\documentclass[12pt]{article}
\setlength{\textwidth}{15cm}
\setlength{\textheight}{20cm}
\setlength{\oddsidemargin}{.5cm}
\setlength{\evensidemargin}{.5cm}



\newtheorem{definition}{Definition}[section]
\newtheorem{assumption}[definition]{Assumption} 
\newtheorem{theorem}[definition]{Theorem} 
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{corollary}[definition]{Corollary}
\newtheorem{remark}[definition]{Remark}
\newtheorem{lemma}[definition]{Lemma}




\def\xone{x_1^*(\gamma_1,w)}
\def\xtwo{x_2^*(\gamma_2,w)}
\def\om{w}
\def\omone{w^1}
\def\omtwo{w^2}
\def\u{\bar{u}}
\def\enddemo{${\bf Q.E.D.}$\\ \medskip} 


\def\EE{\mathop  {\rm EE}\nolimits}
\def\EA{\mathop {\rm   EA}\nolimits}

\def\res{\mathop  {\rm RES}\nolimits}
\def\eb{\mathop {\rm   EB}\nolimits}
%%%%\def\aa{\mathop {\rm   AA}\nolimits}
\def\sa{\mathop {\rm   SA}\nolimits}
\def\pf{\mathop   {\rm   PF}\nolimits}
\def\ps{\mathop   {\rm   PS}\nolimits}
\def\ef{\mathop   {\rm   Eff}\nolimits}
\def\myref#1{(\ref{#1})}

\def\dooWo{v^{1}_{11}-2v^{1}_{12}f' + v^{1}_{22}(f')^2 + v^{1}_{2} f''
+\gamma_{1}(v^{2}_{22}(f')^2 + v^{2}_{2} f'')
}

\def\dotWo{-v^{1}_{12} f' + v^{1}_{22}(f')^2 + v^{1}_{2} f''
-\gamma_{1}(v^{2}_{12}f' - v^{2}_{22}(f')^2-v^{2}_{2} f'')
}

\def\dttWo{
v^{1}_{22}(f')^2 + v^{1}_{2} f''+
\gamma_{1}(v^{2}_{11}-2v^{2}_{12} f' + v^{2}_{22}(f')^2 + v^{2}_{2} 
f'')
}

\def\dotWt{v^{2}_{12} f' - v^{2}_{22}(f')^2 - v^{2}_{2} f''
+\gamma_{2}(v^{1}_{12}f' - v^{1}_{22}(f')^2-v^{1}_{2} f'')
}

\def\dttWt{v^{2}_{11}-2v^{2}_{12} f' + v^{2}_{22}(f')^2 + v^{2}_{2} 
f''
+\gamma_{2}( v^{1}_{22}(f')^2 + v^{1}_{2} f'')
}




\def\r{{\rm I\kern-.20em{R}}}

\def\P{\mathop  {\rm P}\nolimits}

\def\R{\mathop  {\rm R}\nolimits}

\def\proof{{\bf Proof: }}

\def\endproof{${\bf Q.E.D.}$\\ \medskip}



\begin{document}
  \title{On Altruism, Efficiency and Public Goods} \author{Francisco
Marhuenda\footnote{Universidad de Alicante, Departamento de
Fundamentos del An\'alisis Econ\'omico. Ap. 99. E-03080. Spain.  Ph.
and Fax: 34-96/5903614 Email: paco@merlin.fae.ua.es.}\hspace{.2cm} and
\hspace{.2cm} Felipe
P\'erez-Mart\'{\i}\footnote{Instituto de Estudios Superiores de Administraci\'on
(IESA). Av. IESA, San Bernardino, Caracas.  Email:
felipe.perez@iesa.edu.ve}} 
\date{August, 1999\\(Incomplete version)}
\maketitle
  
  \abstract{In this paper we reconsider the problem of altruism and
  efficiency in an economy with public goods. We present a more
  complete and natural model of altruism in an two agent economy with
  public goods than the standard one. We show that welfare comparisons
  can be made, from the point of view of both, selfish and altruistic
  agents, regarding selfish vs. altruistic allocations.  Given a
  degree of altruism (or selfishness), we are able to show uniqueness
  of altruistic equilibria. We also completely characterize the set of
  altruistic equilibria, for all possible degrees of altruism. We then
  show that given a degree of altruism or selfishness which produces an
  inefficient allocation due to market failure, for example,
  altruistic parameters exist such that the new altruistic equilibrium
  is Pareto improving from the point of view of both, the original,
  and the new individuals. We finally illustrate the theory with
  GNU-Linux, the ``open source'' movement.}

\section{Introduction}



One issue that has been unresolved in the economic analysis of public
goods is the one about welfare comparisons when there is altruism. Two
classic papers about altruism and public goods have been
Cornes-Sandler (1984, \cite{CS}) and Andreoni (1989, \cite{A1}). Those
papers have introduced the analysis of altruism in the economic
analysis of public goods. The first deals with ... and the second with
issues of efficient government policies when people are altruists.  A
natural question, though, not addressed by those papers, is if
altruism per se can be considered welfare improving when you have
market (or other mechanism) implementation problems due to the nature
of public goods and asymmetric information (market failure, free
riding\footnote{When there is a public good, the market equilibria
resulting from the private provision of the good is not Pareto
optimal, in general, if the firm can not charge each person a
different price. In fact, a Pareto optimal Lindhal equilibrium, in
which each person is charged a different price for the public good,
requires that each person reveals his or her marginal benefit from
enjoying the public good provided. In equilibrium, the personal price
equals that marginal benefit. Now, that assumption is not natural in
an environment with selfish agents, since they have incentives to
understate their interest in the public good, and free ride on the
rest of the population.}.  Incentive problems with selfish individuals
make the provision of public goods generally impossible or
inefficient. With altruistic agents things might change and they do,
as we show below. On the one hand, the incentive motive to free ride
might desapear, at least in part, since people now care about the
harmful consequences to other people if they do not report
truthfully. On the other hand, depending on the example, altruistic
people might just simply organize themselves in order to provide the
public good, and the final allocation might be efficient or, at least,
Pareto improving with respect to the no provision outcome, or even
with respect to the same-price private provision of the public
good. In this paper we will focus our attention to the last
possibility, since this case is what appears to be happening in some
relevant current situations. 

