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On Divergences between Social Cost and Private Cost
Author(s): Ralph Turvey
Source: Economica , Aug., 1963, New Series, Vol. 30, No. 119 (Aug., 1963), pp. 309-313
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On Divergences between Social Cost and
Private Cost*
The notion that the resource-allocation effects of divergences between
marginal social and private costs can be dealt with by imposing a
tax or granting a subsidy equal to the difference now seems too simple
a notion. Three recent articles have shown us this. First came Professor
Coase’s ” The Problem of Social Cost “, then Davis and Whinston’s
” Externalities, Welfare and the Theory of Games ” appeared, and,
finally, Buchanan and Stubblebine have published their paper
” Externality “.’ These articles have an aggregate length of eighty
pages and are by no means easy to read. The following attempt to
synthesise and summarise the main ideas may therefore be useful.
It is couched in terms of external diseconomies, i.e. an excess of social
over private costs, and the reader is left to invert the analysis himself
should he be interested in external economies.
The scope of the following argument can usefully be indicated by
starting with a brief statement of its main conclusions. The first is that
if the party imposing external diseconomies and the party suffering them
are able and willing to negotiate to their mutual advantage, state
intervention is unnecessary to secure optimum resource allocation.
The second is that the imposition of a tax upon the party imposing
external diseconomies can be a very complicated matter, even in
principle, so that the a priori prescription of such a tax is unwise.
To develop these and other points, let us begin by calling A the
person, firm or group (of persons or firms) which imposes a diseconomy
and B the person, firm or group which suffers it. How much B suffers
will in many cases depend not only upon the scale of A’s diseconomycreating activity, but also upon the precise nature of A’s activity and
upon B’s reaction to it. If A emits smoke, for example, B’s loss will
depend not only upon the quantity emitted but also upon the height
of A’s chimney and upon the cost to B of installing air-conditioning,
indoor clothes-dryers or other means of reducing the effect of the smoke.
Thus to ascertain the optimum resource allocation will frequently require an investigation of the nature and costs both of alternative
activities open to A and of the devices by which B can reduce the
impact of each activity. The optimum involves that kind and scale of
A’s activity and that adjustment to it by B which maximises the
algebraic sum of A’s gain and B’s loss as against the situation where A
* I am indebted to Professor Buchanan, Professor Coase, Mr. Klappholz, Dr.
Mishan and Mr. Peston for helpful comments on an earlier draft.
1 Journal of Law and Economics, Vol. III, October, 1960, Journal of Political Econ-
omy, June, 1962, and Economica, November, 1962, respectively.
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pursues no diseconomy-creating activity. Note that the optimum will
frequently involve B suffering a loss, both in total and at the margin.’
If A and B are firms, gain and loss can be measured in money terms
as profit differences. (In considering a social optimum, allowance has
of course to be made for market imperfections.) Now assuming that
they both seek to maximise profits, that they know about the available
alternatives and adjustments and that they are able and willing to
negotiate, they will achieve the optimum without any government
interference. They will internalize the externality by merger2, or they
will make an agreement whereby B pays A to modify the nature or
scale of its activity.3 Alternatively,4 if the law gives B rights against A,
A will pay B to accept the optimal amount of loss imposed by A.
If A and B are people, their gain and loss must be measured as the
amount of money they respectively would pay to indulge in and prevent
A’s activity. It could also be measured as the amount of money they
respectively would require to refrain from and to endure A’s activity,
which will be different unless the marginal utility of income is constant.
We shall assume that it is constant for both A and B, which is reasonable when the payments do not bulk large in relation to their incomes.5
Under this assumption, it makes no difference whether B pays A or,
if the law gives B rights against A, A compensates B.
Whether A and B are persons or firms, to levy a tax on A which is
not received as damages or compensation by B may prevent optimal
resource allocation from being achieved-still assuming that they can
and do negotiate.6 The reason is that the resource allocation which
maximises A’s gain less B’s loss may differ from that which maximises
A’s gain less A’s tax less B’s loss.
The points made so far can usefully be presented diagrammatically
(Figure 1). We assume that A has only two alternative activities, I and
II, and that their scales and B’s losses are all continuously variable. Let
us temporarily disregard the dotted curve in the right-hand part of the
diagram. The area under A’s curves then gives the total gain to A.
