%Paper: ewp-pe/9403001
%From: Dimitrios Diamantaras <dimitris@astro.ocis.temple.edu>
%Date: Wed, 9 Mar 1994 15:13:05 -0500 (EST)

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\begin{titlepage}
\begin{center}
\vspace*{8ex}
{\huge The Pure Theory of Public Goods:} \\[2ex] 
{\huge Efficiency, Decentralization,} \\[2ex] 
{\huge and the Core\footnotemark[1]} \\[8ex] 
{\Large Dimitrios Diamantaras\footnotemark[4]} \\[2ex] 
{\Large Robert P.~Gilles\footnotemark[3]} \\[4ex] 
{\Large March 1994}
\end{center}

\vspace*{14ex}

Correspondence to: \\
Dimitrios Diamantaras \\
Department of Economics \\
Temple University \\
Philadelphia, PA\ \ 19122

\footnotetext[1]{We wish to thank Andreu Mas-Colell, Hans Haller, Julian 
Manning and Shlomo Weber for their comments on earlier versions and related 
work, and Pieter Ruys for many fruitful discussions.}
\footnotetext[4]{Department of Economics, Temple University, Philadelphia,
Pennsylvania, USA.}
\footnotetext[3]{Department of Economics, Virginia Polytechnic
Institute \& State University, Blacksburg, Virginia, USA.}
\end{titlepage}

\renewcommand{\thefootnote}{\arabic{footnote}}

\begin{abstract}
We extend the results of Mas-Colell (1980) and Weber and Wiesmeth (1991) on 
valuation equilibria and the relationship of cost share equilibria with the 
core.  We allow for any finite number of private goods and a set of public 
projects without any structure.  We show the two welfare theorems for 
valuation equilibrium, the inclusion of the set of cost share equilibria in 
the core, and the nonequivalence of these two sets for an economy with a 
finite number of agents.  In the case that the set of public projects is 
endowed with a topological structure, we provide conditions under which the 
price system needed to decentralize a Pareto efficient allocation as a 
valuation equilibrium is continuous.
\end{abstract}

\section{Introduction}
The first mathematically general approach to the pure theory of public 
goods was made by Mas-Colell (1980).  He studied a model with one private 
good and an abstract set of {\em public projects\/} without any 
mathematical structure.  In particular, he did not assume that it is part 
of a linear space or that there is any ordering at all among public 
projects.  In that paper he also proposed the notions of valuation 
equilibrium and cost share equilibrium.  In a valuation equilibrium in the 
case of a single private good a public project is financed through a system 
of taxes and subsidies, called a {\em valuation\/}.  In a {\em cost share 
equilibrium\/} the subsidies are excluded and the costs of the project are 
distributed among the agents in the economy.

\spr
In relation to these two generalizations of the Lindahl equilibrium concept 
to the case of an economy with an abstract set of public projects, 
Mas-Colell (1980) introduced the appropriate extensions of the standard 
normative notions of Pareto efficiency and the core.

\spr
Mas-Colell (1980) showed that, in the case of a single private good, 
valuation equilibria decentralize Pareto efficient allocations and that the 
set of cost share equilibria is equal to the core.  Weber and Wiesmeth 
(1991) obtained the equivalence of the set of cost share equilibria with 
the core in a more restricted model with one private and one pure public 
good, assuming the public good space to be the nonnegative real half-line. 
They further studied the set of core allocations that are linear cost share 
equilibria, first defined in Mas-Colell and Silvestre (1989).  More 
recently Vasil'ev, Weber and Wiesmeth (1992) and Gilles and Diamantaras 
(1994) also discuss core equivalence in a model where the public goods 
are subject to congestion effects.

\spr
Such decentralization and equivalence results are very important for the 
pure theory of public goods.  The main messages of Mas-Colell (1980) and 
Weber and Wiesmeth (1991) are that (i) valuation equilibria, through the 
notion of a valuation, provide a ``nonlinear price'' for the 
decentralization of the efficient provision of an abstract public project, 
and (ii) cost share equilibria, in which subsidies are not allowed, cover 
the whole core.  The first result generalizes the Lindahlian approach to 
public goods provision to a more abstract setting.  The second result is 
interesting given the well-known examples of the failure of core 
equivalence with Lindahl equilibria in economies with constant returns to 
scale.  However, Mas-Colell (1980) as well as Weber and Wiesmeth (1991) 
worked in a partial equilibrium model, by virtue of their assumption that 
there is only one private good.

\spr
We obtain in this paper a generalization of the results mentioned of 
Mas-Colell (1980) in that we allow for any arbitrary finite number of 
private goods.  We develop the appropriate extension of the definition of 
valuation equilibrium and cost share equilibrium using price systems.  We 
establish the first and second welfare theorems for valuation equilibrium, 
and show the nonequivalence of the core and the set of cost share 
equilibria.

\spr
Diamantaras, Gilles and Ruys (1994) study a model in which trade 
infrastructures are treated as public projects. A public infrastructure 
has a real cost but it reduces transaction costs for agents making more 
trading possible. The model in that paper has a continuum of agents, and 
techniques are used which are similar to some of the ones applied in the 
present paper and in Gilles and Diamantaras (1994).

