). A close look at the Coase theorem. In J. M. Buchanan & B. Monissen (Eds.), The economists’ vision. Essays in modern economic perspectives (pp. 131–162). Frankfurt am Main: Campus Verlag.
Chipman, J. S. (2006). Aggregation and estimation in the theory of demand. In P. Mirowski & D. W. Hands (Eds.), Agreement on demand: Consumer theory in the twentieth century (pp. 106–129). Durham and London: Duke University Press.
Chipman, J. S., & Moore, J. C. (1976). The scope of consumer’s surplus arguments. In A. M. Tang, F. M. Westfield & J. S. Worley (Eds.), Evolution, welfare, and time in economics, essays in honor of Nicholas Georgescu-Roegen (pp. 61–123). Lexington, Mass.: D. C. Heath and Company.
Chipman, J. S., & Moore, J. C. (1980). Compensating variation, consumer’s surplus, and welfare. American Economic Review, 70, 933–949.
Coase, R. H. (1960). The problem of social cost. Journal of Law and Economics, 3, 1–44. Reprinted in Coase (1988), 95–156.
Dutta, P. K., & Radner, R. (2016). Capital Growth in a global warming model: Will China and India sign a climate treaty? Economic Theory. reprinted in this volume.
Edgeworth, F. Y. (1881). Mathematical psychics. London: C. Kegan Paul & Co.
Edgeworth, F. Y. (1891). Osservazioni sulla teoria matematica dell’ -economia politica con riguardo speciale ai principi di economia di Alfredo Marshall; and Ancora a proposito della teoria del baratto. Giorn Economics, Series 2, 2, 233–245, 316–318. Abridged, revised, and corrected English translation in Edgeworth (1925).
Edgeworth, F. Y. (1925). On the determinateness of economic equilibrium. In Papers relating to political economy 3 volumes (Vol. II, pp. 313–319). London: Macmillan and Co., Limited.
Figuiéres, C., & Tidball, M. (2016). Sustainable exploitation of a natural resource: A satisfying use of Chichilnisky criterion. Economic Theory. reprinted in this volume.
Hurwicz, L. (1995). What is the Coase Theorem? Japan World Economy, 7, 49–74.
Karp, L., & Zhang, J. (2016). Taxes versus quantities for a stock pollutant with endogenous abatement costs and asymmetric information. Economic Theory. reprinted in this volume.
Katzner, D. W. (1970). Static demand theory (pp. 47–74). New York: The Macmillan Company.
Laffont, J. -J., & Martimort, D. (2002). The theory of incentives: The principal-agent model. Princeton and Oxford: Princeton University Press.
Lange, O. (1942). The foundations of welfare economics. Econometrica, 10, 215–228.
Lauwers, L. (2016). Intergenerational equity, efficiency, and constructability. Economic Theory. reprinted in this volume.
Lecocq, F., & Hourcade, J. -C. (2016). Unspoken ethical issues in the climate affair. Economic Theory: Insights From a Theoretical Analysis of Negotiation Mandates. reprinted in this volume.
Marshall, A. (1891). Principles of economics (2nd ed.). London: Macmillan and Co. (9th (variorum) ed., Vols. 2). London: Macmillan and Co. Ltd. (1961).
Ostrom, E. (2016). Nested externalities and polycentric institutions: Must we wait for global solutions to climate change before taking action at other scales? Economic Theory. reprinted in this volume.
Pareto, V. (1892). Considerazioni sui principii fondamentali dell’economia politica pura. Giorn Economics, Series 2, 4, 485–512, 5, 119–157.
Polyanin, A. D., Zaitsev, V. F., & Moussiaux, A. (2002). Handbook of first order partial differential equations. London: Taylor & Francis.
Rezai, A., Foley, D. K., & Taylor, L. (2016). Global warming and economic externalities. Economic Theory. reprinted in this volume.
Samuelson, P. A. (1942). Constancy of the marginal utility of income. In O. Lange, F. McIntyre, & T. O. Yntema (Eds.), Studies in mathematical economics and econometrics, in memory of Henry Schultz (pp. 75–91). Chicago: The University of Chicago Press.
Samuelson, P. A. (1954). The pure theory of public expenditure. Review of Economics and Statistics, 36, 387–389.
Samuelson, P. A. (1955). Diagrammatic exposition of a theory of public expenditure. Review of Economics and Statistics, 37, 350–356.
Samuelson, P. A. (1964). Principles of efficiency—discussion. American Economic Review, Papers and Proceedings, 54, 93–96.
Samuelson, P. A. (1969). The pure theory of public expenditure and taxation. In J. Margolis & H. Guitton (Eds.), Public economics (pp. 98–123). London: The Macmillan Press Ltd. and New York: St. Martin’s Press.
Sheeran, K. A. (2006). Who should abate carbon emissions? A Note, Environmental and Resource Economics, 35, 89–98.
Stigler, G. J. (1966). The theory of price (3rd ed.). London: Collier-Macmillan Limited.
Tian, G., & Yang, L. (2009). Theory of negative consumption externalities with applications to economics of happiness. Economic Theory, 39, 399–424.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Chipman, J.S., Tian, G. (2016). Detrimental Externalities, Pollution Rights, and the “Coase Theorem”. In: Chichilnisky, G., Rezai, A. (eds) The Economics of the Global Environment. Studies in Economic Theory, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-319-31943-8_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-31943-8_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31941-4
Online ISBN: 978-3-319-31943-8
eBook Packages: Economics and FinanceEconomics and Finance (R0)