In the referred papers, you could ask, once you assumed altruism,
efficiency and public policy questions which turn out to be very
similar to the ones with selfish individuals. In this paper, on the
other hand, we go back and ask if altruism itself could be interpreted
as a mechanism, a psychological one, used by selfish agents in
order to improve their welfare, given the usual inefficiencies present
when selfish people use a given mechanism. In order to answer this
question, we have to be precise about different possible levels of
altruism. It turns out, as we show below, that not all levels of
altruism produce, using the simplest Nash equilibrium solution
concept, an efficient outcome.

Our question could be posted as: Given an altruistic equilibrium
allocation in an economy with public goods, is it better for selfish
individuals to become altruists?.  We introduce, in order to be able
to make such welfare comparisons, a standard model of altruism, used
in macroeconomic models dealing with intergenerational transfers and
policy issues\footnote{See, for example the seminal paper of Barro
(1974, \cite{Barro}}, as well as in microeconomic models used to
explain relatives behavior\footnote{See, for example Becker (1974,
\cite{Becker}), Stark (1995, \cite{OS}), Arnott and Stiglitz (1991,
\cite{Arnott})}. Our model also has an interesting relationship, which
we make use of, with the central planer characterization of Pareto
optimal allocations. It also turns out that the model is a benchmark
solution to the problem of consistent interdependent preferences (see
Marhuenda and P\'erez-Mart\'{\i} \cite{MP2}), and allows us to consider all
possible degrees of altruism, selfishness being a special case of it
(altruism of  zero degree).


Our model could be thought of as an analysis of the relationship
between two central planners. Generalizing in this way welfare
analysis, we show that, given an inefficient allocation implemented by
any mechanism in an economy populated by individuals with a given
profile of altruism, there is another such profile, with a higher
degree of altruism, that produces a Pareto improving allocation for
both the society with the initial profile, as well as for the society
with the more altruistic profile. The mechanism to implement the
Pareto improving allocation is the simple Nash equilibrium solution
concept.

The plan of the paper is as follows. We first show a couple of results
that allow us to make welfare comparisons between people of different
degrees of altruism. Then we show existence and uniqueness of Nash
equilibrum for an given economy populated by individuals with a given
degree of altruism. Finally, we completely characterize the set of
altruistic equilibria, and show that the set go ``in the right
direction'' in terms of welfare. A corollary is the result described
in the previous paragraph\footnote{This last result complements
another paper of ours (\cite{MP}), in which we treat the case of
incomplete markets, and show that inter-relatives and inter-friends
transfers due to altruism are, in fact, a mechanism to ``complete the
markets'', and improve in people's welfare.}. The uniqueness result,
new in the literature of altruism and public goods, enhances the
already atractive feature of the Nash equilibrium concept as the
simplest mechanism that implements an allocation. As we know,
uniqueness is a very useful feature for predictability, and, for that
reason, makes the theory easily testable.

In the last section, we show that our model can be interpreted as
equivalent to the ones which are standard in the altruism and public
goods literature, for example the one by Andreoni (\cite{A1,
A2}). Since it has more structure, the model is, in fact, a special
case. But, as we will show, it allows a more natural interpretation of
the concept of altruism in the presence of public goods, and avoids
what we consider to be erroneus conceptions in the extreme cases of
pure altruism and pure selfishness. 

In particular,  we are able to use the model in order to
explain, from an economic point of view, the recent ``open source''
software phenomenon, the GNU/Linux movement. We enphasize {\it economic} point
of view, as opposed to a mere ``spiritual'', or non-sustainable one,
since we get that, given a degree of altruism by the Hackers, an
efficient equilibrium is quite robust in the same sense any Nash
equilibrium is: people do not want to move or deviate from there.

Let us examine a little closer this example. Firs of all sofware 
is a pure public good, since its use by
someone does not preclude its use by the rest of the population, even
though that use requires apropriate hardware.  Among  people who
has the required hardware, whith enough iddle disk space, a piece of
sofware is indeed a pure public, if we
assume that its distribution is costless. This is not a farfetched
assumption at present, taking into account what is occurring
nowadays through the internet.  The fact that software is {\it
excludable}, by copyright law does not make it a private good.

Free Software Foundation (FSF), which owns the GNU project, consists
of a group of programmers who have spontaneously asociated themselves
in order to develop a public access sofware. This institution allows
the movement to actually guarantee that the sofware they create is
actually a non-excludable (``free access'', as they call it) public
good. The provision of the good is free of charge, and the person who
gets it is asked to voluntarily provide resourses (money, time,
machine time, skill) in order to make the project
self-sustainable. People who provide resourses, are actually joining
the group, since those ({\it private)}) resourses are an imput in the
production of the public good. The {\it General Public Licence} (GPL)
is the legal instrument used by FSF in order to make sure that the
person who receives the good is not able to sell it for own profit
and/or modify it and distribute it as if that person was the private
owner of the good.  People can modify it, but only if it is an
improvement, or an addition, and the FSF will decide when the
modification proposed is appropriate. If it is, the package, when
updated, will contain that modification, which will then become free
and accesible for all again.  The modification or addition is actually
part of the private imputs to a new pure, non-excludable, public good,
the new version of the piece of software. The ``GNU Manifesto'',
written by Richard Stallman at the beginning of the project, in order
to ask for participation and support, says, among other things:

\begin{quotation}
  
  ``I consider that the golden rule requires that if I like a program
  I must share it with other people who like it.  Software sellers
  want to divide the users and conquer them, making each user agree
  not to share with others.  I refuse to break solidarity with other
  users in this way.  I cannot in good conscience sign a nondisclosure
  agreement or a software license agreement.... So that I can
  continue to use computers without dishonor, I have decided to put
  together a sufficient body of free software so that I will be able
  to get along without any software that is not free.  I have resigned
  from the AI lab to deny MIT any legal excuse to prevent me from
  giving GNU away.''
\end{quotation}


In the last section of the paper, Hackers are shown to exhibit
altruistic behavior, hiden in their ``warm glow'' as authors of very
sound, free of charge, pieces of software. Even though we do not have
a quantitative analysis about the efficiency of their allocations (the
quality of their products, basically), we are be able to apply the
qualitative conclusions of our paper to this example.




