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University ot Illinois
of Urbana-Chain|>aign
Generalized Samuelson Conditions and Welfare
Theorems for Nonsmooth Economies
John P. Conley Dimitrios Diamantaras
Department of Economics Department of Economics
University of IlUnios Temple University
Bureau of Economic and Business Research
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
BEBR
FACULTY WORKING PAPER NO. 93-0167
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
October 1993
Generalized Samuelson Conditions and Welfare
Theorems for Nonsmooth Economies
John P. Conley
Dimitrios Diamantaras
Department of Economics
Generalized Samuelson Conditions and Welfare Theoremsf
for Nonsmooth Economies
John P. Conley*
and
Dimitrios Diamantaras**
Revised: October 1993
t Version 1.0
* Department of Economics, University of Illinois, Champaign, IL 61820
** Department of Economics, Temple University, Philadelphia, PA 19122
Abstract
We give intuitive Samuelson conditions for a very general class of
economies. Smoothness, monotonicity, transitivity and completeness are
not required. We provide necessary and sufficient conditions for all Pareto
efficient allocations, including those on the boundary. We also prove that if
all agents have a cheaper point, the supporting prices fully decentralize the
allocation. Finally, we show first and second welfare theorems as corollaries
to the characterization of efficient allocations.
1. Introduction
Samuelson (1954, 1955) gave the first modern study of economies with pubhc
goods. One of his main results was calculus-based conditions for Pareto efficiency.
These "Samuelson conditions" have since become one of the fundamental tools for
understanding public goods economies. However, his work has several important
limitations. In particular, he did not deal with the issue of corner allocations, in
which at least one type of good is not consumed at all by at least one agent. Given
that this is probably the typical rather than the exceptional case in real life, his
omission is not trivial. Unless we can characterize corner allocations as well, we
must doubt the practical relevance of the studies based on Samuelson conditions.
Later, economists assumed that the most obvious modification of Samuelson's
efficiency conditions would be the correct ones for dealing with corners. However,
as Campbell and Truchon (1988) point out in an important paper, there are cases
where some efficient allocations violate the Sajnuelson conditions, even as modified.
Campbell and Truchon conclude that the Samuelson conditions miss some efficient
allocations, and they provide a different specification of the Samuelson conditions
which they claim are necessary and sufficient for efficiency in economies with one
private good and a finite number of public goods. They assume differentiability of
the utility and cost functions, convexity of preferences and cost, and monotonicity
of preferences.
Unfortunately, the analysis of Campbell and Truchon is limited by their as-
sumption that there is only one private good and their need for differentiability.
The first assumption reduces the relevance of their contribution to an essentially
partial equilibrium domain. Requiring differentiability significantly reduces the class
of economies for which their analysis may be applied. Further, their proof of suffi-
ciency contains an oversight (proof of Lemma 1, page 247), which we explain in the
conclusion. These observations motivate an approach to the problem using convex
analysis, in the standard fashion estabhshed by Arrow (1951) for economies with
private goods only.
Such an analysis was offered by Foley (1970) in the course of formalizing the
general notion of Lindahl equilibrium. However, he requires in his definition that
allocations be in the relative interior of the private goods subspace of the consump-
tion set of each agent. Thus, corner allocations are not dealt with by Foley, either.
Khan and Vohra (1987) generalize Foley (1970) to allow general preferences and
nonconvexities, but their aim is to present the second welfare theorem and they do
not examine the refinements needed to deal with boundary allocations.
In this paper we provide efficiency conditions for economies with a finite number
of private and public goods, without assuming differentiability. We so not require
that commodities be goods. This allows us to consider important real world cases
such as the one in which a public project (a garbage incinerator, for example)
benefits some agents while imposing costs on others. We require only that agents
are not locally satiated. In addition, we do not assume that the preference relations
are complete or transitive. Our analysis deals with corner and interior allocations in
a unified way. Unlike Campbell and Truchon ( 1988), we do not need to appeal to the
Karush-Kuhn-Tucker theorem, and our proofs are simple and geometric in nature.
