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Benevolent preferences and pure public goods
Journal
of Public
Economics
BENEVOLENT
30 (1986) 129-134.
PREFERENCES
North-Holland
AND
PURE
PUBLIC
GOODS
S.Q. LEMCHE* University of Calgary, Calgary, Alberta, Canada T2N IN4 Received July 1984, revised version
received January
1986
Introduction In Samuelson (1954) it was shown that if consumers have ‘selfish’ preferences that satisfy the usual curvature, smoothness, and monotonicity conditions, and there exist both private and ‘pure’ public goods, then an allocation is Pareto optimal if and only if it can be sustained as a Lindahl equilibrium. Here it will be shown that in general Samuelson’s result cannot be expected to hold when each consumer’s preferences depend not only on his own but also on the consumptions of other consumers. In the particular case analyzed here, the consumers’ preferences are defined on the set of allocations for the economy and it is assumed that there exists a relationship, called benevolent interdependence, between each consumer’s preferences and those of others.’ In section 3 it is shown that benevolently interdependent preferences together with the existence of pure public goods implies that the consumers’ preferences are equal in the sense that they yield the same aggregate ordinal ranking of allocations in the economy. (This equality of preferences over allocations does not necessarily obtain in the absence of public goods.) The equality of the consumers’ aggregate preferences is a surprisingly strong implication of assuming benevolent preference interdependence in an economy with pure public goods. Perhaps ‘too strong’ and may suggest that benevolence as defined here in terms of both private and public goods yields more formal structure than one may be prepared to accept. However, it needs to be pointed out that the equality of the consumers’ aggregate preferences does not necessarily mean that the consumers are in all respects identical like ‘perfect clones’. For example, the equality of aggregate pre*The results presented here are based on results contained in my Ph.D. thesis supervised by L. Hurwicz and M.K. Richter of the University of Minnesota. My indebtedness is gratefully acknowledged. The many conversations with and suggestions from G.C. Archibald, C. Blackorby and D. Donaldson have been most helpful as has the comments from John Roberts and an anonymous referee. ‘See section 2. 0047-2727/86/$3.50
0
1986, Elsevier Science Publishers
B.V. (North-Holland)
130
S.Q. Lemche, Benevolent
preferences
ferences does not imply that consumer i’s preferences for i’s private commodities are similar to j’s preferences for j’s private commodities and therefore the , q,:ality of aggregate preferences is consistent with each consumer having preferences that express a unique individuality. The notion of benevolence defined below is closely related to that defined in, for example, Bergstrom (1970, p. 386); the only differences are (1) that the economy specified below has both private and ‘pure’ public goods, while Bergstrom’s economy has only private goods, and (2) that Bergstrom’s requirement that preferences are separable between individuals is unnecessary for our purposes and is therefore not made.’
2. The model To keep matters simple it is assumed that the economy has two consumers (indices i, j= 1,2); one type of private good (denoted by x); and one ‘pure’ public good (denoted by y). Consumer i’s consumption set is C,=X, x Yc R2 (where R2 is the two-dimensional Euclidean space) and the allocation set for the economy is .4=X, xX,x YcR 3. Consumer i’s preference relation, written ki, is a complete preordering on A and is representable by an (ordinal) utility function U’: A+R. The preferences ki are assumed to be classical in the sense that the utility function Vi(.) is monotonely strictly increasing, of A) and such that if quasi-concave, differentiable on A+ (the interior ZE A\A+, then U’(z) > U’(F) for all ZE A+.3 2.1. Private and ‘pure’ public goods With preferences defined on the allocation set of the economy it is important that the notions of private and ‘pure’ public goods, respectively, are clearly distinguished. Here we shall follow the terminology defined by Milleron (1972, table 1, p. 423) and, accordingly, the private goods are characterized as collective concern variables of type D - i.e. private goods with free disposal, exclusion of use, and externality. In addition to the usual private goods characteristics of free disposal and exclusion of use there are externalities associated with private goods because each consumer’s consumption of his private good in general affects the preferences of the other consumer. *Benevolence is similar to, but more restrictive than, the notion of non-malevolence analyzed in Winter (1969), Bergstrom (1970) and Rader (1980). However, if the preferences are continuous and have local non-saturation, then non-malevolence and benevolence are equivalent. The notion of non-paternalism studied in Archibald and Donaldson (1976) is similar to but less restrictive than both benevolence and non-malevolence. defining classical ‘The notation z E A\A+ means that z E A but z 6 A+. This last condition preference implies that such preferences have ‘interior solutions’ ~ i.e. if allocations in A+ are available, then the consumers’ preference maximizing choices are always in A +.
