UNIVERSITE PARIS DAUPHINE Ecole Doctorale

UNIVERSITE PARIS DAUPHINE
Ecole Doctorale EDDIMO
Centre de Recherche CEREMADE
N° attribué par la bibliothèque
THÉORIE DES JEUX À CHAMP MOYEN ET APPLICATIONS ÉCONOMIQUES
Second sujet : Taux d’escompte et développement durable
MEAN FIELD GAMES AND APPLICATIONS TO ECONOMICS
Secondary topic: Discount rates and sustainable development
THÈSE
Pour l’obtention du titre de
DOCTEUR EN SCIENCES : MATHÉMATIQUES APPLIQUÉES
(Arrêté du 7 août 2006)
Présentée et soutenue publiquement par
Olivier GUÉANT
le 30 Juin 2009
JURY
Directeur de thèse : Monsieur Pierre-Louis Lions
Professeur au Collège de France
Rapporteurs :
Suffragants :
Monsieur Roger Guesnerie
Professeur au Collège de France
Monsieur José Scheinkman
Theodore A Wells '29 Professor of Economics at Princeton University
Monsieur Ivar Ekeland
Professor at the University of British Columbia
Director of the Pacific Institute of Mathematical Sciences
Monsieur Jean-Michel Lasry
Professeur associé à l’Université Paris-Dauphine
Monsieur Robert E. Lucas, Jr.
John Dewey Distinguished Service Professor of Economics,
University of Chicago
Monsieur Benoît Perthame
Professeur à l’Université Paris VI
Madame Nancy L. Stokey
Frederick Henry Prince Distinguished Service Professor
University of Chicago
L’université n’entend donner aucune approbation ni improbation aux opinions
émises dans les thèses : ces opinions doivent être considérées comme propres à
leurs auteurs.
Introduction
Cette thèse est composée de deux parties indépendantes. La première, qui se
réfère au sujet principal de la thèse, consiste en une présentation d’applications
de la théorie des jeux à champ moyen, nouvelle branche de la théorie des jeux.
La théorie des jeux à champ moyen a été récemment introduite par J.-M. Lasry
et P.-L. Lions et des applications, notamment en sciences économiques, commencent à poindre. Ci-après, nous présentons des applications économiques variées
avec notamment un modèle de marché du travail, un modèle de gestion de portefeuille et plusieurs applications à la théorie de la croissance. Nous développons
aussi un modèle de répartition de population dont la structure est l’archétype
d’un jeu à champ moyen en temps continu avec un continuum d’états.
La seconde partie de la thèse traite d’un sujet différent et pose la question des
taux d’escompte à utiliser dans les projets de développement durable.
Chaque partie se présente sous la forme d’une suite de chapitres relativement
indépendants et l’objet des paragraphes qui suivent est de présenter les idées
communes à ces différents chapitres.
Jeux à champ moyen
Présentation
La théorie des jeux à champ moyen est une branche nouvelle de la théorie
des jeux qui a été définie par J.-M. Lasry et P.-L. Lions dans trois articles
récents [LL06a, LL06b, LL07a]. Les jeux à champ moyen sont caractérisés par
4 hypothèses :
• Anticipations rationnelles
• Continuum d’agents
• Anonymat des agents
• Interactions de type champ moyen
Les trois premières hypothèses sont standards en théorie des jeux. La première
- hypothèse d’anticipations rationnelles - a été introduite dans les années 1960
et fait maintenant partie à part entière du corpus économique. La seconde hypothèse est souvent utilisée dans les modèles mettant en jeu un grand nombre
d’agents. Il s’agit là d’une approximation largement utilisée pour des raisons
pratiques, de calculabilité par exemple. Dans le cadre de l’introduction de la
théorie des jeux à champ moyen, la limite de jeux à N joueurs, quand N tend
vers l’infini, a été étudiée [LL07a, LL07b] pour justifier l’hypothèse du continuum. La troisième hypothèse a toujours été implicite en théorie des jeux et
consiste à dire que le jeu est insensible à toute permutation des joueurs (un
contre-exemple célèbre est le jeu lié à la concurrence à la Stackelberg où l’on
assigne un ordre aux joueurs).
Venons-en maintenant à la quatrième hypothèse qui est spécifique aux jeux à
champ moyen et porte sur les interactions entre les joueurs. L’idée est qu’un
agent ne peut pas, à lui seul, sauf à considérer les choses marginalement, influencer la distribution des autres agents et donc les stratégies des autres joueurs.
En conséquence, les changements de stratégies ou de caractéristiques d’un nombre fini de joueurs ne sauraient changer l’issu du jeu. Ces idées, et le terme
champ moyen, viennent de la physique des particules bien qu’ici les particules
soient remplacées par des agents rationnels. En physique, une approche classique et ayant portée ses fruits, consiste en effet à considérer que chaque particule
contribue à la constitution d’un champ moyen qui agit sur les particules. On
résume ainsi un problème complexe, de par la combinatoire des interactions, à
un problème plus simple portant sur le couple (x, m) (x étant la caractéristique
d’un agent et m la distribution de ces caractéristiques dans la population), qui
constitue une variable exhaustive (voir annexe). De manière surprenante peutêtre puisque les agents, contrairement aux particules, optimisent un critère, des
équations de la physique apparaissent en théorie des jeux à champ moyen.