\section{The Model}
\label{AltPref}

We consider an economy with two agents, one private and one public
good.  There is no initial endowment of the public good but agent
$i=1,2$ has  $\omega^i$ units of initial endowment of the private good.
We let $\omega=\omega^1+\omega^2$ be the total initial endowment of
private goods.

Preferences are summarized by the following
utility functions $u^1,u^2:\r_+^2\times \r_+\to \r$

\begin{eqnarray} u^1(x_1,x_2,y)&=&v^1(x_1,y)+\gamma_1 v^2(x_2,y) \nonumber \\
  u^2(x_1,x_2,y)&=&v^2(x_2,y)+\gamma_2 v^1(x_1,y), \label{AUF} \end{eqnarray} where
$0\leq \gamma_1,\gamma_2$. Here, $x_i$ represents the consumption of the
private good by individual $i,  i=1,2$, and $y$ is the output (and
consumption) of the pure public good. Selfish
preferences correspond to $\gamma_1=\gamma_2=0$ and a high degree of
altruism is associated with large values of $\gamma_1$ or $\gamma_2$.
Let the function $f:\r\to \r$, be the technology available in society
that  transforms private goods into public goods.



\begin{assumption}
\label{ASST}
Public good technology is represented by a mapping $f:\r_+\to\r_+$,
that is a strictly increasing, strictly concave $C^1$ function, and
$f(0)=0$.
\end{assumption}

Thus, an economy is specified by a vector $(v^1,v^2;f;
\gamma_1,\gamma_2;\omone,\omtwo)$, consisting of the preferences
discussed above, and the initial endowment of the single private good
$w^i$ available to agent $i=1,2$. The resource
constraints for the economy are, then: $ x_1+x_2+z=\omone + \omtwo=\omega$,
and $ y=f(z)$, where $z$ is the amount of input of private goods used
for the production of the public good. 


An allocation $(x_1,x_2)$ is
feasible if $0\leq x_i\leq \omega^i$.  It is understood here that
$x_i$ is the consumption by agent $i=1,2$ of the private good and that
an amount $f(\omega-x_1-x_2)$ of the public good is provided along
with the allocation $(x_1,x_2)$. It is convenient to substitute for
the value of $y$ in preferences. We make this more precise in the following

\begin{assumption} \label{crossint}
The preferences of the agents are represented by utility  functions
\begin{eqnarray*}
        W^1(x_1,x_2;\gamma_1) &=&  v^1(x_1,f(\omega-x_1-x_2)) + 
\gamma_1v^2(x_2,f(\omega-x_1-x_2))\\
W^2(x_1,x_2;\gamma_2) &=& v^2(x_2,f(\omega-x_1-x_2)) + 
\gamma_2v^1(x_1,f(\omega-x_1-x_2))
\end{eqnarray*}
where 
$\gamma_1, \gamma_2\geq 0$
and
$v^i$ are strictly concave functions strictly increasing in both 
arguments  and  such that for $i\neq j$,
\begin{equation}\label{crosspos}v^k_{ij} = {\partial^2 v^k \over \partial x_i \partial 
x_j}(x_1,x_2)\geq 0\end{equation}
and
\begin{equation}\label{interior}
\lim_{x_k\to 0}{\partial v^k\over\partial x_k}(x_1,x_2)=+\infty.
\end{equation}
\end{assumption}


\begin{definition}
A feasible allocation $(x_1,x_2)$ is Pareto Optimal if there is no 
other feasible allocation $(r_1,r_2)$ such that 
$W^i(r_1,r_2;\gamma_i) \geq 
W^i(x_1,x_2;\gamma_i)$ with some inequality strict. We
 let $\P(\gamma_1,\gamma_2)$ denote the set of 
Pareto optimal allocations in the economy $(\gamma_1,\gamma_2)$.
\end{definition}

We represent in the following graph feasible allocations by points
$(x^1,x^2)$ such that $\omega-x^1+x^2\geq 0$. $y$ then can be inferred
as $y=f(\omega-x^1+x^2\geq 0) z\in\r_+\}$. We use a triangle similar
to Kolm's to represent allocations, preferences, Pareto optimal
allocations and equilibria:
\vspace{2cm}

Next result allows us to make preference comparisons. It can be viewed
as a more general approach to central planner welfare analysis.


\begin{proposition}
Consider vectors $\gamma=(\gamma',\gamma'')\geq 0$ such that
$\gamma'\gamma''\leq 1$. Then the Pareto correspondence $\P$ is
decreasing in $\gamma$, i.e. if $(\gamma_1,\gamma_2) \geq
(\gamma_3,\gamma_4)$, then $\P(\gamma_1,\gamma_2)\subset
\P(\gamma_3,\gamma_4)$.
\end{proposition}

\proof 
The Proposition is trivial if $\gamma_1=\gamma_2=0$.
Let us assume, for example, that $\gamma_2,\gamma_4>0$.
An allocation $(x_1,x_2)$ is Pareto optimal in the economy 
$\gamma=(\gamma_1,\gamma_2)$ iff there is an $\alpha\in [0,1]$ such 
that $(x_1,x_2)$ maximizes 
\begin{eqnarray*}
&\alpha W^1(x_1,x_2;\gamma_1) + {(1-\alpha)\over \gamma_2}
W^2(x_1,x_2;\gamma_2)&\\
=&
v^1(x_1,f(\omega-x_1-x_2)) + (\alpha\gamma_1 + {(1-\alpha)\over 
\gamma_2})
v^2(x_2,f(\omega-x_1-x_2)).&
\end{eqnarray*}
Let
$$
\beta = {\alpha\gamma_1 + (1-\alpha)/\gamma_2 -1/\gamma_4\over
\gamma_3-1/\gamma_4}.
$$
Since $\gamma_4\leq\gamma_2$ and $\gamma_1\gamma_2\leq 1$, we have
$$
\alpha\gamma_1\gamma_2\gamma_4 + (1-\alpha)\gamma_4\leq
\alpha\gamma_4 + (1-\alpha)\gamma_4= \gamma_4\leq \gamma_2
$$
so $\alpha\gamma_1 + (1-\alpha)/\gamma_2 -1/\gamma_4\leq 0$. Clearly,
$\gamma_3-1/\gamma_4<0$, since  $\gamma_3\gamma_4\leq 1$. Hence $\beta> 
0$.
On the other hand,
$$
\alpha + {1-\alpha\over \gamma_1\gamma_2}\geq 1
\geq {\gamma_3\over\gamma_1}
$$
so $\alpha\gamma_1 + (1-\alpha)/\gamma_2 -1/\gamma_4\geq
\gamma_3-1/\gamma_4$ and we also have $\beta\leq 1$. 