S.Q. Lemche, Benevolent preferences
131
For an allocation z=(xr, x2, y) EA the component xi (i= 1,2) is called consumer i’s consumption of the private good. A ‘pure’ public good is in Milleron’s terminology an environment variable [Milleron (1972)]. In the context of interdependent preferences a ‘pure’ public good is similar to a private good in that both are collective concern variables but a ‘pure’ public good is different because there is neither free disposal nor any exclusion of use associated with the consumption of a ‘pure’ public good: every consumer always consumes the total amount available. If z= (x,, x2, y) EA is an allocation, then y is the amount of the ‘pure’ public good consumed by both consumers and the vector ci=(xi, y) E Ci is called consumer i’s consumption at z. 2.2. Benevolence In order to define benevolence it is convenient first to introduce the notion of a consumer’s induced preferences. Given 2, E X, let A(XZ) = {z = (x1, x2,y)~A:x2=X2}. Since A(?,) is a subset of A, then the allocations in A(XZ) are ranked according to ki (i= 1,2). First we consider C, (consumer l’s consumption set) and define consumer i’s induced preferences, written kj(XJ, on C, as follows. Definition 1 (induced preferences). For any X, EX~ consumer i’s induced preferences on C,, written 2:(X& are such that for each z=(x1,X2,y) and z’=(x;, X,, y’) in .4(x,) and corresponding c1 =(x1, y) and c’, =(x1, y’) in C,. Cl
Similarly, on Cz.
2: &k;>
if and only if
z 2 i z’.
for any X, EX~ we have consumer
(1)
i’s induced
preferences
kf(XJ
In general the induced preferences kj(XJ on C, will vary both with the consumer (index i= 1,2) and with the value of Xz in X,; similarly kf(Xi) on C, will vary with both i and 3,. Definition 2 (benevolence). The preferences ki (i= 1,2) are benevolent if and only if for each z=(x,, x2, y) E A the induced preferences 2:(x2) and 2:(x,) on C, are equal and the induced preferences zf(x,) and k:(xi) on C, are equal - i.e. for any xj~Xj and any ci,c: E Ci (i, j= 1,2 and ifj). ci 2
‘; (Xj)&
if and only if ci 2; (xj)ci.
(2)
Benevolence implies some quite significant restrictions on the relationship between the consumers’ preferences. In particular, if there are ‘pure’ public
S.Q. Lemche,l3enevolentpreferences
132
goods and the preferences are classical ki (i= 1,2) are in fact equal.
and benevolent,
then the preferences
3. The equality of preferences For i,j=l,2.
z =(x1, x2, y) E A +, write
Proposition Proof.
I.
Benevolence
Benevolence
implies
U;(z) = ~U’(Z)/~X, and
U;(z) = aU’(z)/8y
implies that 2 1 is equal to 2 2 on A+. tha& for i = 1,2,
u;(z)/u;(z) = u;(z)/u,2(2) which, by rearrangement,
for
(3)
yields:
u:(z)/u~(z) = u;(z)/u;(z) = u;(z)/q(z). Further
rearrangement
wYw4
of the left-hand
equality
in (4) leads to
= wwt(4
(5)
which, together with (3) shows that the consumers have equal marginal rates of substitution between all pairs of commodities. Since the preferences are classical, then U’(.) and U”(.) are equal up to monotone transformations on A +. This proves Proposition 1.
4. Pareto optimal and Lindahl equilibrium allocations In Samuelson (1954,1969) it was shown that Lindahl equilibrium is Pareto optimal in the presence of ‘pure’ public goods and with classical but ‘selfish preferences. In a Lindahl equilibrium all consumers pay the same price for each type of private commodity and that price in turn is equal to the price received by producers. With respect to a ‘pure’ public good each consumer pays an individualized (Lindahl) price such that the sum of these prices is equal to the price received by producers. However, if preferences are classical and benevolent (instead of ‘selfish’), then in general a Pareto optimum cannot be sustained as a Lindahl equilibrium.4 To see this, suppose that U = A+R and that U’(z) = U’(z) = U(z), for all z E A+. Furthermore, let F(x, +x2, y) = 0 be the economy’s net (aggregate) production function and assume that F(.) is concave, differentiable and such that F(Z,+%,,j)=O for some Z=(TI,%z,j)~A+. 4The following
owes much to conversations
with D. Donaldson
of U.B.C.