Typiquement, cette quatrième et dernière hypothèse signifie qu’un agent est
réellement atomisé au sein du continuum et n’a qu’un pouvoir marginal. En
conséquence, sa stratégie ne peut porter que sur sa caractéristique propre et la
distribution des caractéristiques.
Maintenant que nous avons défini le cadre général de la théorie des jeux à champ
moyen, nous nous devons d’expliquer en quoi cette nouvelle branche de la théorie
des jeux est utile et l’on se doit d’estimer l’importance des jeux à champ moyen
parmi l’ensemble des jeux.
Le principal intérêt du cadre champ moyen est certainement de simplifier les interactions entre les agents. Il est de fait plus simple de considérer que l’influence
des autres agents sur un joueur donné est résumée par la distribution des caractéristiques des agents que de considérer l’ensemble des interactions possibles
entre les agents deux à deux. De plus, l’hypothèse du continuum d’agents
permet de considérer, en temps continu, des jeux dont la mise en équations
peut le plus souvent se réduire à deux équations aux dérivées partielles (dans
le cas d’un espace d’états continu) ou à un système d’équations différentielles
(dans le cas contraire d’un espace d’états discret). Malgré leur apparente complexité, ces équations ne posent le plus souvent que peu de problèmes, du moins
numériquement. Cette calculabilité est en fait une conséquence de l’hypothèse
champ moyen qui simplifie les interactions entre les agents, la plupart des jeux
à N joueurs étant en effet difficiles à résoudre.
La calculabilité des solutions est donc une contribution importante de la théorie
des jeux à champ moyen à la théorie des jeux.
Concernant maintenant la structure des équations (dans le cas des jeux dynamiques), celle-ci est toujours du type forward/backward c’est-à-dire faite d’une
combinaison d’équations backward (de type Bellman) pour la détermination
des stratégies individuelles et d’équations forward (de type équations de transport) pour décrire l’évolution globale du jeu. Ces couples d’équations forward/backward peuvent être appliqués à de nombreux problèmes comme nous
allons le voir dans la suite.
Il est en effet assez fréquent de rencontrer cette structure forward/backward.
Dans la mesure où les agents décident de leur stratégie par un processus d’induction
à rebours, les équations backward sont naturelles et correspondent à des équations
de Bellman ou de Hamilton-Jacobi. Si l’on considère des jeux dynamiques, il
convient de suivre l’évolution du jeu et pour cela, une équation forward est
nécessaire, équation qui est le plus souvent une équation de transport de type
Kolmogorov-Fokker-Planck.
Cette structure forward/backward a été très peu utilisée par le passé et la raison
pour cela est une question ouverte. Une réponse possible est que ces équations
apparaissent très naturellement lorsque l’on combine à la fois temps continu et
continuum d’agents ce qui est assez rare en théorie des jeux, les jeux à continuum
d’agents étant le plus souvent statiques et les jeux différentiels le plus souvent
à deux joueurs. La raison pour laquelle cette “double continuité” n’a pas été
explorée n’est pas claire. Nous tendons à penser que la théorie développée sur
les fonctions de mesure comme approximations de fonctions symétriques à N
variables (voir l’annexe et [Lio08]) ainsi que les méthodes numériques que nous
présenterons en annexe du chapitre 3, étaient nécessaires pour traiter ces jeux.
Une question reste en suspens, à savoir la taille de la classe des jeux à champ
moyen dans l’ensemble des jeux. Y a-t-il de nombreux jeux à champ moyen
où s’agit-il d’un ensemble restreint de jeux. J.-M. Lasry et P.-L. Lions (voir
[Lio08]) donnent une réponse à cette question, réponse que nous présentons
dans l’annexe. Typiquement, l’on peut montrer, sous des hypothèses techniques
de continuité, qu’un jeu avec un continuum de joueurs vérifiant l’hypothèse
d’anonymat est caractérisé par des interactions de type champ moyen. Ainsi,
les intuitions développées ici peuvent être utilisées de manière très générale.
Comme corollaire, le cadre champ moyen est certainement le bon cadre pour
traiter des jeux dynamiques en présence d’un grand nombre de joueurs.
Applications présentées dans cette thèse
Nous avons dit plus haut que les jeux à champ moyen commençaient à être
utilisés dans des modèles économiques. Dans ce qui suit, nous allons présenter
quatre applications économiques et une application qui n’est pas directement
économique mais liée à des problèmes de répartition de populations se rapprochant de problématiques économiques comme celles des bulles financières,
des mouvements moutonniers ou des choix de technologie des ménages.