\spr
We emphasize that our results form a bridge between the work on economies 
with {\em one\/} private commodity and an abstract set of public projects 
(Mas-Colell 1980) on the one hand and the traditional general equilibrium 
framework for a pure exchange economy on the other hand.  Our welfare 
theorems generalize the traditional results since the latter are obtained 
after the linearization of the set of public projects.  If only one private 
good is assumed, one arrives at the results developed in Mas-Colell (1980).

\section{The Model}
We study an economy in which $A$ is a finite set of economic agents, there 
are $\ell \in \SN$ private commodities, and the commodity space is 
represented by \eucl{\ell}{+}.  We denote by the function $w \colon A \to 
\eucl{\ell}{+} \setminus \{0\}$ the {\em endowment\/} of private 
commodities of the agents in $A$.

\spr
There is a set ${\cal Y}$ of public projects, on which for the time being 
we do not impose any structure.  Each public project has a cost in terms of 
each private good, and we capture this by the vector-valued function $c 
\colon {\cal Y} \to \eucl{\ell}{+}$.  We remark that on the set $\cal Y$ 
one can easily adopt some metric or Euclidean structure.  The first case 
will be employed in Section 4 to discuss continuity properties of price 
systems that decentralize Pareto efficient allocations.  In the case of an 
Euclidean structure we arrive at the classical public goods model, which 
might serve as a reference point in the discussion.

\spr
Each agent $a\in A$ has preferences defined on $\eucl{\ell}{+} \times {\cal 
Y}$, which are represented by a real-valued function $U_{a} \colon 
\eucl{\ell}{+} \times {\cal Y} \to \SR$. 

\spr
An {\bf economy} is a collection $\SE = \left\{A,(U_{a})_{a\in A},w,{\cal 
Y},c\right\}$ such that $\overline{w} \equiv \sum_{a\in A} w(a)$ $\gg c(y)$ 
for all $y\in {\cal Y}$.  Note that this implies that $\overline{w} \gg 0$. 
(We use the vector inequality convention $\geq, >, \gg$.)

\spr
An {\bf allocation} for an economy $\SE$ is a pair $(f,y)$ 
where $f \colon A \to \eucl{\ell}{+}$ and $y
\in {\cal Y}$.  An allocation $(f,y)$ is {\bf feasible} if
\[
\sum_{a\in A} f(a) + c(y) = \overline{w}.
\]
Note that we do not assume free disposal in production, although our 
results hold under it. We denote the set of feasible allocations by $\Phi$.

\section{Efficiency and Valuation Equilibrium}
In this section we define efficient allocations and valuation equilibria 
and we show the two welfare theorems for valuation equilibria.

\begin{defni}
A feasible allocation $(f,y) \in \Phi$ is {\bf Pareto efficient} in 
the economy $\SE$ if there is no allocation $(g,z) \in \Phi$ 
such that
\begin{numm}
\item
for every $a$ in $A$, $U_a(g(a),z) \geq U_a(f(a),y)$,
and
\item
there exists an agent $b\in A$ such that $U_b(g(b),z) > U_b(f(b),y)$.
\end{numm}
\end{defni}

This definition of efficiency is standard. We will need the usual 
price space for the private goods:
\[
S^{\ell-1} := \left\{ q \in 
\eucl{\ell}{+} \left| \sum^{\ell}_{i=1} q_{i} =1 \right.  \right\} .
\]
We can now introduce our first equilibrium concept.

\begin{defni} \label{def:valeq}
A feasible allocation $(f,y) \in \Phi$ is a {\bf valuation equilibrium} for 
$\SE$ if there exist a price system $p \colon {\cal Y} \to S^{\ell-1}$ 
and a valuation function $V \colon A \times {\cal Y} \to \SR$, such that 
\begin{numm}
\item
there is budget neutrality, i.e., $\sum_{a\in A} V(a,y) =
p(y) \cdot c(y)$;
\item
for every agent $a\in A$ the pair $(f(a),y)$ maximizes $U_a$ 
on the budget set
\[ \left\{ (g,z) \in \eucl{\ell}{+} \times {\cal Y}\ \left|\
p(z) \cdot g + V(a,z) = p(z) \cdot w(a)
 \right. \right\}, \]
and for every $z \in {\cal Y}$ we have $V(a,z) \leq p(z) \cdot w(a)$;
\item
$y$ maximizes the surplus $\sum_{a\in A} V(a,z) - p(z) \cdot c(z)$
on ${\cal Y}$.
\end{numm}
\end{defni}

The feature of this definition that deserves special attention is the price 
system.  For each valuation equilibrium we have a price system that values 
private goods contingent on {\em all} public projects, even though, 
naturally, only one public project occurs at this particular equilibrium. 

\spr
In a valuation equilibrium an allocation is supported by a price system as 
well as a valuation.  A valuation can be interpreted as a ``non-linear'' 
price for the public project that is established, while a price system is a 
function that assigns to every potential public project some price vector 
with respect to the private sector of the economy.  It is important to note 
that a valuation equilibrium takes into account the large impact that a 
change of public project has on the private sector, i.e., each change of 
the public project leads to a change in the pricing of private commodities. 
In this sense the valuation equilibrium concept embodies an expression of 
the changes in the private sector of the economy caused by changes in the 
public sector.