The area under B’s curves gives the total loss to B after he has made the
best adjustment possible to A’s activity. This is thus the direct loss as
reduced by adjustment, plus the cost of making that adjustment.
If A and B could not negotiate and if A were unhampered by restrictions of any sort, A would choose activity I at a scale of OR. A scale of
OS would obviously give a larger social product, but the optimum is
clearly activity II at scale OJ, since area 2 is greater than area 1. Now
B will be prepared to pay up to (la+ lb – 2a) to secure this result, while
IBuchanan-Stubblebine, pp. 380-1.
2Davis-Whinston, pp. 244, 252, 256; Coase, pp. 16-17.
‘ Coase, p. 6; Buchanan-Stubblebine agree, p. 383.
4 See previous references.
6 Dr. Mishan has examined the welfare criterion for the case where the only variabJ.e
is the scale of A’s activity, but where neither A nor B has a constant marginal utility
of income; Cf. his paper” Welfare Criteria for External Effects “, American Economic
Review, September, 1961.
6 Buchanan-Stubblebine, pp. 381-3.
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AXs marginal X
rgaarglnal g
|:-t;sI min al gain
t I (urJu2 -fy’Jf,1Y1=F and when IujJx4 ( -) u2lByl/2rJu fyl/frsy s when ul/z4> 0.
In (9), X, and Yj are used to designate, respectively, the activities of
A and B in consuming or in utilising some numeraire commodity or
service that, by hypothesis, is available on identical terms to each of
them. As is indicated by the transposition of signs in (9), the conditions
for Pareto relevance differ as between external diseconomies and
economies. This is because the ” direction ” of change desired by A on
the part of B is different in the two cases. In stating the conditions for
Pareto relevance under ordinary two-person trade, this point is of no
significance since trade in one good flows only in one direction. Hence,
absolute values can be used.
The condition, (9), states that A’s marginal rate of substitution
between the activity, Yl, and the numeraire activity must be greater
than the ” net ” marginal rate of substitution between the activity and
the numeraire activity for B. Otherwise, ” gains from trade ” would
not exist between A and B.
Note, however, that when B has achieved utility-maximising equilibrium,
(10) U1/uYBj =f/B IfB
That is to say, the marginal rate of substitution in consumption or
utilisation is equated to the marginal rate of substitution in production
or exchange, i.e., to marginal cost. When (10) holds, the terms in the
brackets in (9) mutually cancel. Thus, potentially relevant marginal
externalities are also Pareto-relevant when B is in utility-maximisimg
equilibrium. Some trade is possible.
Pareto equilibrium is defined to be present when,
(11) (- )ujI/UX = [u1/uB _fB/If”], and when uA /uAx 0.
Condition (11) demonstrates that marginal externalities may continue
to exist, even in Pareto equilibrium, as here defined. This point may be
shown by reference to the special case in which the activity in question
may be undertaken at zero costs. Here Pareto equilibrium is attained
when the marginal rates of substitution in consumption or utilisation
for the two persons are precisely offsetting, that is, where their interests
are strictly opposed, and not where the left-hand term vanishes.
What vanishes in Pareto equilibrium are the Pareto-relevant externalities. It seems clear that, normally, economists have been referring
only to what we have here called Pareto-relevant externalities when
1 We are indebted to Mr. M. McManus of the University of Birmingham for
pointing out to us an error in an earlier formulation of this and the following
similar conditions.
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they have, implicitly or explicitly, stated that external effects are not
present when a position on the Pareto optimality surface is attained.1
For completeness, we must also consider those potentially relevant
infra-marginal externalities. Refer to the discussion of these as summarised in (8) above. The question is now to determine whether or not,
A, the externally affected party, can reach some mutually satisfactory
agreement with B, the acting party, that will involve some discrete
(non-marginal) change in the scope of the activity, Y1. If, over some
range, any range, of the activity, which we shall designate by AY1, the
rate of substitution between Y1 and Xj for A exceeds the ” net” rate
of substitution for B, the externality is Pareto-relevant. The associated
changes in the utilisation of the numeraire commodity must be equal
for the two parties. Thus, for external economies, we have
(12) /j f>()[/Y // -/ Af7A , and the
same with the sign in parenthesis transposed for external diseconomies.