Finally, we see that 
$\alpha W^1(x_1,x_2;\gamma_1) + {(1-\alpha)\over \gamma_2}
W^2(x_1,x_2;\gamma_2) =
\beta W^1(x_1,x_2;\gamma_3) + {(1-\beta)\over \gamma_4}
W^2(x_1,x_2;\gamma_4)$ so
$(x_1,x_2)$ is Pareto optimal in the economy 
$\gamma=(\gamma_3,\gamma_4)$ as well.
\endproof

Even tough we have the possibility of welfare comparisons now, there
is a slight complication. Someone might argue that even though an
increase in the altruistic profile of a society keeps some of the
previous efficient allocations still efficient, society might be
losing some key efficient allocations. Next result shows that it is not
the case. 

\begin{proposition}\label{BETTER}
Consider vectors $\gamma=(\gamma',\gamma'')\geq 0$ such that
$\gamma'\gamma''\leq 1$. Let $(x_1,x_2)$ be an inefficient allocation
for society $(\gamma_1,\gamma_2)$, and let
$\P(\gamma_1,\gamma_2;x_1,x_2)$ be the Pareto efficient allocations
preferred by both individuals to $(x_1,x_2)$ Then there exists an
economy  $(\gamma_3,\gamma_4)\geq  (\gamma_1,\gamma_2)$ such that 
$\P(\gamma_1,\gamma_2;x_1,x_2) \subset \P(\gamma_3,\gamma_4)$.
\end{proposition}

\proof 
To be provided.


\begin{definition}
  \rm A feasible allocation $(x_1,x_2)$ is an Altruistic Agreement if
  $\gamma_1\gamma_2\leq 1$, $x_1$ is a solution to
\begin{equation}
\label{re1}
\max_{0\leq \xi\leq w^1}W^1(\xi,x_2;\gamma_1)
\end{equation}
and $x_2$ 
is a solution to 
\begin{equation}
\label{re2}
\max_{0\leq \xi\leq w^2}W^2(x_1,\xi;\gamma_2).
\end{equation}
\end{definition}
This is the application of the Nash equilibrium concept to our
model. It turns out that, in general, there is an inherent conflict
among the agents of this economy, in spite of their altruism. Only in
the case $\gamma_1\gamma_2= 1$, what each agent wants for himself of
the private good coincide exactly with what the other agent wants for
him (the peaks for equations ??  coincide for both agents), and the
preferred quality of the public good also coincide\footnote{In fact,
if the maximization problems (\ref{re1}) and (\ref{re2}) were
performed over the two first arguments, an internal single peak
vector would be obtained as solution, given our assumptions. The
results of Marhuenda and P\'erez-Mart\'{\i} (\cite{MP}) that says that
the peaks coincide when $\gamma_1\gamma_2=1$ apply here too.}. For
the other cases, we have to use the Nash equilibrium concept in order
to guarantee that they agree upon a common quality of the (common)
public good. Now, in the case $\gamma_1\gamma_2< 1$, Nash equilibrium
entails the kind of disagreement that says that each agent, if he were
to choose how to allocate the endowments of all the economy, he would
prefer to give less of his own private endowment for the production of
the public good, and more of the private endowment of the other
individual for that purpose. In the case $\gamma_1\gamma_2> 1$, a Nash
equilibrium would prescribe an allocation such that, if each agent
were to do a similar choice, he would prefer to give more to the
public good, and the other agent to give less, of their own resources,
than the amounts prescribed in the proposed equilibrium. As we will
see, the case $\gamma_1\gamma_2= 1$ generates an equilibrium that is a
Pareto allocation for both economies, the one with selfish, as well as
the one with altruistic people. Furthermore, the Pareto allocation for
the altruistic economy is, in that case, the unique possible Pareto
allocation.

\begin{proposition}
\label{existence}
Given $\gamma_1,\gamma_2\geq 0$, $\gamma_1\gamma_2\leq 1$, there is 
(at least) an Altruistic Agreement in the economy 
$(\gamma_1,\gamma_2)$.
\end{proposition}

\proof
It is easy to verify that the mappings $W^i$ are strictly concave and 
thus,
given $\theta\in [0,w^2]$, there is a unique solution 
$\varphi_1(\theta,\gamma_1)\in [0,w^1]$ to the maximization problem
$\max_{0\leq \xi\leq w^1}W^1(\xi,\theta;\gamma_1)$. This defines a mapping
$\varphi_1:[0,\omega^2] \to [0,\omega^1]$ representing the best 
response (in terms of consumption of the private good) of  player~1 
given that agent~2 consumes $\theta$. By the Theorem of the Maximum, 
the function $\varphi_1$ is continuous.
Similarly, there is a continuous mapping $\varphi_2:[0,\omega^1] \to 
[0,\omega^2]$ such that $\varphi_2(\theta,\gamma_2)\in [0,w^2]$ is the 
unique solution  to the maximization problem
$\max_{0\leq \xi\leq w^2}W^2(\theta,\xi;\gamma_2)$.
Note that $(x_1,x_2)$ is a Altruistic Agreement if and 
only if
$\varphi_1(x_2;\gamma_1)=x_1$ and  $\varphi_2(x_1,\gamma_2)=x_2$.