We develop the most general from of the Samuelson conditions in a simple and
operational form, and we further show the existence of fully (Lindahl) supporting
prices at any Pareto efficient allocation, for all agents who are allowed a cheaper
point by the Samuelson prices corresponding to the allocation. As corollaries to
these efficiency conditions we show first and second welfare theorems.
2. The Model
We consider an economy with L private goods and M public goods, / individual
consumers, and F firms. We use the convention J = {1, . . . , /}, and similarly for
£, ^A and J-. Superscripts are used to represent firms and consumers and subscripts
to represent goods.
Each agent i E X is characterized by an endowment u;' G R^, and a preference
relation >-' over the consumption set C = R^"^ . A typical consumption bundle
will be written (x, y ) where x is a bundle of private goods, and y is a bundle of public
goods. We remark that assuming the consumption set to be the nonnegative orthant
is not less general than Campbell and Truchon's introduction of a nonnegative lower
bound for the consumption of the private good by each agent, since we can always
translate the preferences in order to make this lower bound zero. It is also possible
to generalize the results in this paper to bounded below, convex consumption sets
at the cost of complicating the proofs.
We make the following assumptions on >-' for all i G T.
Al) >-' is irreflexive.
A2) >-' is continuous (the strict upper and lower preferred sets are open).
A3) If (x,j/) )>-' (x,y), then for all A G (0, 1), A(x,y) + (1 - A)(x, y) )-' (x, y).
(Weak convexity)
A4) For all (x,y) G C' and all e > there exists (x,y) G C such that || (x,y) —
(x,y) II < € and (x,y) >-' (x,y). ^ (Local nonsatiation)
We normalize supporting prices to sum to one, but do not assume that prices
are positive:
The three kinds of vector inequalities are represented by >, >, and ^.
I e m I )
Define the marginal rate of substitution correspondence for consumer z, MRS' :
C ^^ n, by:
MRS'(x,y) =
{{P.q) e n | {p.q') ■ (x,y) < (p,g') • (i,y) V(i,y) G C s.t. (f,y) )^' (x,y)} .
Define also the weak marginal rate of substitution correspondence for consumer i,
WMRS' : C ^^ n, by:
WMRS'(j,y) =
{(p,?) 6 n I (p,5') • (x,y) < (p,g') • {i,y)y{i,y) e C s.t. (i,y) x' (x,y)} .
Note that the marginal rate of substitution set MRS'(a:,y) is always a subset of
the weak marginal rate of substitution set WMRS'(x,y), and MRS'(i:,y) can be
empty. The WMRS correspondence is never empty-valued, as we indicate in our
proof. Also note that if (p, q) G MRS'( j, y) and the agent has income (p, q' ) • (x, y),
then (x,y) is a preference maximizing choice over the budget set. On the other
hand if (p, g) G WMRS'(x,y) then we are only guaranteed that {x,y) minimizes
expenditure over the set of consumption bundles that are not inferior to (x,y).
[Figure 1 here]
In the example depicted in Figure 1, the agent's indifference curves intersect
the public good axis with an vertical slope, and terminate at their intersection with
this axis. Otherwise, the preferences are standard, satisfying all of the assumptions
A and furthermore all other assumptions commonly made on preferences. At every
point on the public good axis, the weak marginal rate of substitution correspondence
has a singleton value of (1,0).'^ In other words, the vertical axis supports the
preferred set. Since the WMRS correspondence contains the MRS correspondence,
and the unique line of support intersects the preferred set, the marginal rate of
substitution correspondence is empty-valued.
We represent each firm / E /" by a production set P-^ C R^ x R^. A typical
production plan will be written {z,y), where ^ is a net output vector of private
goods and y is a the output vector of pubhc goods.
Define the marginal rate of transformation correspondence for P-' , MRT-^ :
P^ —*—^ n, as follows:
MRT^(.^y) =
The comprehensive hull of a set in R^ x R^ is defined as follows:
comp(Z)= {{z,y) G R^ x R^^ | 3(i, y) G Z s.t. (z, y) <(£,y)}.