The difference to be noted between (12) and (9) is that, with inframarginal externalities, potential relevance need not imply Pareto
relevance. The bracketed terms in (12) need not sum to zero when B
is in his private utility-maximising equilibrium.
We have remained in a two-person world, with one person affected
by the single activity of a second. However, the analysis can readily
be modified to incorporate the effects of this activity on a multi-person
group. That is to say, B’s activity, Y1, may be allowed to affect several
parties simultaneously, several A’s, so to speak. In each case, the
activity can then be evaluated in terms of its effects on the utility of e
person. Nothing in the construction need be changed. The only stage
in the analysis requiring modification explicitly to take account of the
possibilities of multi-person groups being externally affected is that
which involves the condition for Pareto relevance and Pareto equilibrium.
For a multi-person group (A1, A2, …., A), any one or all of whom
may be externally affected by the activity, Y1, of the single person, B,
the condition for Pareto relevance is,
(9A) (- ) 27 ujl/u?> turl/ur -fy1/frJ, y when uAi/ux< 0, and, E url/2Xs> (- Y,/r-y/ rsy Yi when uAy/lu>x.
evaluation of the activity by B. Again, in prvate equilibrium for B,
1This applies to the authors of this paper. For recent discussion of external
effects when we have clearly intended only what we here designate as Pareto-relevant,
see James M. Buchanan, ” Politics, Policy, and the Pigovian Margins “, Economica,
vol. xxvix (1962), pp. 17-28, and, also, James M. Buchanan and Gordon Tullock,
The Calculus of Consent, Ann Arbor, 1962.
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marginal externalities are Pareto-relevant, provided that we neglect
the important element involved in the costs of organising group decisions. In the real world, these costs of organising group decisions
(together with uncertainty and ignorance) will prevent realisation of
some “6 gains from trade “-just as they do in organised markets. This
is as true for two-person groups as it is for larger groups. But this does
not invalidate the point that potential ” gains from trade” are available. The condition for Pareto equilibrium and for the infra-marginal
case summarised in (11) and (12) for the two-person model can readily
be modified to allow for the externally affected multi-person group.
The distinctions developed formally in Section I may be illustrated
diagrammatically and discussed in terms of a simple descriptive
example. Consider two persons, A and B, who own adjoining units of
residential property. Within limits to be noted, each person values
privacy, which may be measured quantitatively in terms of a single
criterion, the height of a fence that can be constructed along the common boundary line. We shall assume that B’s desire for privacy holds
over rather wide limits. His utility increases with the height of the
fence up to a reasonably high level. Up to a certain minimum height,
A’s utility also is increased as the fence is made higher. Once this
minimum height is attained, however, A’s desire for privacy is assumed
to be fully satiated. Thus, over a second range, A’s total utility does not
change with a change in the height of the fence. However, beyond a
certain limit, A’s view of a mountain behind B’s property is progressively obscured as the fence goes higher. Over this third range, therefore, A’s utility is reduced as the fence is constructed to higher levels.
Finally, A will once again become wholly indifferent to marginal
changes in the fence’s height when his view is totally blocked out.
We specify that B possesses the sole authority, the only legal right,
to construct the fence between the two properties.
The preference patterns for A and for B are shown in Figure 1, which
is drawn in the form of an Edgeworth-like box diagram. Note, however,
that the origin for B is shown at the upper left rather than the upper
right corner of the diagram as in the more normal usage. This modification is necessary here because only the numeraire good, measured
along the ordinate, is strictly divisible between A and B. Both must
adjust to the same height of fence, that is, to the same level of the
activity creating the externality.
As described above, the indifference contours for A take the general
shape shown by the curves aa, a’a’, while those for B assume the shapes,
bb, b’b’. Note that these contours reflect the relative evaluations, for
A and B, between money and the activity, Y1. Since the costs of undertaking the activity, for B, are not incorporated in the diagram, the
” contract locus ” that might be derived from tangency points will
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undertaken at zero costs.
Figure 2 depicts the marginal evaluation curves for A and B, as
derived from the preference fields shown in Figure 1, along with some
incorporation of costs. These curves are derived as follows: Assume
Figure 1
Height of Fence ->
(B’s Activity)
an initial distribution of” money ” between A and B, say, that shown
at M on Figure 1. The marginal evaluation of the activity for A is then
derived by plotting the negatives (i.e., the mirror image) of the slopes
of successive indifference curves attained by A as B is assumed to
increase the. height of the fence from zero. These values remain positive for a range, become zero over a second range, become negative
for a third. and. finally. return to zero again.”