Pour commencer, nous présentons deux modèles économiques simples. Le premier est un modèle de marché du travail qui nous permet de décrire la réponse
dynamique du marché du travail à un choc technologique. Il s’agit d’un jeu à
champ moyen à 2 états, caractérisé par un système d’équations différentielles ordinaires forward/backward. Le second modèle, qui constitue le chapitre suivant,
est un modèle financier qui tâche de comprendre l’influence de la concurrence
entre les gérants d’actifs sur les choix de portefeuille. A la différence des autres
modèle, il s’agit là d’un jeu à champ moyen sans composante dynamique et donc
simplement backward. Le but est ici de montrer combien sont importants les
effets de la concurrence entre agents et que le modèle de Markovitz n’est pas
aussi robuste qu’il y parait, notamment pour traiter des choix de portefeuille
opérés par des firmes de gestion d’actifs.
Après ces deux modèles purement économiques, nous considérons un modèle
utilisant l’archétype d’un jeu à champ moyen en temps continu avec un continuum d’états. On a dans ce modèle, qui est un modèle de répartition de
population les deux EDP évoquées plus haut : Hamilton-Jacobi-Bellman et
Kolmogorov-Fokker-Planck. Des solutions explicites sont présentées puis la
lumière est mise sur différentes notions de stabilité. Ces notions de stabilité sont
utiles pour traiter numériquement des jeux à champ moyen et deux résolutions
numériques sont proposées en annexe de ce troisième chapitre.
Après ce modèle archétypal, nous présentons deux applications à la théorie de
la croissance via deux modèles de croissance que d’aucuns qualifieraient d’antiSchumpeterienne. Les deux modèles en question sont techniquement proches
bien que le premier traite du capital humain et le second de productivité industrielle. La croissance y est le fruit d’un processus qui fait intervenir toute la
population et pas seulement une fraction de celle-ci comme dans les modèles
habituels de croissance par R&D. L’aspect champ moyen est comparable à
celui du modèle du chapitre 3 à ceci près que l’on s’inscrit ici dans un cadre
déterministe et non plus stochastique.
Taux d’escompte
Les jeux à champ moyen constitue le principal sujet de cette thèse. Toutefois,
un second sujet a été étudié au cours de cette thèse et il est l’objet de cette
seconde partie sur les taux d’escompte.
Les taux d’escompte à utiliser dans les problèmes économiques sont le plus souvent arbitraires ou, au mieux, basés sur des analyses empiriques (voir [Wei01]).
La question du taux d’escompte est particulièrement problématique lorsqu’il
s’agit de problème de long terme puisqu’une escompte à taux constant positif “écrase” fortement le futur, rendant tout revenu dans un futur lointain
négligeable devant un revenu présent. Le premier chapitre est dédié à l’étude du
bon taux d’escompte à utiliser pour des problèmes de développement durable.
Ce chapitre est proche d’un article à paraı̂tre en collaboration avec R. Guesnerie et J.-M. Lasry et diffère d’un texte publié précédemment ([GLZ07]) dans
“Les Cahiers de la Chaire de Finance et Développement Durable”. La principale
contribution est d’introduire les taux écologiques et d’étudier leurs propriétés
de moyen et long terme. Une de nos conclusions est que des taux faibles, voire
négatifs, devraient être utilisés pour des projets de développement durable.
Une autre question qui est apparue en parallèle de la précédente est la possibilité de considérer des taux d’escompte psychologiques négatifs dans les modèles
économiques. C’est en effet pour des raisons mathématiques (de convergence) et
non pour des raisons économiques que l’on utilise d’habitude des taux positifs
dans le modèle de Ramsey. Le second chapitre de cette seconde partie mon-
tre comment l’introduction de stochasticité peut permettre d’utiliser des taux
d’escompte psychologiques plus faibles et même négatifs dans un modèle à la
Ramsey. Cette partie doit être vue comme un complément plutôt que comme
un chapitre à part entière mais a été incluse en tant que tel en raison de la
proximité avec le chapitre sur les taux écologiques.
References
[GLZ07] O. Guéant, J.-M. Lasry, and D. Zerbib. Autour des taux d’intérêt
écologiques. Cahiers de la Chaire Finance et Développement Durable,
(3), 2007.
[Lio08]
P.-L. Lions.
Théorie des jeux à champs moyen et applications.
Cours au Collège de France, http://www.college-defrance.fr/default/EN/all/equ der/cours et seminaires.htm, 2007-2008.
[LL06a] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen i. le cas stationnaire.
C. R. Acad. Sci. Paris, 343(9), 2006.
[LL06b] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen ii. horizon fini et
contrôle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
[LL07a] J.-M. Lasry and P.-L. Lions. Mean field games. Japanese Journal of
Mathematics, 2(1), Mar. 2007.
[LL07b] J.-M. Lasry and P.-L. Lions. Mean field games. Cahiers de la Chaire
Finance et Développement Durable, (2), 2007.