For all / E ^ we assume:
Bl) P^ is a nonempty, closed set.
B2) P^ is a convex set.
B3) Pf = comp(P^) (Free disposal).
We define the global production set in the usual way:
P= {{z,y) eR^ x^^
'+
(z,y) = J2i^^^y^) and (.~^y^) € P^ V/ e J^ i ,
and we define the aggregate marginal rate of transformation correspondence MRT :
p ^^ n by
MRT(^,y) = |(p,g) 6 n I {p,Y^q')-{z,y) > (p, ^ g') • (f, y) V (i,y) e P
Properly speaking, we should have indicated all the elements of the supporting vector here, in
accordance with the definition of MRS and WMRS. However, in all discussions of examples we
only indicate the components relating to the goods consumed by the agent in question, to enhance
clarity.
We make the additional assumption:
B4) P is closed.
Notice that P inherits convexity and comprehensiveness from the individual P^
sets.
An allocation is list a = ((x\ y^ ),..., (j^, y^), (c^, y^ ) ... (r^, y^)) G C^ x • • ■ x
C X P^ X ••• X P^ . Let .4 denote the set of feasible allocations:
A = la e C^ X ■ ■ ■ X C^ X P^ X ■ ■ ■ X P^
Y^z^ = X](w' - x') and ^ y^ = y' V ? G T
/ ' /
The set of Pareto efficient allocations is defined as
PE =
{ae A\^ae As.t. Vi G J, (x',y') ^' (x',y') and 3je Is.t. (F,y'') y^ {x\y^)} .
Let A^~^ denote the / — 1 dimensional simplex:
A^-^ = i ^ G RM J^ ^' = 1, and ^' > V z G J > .
We denote a profit share system for a private ownership economy by ^ =
{9^ , . . .9^ , . . .6^) G A^~^ X ... X A^~^ = where ^''-^ is interpreted as consumer
z's share of the profits of firm /.
An allocation and price vector (a,p, g) G A x 11 is said to be a Lmdahl equi-
librium relative to the endowment uj G R^^^ and profit shares ^ G if and only
if:
a. for all / G T", (p,E.9') € MRT^zf^yf).
b. iorsil\^eIAp.q)eMRS\x\y')aIid{p,q')■{x\y) = p■u' + Zf^''^iP^T.^Q)■
Note that given the definitions of MRS' and MRT-^, and the fact that local
nonsatiation implies that each agent will exhaust his income, these are equivalent
to profit and preference maximization, respectively. Feasibility is already required
by the definition of an allocation. Define the Lindahl equilibrium allocation corre-
spondence LE : R^^^ X — >— ^ A as follows:
LE{u,d) =
{a G A \ for some (p, q) E H, (a,p, q) is a Lindahl equilibrium for uJ and 8}.
3. Results
We start with a simple statement of our main results. The following two
conditions are necessary for an allocation a G A to be Pareto efficient:
There exists a price vector (p, g^ , . . . , g" ) £ 11 such that
a. for allfeJ" (p, Ef=i ?') ^ MKT^{zf,yf),
h. for all I €l, {p,q) eWMRS\x\y).
Alternatively, for a E A, if for all i E X there is a cheaper point than {x\y) in
C (this would be true for example if every agent's consumption bundle was in the
interior of his consumption set), the following conditions are necessary for a to be
a Pareto efficient allocation:
There exists a price vector (p, g^ , . . . , g") E IT such that
a. for allfeJ" (p, ELi ?') € MRT^(.-/, y^),
b. for all I E I, {p,q) E MRS*(x',y).
Finally, in both cases the following two conditions are sufficient for an allocation
a G A to be Pareto efficient :
There exists a price vector (p, g^ , . . . , g" ) G 11 such that
a. for allfeJ" (p^ELi?') ^ MRT^(c/,y/),
b. for all I £ J, {p,q) E MRS'{x\y).
Notice that if we assume differentiability, then MRT-* and MRS' are singletons and
we have the familiar Samuelson conditions.