I For an early use of marginal evaluation curves, see J. R. Hicks, ” The Four
Consumer’s Surpluses “, Review of Economic Studies, vol. xi (1943), pp. 3141.
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B’s curves of marginal evaluation are measured downward from the
upper horizontal axis or base line, for reasons that will become apparent.
The derivation of B’s marginal evaluation curve is somewhat more
complex than that for A. This is because B, who is the person authorised to undertake the action, in this case the building of the fence,
must also bear the full costs. Thus, as B increases the scope of the
activity, his real income, measured in terms of his remaining goods and
services, is reduced. This change in the amount of remaining goods and
t< Figure 2 ME A '4 S .% a~~~~m ME MEB X , +S - NMEB= MSCA MC services will, of course, affect his marginal evaluation of the activity in question. Thus, the marginal cost of building the fence will determine, to some degree, the marginal evaluation of the fence. This necessary interdependence between marginal evaluation and marginal cost complicates the use of simple diagrammatic models in finding or locating a solution. It need not, however, deter us from presenting the solution diagrammatically, if we postulate that the marginal evaluation curve, as drawn, is based on a single presumed cost relationship. This done, we may plot B's marginal evaluation of the activity from the negatives of the slopes of his indifference contours attained as he constructs the fence to higher and higher levels. B's marginal evaluation, shown in Figure 2, remains positive throughout the range to the point H6, where it becomes zero. This content downloaded from on Thu, 03 Feb 2022 04:05:12 UTC All use subject to 380 ECONOMICA [NOVEMBER The distinctions noted in Section I are easily related to the construction in Figure 2. To A, the party externally affected, B's potential activity in constructing the fence can be assessed independently of any prediction of B's actual behaviour. Thus, the activity of B would, (1) exert marginal external economies which are potentially relevant over the range OH1; (2) exert infra-marginal external economies over the range HH12, which are clearly irrelevant since no change in B's behaviour with respect to the extent of the activity would increase A's utility; (3) exert marginal external diseconomies over the range H2H4 which are potentially relevant to A; and, (4) exert infra-marginal external economies or diseconomies beyond H4, the direction of the effect being dependent on the ratio between the total utility derived from privacy and the total reduction in utility derived from the obstructed view. In any case, the externality is potentially relevant. To determine Pareto relevance, the extent of B's predicted performance must be determined. The necessary condition for B's attainment of " private " utility-maximising equilibrium is that marginal costs, which he must incur, be equal to his own marginal evaluation. For simplicity in Figure 2, we assume that marginal costs are constant, as shown by the curve, MC. Thus, B's position of equilibrium is shown at HB, within the range of marginal external diseconomies for A. Here the externality imposed by. B's behaviour is clearly Pareto-relevant: A can surely work out some means of compensating B in exchange for B's agreement to reduce the scope of the activity-in this example, to reduce the height of the fence between the two properties. Diagrammatically, the position of Pareto equilibrium is shown at H3 where the marginal evaluation of A is equal in absolute value, but negatively, to the "net" marginal evaluation of B, drawn as the curve NMEB. Only in this position are the conditions specified in (11), above, satisfied.' m Aside from the general classification of externalities that is developed the approach here allows certain implications to be drawn, implications that have not, perhaps, been sufficiently recognised by some welfare economists. The analysis makes it quite clear that externalities, external effects, may remain even in full Pareto equilibrium. That is to say, a position 1 This diagrammatic analysis is necessarily oversimplified in the sense that the Pareto equilibrium position is represented as a unique point. Over the range between the " private " equilibrium for B and the point of Pareto equilibrium, the sort of bargains struck between A and B will affect the marginal evaluation curves of both individuals within this range. Thus, the more accurate analysis would suggest a " contract locusV" of equilibrium points. At Pareto equilibrium, however, the condition'shown in the diagrammatic presentation holds, and the demonstration of this fact rather than the location of the solution is the aim of this diagrammatics. This content downloaded from on Thu, 03 Feb 2022 04:05:12 UTC All use subject to 1962] EXTERNALITY 381 may be classified as Pareto-optimal or efficient despite the fact that, at the marginal, the activity of one individual