[Wei01] M. Weitzman. Gamma discounting. The American Economic Review,
91(1), Mar. 2001.
Introduction
This dissertation is divided in two independent parts. The first one deals with
the main subject of the PhD which is to present applications of a rather new
branch of game theory called mean field games theory. This new branch has
developed after the publication of three seminal articles by J.-M. Lasry and P.L. Lions and applications, mainly in economics, begin to be exhibited. In this
dissertation, we present economic applications in various fields such as labor
economics, finance or growth theory. Also, we develop a model of population
repartition using the archetypal structure of a mean field game in continuous
time. The secondary topic of the dissertation is the subject of the second part
and has to do with the the discount rates used in economics, mainly for sustainable development issues. Each part consists in a collection of nearly independent
chapters and we are going to present in the next paragraphs the general ideas
and theories that are common to most of them.
Mean Field Games
Presentation
Mean field games theory is a new branch of game theory that has been defined
and developed recently in the seminal papers by J.-M. Lasry and P.-L. Lions
[LL06a, LL06b, LL07a]. Mean field games are characterized by 4 hypotheses:
• Rational expectations
• Continuum of agents
• Agents anonymity
• Social interactions of the mean field type
The first three hypotheses are common in game theory. The first one - the
rational expectation hypothesis - has been introduced in the 60’s and is now
well accepted among game theorists. The second hypothesis is often used to
model games with a large number of players. It’s a rather well accepted approximation that has been used for tractability purposes and here, for mean
field games, the limit of a game with N players as N goes to infinity has been
studied in [LL07a, LL07b] to support this hypothesis. The third hypothesis has
always been implicit in game theory but is worth recalling. Basically, it says
that agents are anonymous in the sense that any permutation of the agents does
not change the outcome of the game.
The fourth hypothesis is specific to mean field games and is an hypothesis on
interactions between players. The main idea is that a given agent cannot influence by herself (but marginally) the distribution of the population of players and
therefore the strategies of others. By her behavior, however, a given agent contributes marginally to the statistics that are used by agents to decide upon their
strategies. A consequence of this hypothesis (and one can see in the appendix
that it is equivalent in a certain sense) is that changes either for characteristics
or for strategies that concern any finite number of players do not change the
outcome of the game. These ideas and the name mean field come from particle physics but, here, the particles are replaced by rational agents. In physics
indeed, it’s really common to consider interactions between particles with the
following paradigm: each particle creates a field and the mean field influences
particles. Surprisingly perhaps, in spite of the optimization made by rational
agents, equations from physics can be derived for mean field games.
Typically, this fourth assumption means that an agent is really atomized in the
continuum and has no power but a marginal one. Consequently, an agent only
uses strategies that depend on the distribution of players’ characteristics1 and
this is the main specific hypothesis of the mean field games theory.
Now that we have defined the general framework of mean field games, we shall
explain why this class of games is interesting and then estimate the size of this
class amongst all games.
The main interest of mean field games is certainly to simplify the interactions
between agents. It’s arguably simpler indeed to consider that the influence of
other players on a given agent is summed up by the distribution of players’
characteristics than to deal with the huge set of possible interactions between
individuals. Also, the continuum hypothesis, in addition to the mean field hypothesis, allows us to consider differential games that can often be reduced to
two partial differential equations in the case where the state space is not discrete and to systems of ordinary differential equations otherwise (these systems
being in some sense discrete approximations of the PDEs). These equations are
most of the time tractable, at least numerically. Tractability is in fact a natural
consequence of the mean field hypothesis that simplifies interactions between
agents. Most of the games with N players were untractable and so even numerically. The mean field hypothesis, in spite of the a priori complexity of the
differential equations, allows for numerical resolution as in physics and this is
an important contribution of the theory.
Also, concerning the structure of the equations (in the case of dynamical games),
it is always a forward/backward structure that combines backward (Bellman)
equations for the determination of individual strategies and forward (transport)
equations to describe the global evolution of the game. These coupled forward/backward equations can be applied to numerous problems as it will be
shown in the dissertation.
It’s indeed quite common to have such a structure. Since players find their
optimal strategy using backward induction processes, the backward equation is
natural and corresponds to a Bellman equation or an Hamilton-Jacobi-Bellman
equation (HJB equation). Also, to take into account the evolution of the game, a
forward equation is needed and is often of the Kolmogorov-Fokker-Planck type.
This forward/backward structure has not really been exhibited before as far
1 In other words, the couple (x, m) is sufficient to explain interaction where x is a personal
characteristics and m the distribution of those characteristics in the population.
as we know and the reason for that is an open question. A possible answer
is perhaps that the equations appeared naturally for differential games with a
continuum of agents in a continuous time framework. This type of games with
both continuous time and continuums of players are quite rare in the literature
since game theorists often focused either on continuous time (differential) game
theory with a finite number of players or on static games with a continuum of
agents. The reason why game theorists did not consider both continuous time
and continuums of agents is not clear. Our belief is that the theories developed
on functions of measures, to prove that mean field games with a continuum of
players were good approximations of games with finite number of players (see
appendix and [Lio08]) and also the numerical recipes we will present in the appendix of Chapter 3, were, in a way, necessary to well study these games.