We begin our demonstration of these claims by showing that private goods
prices must be nonnegative.
Lemma 1. For all (z^y) G P and all p such that there exists q with {p,q) G
MRT(2,y), p>0.
Proof /
Suppose not; then for some {z,y) G P and p,q such that {p,q) G MRT(-,y),
there is a private good ^ G £ such that p( < 0. By free disposal, for all 6 >
{zi....,ze-S....,ZL,y)eP. But (p, X:. g')(~~i, • • • , ~f-^ • • • ,~L, y) >(p, E. ?')(- y)'
contradicting the definition of MRT(^,y).
The following lemma states given an allocation a G A and prices {p,q) G 11,
{z,y) majcimizes profits over the global production set P at prices (p, E' ?') ^^ ^^^
only if {z\y') maximizes the profits of each firm / G ^ at these prices. This allows
us to state the subsequent theorems in terms of maximizing profits over the global
production set instead of going to the extra step of considering each firm.
Lemma 2. Given {z,y) G P and {p,q) G 11, {p,q) ■ {z,y) > {p,q) ■ (2, y) for all
i^iV) € P if and only if for all f E ^ there exists {z^ , y^) such that {p,q)-{z^ ,y^) >
{p,q)-{zf,yf)forall{zf,yf)ePf andZfi^^,y^) = i^^y)-
Proof /
1. Necessity: Suppose not, then for all / ^ ^ there exists {z^ ,y^) such that
[p.q) ■ (~^y^) > ip^q) ■ (5^-y^) for all (r/,y/) G P^ and E/(~^/) = (-'?/).
but for some (5, y) G P, (p, 5) • (c, y) < (p, g) • (c, y). By definition, there exists
a collection of production vectors (5-^,y-^) 6 P-'^ such that X]/^^'^- ^■'^) ~ i-'U)-
However, by hypothesis for all / G J^, {p,q)-{z^ ,y^ ) > (p, q)-{z^ ,y^ ). But this
implies (p,g)-(2,y) = E/(P' ?) ' (-^' y^) ^ (p>9)-(^'y) = Ylfip^q) ' i^^ ^y^ )^ a
contradiction.
2. Sufficiency: Suppose not, then there exists {z,y) G P such that (p, ^) • (c,y) >
(p, 5) -(r.y) for all {z,y) G P, and plan for each firm {z^,y^) G P^ such that
^ J^:-'^, y-'^) = {z, y), but for some /' G ^, there exists (z^ ,y^)E P^ such that
(p, g)-(2/' , y/' ) < (p, q)-{zf' , y^' ). But then E/^/'(-~^. y^) + (^-^' , y^' ) G P, and
iP^q)- T^f^fi^^ ^y^) + {P^q) ■{^^\y^') > iP^q) -Jlfi^^^y^)^ ^ contradiction.
•
We now give the first necessity theorem.
Theorem 1. If a G A is a Pareto efficient allocation, then there exists a price
vector {p,q\...,q'') G H such that (a) (p,^!.?') ^ MRT(X;,(-?^' -^''),y) and, (h)
for all I G J, (p,9) G WMRS'(x',y).
Proof /
Following Foley, we define an artificial production set in which public goods
are treated as strictly jointly produced private goods:
AP = {{z, y\...,y')\y'=...y' = yaxid{z,y)eP}.
AP is closed, convex, and nonempty as a consequence of P possessing these
properties. Next we define the socially preferred set of the allocation a:
I
SP{a)= {{z,y\...,y^)eR^xR
rJ\ r- TO L ^. IQ) [ M
= E(i'--')-
1=1
V ^ 6 J, V (x',y') G C',(x.,y) >^' {i\y') and 3; G I s.t. (i^y^) >^^ (x,,y) ^.