\spr
Before we state our generalization of the first welfare theorem to our
setting we introduce some terminology.
A utility function $U_{a}$ is {\em monotone\/} if for all $f,g\in 
\eucl{\ell}{+}$ and all $y\in {\cal Y}$ with $f \gg g$, $U_{a}(f,y) > 
U_{a}(g,y)$. Furthermore, $U_a$ is {\em strictly monotone\/} if for all 
$f,g\in \eucl{\ell}{+}$ and all $y\in {\cal Y}$ with $f > g$, $U_{a}(f,y) 
> U_{a}(g,y)$.

\begin{thm} \label{ve-po}
If for all agents $a \in A$ the utility function $U_a$ is monotone, then 
every valuation equilibrium in $\SE$ is Pareto efficient.
\end{thm}

{\sc Proof}
 
Let $(f,y) \in \Phi$ be a valuation equilibrium with price system $p$ and 
valuation function $V$. We must show that $(f,y)$ is Pareto efficient.

\medskip
Suppose to the contrary that $(f,y)$ is not Pareto efficient.  Then there 
exists an allocation $(g,z) \in \Phi$ with for all $a$ in $A$
\[ U_a(g(a),z) \geq U_a(f(a),y), \]
and there is $b\in A$, such that
\[ U_b(g(b),z) > U_b(f(b),y). \]
Since $(g,z) \in \Phi$ it follows that
\begin{equation} \label{eq1}
\sum_{a\in A} g(a) + c(z) = \overline{w}.
\end{equation}
Condition (ii) of the definition of a valuation equilibrium and the 
monotonicity of the utility functions imply that for all $a$ in $A$ 
we have that $p(z) \cdot g(a) + V(a,z) \geq p(z) \cdot w(a)$ and for all 
$b$ we have $p(z) \cdot g(b) + V(b,z) > p(z) \cdot w(b)$.  Hence,
% \mathindent 0mm
\begin{equation} \label{eq2}
p(z) \cdot \sum_{a\in A} g(a) +
\sum_{a\in A} V(a,z) > p(z) \cdot \overline{w}.
\end{equation}
\mathindent 2em
Condition (iii) of the definition of valuation equilibrium now implies that
\begin{equation} \label{eq3}
\sum_{a\in A} V(a,y) - p(y) \cdot c(y) \geq \sum_{a\in A} V(a,z) - p(z) 
\cdot c(z) 
\end{equation}
Since equation (\ref{eq1}) can be written as
\begin{equation} \label{eq4}
\sum_{a\in A} (w(a)-g(a)) - c(z) =0,
\end{equation}
we conclude that
\[ \begin{array}{lll}
0 & = \sum_{a\in A} V(a,y) - p(y) \cdot c(y) \geq & \mbox{by
(\ref{eq3})} \\
& \geq \sum_{a\in A} V(a,z) - p(z) \cdot c(z) > & \mbox{by (\ref{eq2})} \\
& > p(z) \cdot \sum_{a\in A} (w(a)-g(a)) - 
p(z) \cdot c(z) = \quad & \mbox{by (\ref{eq4})} \\
& = 0. &
\end{array} \]
This is a contradiction.
\qed\

\bigskip
For the second welfare theorem we need the following condition, which is an 
extension and strengthening of the indispensability condition of Mas-Colell 
(1980).

\begin{defni}[Essentiality condition] \ \ \\
The space of public projects ${\cal Y}$ satisfies the essentiality condition 
in $\SE$ if it satisfies the following two conditions:
\begin{numm}
\item
For every agent $a \in A$, each bundle $f \in 
\eucl{\ell}{+}$, and all potential public projects $y,z \in {\cal Y}$ 
there exists a bundle $g \in \eucl{\ell}{+}$ such that $U_a(g,z) > 
U_a(f,y)$.
\item
For every agent $a \in A$, every $x\in\eucl{\ell}{+}\setminus\{0\}$ and 
all public projects $y,z\in {\cal Y}$, $U_{a}(x,y) > U_{a}(0,z)=0$.
\end{numm}
\end{defni}

The first condition states that all public projects in principle can be 
compensated by sufficiently large amounts of private goods.  In the second 
condition, setting $U_{a}(0,z) = 0$ is just a normalization; the important 
part of this latter statement is that $U_{a}(0,y) = U_{a}(0,z)$ for all 
projects $y,z \in {\cal Y}$.  This is similar to the 
indispensability condition of Mas-Colell (1980).