A question remains about the size of the class of mean field games: are there
many mean field games or is it a real restriction for a game to be of the mean
field type? An answer to this question has been given by J.-M. Lasry and P.L. Lions (see [Lio08]) and we present it formally in the appendix. Basically,
what is shown is that under some natural continuity conditions, a game with a
continuum of players that satisfies the anonymity hypothesis is characterized by
interactions that are of the mean field type. A consequence is that the intuitions
developed here can be used for a large set of games. Another corollary is that
the mean field framework is certainly the right viewpoint to deal with dynamical
games with a continuum of players.
Applications presented in the dissertation
As we said before, mean field games start to be used in economic models. Here
we are going to present four economic applications in various fields and an application that is not pure economics since it deals with population repartition
problems but can be related to economics because of the close relations with
herd behavior or financial bubbles.
To start, we present two economic models that embed simple mean field games.
The first chapter is dedicated to a labor market model that allows us to describe
the dynamical response of the labor market to a technological shock. This mean
field game is characterized by a discrete state space and therefore by a system
of ordinary differential equations, half of them being forward, the other half
being backward. The second model we present is a financial one. The main
goal is to understand the influence of competition between asset managers on
their portfolio choices. The model is of the mean field type but is different from
all the other models presented in the dissertation since it is a static model that
only embeds a backward type reasoning. The main contribution of the model is
to show how important competition effects can be and to show that Markovitz’s
portfolio theory is not robust to the introduction of competition and therefore
only applies to single individuals and may not be well adapted for asset management companies.
After these two models that are purely economic, we focus on the archetype of
a mean field game in continuous time characterized by a backward HamiltonJacobi-Bellman partial differential equation and a forward Kolmogorov-FokkerPlanck partial differential equation. This part is applied to agents who want
to live as a community in a random environment. Equilibrium distributions
are exhibited for both one population and several populations with complex
interactions. Stability notions for mean field games are discussed and theorems
of stability are proved on the linearized problem. Numerical methods are presented in the appendix of this chapter.
Once we have presented the archetypal mean field game applied to a simple,
though technical, problem, we focus on economic growth with two models that
introduce an anti-Schumpeterian theory of growth. Our two models are technically similar, even though one is based on human capital and the other on
firm cost reduction, and consist in a growth process where all the working force
participates to growth and not only researchers close to the technological frontier. In terms of mean field games, these models are close to the model used
to deal with population issues but do not embed randomness except a common
randomness that is orthogonal to the randomness used earlier.
Discount rates
Mean field games are the main topic of the dissertation but another subject
has been dealt with in this PhD: discount rates. The discount rates to be used
in economic problems are indeed often chosen arbitrary or are, at best, based
on some empirical analysis (see [Wei01]). This issue is really relevant when it
comes to problems that have to do with the long run since any exponential
discounting at positive constant rate makes the long run future unimportant
compared to present. The first chapter in this part is dedicated to the discount
rates to be used for sustainable development issues. This chapter is really close
to an article to be published in collaboration with Roger Guesnerie and slightly
differs from a paper published earlier ([GLZ07]) in “Les Cahiers de la Chaire
de Finance et Développement Durable”. The main contribution is to introduce
the so-called ecological discount rates and to study the properties of these rates
both asymptotically and in the medium run. One of the conclusions is that
very small rates, even negative, should be used for projects that have to do with
sustainable development.
Another question that appeared as a parallel to the first issue is whether or not
it was possible to use negative discount rates in simple economic models. It’s
arguably for mathematical reasons (convergence issues) and not for economic
reasons that positive discount rates are used in the Ramsey model. The second
chapter of this second part shows how a random environment can allow for
smaller or negative discount rates in an à la Ramsey framework. It is to be seen
more as an appendix than as a proper chapter but it was included because of
the close links with the first chapter of the second part on ecological discount
rates.
References
[GLZ07] O. Guéant, J.-M. Lasry, and D. Zerbib. Autour des taux d’intérêt
écologiques. Cahiers de la Chaire Finance et Développement Durable,
(3), 2007.
[Lio08]
P.-L. Lions.
Théorie des jeux à champs moyen et applications.
Cours au Collège de France, http://www.college-defrance.fr/default/EN/all/equ der/cours et seminaires.htm, 2007-2008.
[LL06a] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen i. le cas stationnaire.
C. R. Acad. Sci. Paris, 343(9), 2006.
[LL06b] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen ii. horizon fini et
contrôle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
[LL07a] J.-M. Lasry and P.-L. Lions. Mean field games. Japanese Journal of
Mathematics, 2(1), Mar. 2007.