Now, the function  $\varphi_2\circ\varphi_1:[0,\omega^2] \to 
[0,\omega^2]$
is also continuous. By the fixed point theorem,
there is $\theta\in [0,\omega^2]$ such that 
$\varphi_2\circ\varphi_1(\theta,\gamma_1) =\theta$. Thus, $x_1 
=\varphi_1(\theta,\gamma_1)$ and $x_2 = \theta$ is a Altruistic Agreement.
\endproof


We make the necessary assumptions, so that the solutions the the
problems~\ref{re1} and~\ref{re2} are determined by the first order
conditions.  In fact, the mappings $\varphi_1$ and $\varphi_2$
introduced in the proof of Proposition~\ref{existence} are defined
implicitly by the equations
$D_1W^1(\varphi_1(\theta,\gamma_1),\theta;\gamma_1)=0$ and
$D_2W^2(\theta,\varphi_2(\theta,\gamma_2);\gamma_2)=0$ and so are
$C^1$.  Note that $\varphi_1$ and $\varphi_2$ depend also on
$\gamma_1$ and $\gamma_2$, respectively.

\begin{lemma}
\label{decrease}

\begin{enumerate}
\item
For $\gamma_1, \gamma_2$ fixed, $\varphi_i$, $i=1,2$ is strictly 
decreasing in
$\theta$.
\item
For $\theta$ fixed, $\varphi_1$  is strictly decreasing in $\gamma_1$, 
$\varphi_2$  is strictly decreasing in $\gamma_2$.
\end{enumerate}
\end{lemma}

\proof
By implicit differentiation, we have that 
$\varphi_1'=\partial \varphi_1/\partial\theta$ is derived from equation
$D_{11}W^1(\varphi_1(\theta,\gamma_1),\theta)\varphi_1'(\theta) +
D_{12}W^1(\varphi_1(\theta,\gamma_1),\theta)=0$. Thus, 
$$\varphi_1'(\theta,\gamma_1)=-{D_{12}W^1(\varphi_1(\theta,\gamma_1),\theta)\over 
D_{11}W^1(\varphi_1(\theta,\gamma_1),\theta)}.$$
A simple computation shows (using assumptions \ref{ASST} and \ref{crossint}) that
$$D_{11}W^1=\dooWo<0$$
and 
$$D_{12}W^1=\dotWo < 0$$
so $\varphi_1'(\theta,\gamma_1) <0$ and $\varphi_1$ is decreasing in $\theta$.

By differentiating now implicitly 
$D_{1}W^1(\varphi_1(\theta,\gamma_1),\theta;\gamma_1)=0$ with respect to 
$\gamma_1$ we have
$$D_{11}W^1{\partial \varphi_1 \over\partial \gamma_1} +
{\partial \over\partial \gamma_1}D_{1}W^1=0$$
which is the same as
$$D_{11}W^1{\partial \varphi_1 \over\partial \gamma_1} =
v^2_2f'$$
But, $D_{11}W^1<0$ and $v^2_2f'>0$, so ${\partial \varphi_1 \over\partial 
\gamma_1}<0$. A similar  argument applies to $\varphi_2$.
\endproof


\begin{proposition}
\label{unique}
Given $\gamma_1,\gamma_2> 0$, $\gamma_1\gamma_2\leq 1$, 
the Altruistic Agreement of Proposition~\ref{existence} 
is unique.
\end{proposition}

\proof
By Lemma~\ref{decrease}, for each $\gamma_1$ fixed,
the function $\varphi_1$ is monotonic in $\theta$,
and thus, it has an inverse $\psi_1=\varphi_1^{-1}$. An altruistic Agreement is determined by a point $\theta$ such that 
$\psi_1(\theta,\gamma_1)= \varphi_2(\theta,\gamma_2)$. 

We claim now that 
$\psi_1'(\theta,\gamma_1) <\varphi_2'(\theta,\gamma_2)$
for $0<\gamma_1\gamma_2\leq 1$  and all 
$\theta$ in the domain of $\psi_1$.
 To see this fix $\gamma_1,\gamma_2$ such that 
$0<\gamma_1\gamma_2\leq 1$.
 Let $\gamma_1'=1/\gamma_2$. Note that $\gamma_1 \leq \gamma_1'$.
 After some computation we have that
 \begin{eqnarray*}
 \varphi_2'(\theta,\gamma_2)     & = & 
-{D_{12}W^2(\varphi_1(\theta,\gamma_1),\theta)\over 
D_{22}W^2(\varphi_1(\theta,\gamma_1),\theta)}  \\
         & = & {\dotWt\over \dttWt}  
 \end{eqnarray*}
 
 Since $W^1$ is strictly concave,
the determinant 
$$\left|
\matrix{D_{11}W^1 & D_{12}W^1\cr
D_{12}W^1 & D_{22}W^1\cr}
\right| = D_{11}W^1 D_{22}W^1 -(D_{12}W^1)^2 
$$
is strictly positive. Thus,
\begin{eqnarray*}
\psi_1'(\theta,\gamma_1)         & = & - 
{D_{11}W^1(\varphi_1(\theta,\gamma_1),\theta)\over 
D_{12}W^1(\varphi_1(\theta,\gamma_1),\theta)} \\
         & < & -{D_{12}W^1(\varphi_1(\theta,\gamma_1),\theta)\over 
D_{22}W^1(\varphi_1(\theta,\gamma_1),\theta)}  \\
         & = & -{\dotWo \over \dttWo}\\
         & < & {v^{1}_{12} f' - v^{1}_{22}(f')^2 - v^{1}_{2} f''
+\gamma_1'(v^{2}_{12}f' - v^{2}_{22}(f')^2-v^{2}_{2} f'')
 \over 
 v^{1}_{22}(f')^2 + v^{1}_{2} f''+
\gamma_1'(v^{2}_{11}-2v^{2}_{12} f' + v^{2}_{22}(f')^2 + v^{2}_{2} 
f'')}\\
                 & = & {\dotWt\over \dttWt}\\
                                 & = & \varphi_2'(\theta,\gamma_2)
\end{eqnarray*}
where the last inequality follows because
$v^{2}_{12}f' - v^{2}_{22}(f')^2-v^{2}_{2} f'' > 0$,
$v^{2}_{11}-2v^{2}_{12} f' + v^{2}_{22}(f')^2 + v^{2}_{2} f''<0$
and $\gamma_1 \leq \gamma_1'$. The claim is now proved.