The socially preferred set inherits convexity, and by continuity and nonsatiation it
has nonempty interior.
a. Since a is Pareto efficient by assumption, SP{a) fl AP = 0. Then by the
Minkowski Separation Theorem (Takayama (1985, p. 44)), there exists a price
vector {p,q^ , . . . ,q^ ) ^ with || p || < oo, and a scalar r, such that:
(i) For all (5,y\...,y^) € AP, p- z + ^^q' -y <r (where f = ■ ■ ■ = y^ = y.)
(ii) For all (s, y\ . . . , y^) G closure( 5P(a)), p • i + E, g' • y > r.
By continuity and nonsatiation, (c,y\...,y ) 6 closure(5P(a)). By hypoth-
esis, (2, y^, . . . , y^) G AP. It follows from (i) and (ii) that p- z + X],^' ' H = ^■
Therefore, for all (i,y) G P:
P- z + ^q' -y =^r>p-z + ^q' -y.
Since it is possible to renormalize these prices to be elements of H, this establishes
part (a) of the theorem.
b. Now suppose that part (b) is false. Then for all j 6 I there exists ( j-' , y-' ) G C-'
such that [x^ ,y^) y^ (a:-', y) and (p, 9-') • [x^ ,y^) < {p,q^) • [x^ ,y). Hence,
^(x' - 0.') + (x^ - u;^ ), yS . . . , y^ . . . , y^ ) G SP{a)
and
p ■ Y^{x' -Lo') + p-ixJ -LjJ) + Yl 9' -y' ^<i^ -y^
<(p,9\---,/)-fe(^'-^')'J/''---'^J'
a contradiction to (ii) above.
10
As a corollary to this we state a version of the second welfare theorem. In
particular, we show that we can decentralize any Pareto efficient allocation through
prices for some set of endowments and profit shares such that the production of each
firm is profit maximizing under the prices, and each agent's consumption bundle
minimizes expenditure over the weakly preferred set. This is not quite the same
thing as decentralizing the allocation as a Lindahl equilibrium since agents are not
necessarily maximizing preferences over the budget set. To get this stronger result,
slightly stronger conditions as needed. We show this below. See Debreu (1959) for
details.
Corollary 1.1 (weak second welfare theorem) If a E A is a Pareto efficient allo-
cation, then there exists a price vector (p, g^ , . . . , g") € 11 an endowment vector lj,
and a proEt share system 6 such that (a) (p, ^^ q*) € MRT(^j(x' — u,?'), y), (b) for
allieI,andall{i\y)y^{x\y),{p.q')-ii\y)<p-^'+Zf^''^iP^E,<l)-i^^^y^)
ai^d, (c) ^^iOt = ^,'^t
Proof /
We know by Theorem 1, there exist prices {p,q) G H such that (a)(p, ^^ g') G
MRT(X;,(j:' -u;'),y) and (b) for all i £ I, {p,q') € WMRS'(x', j/). Notice that:
^(p,g').(x',y') =
In words, the total cost of consumption equals the value of the endowment plus
the profit shares. It only remains to show that we can redistribute endowment and
profit shares in a way that satisfies (c) such that the cost of the Pareto efficient
11
consumption bundle (x',y) for each agent equals the implied income under these
prices. But clearly this is possible since total income to society does not change
when we vary the distribution, we can continuously vary the income distribution
over the full range of possibilities, and we know from the above that there is exactly
enough endowment so that when it is fully distributed, society's budget balances.
Next we give a second necessity theorem. We strengthen the hypothesis to
require that all agents have a cheaper point in the consumption set. This allows
us to conclude that there will exist supporting prices in the MRS correspondence
of each agent, instead of just the WMRS. This means that the prices are fully
decentralizing.
Theorem 2. If a E A is a Pareto efficient allocation, then for every i E X sucii that
(a) p ■ x' > 0, or (b) 3m s.t. ql^ < 0, or (c) 3 m s.t. q]^ > and ym > 0, where p, q'
are the prices established by Theorem 1, we have that {p,q) E MRS'(j',y).