externally affects the utility of another individual. Figure 2 demonstrates this point clearly. Pareto equilibrium is attained at H3, yet B is imposing marginal external diseconomies on A. This point has significant policy implications for it suggests that the observation of external effects, taken alone, cannot provide a basis for judgment concerning the desirability of some modification in an existing state of affairs. There is not a prima facie case for intervention in all cases where an externality is observed to exist.' The internal benefits from carrying out the activity, net of costs, may be greater than the external damage that is imposed on other parties. In full Pareto equilibrium, of course, these internal benefits, measured in terms of some numeraire good, net of costs, must be just equal, at the margin, to the external damage that is imposed on other parties. This equilibrium will always be characterised by the strict opposition of interests of the two parties, one of which may be a multi-person group. In the general case, we may say that, at full Pareto equilibrium, the presence of a marginal external diseconomy implies an offsetting marginal internal economy, whereas the presence of a marginal externa economy implies an offsetting marginal internal diseconomy. In " private " equilibrium, as opposed to Pareto equilibrium, these net internal economies and diseconomies would, of course, be eliminated by the utility-maximising acting party. In Pareto equilibrium, these remain because the acting party is being compensated for " suffering" internal economies and diseconomies, that is, divergencies between " private " marginal costs and benefits, measured in the absence of compensation. As a second point, it is useful to relate the whole analysis here to the more familiar Pigovian discussion concerning the divergence between marginal social cost (product) and marginal private cost (product). By saying that such a divergence exists, we are, in the terms of this paper, saying that a marginal externality exists. The Pigovian terminology tends to be misleading, however, in that it deals with the acting party to the exclusion of the externally affected party. It fails to take into account the fact that there are always two parties involved in a single externality relationship.2 As we have suggested, a marginal externality is Pareto-relevant except in the position of Pareto equilibrium; gains from trade can arise. But there must be two parties to any trading arrangement. The externally affected party must compensate the acting party for modifying his behaviour. The Pigovian terminology, through its concentration on the decision-making of the acting party alone, tends to obscure the two-sidedness of the bargain that must be made. 1 Cf. Paul A. Samuelson, Foundations of Economic Analysis, Cambridge, Mass., 1948, p. 208, for a discussion of the views of various writers. 2This criticism of the Pigovian analysis has recently been developed by R. H. Coase; see his " The Problem of Social Cost ", Journal of Law and Economics, vol. m (1960), pp. 1-44. This content downloaded from on Thu, 03 Feb 2022 04:05:12 UTC All use subject to 382 ECONOMCA [NOVEMBR To illustrate this point, assume that A, the externally affected party in our model, successfully secures, through the auspices of the " state ", the levy of a marginal tax on B's performance of the activity, Y1. Assume further that A is able to secure this change without cost to himself. The tax will increase the marginal cost of performing the activity for B, and, hence, will reduce the extent of the activity attained in B's " private" equilibrium. Let us now presume that this marginal tax is levied "correctly " on the basis of a Pigovian calculus; the rate of tax at the margin is made equal to the negative marginal evaluation of the activity to A. Under these modified conditions, the effective marginal cost, as confronted by B, may be shown by the curve designated as MSCB in Figure 2. A new " private " equilibrium for B is shown at the quantity, H3, the same level designated as Pareto equilibrium in our earlier discussion, if we neglect the disturbing interdependence between marginal evaluation and marginal costs. Attention solely to the decision calculus of B here would suggest, perhaps, that this position remains Pareto-optimal under these revised circumstances, and that it continues to qualify as a position of Pareto equilibrium. There is no divergence between marginal private cost and marginal social cost in the usual sense. However, the position, if attained in this manner, is clearly neither one of Pareto optimality, nor one that may be classified as Pareto equilibrium. In this new " private" equilibrium for B, (13) uYu- By BY _ X y4 Y where uy,/uA, represents the marginal tax imposed on B as he performs the activity, Y1. Recall the necessary condition for Pareto relevance defined in (9) above, which can now be modified to read, (9B) (-) UAL/XA,> [uR/z4ju -fi/f y+Ul/ux] y when AJUA
and UAUX (, when uj juX>O.