\spr
We note that in the standard case of pure public goods (${\cal Y} = 
\eucl{m}{+}$), continuity of preferences, indispensability of the private 
goods and strict monotonicity in one private good together imply 
essentiality.

\begin{thm} \label{po-ve}
Let for every agent $a \in A$ the utility function $U_a$ be continuous,
quasi-concave and strictly monotone on \eucl{\ell}{+}.  Assume that ${\cal 
Y}$ satisfies the essentiality condition.  Then every Pareto efficient 
allocation in $\SE$ can be supported as a valuation equilibrium.
\end{thm}

{\sc Proof}
 
Let $(f,y)$ be a Pareto efficient allocation in $\SE$, and let $a\in A$ and 
$z\in{\cal Y}$ be arbitrary.  We define
\[
F(a,z) := \left\{ g\in\eucl{\ell}{+} \left| U_{a}(g,z) > U_{a}(f(a),y) 
\right.\right\}.
\]
Note that $F(a,z) \neq \emptyset$ for all $a \in A$ and $z \in {\cal Y}$ by 
part (i) of the essentiality condition.  Moreover, from the assumptions on 
$U_{a}$ it follows that $F(a,z)$ is open, convex, and bounded from below. 
Let, for every $z\in {\cal Y}$,
\[
F(z) := \sum_{a\in A} F(a,z) + c(z) - \overline{w}.
\]
$F(z)$ is nonempty.  Moreover, 
from the above it follows that $F(z)$ is also open, convex, and bounded 
from below.  Finally, because $(f,y)$ is efficient, we have that
\[
\eucl{\ell}{-} \cap F(z) = \emptyset.
\]
By Minkowski's separation theorem, there exists a hyperplane separating 
\eucl{\ell}{-} from $F(z)$.  Hence, there exists a normal vector $p(z)\in 
S^{\ell-1}$ such that $p(z)\cdot F(z) > 0$.  Obviously, this defines a 
function $p \colon {\cal Y} \to S^{\ell-1}$.  We now show that $p$ satisfies 
the conditions as required in Definition~\ref{def:valeq}.

\medskip
Let for every agent $a\in A$ and every $z\in {\cal Y}$ the bundle 
$x(a,z) \in \eucl{\ell}{+}$ be chosen such that, in case $z\neq y$,
\begin{numm}
\item
$p(z)\cdot x(a,z) = \inf\ p(z)\cdot F(a,z)$;
\item
$U_{a}(x(a,z),z) \geq U_{a}(f(a),y)$,
\end{numm}
and in case $z=y$, $x(a,z) = x(a,y) = f(a)$.
Clearly, such bundles exist. Finally, we define a valuation function 
$V \colon A \times {\cal Y}\to \eucl{}{}$ by
\[
V(a,z) := p(z)\cdot w(a) - p(z)\cdot x(a,z).
\]
Note that $V(a,z)$ is finite and $V(a,z) \leq p(z)\cdot w(a)$ for 
all $a\in A$ and all $z\in {\cal Y}$ by definition.

We now check the three requirements of Definition~\ref{def:valeq}.

\medskip
{\sc Condition (i)}

By the feasibility of $(f,y)$ and the definition of $V$,
\[
\sum_{a\in A} V(a,y) = p(y)\cdot \overline{w} - p(y)\cdot \sum_{a\in A} 
f(a) = p(y)\cdot c(y). 
\]

\medskip
{\sc Condition (iii)}

By construction, $p(z)\cdot \inf F(z) \geq 0$, and, hence, for all 
$z\neq y$:
\[
p(z)\cdot \sum_{a\in A} x(a,z) + 
p(z)\cdot c(z) \geq  p(z)\cdot \overline{w}.
\]
(The inequality becomes weak because $x(a,z)$ is at the infimum.)
{}From this we obtain
\[
\sum_{a\in A} V(a,z) = p(z)\cdot \overline{w} 
 - p(z)\cdot \sum_{a\in A} x(a,z) \leq p(z)\cdot c(z),
\]
while
\[
\sum_{a\in A} V(a,y) = p(y)\cdot \overline{w} 
 - p(y) \cdot \sum_{a\in A} x(a,y) = p(y)\cdot c(y),
\]
by feasibility.
Thus, (iii) is shown.

\medskip
{\sc Condition (ii)}

Note that, by part (ii) of the essentiality condition and the continuity 
and strict monotonicity of preferences, it follows for every $a\in A$ that 
$U_{a}(x(a,z),z) = U_{a}(f(a),y)$ for all $z\in{\cal Y}$.
For any $(g,z)\in\eucl{\ell}{+}\times{\cal Y}$ with $U_{a}(g,z) \geq 
U_{a}(f(a),y) = U_{a}(x(a,z),z)$ we have
\begin{equation}
	p(z)\cdot g + V(a,z) = p(z)\cdot g + p(z)\cdot w(a) - p(z)\cdot x(a,z) 
	\geq p(z)\cdot w(a),
	\label{first}
\end{equation}
since $p(z)\cdot g \geq p(z)\cdot x(a,z)$ by the definition of $x(a,z)$.

