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2016 | OriginalPaper | Buchkapitel

Detrimental Externalities, Pollution Rights, and the “Coase Theorem”

verfasst von : John S. Chipman, Guoqiang Tian

Erschienen in: The Economics of the Global Environment

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel Global Warming and Economic Externalities

Nächstes Kapitel Taxes Versus Quantities for a Stock Pollutant with Endogenous Abatement Costs and Asymmetric Information

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.

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.

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.

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).

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.

The symbols x and y need to be interchanged to reconcile the present notation with Edgeworth’s.

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.

See Fig. 2 above, also Fig. 1 in Hurwicz (1995, p. 57). But this condition is violated in Fig. 2 of Samuelson (1969, p. 113).

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.

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).

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Titel: Detrimental Externalities, Pollution Rights, and the “Coase Theorem”
verfasst von: John S. Chipman
Guoqiang Tian
Verlag: Springer International Publishing
Buch: The Economics of the Global Environment
Print ISBN: 978-3-319-31941-4

Electronic ISBN: 978-3-319-31943-8

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-31943-8_21

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