In (9B), Y1 represents the ” private ” equilibr
mined by B, after the ideal Pigovian tax is imposed. As before, the
bracketed terms represent the ” net” marginal evaluation of the
activity for the acting party, B, and these sum to zero when equilibrium is reached. So long as the left-hand term in the inequality re-
mains non-zero, a Pareto-relevant marginal externality remains,
despite the fact that the full ” Pigovian solution ” is attained.
The apparent paradox here is not difficult to explain. Since, as
postulated, A is not incurring any cost in securing the change in B’s
behaviour, and, since there remains, by hypothesis, a marginal diseconomy, further ” trade ” can be worked out between the two parties.
Specifically, Pareto equilibrium is reached when,
(JJA) (-)U4,1/U4 =[UyIIUPf _f_ /f l ?ullUAU] when uA1JuA O.
Diagrammatically, this point may be made with reference to
Figure 2. If a unilaterally imposed tax, corresponding to the marginal
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evaluation of A, is placed on B’s performance of the activity, the new
position of Pareto equilibrium may be shown by first subtracting the
new marginal cost curve, drawn as MSCB, from B’s marginal evaluation curve. Where this new ” net” marginal evaluation curve, shown
as the dotted curve between points H3 and K, cuts the marginal evaluation curve for A, a new position of Pareto equilibrium falling between
H2 and H3 is located, neglecting the qualifying point discussed in
Footnote 1, page 380.
The important implication to be drawn is that full Pareto equilibrium
can never be attained via the imposition of unilaterally imposed taxes
and subsidies until all marginal externalities are eliminated. If a taxsubsidy method, rather than ” trade “, is to be introduced, it should
involve bi-lateral taxes (subsidies). Not only must B’s behaviour be
modified so as to insure that he will take the costs externally imposed
on A into account, but A’s behaviour must be modified so as to insure
that he will take the costs ” internally ” imposed on B into account.
In such a double tax-subsidy scheme, the necessary Pareto conditions
would be readily satisfied.’
In summary, Pareto equilibrium in the case of marginal externalities
cannot be attained so long as marginal externalities remain, until and
unless those benefiting from changes are required to pay some ” price”
for securing the benefits.
A third point worthy of brief note is that our analysis allows the
whole treatment of externalities to encompass the consideration of
purely collective goods. As students of public finance theory will have
recognised, the Pareto equilibrium solution discussed in this paper is
similar, indeed is identical, with that which was presented by Paul
Samuelson in his theory of public expenditures.2 The summed marginal
rates of substitution (marginal evaluation) must be equal to marginal
costs. Note, however, that marginal costs may include the negative
marginal evaluation of other parties, if viewed in one way. Note, also,
that there is nothing in the analysis which suggests its limitations to
purely collective goods or even to goods that are characterised by
significant externalities in their use.
Our analysis also lends itself to the more explicit point developed in
Coase’s recent paper.3 He argues that the same ” solution ” will tend
to emerge out of any externality relationship, regardless of the structure
of property rights, provided only that the market process works
smoothly. Strictly speaking, Coase’s analysis is applicable only to
inter-firm externality relationships, and the identical solution emerges
only because firms adjust to prices that are competitively determined.
In our terms of reference, this identity of solution cannot apply because
of the incomparability of utility functions. It remains true, however,
1 Although developed in rather different terminology, this seems to be closely in
accord with Coase’s analysis. Cf. R. H. Coase, loc. cit.
2 Paul A. Samuelson, ” The Pure Theory of Public Expenditure “, Review of
Economics and Statistics, vol. xxxvi (1954), pp. 386-9.
8 R. H. Coase, loc. cit.
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that the basic characteristics of the Pareto equilibrium position remain
unchanged regardless of the authority undertaking the action. This
point can be readily demonstrated, again with reference to Figure 2.
Let is assume that Figure 2 is now redrawn on the basis of a different
legal relationship in which A now possesses full authority to construct
the fence, whereas B can no longer take any action in this respect. A
will, under these conditions, ” privately ” construct a fence only to the
height Ho, where the activity clearly exerts a Pareto-relevant marginal
external economy on B. Pareto equilibrium will be reached, as before,
at H3, determined, in this case, by the intersection of the ” net ”
marginal evaluation curve for A (which is identical to the previously