Let $z\in{\cal Y}$.
By Condition (iii), as shown above, 
\[ \sum_{a\in A}^{}V(a,z) \leq p(z)\cdot c(z). \]
By assumption, $\overline{w} \gg c(z)$. Thus, 
since $p(z) > 0$, it follows that
\[ p(z) \cdot \overline{w} > p(z) \cdot c(z) \geq \sum_{a\in A}^{}V(a,z). \]
Therefore, there exists an $a\in A$ with $p(z)\cdot w(a) > V(a,z)$. For 
any $\tilde{g}\in \eucl{\ell}{+}$ such that $p(z)\cdot \tilde{g} + V(a,z) 
< p(z)\cdot w(a)$, it follows from~(\ref{first}) that $U_{a}(\tilde{g},z) 
< U_{a}(x(a,z),z)$. Since every $\hat g\in \eucl{\ell}{+}$ with
\[ p(z)\cdot \hat{g} + V(a,z) = p(z)\cdot w(a) \]
is the limit of a sequence $(\hat{g}_{n})_{n \in\SN}$ in \eucl{\ell}{+} with
\[ p(z)\cdot \hat{g}_{n} + V(a,z) < p(z)\cdot w(a), \]
we must have 
\[ U_{a}(\hat{g},z) \geq U_{a}(x(a,z),z) = U_{a}(f(a),y). \]
Since $z$ was chosen arbitrarily, $(f(a),y)$ maximizes $U_{a}$ on 
\[ \left\{ (g,z) \in \eucl{\ell}{+} \times {\cal Y}\ \left|\
p(z) \cdot g + V(a,z) = p(z) \cdot w(a) \right. \right\}. \]
By strict monotonicity of preferences, this implies that $p(z) \gg 0$. 
Since $z$ was chosen arbitrarily, this holds for every $z\in{\cal Y}$.

\medskip
Consider now an agent $a\in A$ and a project $z\in {\cal Y}$ such that 
$p(z)\cdot w(a) = V(a,z)$. Then $p(z)\cdot x(a,z) = 0$ by the definition 
of $V(a,z)$. Since $p(z)\gg 0$, we have $x(a,z) = 0$. By part (ii) of the 
essentiality condition and $U_{a}(f(a),y) = U_{a}(x(a,z),z) = 0$, we have 
$f(a) = 0$. Therefore, $x(a,z) = 0$ for every $z\in {\cal Y}$. Thus, 
$U_{a}$ is maximized by $(f(a),y)$ on
\[ \left\{ (0,z) \left| z\in {\cal Y}\right.\right\} =
 \left\{ (g,z) \in \eucl{\ell}{+} \times {\cal Y}\ \left|\
p(z) \cdot g + V(a,z) = p(z) \cdot w(a)
 \right. \right\}. \] 
This shows condition (ii).
\qed\


\section{Continuity of the price system}

Next we consider under what conditions the price system $p \colon {\cal Y} 
\to S^{\ell-1}$ of a valuation equilibrium can be chosen to be a continuous 
function.  This seems to be a very natural requirement once the set of 
public projects is endowed with a topological structure.

\spr
We will assume that ${\cal Y}$ is endowed with a topology generated by some 
well-chosen metric $d \colon {\cal Y}\times {\cal Y} \to 
\eucl{\ell}{+}$, and that the cost function $c$ is continuous with respect 
to this topology. We first state our continuity theorem for a truncated 
economy.

\begin{thm} \label{po-ve:cont}
Let $k \in \SN$ be some positive integer such that $k > \overline{w}_{j}$ 
for every $j\in \{1,\ldots,\ell\}$.
Let $\SE^k$ be a truncated economy such that the 
following conditions are satisfied:
\begin{numm}
\item
The set of public projects $\cal Y$ is endowed with a topology generated 
by a metric $d \colon {\cal Y} \times {\cal Y} \to \SR$.
\item
For all $a\in A$, $U_{a}$ is continuous, quasi-concave and strictly 
monotonic on \eucl{\ell}{+}.
\item
The cost function $c$ is continuous on $({\cal Y},d)$ and $c(z) \ll 
\overline{w}$ for all $z\in {\cal Y}$.
\item
The set of public projects ${\cal Y}$ satisfies the essentiality condition.
\item
The consumption space of every agent $a\in A$ is restricted to $\eucl{\ell}{+} 
\cup \{z\in\eucl{\ell}{} \mid z\leq ke\}$, where $e = (1, \ldots ,1) \in 
\eucl{\ell}{+}$ denotes the vector with unity in every component.
\end{numm}
Then every Pareto efficient allocation in $\SE^k$ can be supported 
as a valuation equilibrium with a continuous price system. 
\end{thm}

{\sc Proof}

In the sequel we denote by $K$ the set $\{z\in\eucl{\ell}{} \mid z\leq ke\}$.

We show that there exists a price system $p \colon {\cal Y} \to S^{\ell-1}$ 
which has the same properties as the one constructed in the proof of 
Theorem~\ref{po-ve} and is, in addition, continuous with respect to
the metric space $({\cal Y},d)$.

Let $(f,y)$ be a Pareto efficient allocation in 
$\SE^{k}$ and let $a\in A$ and $z\in{\cal Y}$ be arbitrary. Define
\[
F^{k}(a,z) := \left\{ g\in \eucl{\ell}{+}\cap K \mid U_{a}(g,z) > 
U_{a}(f(a),y) 
\right\}.
\]

\medskip
{\bf Claim 1. }
The correspondence $\sum_{a\in A} F^{k}(a,z)$ has a closed graph.

\medskip
{\bf Proof of Claim 1. }

Note that $F^{k}(a,\cdot)$ has a closed graph for all $a\in A$ because 
$U_{a}(g,\cdot)$ is a continuous function on ${\cal Y}$, for all $a\in A$. 
Further, because of the boundedness of the range of $F^{k}$, there exists 
a function $h:A\to\SR$ such that $\|\,F^{k}(a,z)\,\| < h(a)$ 
for every $a\in A$.  By, e.g., Proposition D.8 of Hildenbrand (1974, page 73) 
it follows that $\sum_{a\in A} F^{k}(a,z)$ is closed at $z$, for all 
$z\in{\cal Y}$. Since $z\in{\cal Y}$ is arbitrary, Claim 1 is shown.
\hfill QED.