Suppose now that there are two 
Altruistic Agreements $\theta_1<\theta_2$ such that
$\psi_1(\theta_1)= \varphi_2(\theta_1)$ and
$\psi_1(\theta_2)= \varphi_2(\theta_2)$. Then 
$\psi_1(\theta)- \varphi_2(\theta,\gamma_2)$ has two zeros in the interval
$[\theta_1,\theta_2]$, so there is $\theta_3\in (\theta_1,\theta_2)$
such that $\psi_1'(\theta_3) =\varphi_2'(\theta_3)$ which contradicts 
the claim.
\endproof

In the following graph, in which the Pareto frontier for selfish
agents is shown, we illustrate two Altruistic Agreements as the
crossings of the reaction curves $\psi$ and $\varphi$ for different
gamma vectors. There, $\psi'<\varphi<0$. In the case
$\gamma_1\gamma_2=1$, $\psi(\;,\gamma_1)$ and $\varphi(\;,\gamma_2)$
cross exactly at the Pareto frontier of the selfish agents $(P(0,0))$,
and $\P(\gamma_1,\gamma_2)$ consists of just that point, which is the
unique Altruistic Agreement. We have also represented the case of
$(\gamma_1,\gamma'_2)$, with $\gamma'_2<\gamma_2)$, so that
$\gamma_1\gamma'_2<1$. Here $\psi(\;,\gamma_1)$ and
$\varphi(\;,\gamma'_2)$ cross at an inefficient allocation, above
$P(0,0)$. In this case, $P(\gamma_1,\gamma'_2)$ is the segment of
$P(0,0)$ that goes from the point where $\varphi(\;,\gamma'_2)$
crosses it to $\P(\gamma_1,\gamma_2)$.
\vspace{2cm}

In order to fully characterize the set of all possible Altruistic
Agreements (agreements for all possible gamma combinations), we analyze
 the structure of the mapping
$$(\gamma_1,\gamma_2)\mapsto \R(\gamma_1,\gamma_2).$$
It  assigns to every economy $(\gamma_1,\gamma_2)$ such that 
$\gamma_1,\gamma_2>0$ $\gamma_1\gamma_2\leq 1$ its unique 
Altruistic Agreement.
First note that $\R(\gamma_1,\gamma_2)$ is given implicitly by the 
system of equations $D_1W^1=D_2W^2=0$. The Jacobian is
$D_{11}W^1 D_{22}W^2 - D_{12}W^1 D_{12}W^2 >0$ if 
$\gamma_1,\gamma_2 > 0$. Thus, $\R(\gamma_1,\gamma_2)$ is a $C^1$ 
function of $(\gamma_1,\gamma_2)$.
By Proposition~\ref{decrease}, the function $\R$ defines a 
diffeomorphism from the open set $A=\{(\gamma_1,\gamma_2)\in\r^2:
\gamma_1,\gamma_2>0, \gamma_1\gamma_2<1\}$ into the set of feasible 
allocations which extends to the boundary
$\partial A=\{(\gamma_1,\gamma_2)\in\r^2_+:\gamma_1\gamma_2=0\}\cup
\{(\gamma_1,\gamma_2)\in\r^2_+:\gamma_1\gamma_2=1\}$.
Let $B=\{(\gamma_1,\gamma_2)\in\r^2_+:\gamma_1\gamma_2=1\}$.

\begin{proposition} \label{SCOPE}
$\P(0,0) =\R(B)\cup\{ (\bar{x}_1,0)\} \cup\{(0,\bar{x}_2) \}$, where,
for $i=1,2$, $\bar{x}_i$ maximizes $$\max_{0\leq z\leq w}
v_i(z,f(\omega-z)$$ .
\end{proposition}

% \proof
% If $\gamma_1\gamma_2=1$ then the maximization problems~\ref{re1} and~\ref{re2}
% are equivalent and have a unique solution.
% \endproof

 It follows that $\R(A)$ is a region of $\r^2$ with a non--empty
 interior whose boundary is determined by $\P(0,0)$ and the graphs of
 the mappings $\varphi_1(\theta,0), \varphi_2(\theta,0)$. $\R(A)$ can
 be seen in the following graph. In general, a market equilibrium
 under-produces public goods, and generate allocations above
 $P(0,0)$. A straightforward consequence of the characterization of
 $\R(A)$ and of Proposition \ref{BETTER} is that, given a market
 allocation of public goods which is inefficient, there are altruistic
 parameters that improves upon that allocation in the Pareto
 sense. Formally:

\begin{corollary}
Consider vectors $\gamma=(\gamma',\gamma'')\geq 0$ such that
$\gamma'\gamma''\leq 1$. Let $(x_1,x_2)$ be an inefficient allocation
for society $(\gamma_1,\gamma_2)$. Then there exists an
economy  $(\gamma_3,\gamma_4)\geq  (\gamma_1,\gamma_2)$ such that, if
$(x_3,x_4)$  is the Altruistic Equilibrium that corresponds to economy
$(\gamma_3,\gamma_4)$, this allocation is Pareto preffered to
$(x_1,x_2)$ in both economies.
\end{corollary}
  

\section{Example: Efficiency of ``warm glow'' Giving}

In our original model, preferences for person 1 where given by utility
function

\begin{equation}u^1(x_1,x_2,y)=v^1(x_1,y)+\gamma_1 v^2(x_2,y)
  \label{PREFMP} \end{equation}