S.Q. Lemche, Benevolent
preferences
133
An allocation 2 = (a,, &, 9) E A is Pareto optimal Definition 3 (Pareto optimality). if F(j?l+iZ,j)=O and there is no z=(x,,x,,y)~A such that F(x,+x,,y)=O and U’(z) 2 U’(3) (i= 1,2) with strict inequality for some i. If the preferences are equal and the utility representation U(z) is common to both consumers, then Z = (a,, A,, 9) E A is Pareto optimal if and only if B maximizes U(z) subject to F(x,+x,,y)=O. With classical preferences the first-order imply that if h E A + is Pareto optimal, then
conditions
for Pareto
U,o=U,o=F,(%+~,,9)>0 u,(;)
u,(4
F,@,+L 9)
optimality
(6)
’
Thus, if &E A + is Pareto optimal, then each consumer has the same marginal rate of substitution between ‘his own’ private good and the ‘pure’ public good and, moreover, this marginal rate of substitution in consumption is equal to the marginal rate of substitution between the private and the public goods in production. Definition 4 rium if F(R, private good ‘pure’ public
(Lindahl equilibrium). An allocation z^E A is a Lindahl equilib+z?~, 9) =0 and there exists a (numeraire) price P,= 1 of the and (relative) individualized Lindahl prices Pi>0 (i= 1,2) of the good in terms of the private good such that:
(i) (utility maximization) then xi - z?~+ Pi(y - 9) > 0;
for j# i (i,j=
(ii) (profit maximization) (a,+a,)+(P:+P3)(y-9)~0.
if ZE A and
Proposition
2.
Lindahl equilibrium
1,2) if z E A(kj)
and
F(x, +x,, y) =O, then
is not Pareto
U’(z) > U’(2), (x1 +x,) -
optimal.
Proof: From the assumptions that F(x, +x,, y) =0 for some ZE A+ and that preferences are classical it follows that if &E A is a Lindahl equilibrium then SEA, and
9uj(s)=PL>O
(i=1,2)
(7)
and
F,(% + L9) =p;+p;>o.
F,(% + %,9)
03)
S.Q. Lemche, Benevolent preferences
134
Because preferences are equal and the utility representation to both consumers, then (7) can be re-written as: LJ (2)
-=I$>0
U(z) is common
(i= 1,2).
ui(2)
Combined,
(8) and (9) lead to
which is inconsistent with the equalities (6) that must obtain if 2 is Pareto optimal. Thus, a Lindahl equilibrium cannot be Pareto optimal and Proposition 2 has been proved.
5. Conclusion A Lindahl equilibrium is not Pareto optimal (efficient) because it does not recognize that the private goods are ‘collective concern variables’ and fails to price the externalities associated with private goods when the preferences are (benevolently) interdependent. In other words, the inefficiency of a Lindahl equilibrium is caused by a pricing rule for the private goods which, on the one hand, fails to charge each consumer for all goods that affect his preferences and, on the other hand, fails to divide properly the costs of each private good among all those whose preferences are affected by it. The result is that the Lindahl equilibrium price of the private good is ‘too high’, and those of the public good ‘too low’, for such an equilibrium to be efficient.
References Archibald, G.C. and David Donaldson, 1976, Non-paternalism and the basic theorems of welfare economics, The Canadian Journal of Economics IX, no. 3,492-507. Bergstrom, T.E., 1970, A ‘Scandinavian consensus’ solution for efficient income distribution among non-malevolent consumers, Journal of Economic Theory 2, 3833398. Lemche, S.Q., 1978, The first optimality theorem with interdependent preferences and public commodities, Unpublished Ph.D. dissertation, University of Minnesota. Milleron, J.C., 1972, Theory of value with public goods: A survey article, Journal of Economic Theory 5, 419477. Rader, J.T., 1980, The second theorem of welfare economics when utilities are interdependent, Journal of Economic Theory 23,42@424. Samuelson, P.A., 1954, The pure theory of public expenditure, Review of Economics and Statistics XXXVI, 3877389. Samuelson, P.A., 1969, Pure theory of public expenditure and taxation, in: J. Margolis and H. Guitton, eds., Public Economics. Proceedings of I.E.A. Conference (Macmillan and St. Martin’s Press). Winter, S.G., 1969, A simple remark on the second optimality theorem of welfare economics, Journal of Economic Theory 1, 999103.
of Public
Economics
BENEVOLENT
30 (1986) 129-134.