\medskip
As in the proof of Theorem~\ref{po-ve}, define, for all $z\in {\cal Y}$,
\[
F^{k}(z) \equiv \sum_{a\in A} F^{k}(a,z) + c(z) - \overline{w}.
\]
By Claim 1 and the continuity of the function $c$ on ${\cal Y}$, the 
correspondence $F^{k}$ has a closed graph.
Define now the correspondence $T \colon {\cal Y} \to \eucl{\ell}{}$ by
\[
T(z) \equiv \left\{p\in\eucl{\ell}{} \mid p\cdot F^{k}(z) > 0\right\}.
\]
The correspondence $T$ assigns to every potential public project the set
of {\em all\/} normal vectors of separating hyperplanes of $F^k(\cdot)$
and \eucl{\ell}{-}. Following the proof of Theorem \ref{po-ve} we know
that any selection in $T$ corresponds to a valuation equilibrium
supporting the Pareto efficient allocation $(f,y)$.
We now show that there exists a continuous selection in $T$, thus 
establishing the assertion.

Note that by application of Minkowski's separation theorem (see the proof of 
Theorem \ref{po-ve}) (i) $T(z)$ is a non-empty convex cone, and by 
monotonicity (ii) $T(z)\subset \eucl{\ell}{+}$ for all $z\in{\cal Y}$.

We now adapt a result from Aubin (1993, pages 291, 338):

\medskip
{\bf Claim 2. }
If $T(z)^{-}\equiv \{q\in\eucl{\ell}{+} \mid p\cdot q \leq 0 \mbox{ for all } 
p \in T(z) \}$ has a closed graph, then $T$ is lower semi-continuous.

\medskip
{\bf Proof of Claim 2. }

Consider a sequence $z_{n}\to z$ and take some $p\in T(z)$.  Suppose that 
$\Pi_{T(z_{n})}$ is the orthogonal projector onto $T(z_{n})$.  It suffices 
to show that $p_{n}\equiv \Pi_{T(z_{n})}(p)$ converges to $p$.  Then 
$\pi_{n} = p-p_{n}$ is the projector of $p$ onto $T(z_{n})^{-}$.  Thus 
$\pi_{n}\cdot p_{n}=0$ and $\|\,\pi_{n}\,\| \leq \|\,p\,\|$. 
Consequently, since \eucl{\ell}{+} is finite-dimensional, a subsequence 
(keeping the notation $\pi_{n}$) converges to $\bar\pi$.  This element 
$\bar\pi$ belongs to $T(z)^{-}$, since the graph of $T(z)^{-}$ is closed by 
hypothesis, and satisfies $\bar\pi\cdot (p-\bar\pi)=\lim_{n\to 
\infty}\pi_{n}\cdot p_{n}=0$.  Thus $\|\,\bar\pi\,\|^{2}=\bar\pi\cdot p 
\leq 0$ since $\bar\pi\in T(z)^{-}$ and $p\in T(z)$.  Consequently, 
$\bar\pi=0$ and $p_{n}\to p = p-\bar\pi$.
\hfill QED. 

\medskip
{\bf Claim 3. }
The correspondence $T(\cdot)^{-}$ has a closed graph.

\medskip
{\bf Proof of Claim 3. }

Take a sequence $(z_{n},q_{n}) \to (z,q)$ with 
$z_{n}\in {\cal Y}$, $q_{n}\in T(z_{n})^{-}$ for all $n$. This means that 
$p_{n}\cdot q_{n} \leq 0$ for all $p_{n}\in T(z_{n})$ by the definition 
of $T(\cdot)^{-}$. We must show that $p\cdot q \leq 0$ for all $p\in T(z)$.

Suppose the contrary holds.  Then there exists $p\in T(z)$ such that 
$p\cdot q > 0$.  Since $q_{n}\to q$, $p_{n}\cdot q \leq 0$ for all 
$p_{n}\in T(z_{n})$, we have that $p_{n}\cdot q_{n} \leq 0$.  Therefore, 
for $n$ large enough, $p\notin T(z_{n})$.  But $p\in T(z)$.  This implies 
that for large enough $n$, there exists $x_{n}\in F^{k}(z_{n})$ such that 
$p\cdot x_{n}\leq 0$.  However, $p\cdot F^{k}(z) > 0$.  The sequence 
$(x_{n})$ is bounded (because of the definition of the range of $F^{k}$), 
therefore it has a converging subsequence.  Keeping notation and switching 
to the subsequence, $x_{n}\to x$.

By the closedness of the correspondence $F^{k}$, which, as noted, follows 
from Claim 1, $x\in F^{k}(z)$. Therefore, $p\cdot x > 0$. But we also have 
$p\cdot x_{n}\leq 0$ for large enough $n$, which is a contradiction. 
\hfill QED. 