[LL07b] J.-M. Lasry and P.-L. Lions. Mean field games. Cahiers de la Chaire
Finance et Développement Durable, (2), 2007.
[Wei01] M. Weitzman. Gamma discounting. The American Economic Review,
91(1), Mar. 2001.
Part I
Mean Field Games
Part I - Chapter 1
A Two-State Mean Field Game for the Labor Market
A Two-State Mean Field Game for the Labor
Market
Abstract
This part presents a mean field game with two states that arises in a
simplified view of the labor market. In a full-employment framework, the
model allows for a more progressive adaptation to technological shocks
than the usual neoclassical models do. This model is totally orthogonal
to many models of the labor market that study the influence of technical
change (SBTCs, ...) since the focus is on the dynamical absorption of a
technological shock. It’s noteworthy that these dynamical aspects can be
studied thanks to the forward/backward approach.
Résumé
Dans cette partie, nous présentons un jeu à champ moyen à 2 états
pour modéliser simplement le marché du travail. Dans un cadre de plein
emploi, nous modélisons l’adaptation dynamique du marché du travail
à des chocs de productivité, adaptation plus progressive que dans de
nombreux modèles néoclassiques. L’originalité de ce modèle réside dans
l’étude de la dynamique de transition qui nécessite une approche forward/backward.
Introduction
Technological shocks have been responsible for massive migrations in history. These migrations occurred geographically, from rural to urban areas,
and are still at work in less developed countries. They also occur now in
the labor market of developed countries, where workers are attracted by
services and leave industrial activities.
If we consider two sectors where technological shocks can arise (let’s say
the financial sector and the traditional, though technology demanding, industrial sector) it’s interesting to understand the choices of young workers
when they decide to start their career in one or the other sector. This is
the topic of this first “mean field game” model with only two states.
Contrary to the mean field games theory and to most of the examples we
will develop later on that involve partial differential equations, we chose
to focus in this first application on the origins of mean field games and
we consider systems of ordinary differential equations that have the same
characteristics as the mean field games PDEs, half of the equations being
forward, the other half being backward. These systems arise naturally in
dynamical economic models where the decisions are made using an induction process that factors in some anticipations about the future behavior
of economic agents or the future economic situation. This is exactly the
framework of the labor market for young workers since they have to choose
their first job so as to maximize (this may not be the only criterion) their
future wages; these wages depending on the economic choices of the other
workers.
In the first section, we present the model. The second section is dedicated to its resolution in the stationary case. Finally, we come to the
dynamical properties of the model and give in the last section an application to productivity or technological shocks.
1
The model
1.1
Presentation of the framework
In what follows, we consider a continuum of individuals (of size 1) that
is going to be on the labor market. This labor market is reduced to 2
sectors and each individual will have to make a decision about the sector
in which she is going to work. For that matter, each person’s utility is
characterized by two components. First of all, there is a monetary part
that simply is the amount of money the individual expects to make. Second, every single person has an intrinsic utility that makes her prefer one
or the other sector.
Contrary to usual theories we assume that salaries are not identical across
sectors because of the impossibility to look for a job in one sector while
working in the other. Basically, we develop two scenarii that are going to
lead to the same model:
• We can have a long-term viewpoint in which we consider a flow of
generations. In each generation (of constant size λ), people choose
a job at the beginning and stay in the same job until they “retire”,
the retirement date being random and following a poisson process
of intensity λ. At each date t, a new generation enters the market
and the same quantity of individuals leaves it (this hypothesis can
be thought of as a full-employment hypothesis combined with the
absence of demographic growth).
• We can alternatively have in mind a short-term model in which there
is no unemployment but some turnover on the labor market. In
other words, people can be fired but find another job instantaneously.
They are indeed fired with an instantaneous probability λ and can
decide in which sector they want their new job1 .
1 This myopic behavior of agents that cannot change their job without being fired can be
justified by a very high cost of looking for a new job while working.
1.2
The setup
First, let’s consider the two sectors hereafter denoted 1 and 2. The production function of the two sectors are:
Y1 = A1 N1α Y21−α
Y2 = A2 N2β Y11−β
where the A’s stand for the technological coefficients and the N ’s for the
labor force dedicated to each sector.
In other words, the two sectors are interdependent and this dependence
will have some influence on the wage structure since the wages will be
function of the repartition of the labor force across the two sectors.
These wages can be computed using two different hypotheses. We can
indeed suppose that the effect of the sector 2 on the wage in sector 1
is endogenous or suppose it is not. Whatever the hypothesis, the functional form of the wage is the same, up to a multiplicative constant. Here
we consider the case where there is endogenization and this leads to the
following property:
Proposition 1 (Wages).
Aγ1 N1αγ−1 N2
(1−β)γ
Aγ2 N2βγ−1 N1
w2 (N1 , N2 ) = βγA1
where γ =
β(1−α)γ
(1−α)γ
w1 (N1 , N2 ) = αγA2
α(1−β)γ
1
.