PREFERENCES
North-Holland
AND
PURE
PUBLIC
GOODS
S.Q. LEMCHE* University of Calgary, Calgary, Alberta, Canada T2N IN4 Received July 1984, revised version
received January
1986
Introduction In Samuelson (1954) it was shown that if consumers have ‘selfish’ preferences that satisfy the usual curvature, smoothness, and monotonicity conditions, and there exist both private and ‘pure’ public goods, then an allocation is Pareto optimal if and only if it can be sustained as a Lindahl equilibrium. Here it will be shown that in general Samuelson’s result cannot be expected to hold when each consumer’s preferences depend not only on his own but also on the consumptions of other consumers. In the particular case analyzed here, the consumers’ preferences are defined on the set of allocations for the economy and it is assumed that there exists a relationship, called benevolent interdependence, between each consumer’s preferences and those of others.’ In section 3 it is shown that benevolently interdependent preferences together with the existence of pure public goods implies that the consumers’ preferences are equal in the sense that they yield the same aggregate ordinal ranking of allocations in the economy. (This equality of preferences over allocations does not necessarily obtain in the absence of public goods.) The equality of the consumers’ aggregate preferences is a surprisingly strong implication of assuming benevolent preference interdependence in an economy with pure public goods. Perhaps ‘too strong’ and may suggest that benevolence as defined here in terms of both private and public goods yields more formal structure than one may be prepared to accept. However, it needs to be pointed out that the equality of the consumers’ aggregate preferences does not necessarily mean that the consumers are in all respects identical like ‘perfect clones’. For example, the equality of aggregate pre*The results presented here are based on results contained in my Ph.D. thesis supervised by L. Hurwicz and M.K. Richter of the University of Minnesota. My indebtedness is gratefully acknowledged. The many conversations with and suggestions from G.C. Archibald, C. Blackorby and D. Donaldson have been most helpful as has the comments from John Roberts and an anonymous referee. ‘See section 2. 0047-2727/86/$3.50
0
1986, Elsevier Science Publishers
B.V. (North-Holland)
130
S.Q. Lemche, Benevolent
preferences
ferences does not imply that consumer i’s preferences for i’s private commodities are similar to j’s preferences for j’s private commodities and therefore the , q,:ality of aggregate preferences is consistent with each consumer having preferences that express a unique individuality. The notion of benevolence defined below is closely related to that defined in, for example, Bergstrom (1970, p. 386); the only differences are (1) that the economy specified below has both private and ‘pure’ public goods, while Bergstrom’s economy has only private goods, and (2) that Bergstrom’s requirement that preferences are separable between individuals is unnecessary for our purposes and is therefore not made.’
2. The model To keep matters simple it is assumed that the economy has two consumers (indices i, j= 1,2); one type of private good (denoted by x); and one ‘pure’ public good (denoted by y). Consumer i’s consumption set is C,=X, x Yc R2 (where R2 is the two-dimensional Euclidean space) and the allocation set for the economy is .4=X, xX,x YcR 3. Consumer i’s preference relation, written ki, is a complete preordering on A and is representable by an (ordinal) utility function U’: A+R. The preferences ki are assumed to be classical in the sense that the utility function Vi(.) is monotonely strictly increasing, of A) and such that if quasi-concave, differentiable on A+ (the interior ZE A\A+, then U’(z) > U’(F) for all ZE A+.3 2.1. Private and ‘pure’ public goods With preferences defined on the allocation set of the economy it is important that the notions of private and ‘pure’ public goods, respectively, are clearly distinguished. Here we shall follow the terminology defined by Milleron (1972, table 1, p. 423) and, accordingly, the private goods are characterized as collective concern variables of type D - i.e. private goods with free disposal, exclusion of use, and externality. In addition to the usual private goods characteristics of free disposal and exclusion of use there are externalities associated with private goods because each consumer’s consumption of his private good in general affects the preferences of the other consumer. *Benevolence is similar to, but more restrictive than, the notion of non-malevolence analyzed in Winter (1969), Bergstrom (1970) and Rader (1980). However, if the preferences are continuous and have local non-saturation, then non-malevolence and benevolence are equivalent. The notion of non-paternalism studied in Archibald and Donaldson (1976) is similar to but less restrictive than both benevolence and non-malevolence. defining classical ‘The notation z E A\A+ means that z E A but z 6 A+. This last condition preference implies that such preferences have ‘interior solutions’ ~ i.e. if allocations in A+ are available, then the consumers’ preference maximizing choices are always in A +.