\medskip
By combination of Claims 2 and 3 it follows that the correspondence $T$ is 
lower semi-continuous.  Hence, we may apply Michael's selection theorem 
(see, e.g., Klein and Thompson 1984, Theorem~8.1.8).  The proof of 
Theorem~\ref{po-ve} shows that every selection $p$ from $T$ is a price 
system corresponding to a valuation equilibrium.  Applying Michael's 
selection Theorem, we conclude that there exists a continuous selection in 
$T$.  By repeating the proof of Theorem \ref{po-ve} for this continuous 
selection we establish the assertion, i.e., the allocation $(f,y)$ indeed 
can be supported by a continuous price system $p \colon {\cal Y} \to 
S^{\ell-1}$ as a valuation equilibrium.
\qed\

\bigskip
The extension of Theorem \ref{po-ve:cont} to a nontruncated economy is 
evident as $k$ above becomes large enough. We refer to Debreu (1959) for
the formal techniques to be used. The proof of the following corollary is
therefore omitted.

\begin{coro} \label{po-ve:cont-corr}
Let $\SE$ be an economy such that $\cal Y$ is endowed with a topology 
generated by a metric $d$, for all agents $a \in A$ preferences are 
continuous, quasi-concave and strictly monotonic on \eucl{\ell}{+}, the 
cost function is continuous on $({\cal Y},d)$, and $\cal Y$ satisfies the 
essentiality condition.  Then every Pareto efficient allocation in $\SE$ 
can be supported as a valuation equilibrium with a continuous price 
system.
\end{coro}

\section{Cost Sharing and the Core}

Following Mas-Colell (1980) we now turn to 
the question whether we are able to give a plausible extension of the 
notion of a cost share equilibrium to our setting, in such a way as to 
preserve core equivalence.  A {\em cost share\/} equilibrium is a special 
kind of valuation equilibrium, in which the valuation function is simply a 
distribution of the costs of the public project over all agents in the 
economy; subsidies are not allowed. It turns out that the plausible 
extension of cost share equilibrium is immediate, and all cost share 
equilibria are in the core, but the core may be strictly larger than the 
set of cost share equilibria. Hence, the core equivalence results in 
Mas-Colell (1980) and Weber and Wiesmeth (1991) depend crucially on the 
assumption that there is only one private good.
 
\begin{defni} \label{cse}
A feasible allocation $(f,y) \in \Phi$ is a {\bf cost share equilibrium} in 
$\SE$ if there exist a price system $p \colon {\cal Y} \to S^{\ell-1}$ 
and a non-negative valuation function $V \colon A \times {\cal Y} 
\to \SR_+$ such that
\begin{numm}
\item
there is no surplus, i.e., $\sum_{a\in A} V(a,y) = p(y) \cdot c(y)$ ;
\item
for every agent $a \in A$, $(f(a),y)$ maximizes 
$U_{a}$ on the budget set
\[ \left\{ (g,z) \in \eucl{\ell}{+} \times {\cal Y}\ \left| \
p(z) \cdot g + V(a,z) = p(z) \cdot w(a)
\right. \right\} \]
and for every $z \in {\cal Y} :\ V(a,z) \leq p(z) \cdot w(a)$ ;
\item
$y$ maximizes $\sum_{a\in A} V(a,z)\,d\mu - p(z) \cdot c(z)$ over $z\in 
{\cal Y}$.
\end{numm}
\end{defni}

A cost share equilibrium is a valuation equilibrium in which there are no 
subsidies.  In the sequel we will use the abbreviation CSE to indicate cost 
share equilibria.

\bigskip
We now define the core for our framework.  In this setting a core 
allocation is defined straightforwardly as an extension of the one proposed 
by Mas-Colell (1980).  In Roberts (1974), Wooders (1989), Vasil'ev, Weber 
and Wiesmeth (1992), and Gilles and Diamantaras (1994) alternative 
formulations of the core are proposed and core equivalence theorems are 
established.  From these studies it follows that such equivalence results 
hold only for a modified core concept.  In this paper we show that there is 
no equivalence of the core and the set of CSE, a not unexpected result. 
This shows that the core equivalence results of Mas-Colell (1980) and Weber 
and Wiesmeth (1991) for finite economies rely solely on the assumption that 
$\ell = 1$.

\begin{defni}
A feasible allocation $(f,y) \in \Phi$ is in the {\bf core} of 
$\SE$ if there are no coalition $E \subset A$, public project 
$z \in {\cal Y}$, and allocation of private goods $g \colon E \to 
\eucl{\ell}{+}$ such that the following requirements are satisfied:
\begin{numm}
% \item
% $\mu(E) >0$;
\item
$(g,z)$ is feasible for $E$, i.e.,
\[ \sum_{a\in E} g(a) + c(z) = \sum_{a\in E} w(a), \ \ \mbox{ and} \]
\item
for all members $a$ of $E$:
\[ U_a(g(a),z) > U_a(f(a),y). \]
\end{numm}
\end{defni}

The core of the economy $\SE$ will be indicated by ${\cal C}(\SE) 
\subset \Phi$.

