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Full text of "Generalized Samuelson conditions and welfare theorems for noonsmooth economies"

UNIVERSITY OF 

ILLINOIS LiHRARY 

AT URBANA-CHAivlPAIGN 

BOOKSTACKS 



Digitized by the Internet Archive 

in 2012 with funding from 

University of Illinois Urbana-Champaign 



http://www.archive.org/details/generalizedsaniue167conl 



Faculty Working Paper 93-0167 



330 The Library of the 

B385 

1993:167 CX)PY 2 JAN C 1994 

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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