S.Q. Lemche, Benevolent preferences
131
For an allocation z=(xr, x2, y) EA the component xi (i= 1,2) is called consumer i’s consumption of the private good. A ‘pure’ public good is in Milleron’s terminology an environment variable [Milleron (1972)]. In the context of interdependent preferences a ‘pure’ public good is similar to a private good in that both are collective concern variables but a ‘pure’ public good is different because there is neither free disposal nor any exclusion of use associated with the consumption of a ‘pure’ public good: every consumer always consumes the total amount available. If z= (x,, x2, y) EA is an allocation, then y is the amount of the ‘pure’ public good consumed by both consumers and the vector ci=(xi, y) E Ci is called consumer i’s consumption at z. 2.2. Benevolence In order to define benevolence it is convenient first to introduce the notion of a consumer’s induced preferences. Given 2, E X, let A(XZ) = {z = (x1, x2,y)~A:x2=X2}. Since A(?,) is a subset of A, then the allocations in A(XZ) are ranked according to ki (i= 1,2). First we consider C, (consumer l’s consumption set) and define consumer i’s induced preferences, written kj(XJ, on C, as follows. Definition 1 (induced preferences). For any X, EX~ consumer i’s induced preferences on C,, written 2:(X& are such that for each z=(x1,X2,y) and z’=(x;, X,, y’) in .4(x,) and corresponding c1 =(x1, y) and c’, =(x1, y’) in C,. Cl
Similarly, on Cz.
2: &k;>
if and only if
z 2 i z’.
for any X, EX~ we have consumer
(1)
i’s induced
preferences
kf(XJ
In general the induced preferences kj(XJ on C, will vary both with the consumer (index i= 1,2) and with the value of Xz in X,; similarly kf(Xi) on C, will vary with both i and 3,. Definition 2 (benevolence). The preferences ki (i= 1,2) are benevolent if and only if for each z=(x,, x2, y) E A the induced preferences 2:(x2) and 2:(x,) on C, are equal and the induced preferences zf(x,) and k:(xi) on C, are equal - i.e. for any xj~Xj and any ci,c: E Ci (i, j= 1,2 and ifj). ci 2
‘; (Xj)&
if and only if ci 2; (xj)ci.
(2)
Benevolence implies some quite significant restrictions on the relationship between the consumers’ preferences. In particular, if there are ‘pure’ public
S.Q. Lemche,l3enevolentpreferences
132
goods and the preferences are classical ki (i= 1,2) are in fact equal.
and benevolent,
then the preferences
3. The equality of preferences For i,j=l,2.
z =(x1, x2, y) E A +, write
Proposition Proof.
I.
Benevolence
Benevolence
implies
U;(z) = ~U’(Z)/~X, and
U;(z) = aU’(z)/8y
implies that 2 1 is equal to 2 2 on A+. tha& for i = 1,2,
u;(z)/u;(z) = u;(z)/u,2(2) which, by rearrangement,
for
(3)
yields:
u:(z)/u~(z) = u;(z)/u;(z) = u;(z)/q(z). Further
rearrangement
wYw4
of the left-hand
equality
in (4) leads to
= wwt(4
(5)
which, together with (3) shows that the consumers have equal marginal rates of substitution between all pairs of commodities. Since the preferences are classical, then U’(.) and U”(.) are equal up to monotone transformations on A +. This proves Proposition 1.
4. Pareto optimal and Lindahl equilibrium allocations In Samuelson (1954,1969) it was shown that Lindahl equilibrium is Pareto optimal in the presence of ‘pure’ public goods and with classical but ‘selfish preferences. In a Lindahl equilibrium all consumers pay the same price for each type of private commodity and that price in turn is equal to the price received by producers. With respect to a ‘pure’ public good each consumer pays an individualized (Lindahl) price such that the sum of these prices is equal to the price received by producers. However, if preferences are classical and benevolent (instead of ‘selfish’), then in general a Pareto optimum cannot be sustained as a Lindahl equilibrium.4 To see this, suppose that U = A+R and that U’(z) = U’(z) = U(z), for all z E A+. Furthermore, let F(x, +x2, y) = 0 be the economy’s net (aggregate) production function and assume that F(.) is concave, differentiable and such that F(Z,+%,,j)=O for some Z=(TI,%z,j)~A+. 4The following
owes much to conversations
with D. Donaldson
of U.B.C.