1−(1−α)(1−β)
Proof:
We have:
1−α
Y1 = A1 N1α Y21−α = A1 N1α A2 N2β Y11−β
1−(1−β)(1−α)
⇒ Y1
β(1−α)
= A1 N1α A1−α
N2
2
(1−α)γ
⇒ w1 (N1 , N2 ) = αγA2
(1−α)γ
⇒ Y1 = Aγ1 N1αγ A2
β(1−α)γ
N2
β(1−α)γ
Aγ1 N1αγ−1 N2
The result is obtained by symmetry for the salary in sector 2.
Now that we have dealt with the two sectors, let’s come to the individuals. The monetary component of utility that is given by working into
the sector i is:
Z t+T
ui (t) = E
wi (N1 (s), N2 (s))e−r(s−t) ds
t
where T is the random variable that corresponds to the time spent in the
job. This variable is supposed to follow an exponential law of intensity λ
and we have therefore:
Z ∞
wi (N1 (s), N2 (s))e−(r+λ)(s−t) ds
ui (t) =
t
Now, let’s present the second component of agents’ utility. Each agent
has an intrinsic preference for one sector or the other and this preference
is measured by a coefficient µ that represents the additional utility the
sector 1 automatically gives in comparison with the sector 2. We suppose
that the µ’s are distributed according to a standard normal distribution.
We can now write the dynamics of the population in each sector:
Proposition 2 (Dynamics of the population).
Ṅ1 (t) = −λN1 (t) + λF (u1 (t) − u2 (t))
Ṅ2 (t) = −λN2 (t) + λF (u2 (t) − u1 (t))
where F is the cumulative distribution function of a standard normal variable.
Proof:
For simplicity, let’s reason on an infinitesimal interval [t, t + dt]. A
proportion λdt of people retire or are fired and this is true for both sector
1 and sector 2. There is a population of size λdt entering the whole market
and the proportion that choose sector 1 is given by:
P (u1 (t) + µ ≥ u2 (t)) = F (u1 (t) − u2 (t))
This gives the result.
We see that the variables u’s only appear as differences. Therefore, it’s
interesting to introduce the variable ∆u = u1 − u2 . Also, since ∀t, N1 (t) +
N2 (t) = 1 the problem we have to solve is simplified to the following
equations:
Proposition 3 (The ODEs). The problem can be reduced to the following
equations:
Ṅ1 (t) = −λN1 (t) + λF (∆u(t)),
N1 (0) given
d∆u
(t) = (r + λ)∆u(t) − [w1 (N1 (t), 1 − N1 (t)) − w2 (N1 (t), 1 − N1 (t))]
dt
lim e−(r+λ)t ∆u(t) = 0
t→+∞
Proof:
The only thing to verify is that the equations for ∆u indeed lead to
the integral form presented above. But:
h
i
d e−(r+λ)t ∆u(t)
= −e−(r+λ)t [w1 (N1 (t), 1 − N1 (t)) − w2 (N1 (t), 1 − N1 (t))]
dt
Because of the transversality condition we get:
−e−(r+λ)t ∆u(t) = −
∞
Z
e−(r+λ)s [w1 (N1 (s), 1 − N1 (s)) − w2 (N1 (s), 1 − N1 (s))] ds
t
∞
Z
∆u(t) =
t
e−(r+λ)(s−t) [w1 (N1 (s), 1 − N1 (s)) − w2 (N1 (s), 1 − N1 (s))] ds
2
Resolution
We have to solve two coupled ordinary differential equations with an initial
condition for N1 and a “final” condition for ∆u. This is typical of the forward/backward systems involved in mean field games. We are first going
to solve the stationary problem and we will then come to the dynamical
problem.
2.1
Stationary solution
The stationary problem is easy to solve:
Proposition 4. There is a unique stationary solution and this solution
is characterized by N1∗ the unique solution of:
1
N1∗ = F
(w1 (N1∗ , 1 − N1∗ ) − w2 (N1∗ , 1 − N1∗ ))
r+λ
Proof:
Any stationary solution has to verify:

∗
∗
∗
∗
∗
 (r + λ)∆u = [w1 (N1 , 1 − N1 )) − w2 (N1 , 1 − N1 )]

N1∗ = F (∆u∗ )
Therefore, if we introduce the expression of ∆u∗ from the first equation, in the second equation we get:
1
∗
∗
∗
∗
∗
(w1 (N1 , 1 − N1 ) − w2 (N1 , 1 − N1 ))
N1 = F
r+λ
This equation has a unique solution because the left-hand side is increasing and goes from 0 to 1 whereas the right-hand side is decreasing
and goes from 1 to 0.
A question that appears now is whether or not dynamical solutions
exist and if they converge towards the unique stationary solution.