\spr
Our first result extends the insight that CSE always are in the core --- as 
shown in Mas-Colell (1980) --- to our framework.

\begin{thm} \label{cse-wc}
Let for all agents $a \in A$ the utility function $U_{a}$ be monotone. 
Then every cost share equilibrium is in the core of $\SE$.
\end{thm}

{\sc Proof}

Suppose that there is a coalition \( E \subset A \), a public project \( 
z \in {\cal Y} \), and an allocation of private goods \( g \colon E \to 
\eucl{\ell}{+} \) such that
\[
\sum_{a\in E} g(a) + c(z) = \sum_{a\in E} w(a),
\]
and, for all all $a\in E$,
\[
U_{a}(g(a),z) > U_{a}(f(a),y).
\]
By condition (ii) of the definition of CSE, for all \( a \in E \),
\[
p(z) \cdot g(a) + V(a,z) > p(z) \cdot w(a).
\]
It follows that
\begin{equation}\label{star}
p(z) \cdot \sum_{a\in E} g(a) + 
\sum_{a\in E} V(a,z) > p(z) \cdot \sum_{a\in E} w(a).
\end{equation}
Now we have the following chain:
\begin{eqnarray*}
0 & = &     \sum_{a\in A} V(a,y) - p(y) \cdot c(y)\\
  & \geq & \sum_{a\in A} V(a,z) - p(z) \cdot c(z)\\
  & \geq & \sum_{a\in E} V(a,z) - p(z) \cdot c(z)\\
  & > & p(z) \cdot \sum_{a\in E} w(a) - p(z) \cdot \sum_{a\in E} g(a) - p(z) 
  \cdot c(z).
\end{eqnarray*}
In this chain the first equality follows from condition (i) of the definition 
of CSE and the first weak inequality follows from condition (iii) of the same 
definition.
The second weak inequality follows from \( V(\cdot,\cdot) 
\geq 0 \), and the strict inequality follows from~(\ref{star}) above.
However, the first and last items in the chain contradict the feasibility of 
improving by coalition \( E \) because \( p(z) > 0 \) by monotonicity.
\qed\

\bigskip
The failure of the converse to hold in economies with multiple private
commodities is shown next by example. The example 
also illustrates how to compute a valuation equilibrium and how there may 
not be much choice in the price system and valuation functions that 
support a given efficient allocation.

\begin{thm}\label{th:noequiv}
There exists well-behaved economies $\SE$ such that the core of $\SE$ is 
strictly larger than the set of cost share equilibria of $\SE$.
\end{thm}

{\sc Proof}

Let $A=\{a,b\}$, $\ell=2$, ${\cal Y} = \{y,z\}$, $w(a) = (3,0)$, $w(b) = 
(0,3)$, $c(y) = (1,1)$, and $c(z) = (0,2)$.  The utility functions are 
$U_{a}(f_{1},f_{2},y) = U_{b}(f_{1},f_{2},y) = 3\sqrt{f_{1}} + 
\sqrt{f_{2}}$ and $U_{a}(f_{1},f_{2},z) = U_{b}(f_{1},f_{2},z) = 
\sqrt{f_{1}} + \sqrt{f_{2}}$ for every commodity bundle $f = (f_1,f_2)
\in \eucl{2}{+}$.

\spr
Consider the allocation $(f,y)$ with $f(a) = f(b) = (1,1)$. It is a 
feasible allocation, because $f(a)+f(b)+c(y) = (3,3) = w(a)+w(b)$.

\spr
We claim that $(f,y)$ is a core allocation. First, it is clearly Pareto 
efficient. Indeed, as long as $y$ is chosen, clearly there can be no 
improvement for any agent without harming the other; if $z$ is chosen, 
one of the agents must become worse off relative to $(f,y)$. Second, no 
agent can unilaterally improve on $(f,y)$: agent $a$ cannot undertake any 
project alone because he does not have any amount of private good 2, 
while agent $b$ can produce $z$ but this leaves her with a utility of 1, 
less than the 4 attained at $(f,y)$.

\spr
There is a price system and valuation functions that turn the allocation 
$(f,y)$ into a valuation equilibrium: $p(y) = (\sfrac{3}{4},\sfrac{1}{4})$, 
$p(z) = (\half,\half)$, and $V(a,y) = \sfrac{5}{4}$, $V(b,y) = 
-\sfrac{1}{4}$, $V(a,z) = \half$, 
$V(b,z) = \half$. It is easy to check that these make $(f,y)$ a valuation 
equilibrium, and it is clear that it is not a cost share equilibrium, 
since $V(b,y) < 0$. The question then is whether it is possible to find 
another price and valuation system supporting this allocation, in which 
no valuation is negative.

\spr
Note that $p(y)$ must remain the same in any valuation equilibrium, 
because the utility functions are smooth and must be supported by $p(y)$ 
at $(1,1)$. Let us try to construct a valuation equilibrium with $V(b,y) 
\geq 0$. This inequality implies $V(a,y) \leq 1$. From $a$'s budget we 
see that this implies $\sfrac{3}{4} f_{1}(a) + \sfrac{1}{4} f_{2}(a) 
\geq \sfrac{5}{4}$; now this 
implies that agent $a$ obtains a utility equal to at least $2\sqrt{5}$ by 
demanding a vector $(g_{1}(a),g_{2}(a)) = (x,x)$ with $x\geq 
\sfrac{5}{4}$. This 
means that the allocation $(f,y)$ is not a valuation equilibrium.
\qed\

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\end{document}