Proof /
(a) Suppose that for some i E T, p- x^ > and (p, g) ^ MRS'(j:',y). The latter
implies that there exists (x', y') E C' such that (x',y') >-' {x\y) and (p, g')-(x', y) >
(p, g') • {x\ y'). Since p > by Lemma 1, and x' > because {x\ y) E C\ p- x' >
implies that there exists £ E C such that pe > and x\ > 0.
Denote the open line segment between two points as follows:
L{{x\y),{x\y')) =
{(f',y') I 3X E (0,1) and (x\y') = \{x\y) + (1 - A)(x',y')} .
By the convexity of preferences and the linearity of the budget constraint, for all
(i',y') € i:((x',y),(x',y')), we have (i',y') X' (x',y) and (p,g') • (x',y) > {p,q') •
(x*,y'). For (x',y') close enough to (x',y) (A close enough to 1), x^ > 0. By the
12
continuity of preferences, there exists e > such that (x\ x\ — e, . . . , i'^, y' ) >-
(x', y). Since pe > 0, there follows {p,q^) • {x\,. . . ,x\ — e, . . . , x\,y^) < (p, g') •
{i\y^) < (p. ?') • (J^*' y), leading to a contradiction to (ii) in the proof of Theorem 1
in the same manner as in that proof.
(b) Suppose now that for some i £ J, 3m s.t. ql^ < and {p,q^) ^ MRS'(x',y).
The latter implies that there exists (x',y') £ C such that (j',y') x' (x\y) and
(p, g') • {x\ y) > [p, q^) ■ {x\ y'). By the continuity of preferences, there exists e >
such that {x\y\, . . . ,y\^ +e,...,y)^) )^' (j',y). Since g^ < 0, this leads to the
same contradiction as before.
(c) Finally, suppose that for some i € X, 3m s.t. q\^ > Q and ym > and
(PiQ^) i MRS'(x',y). The latter imphes that there exists (x',y') E C such that
(x', y') >-' {x\y) and (p, g') • (x',y) > (p, g') • (x', y'). We can now mimic the proof
of (a) above, with y^ in the place of x\ and 5^ in the place of pi.
The reason that the extra assumption is required to obtain the full support
is illustrated in the following example. Consider an economy with two agents, one
private and one public good, one firm with one-to-one linear technology, and endow-
ment of one unit of the private good for each agent. Agent 1 has preferences exactly
as in Figure 1, and agent 2 has translated Cobb-Douglas preferences such that the
slope of agent 2's indifference curve at (x^, y^ ) = (1/2,3/2) is —1. Then the alloca-
tion (x^y^x^y^2^y^) = (0, |,|, |,-|,|) is Pareto efficient, but WMRS^O, |)
contains only the vector (1,0), which intersects the strictly preferred set of agent 1.
Therefore, the Samuelson prices arising from Theorem 1 are not separating prices,
and this failure occurs for agent 1 who violates all three of the conditions of Theorem
2. This allows us to state a stronger second welfare theorem.
Corollary 2.1 ("strong second welfare theorem) If a G A is a Pareto efficient allo-
cation such that for all agents i E 2", (x', y) is in the interior of C\ then there exists
13
a price vector {p,q^ , . . . ,q"^) E U. an endowment vector u, and a profit share system
9 such that a € LE{Cj,d)'and Xlj^^i — Z]i*^'-
Proof /
Since for all agents z G X, (x' , y) is in the interior of C, the hypothesis of Theo-
rem 2 is satisfied. Therefore, there exist prices (p, g) € 11 such that (a)(p, ^,_i «?' ) G
MRT(^f^i(x' -^'),y) and (b) for all i G I, (p,g) € MRS'(x^y). But (a) means
all firms profit maximize under the prices, and (b) means each consumer i chooses
(x',y) when he maximizes his preferences while having income {p,q') ■ {x\y). But
we know from the argument given in the proof of Corollary 1.1 that it is possible to
divide endowments and profits so that each agent has exactly his income, and the
social endowment is exactly exhausted.
Now we give our sufficiency theorem.
Theorem 3. If a £ A is an allocation and there exists a price vector {p,q^ , . . . ,q )
such that (a) ip^YlUi^') ^ MRT(X;[=i(a:' -u^'),y) and (b) for all i G I, (p,g) G
MRS*(x',y), then a is Pareto efficient.