Let $z_i=w^i-x_i, i=1,2.$ So $z_i$ is the amount of private good that
person $i$ contributes to the provision of the public good. If person
1 has enough knowledge to realize that $y=f(z)$, where $z=z_1+z_2$,
then, since $y=f(z_1+z_2)=f(z_1+w^2-x_2)$, and $f$ is invertible, he
can figure that $x_2=w^2+z_1-f^{-1}(y)$. Having this relations in mind,
person 1 realizes that, from (\ref{PREFMP}),
\begin{eqnarray}
u^1(x_1,x_2,y)&=&v^1(x_1,y)+\gamma_1 v^2(x_2,y)\nonumber\\
&=&v^1(x_1,y)+\gamma_1 v^2(w^2+z_1-y,y)\nonumber\\
&\equiv&G^1(x_1,y,z_1),\label{PREFA}
\end{eqnarray}
which is an equivalent formulation to the one of Andreoni (\cite{A2}).
Now, in Andreoni's formulation, people do not seem to realize directly
that their amount of gifts to the public good affect the level of
private consumption of others. Our model provides an interpretation
regarding where the ``warm glow'' might come from: it certainly has to
do with the relationship of the person in question with
others\footnote{After all, our main motivation is why people are
altruists in the first place.  Even though we do not show the process
of learning or evolution of altruistic preferences here, the paper
argues that a justification for altruism that it makes (selfish or
altruist) people better off under certain circumstances.}. In our
model, the warm glow comes from the satisfying fact that own gifts to
public good enhance others people's pleasure of consuming both,
private and public kind of goods. In fact, given an amount of public
good, an increase in own gifts increases consumption of the private
goods by others, since your labor, say, is a substitute for the labor
of others. On the other hand, given an amount of consumption of
the private good by others, an increase in own gift to public good
increases everybody's consumption of the public good.



          

\section{Final remarks}

We have shown that, indeed, we can interpret altruism in an economy
with public goods as a mechanism people use in order to improve their
welfare. We have not talked about the evolution process that might
generate changes in preferences towards altruism. Our models shows, as
it should be obvious, that if there a fitness function, it should be
referred to society, and not necessarily to individuals. In fact, they
have an incentive to deviate and stay selfish if they know the other
members of society are going to evolve towards altruism. Now,
societies with a right enough profile of altruism might be more
successful than others due to the fact that they are more efficient in
the absence of a mechanism that implements efficient outcomes for the
selfish society members. A certain degree of reciprocity might be
achieved, to complement society fitness, if there is a certain
threshold in consumption below which individuals desapear from society
and weakens society economic potential.

We want to pursue this line of research, extending, at the same time,
the size of society to any finite natural number $n>2$. In order to
extend the number of society members, a complication shows up: if we
require that individuals care for every other person in society, the
assumed level of rationality regarding amount of information and
computational ability becomes huge and not realistic. The model would
have the same criticism of impracticality of the central planner
solutions, only that this time the criticism would have to be
multiplied, at least, by the number $n$. To keep things simple, we are
working on an extension that requires that any individual knows and
cares only about his neighbor. It turns out that evolution towards
altruism has been shown\footnote{See Ok and Vega-Redondo  (1999,
\cite{OVR}).} to be robust only in the cases of complete
information. Individualistic preferences become evolutively robust in
presence of incomplete information. Our requirement for extension of
the model has, then, two advantages: it is realistic (people know and
care, at least, about their neighbors), and it makes the model
evolutionarily stable.


We make now some remarks comparing definitions of altruism. We think
our definition is more appropriate than Andreoni's, the standard one in
the literature of altruism and public goods.  For Andreoni, when $G^1$
does not depend on $z_1$, person 1 is a ``pure altruist''. In our
model, this case corresponds to having $\gamma_1=0$, the case when the
person is a ``pure selfish'' individual. The rationale Andreoni
presents for this is that when one cares about the public good one
consumes, one is caring (indirectly) about the others who consume
it. But that is not a necessity, as the related case of externalities
shows.  In our model we see that the fact that an individual cares
about public goods (when $y$ appears in the ``pleasure'' function
$v(x,y)$) doesn't really mean that he is an altruist in the sense of
caring for others. Indeed, even if he does not provide anything to its
production (implying that only other persons are experiencing
desutility because of their effort), he still enjoys consuming it.

Similarly, when $G^1$ does not depend on $y$, person 1 is a ``pure
selfish'' individual for Andreoni. Our interpretations coincide when
individual 1 does not care for the public good and individual 2 does,
but $\gamma_1=0$. Now, in our model, this might happen when both
agents do not care for the public good, even though agent 1 is an
altruist. This is not a relevant case for an economy with public
goods, though. But our model allows for a more insightful case: when
individual 1 does not care for the public good and individual 2 does,
and $\gamma_1>0$. In this case, the more of the private good person 1
gives away, the more person 2 benefits. Now, the action of giving
away resources lowers person 1 consumption, of course. So person 1 is
improving person 2's welfare at his own expense. Can this be called
pure selfishness?. If anything this is pure altruism.

In this last case, Andreoni's interpretation comes from considering
that if one cares for what one gives, directly, what one is caring
about is not what the others get from it, but about the fact that
``one feels good'' about it. We disagree with this interpretation,
since there might be a reason, explicit or not, for that
feeling. Indeed, one of the purposes of our paper is to show that
``true altruism'', which means caring about others, has probably to do
with selfish individuals looking for a way of improving their own
welfare in the presence of a market failure, say. As we said above,
our model of altruism also gives an interpretation about where the
warm glow might come from, and that interpretation makes sense. Now,
Andreoni's results apply to what he really thinks happens in reality,
the case of ``impure altruism'', our normal altruism case. So, both
interpretations might be said to coincide in all the cases, except in
the extreme cases, where ours is more natural. In this sense, the
extra structure of our model also has a payoff . 

We finally remark that the warm glow function $G$ allows us to include
in our model the case of software Hackers. By Hackers we mean not only
the expert Hackers, in the usual sense, but also the many volunteers
who provide resources in any amount for the common good (like
reporting bugs). Apart from the fact that the model stresses the
possible economic robustness of the open source movement, the paper
opens a window to consider in a new light the copyright laws. It might
be inefficient to enforce a copyright law implying exclusion of a
public good when it has been produced by a spontaneous altruistic
movement\footnote{Think of the case of UNIX, produced originally by
what we should well consider an altruistic group, appropriated by some
other people, retarding the Linux phenomenon.}. So property rights of
law-excludable public goods should be reconsidered in at least some
cases, in light of efficiency considerations. Also, public policy
considerations regarding economic growth due to technological
development might include the alternative of preference formation and
support to altruistic groups that produce such law-excludable public
goods.