S.Q. Lemche, Benevolent
preferences
133
An allocation 2 = (a,, &, 9) E A is Pareto optimal Definition 3 (Pareto optimality). if F(j?l+iZ,j)=O and there is no z=(x,,x,,y)~A such that F(x,+x,,y)=O and U’(z) 2 U’(3) (i= 1,2) with strict inequality for some i. If the preferences are equal and the utility representation U(z) is common to both consumers, then Z = (a,, A,, 9) E A is Pareto optimal if and only if B maximizes U(z) subject to F(x,+x,,y)=O. With classical preferences the first-order imply that if h E A + is Pareto optimal, then
conditions
for Pareto
U,o=U,o=F,(%+~,,9)>0 u,(;)
u,(4
F,@,+L 9)
optimality
(6)
’
Thus, if &E A + is Pareto optimal, then each consumer has the same marginal rate of substitution between ‘his own’ private good and the ‘pure’ public good and, moreover, this marginal rate of substitution in consumption is equal to the marginal rate of substitution between the private and the public goods in production. Definition 4 rium if F(R, private good ‘pure’ public
(Lindahl equilibrium). An allocation z^E A is a Lindahl equilib+z?~, 9) =0 and there exists a (numeraire) price P,= 1 of the and (relative) individualized Lindahl prices Pi>0 (i= 1,2) of the good in terms of the private good such that:
(i) (utility maximization) then xi - z?~+ Pi(y - 9) > 0;
for j# i (i,j=
(ii) (profit maximization) (a,+a,)+(P:+P3)(y-9)~0.
if ZE A and
Proposition
2.
Lindahl equilibrium
1,2) if z E A(kj)
and
F(x, +x,, y) =O, then
is not Pareto
U’(z) > U’(2), (x1 +x,) -
optimal.
Proof: From the assumptions that F(x, +x,, y) =0 for some ZE A+ and that preferences are classical it follows that if &E A is a Lindahl equilibrium then SEA, and
9uj(s)=PL>O
(i=1,2)
(7)
and
F,(% + L9) =p;+p;>o.
F,(% + %,9)
03)
S.Q. Lemche, Benevolent preferences
134
Because preferences are equal and the utility representation to both consumers, then (7) can be re-written as: LJ (2)
-=I$>0
U(z) is common
(i= 1,2).
ui(2)
Combined,
(8) and (9) lead to
which is inconsistent with the equalities (6) that must obtain if 2 is Pareto optimal. Thus, a Lindahl equilibrium cannot be Pareto optimal and Proposition 2 has been proved.
5. Conclusion A Lindahl equilibrium is not Pareto optimal (efficient) because it does not recognize that the private goods are ‘collective concern variables’ and fails to price the externalities associated with private goods when the preferences are (benevolently) interdependent. In other words, the inefficiency of a Lindahl equilibrium is caused by a pricing rule for the private goods which, on the one hand, fails to charge each consumer for all goods that affect his preferences and, on the other hand, fails to divide properly the costs of each private good among all those whose preferences are affected by it. The result is that the Lindahl equilibrium price of the private good is ‘too high’, and those of the public good ‘too low’, for such an equilibrium to be efficient.
References Archibald, G.C. and David Donaldson, 1976, Non-paternalism and the basic theorems of welfare economics, The Canadian Journal of Economics IX, no. 3,492-507. Bergstrom, T.E., 1970, A ‘Scandinavian consensus’ solution for efficient income distribution among non-malevolent consumers, Journal of Economic Theory 2, 3833398. Lemche, S.Q., 1978, The first optimality theorem with interdependent preferences and public commodities, Unpublished Ph.D. dissertation, University of Minnesota. Milleron, J.C., 1972, Theory of value with public goods: A survey article, Journal of Economic Theory 5, 419477. Rader, J.T., 1980, The second theorem of welfare economics when utilities are interdependent, Journal of Economic Theory 23,42@424. Samuelson, P.A., 1954, The pure theory of public expenditure, Review of Economics and Statistics XXXVI, 3877389. Samuelson, P.A., 1969, Pure theory of public expenditure and taxation, in: J. Margolis and H. Guitton, eds., Public Economics. Proceedings of I.E.A. Conference (Macmillan and St. Martin’s Press). Winter, S.G., 1969, A simple remark on the second optimality theorem of welfare economics, Journal of Economic Theory 1, 999103.