Proof /
Suppose that the hypotheses of the Theorem are met but a is not Pareto
efficient. Then there exists a (Pareto dominant) feasible allocation a E A such that
for no z G J is it the case that (x',y') >-' (x',y') and for some j G X, we have
(x-^, y-') >-^ (x',y'). Then by (b) and summing up over all agents,
^p-x' + ^g'-y >^p-x'+^g'-y. • (i)
till
But by (a),
11 t i
14
Since (ii) contradicts (i), the proof is finished.
Finally, we get the first welfare theorem as an immediate consequence of this.
Corollary 3.1 If a G LE{u},d), a is Pareto efficient.
Proof /
By the definition of Lindahl equilibrium, there exists a price vector {p,q^ , . . . ,q^ )
such that (a) (p,Ef=i9') ^ ^'IH-T(Ef=i('^' -^').y) and (b) for all i G I, {p,q) G
MRS'(x',y). But then by Theorem 3, a is Pareto efficient.
4. Some comments on the literature
The oversight in Campbell and Truchon ( 1988) occurs on page 247, in the proof
of Lemma 1. The first inequality of the last sequence of inequalities in that proof
does not hold, because some the dx, may be negative.
Condition GOC in Campbell and Truchon (1988) attempts to deal with agents
pushed against the public good axis by weighting their (unique, under their dif-
ferentiability assumption) MRS by a nonnegative coefficient that can be less than
unity. In such a case, if we allow the coefficient to run from to 1, we trace our
WMRS, as shown in Figure 2.
[Figure 2 here]
Khan and Vohra (1987) have general assumptions on preferences and they allow
for nonconvexities in preferences and production, but they require all public goods
15
to be desired by all agents, so that the example with the incinerator mentioned in our
introduction falls outside their coverage. They prove a version of the second welfare
theorem employing a notion of supporting vector set equivalent, under convexity,
to our WMRS(x',y) (Khan and Vohra 1987, page 236).
Finally, two notes on papers that are tangentially relevant. Saijo (1990) ad-
dresses a quite different point arising from Campbell and Truchon than we do;
namely, he shows that the robustness of boundary Pareto efficient allocations ob-
served by Cambpell and Truchon is not a phenomenon specific to public good
economies, since it also happens in exchange economies. Manning (1993, chap-
ter 3) contains an extension of Foley's (1970) results to economies with local public
goods, using assumptions based on Foley's, such as constant returns to scale and
ruling out the private goods boundary.
5. References
Arrow, K. (1951): "An extension of the basic theorems of classical welfare eco-
nomics," in J. Neyman (ed.), by Proceedings of the Second Berkeley Sympo-
sium on Mathematical Statistics and Probability.
Berkeley : University of California Press, pp. 507-532/
Campbell, D.E., and M. Truchon (1988): "Boundary optima and the theory
of public goods supply," Journal of Public Economics, 35, 241-249.
Debreu, G. (1959): The theory of value. Yale University Press.
Foley, D.K. (1970): "Lindahl's solution and the core of an economy with pubhc
goods," Econometrica, 38(1), 66-72.
Khan, M.A. and R. Vohra (1987): "An extension of the second welfare theo-
rem to economies with nonconvexities and public goods," Quarterly Journal of
16
Economics, 223-241.
Manning, J.R.A. (1993): Local public goods: a theory of the first best. University
of Rochester Doctoral Dissertation.
Saijo, T. (1990): "Boundary optima and the theory of pubHc goods supply: a
comment," Journal of Public Economics, 42, 213-217.
Samuelson, P. (1954): "The pure theory of public expenditure," Review of Eco-
nomics and Statistics, 36, 387-389.
Takayama, A. (1985): Mathematical Economics. Cambridge University Press,
2nd edition.
17
Figure 1
Campbell+Truchon's MRS
*: The horizontal edge of WMRS is not in MRS
Figure 2
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