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\bibitem{A2} Andreoni, James, ``Impure Altruism and Donations to
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  Journal}, 1990, v.100, pp. 464-477.

\bibitem{Barro} J. R. Barro, ``Are government bonds net wealth?'', 
{\it Journal of Political Economy}, 1095--1117, 1974.

\bibitem{Becker} G. S. Becker, ``A theory of social interactions'', 
{\it Journal of Political Economy}, 1063--1093, 1974.

\bibitem{Borcer} Border, K. C. {\it Fixed Point Theorems with
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\bibitem{CS} Cornes, Richard, and Sandler, Todd. ``Easy Riders, Joint
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\bibitem{MP1} Marhuenda, Francisco and Felipe P\'erez-Mart\'{\i}, ``Altruism and
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\bibitem{MP2} Marhuenda, Francisco and Felipe P\'erez-Mart\'{\i},
``Consistency of interdependent preferences'', Universidad Carlos III
de Madrid working paper. September 1999.

\bibitem{MC} Mas-Colell, A., Whinston, M. and J. G. {\it
Microeconomic Theory} Oxford University Press, 1995.

\bibitem{OVR} Ok, Efe and Vega-Redondo, Fernando, ``On the Evolution
of Individualistic Preferences: Complete versus Incomplete Information
Scenarios''. University of Alicante working paper, March 1999.

\bibitem{OS} Stark, O. {\it Altruism and Beyond.} Cambridge University
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\end{thebibliography}


\end{document}

\begin{definition}{[$\sa$]} 
An  allocation $(x^1,x^2,y)$ is an
Altruistic Allocation (AA)  
if there are $\gamma_1,\gamma_2\in [0,\infty)$ satisfying
$\gamma_1\gamma_2\leq 1$ and such that 
 $x^1_{s}=\varphi_1(\gamma_1,\gamma_2,\om^1_s,\om^2_s)$, 
 $x^2_{s}=w_s-c^1_{s}=\varphi_2(\gamma_1,\gamma_2,\om^1_s,\om^2_s)$,
 $y=f(w-(x_1+x_2))$
where $\varphi=(\varphi_1,\varphi_2)$ is defined
 in equation~(\ref{??}). 
The Scope of Altruistic Preferences ($\sa$)
is the set of all Altruistic Allocations\end{definition} 


\begin{definition}
Persons 1 and 2 reach and Exact Altruistic Agreement (EAA) if 
 $x^i_i(\gamma_i,w)= x^j_i(\gamma_j,w)$ and
$y^i= y^j$, for $i,j=1,2, i\neq j$. 
\end{definition}

In view of the equality $x^1_1+x^1_2+\xi^1=w= x^2_2+x^2_1+\xi^2$,
where $\xi^1$ and $\xi^2$ are the amounts of private goods to produce
$y^1= f(\xi^1)$, and $y^2 =f(\xi^2)$, we have that $x^1_1=x^2_1$ if
and only if $x^2_2=x^1_2$ and $y^1=y^2$.

\begin{lemma} \label{YPF} 
Agents reach an Exact 
Altruistic  Agreement if and only if $\gamma_1\gamma_2= 1$. 

\end{lemma}

{\bf Proof:}
The first order conditions for $x^i_j, y^i, y^j$ are
\begin{eqnarray}
v^1_1(x^1_1,y^1)&=&\gamma_1 v^2_1(x^1_2,y^1), \label{EQ1}\\
v^1_1(x^1_1,y^1)&=&[v^1_2(x^1_1,y^1)+\gamma_1
v^2_2(x^1_2,y^1)]f'(w-(x^1_1+x^1_2))\label{EQ2}\\
y^1&=&f(w-(x^1_1+x^1_2)),\label{EQ3}\mbox{ and }\\
v^2_1(x^2_2,y^2)&=&\gamma_2 v^1_1(x^2_1,y^2),\label{EQ4}\\
v^2_1(x^2_2,y^2)&=&[v^2_2(x^2_1,y^2)+\gamma_2
v^1_2(x^2_1,y^2)]f'(w-(x^2_1+x^2_2)),\label{EQ5}\\
y^2&=&f(w-(x^2_1+x^2_2)).\label{EQ6}
\end{eqnarray}

Let $\gamma_1\gamma_2=1$. Then, $v^1_1(x^1_1,y^1)=\frac{1}{\gamma_2}
v^2_1(x^1_2,y^1)$, using (\ref{EQ1}). Then
\begin{equation}v^2_1(x^1_2,y^1)=\gamma_2
  v^1_1(x^1_1,y^1).\label{EQQ1}\end{equation} 
Also $v^1_1(x^1_1,y^1)=\frac{1}{\gamma_2}[\gamma_2v^1_2(x^1_1,y^1)+
v^2_2(x^1_2,y^1)]f'(w-(x^1_1+x^1_2))$, by (\ref{EQ2}). So that
\begin{eqnarray}
  \gamma_2v^1_1(x^1_1,y^1)&=&[ v^2_2(x^1_2,y^1)+
  \gamma_2v^1_2(x^1_1,y^1)]f'(w-(x^1_1+x^1_2))\nonumber \\
&=&v^2_1(x^1_2,y^1) \quad \mbox{ (using (\ref{EQQ1}))}\label{EQQ2}
\end{eqnarray} In addition we have
\begin{equation}y^1=f(w-(x^1_1+x^1_2)).\label{EQQ3}\end{equation}
But (\ref{EQQ1})--(\ref{EQQ3}) define uniquely $(x^2_2,x^2_1,y^2)$, by
(\ref{EQ4})--(\ref{EQ6}).

Now, assume that there is exact agreement, so that
$(x^1_1,x^1_2,y^1)=(x^2_1,x^2_2,y^2)\equiv(x_1,x_2,y)$. Then
\begin{eqnarray}
\gamma_2
\gamma_1v^2_1(x_2,y)&=&\gamma_2v^1_1(x_1,y) \mbox{ by (\ref{EQ1})}\nonumber\\
&=&v^2_1(x_2,y). \mbox{ by (\ref{EQ4})}
\end{eqnarray}
It follows that $\gamma_1\gamma_2=1$.
${\bf Q.E.D.}$\\[.5cm]