2.2
Dynamical properties
Let’s consider the problem without the terminal condition on ∆u (the
transversality condition):
d∆u
(t)
dt
dN1
(t)
dt
= (r + λ)∆u(t) − [w1 (N1 (t), 1 − N1 (t)) − w2 (N1 (t), 1 − N1 (t))]
= −λN1 (t) + λF (∆u(t))
This dynamical system can be studied using the classical methods of
2D dynamical systems and with the help of a phase diagram.
We enounce a qualitative proposition that can be verified easily:
Proposition 5 (Phase diagram). The phase diagram associated to the
dynamical system is of the following form:
Figure 1: Phase diagram for two identical sectors with α = β = 0.5 and A1 =
A2 = 1
Proposition 6 (Saddle point). The behavior in the neighborhood of the
stationary solution is those of a saddle point.
Proof:
We just need to linearize the system around the stationary solution
(N1∗ , ∆u∗ ).
d∆u
(t)
dt
dN1
(t)
dt
= (r + λ)∆u(t) − [∂1 w1 − ∂2 w1 − ∂1 w2 + ∂2 w2 ] (N1∗ , 1 − N1∗ )N1 (t)
= −λN1 (t) + λ∆u(t)F 0 (∆u∗ )
Then, we need to study the eigenvalues of the following matrix:
M=
r+λ
λF 0 (∆u∗ )
− [∂1 w1 − ∂2 w1 − ∂1 w2 + ∂2 w2 ] (N1∗ , 1 − N1∗ )
−λ
We have
det(M ) = −λ(r + λ) +λ F 0 (∆u∗ ) [∂1 w1 − ∂2 w1 − ∂1 w2 + ∂2 w2 ] (N1∗ , 1 − N1∗ )
| {z }
| {z } |
{z
}
<0
>0
<0
.
Hence, det(M ) < 0 and we can deduce that the two eigenvalues are of
opposite signs. This defines a saddle point.
If we consider the dynamical system defined by the two differential
equations and if we replace the transversality condition by a fake initial
condition on ∆u, we can see that there are three possible trajectories: trajectories that diverge towards the top right “corner”, with ∆u(t) → +∞
and N1 (t) → 1, trajectories that diverge towards the bottom left “corner”, with ∆u(t) → −∞ and N1 (t) → 0, and a unique trajectory that
converges towards the stationary equilibrium (this is the stable variety
theorem for a hyperbolic dynamical system). Now, we are going to prove
that the unique trajectory compatible with the transversality condition is
the trajectory that converges towards the stationary equilibrium.
Proposition 7 (Compatible Trajectory). The only trajectory compatible
with the transversality condition is the trajectory that converges towards
the stationary equilibrium.
Proof:
Imagine that the transversality condition is verified on a trajectory
that diverges towards the top right “corner” (the case of the bottom left
“corner” is the same).
There exists t0 such that ∆u and N1 are both increasing for t ≥ t0 .
Therefore, we can write:
∞
Z
∆u(t) =
e−(r+λ)(s−t) [w1 (N1 (s), 1 − N1 (s)) − w2 (N1 (s), 1 − N1 (s))] ds
t
∞
Z
⇒ ∆u(t) ≤
e−(r+λ)(s−t) [w1 (N1 (t), 1 − N1 (t)) − w2 (N1 (t), 1 − N1 (t))] ds
t
1
[w1 (N1 (t), 1 − N1 (t)) − w2 (N1 (t), 1 − N1 (t))] ds
r+λ
1
⇒ ∆u(t0 ) ≤ ∆u(t) ≤
[w1 (N1 (t), 1 − N1 (t)) − w2 (N1 (t), 1 − N1 (t))]
r+λ
1
⇒ ∆u(t0 ) ≤ lim
[w1 (N1 (t), 1 − N1 (t)) − w2 (N1 (t), 1 − N1 (t))] = −∞
t→+∞ r + λ
This is absurd and therefore the divergent trajectories are not compatible with the transversality condition. The only trajectory that could
be compatible is the strategy that converges towards the stationary solution. This trajectory is indeed compatible since limt→+∞ e−(r+λ)t ∆u(t) =
0.
⇒ ∆u(t) ≤
If the phase diagram is not well suited to study the problem because
the dynamical system is a forward/backward system, we have shown that
it can be used anyway to solve it, though not directly.
Whatever the initial condition N1 (0), we have shown that the economy
converges to its unique equilibrium. This situation allows us to study what
happens when there is an exogenous change in the long term equilibrium,
typically when there is a productivity shock that changes A1 and/or A2 .
3
Productivity shocks
Productivity shocks occur frequently and are responsible for change in
wages and change in the structure of the economy. Here, we can model
the influence of productivity shocks quite easily and we can compare the
influence in this model and the influence in the basic economic model
where wages are equalized across sectors. For this comparison to be possible we need to start with two identical sectors at equilibrium and we are
going to consider a productivity shock in sector 2 where A2 will change
from 1 to 2 instantaneously. What happens is that there is a new equilibrium in the two models. For our model, there is a (negative) jump in
∆u and the economy converges slowly toward its new equilibrium with
N1∗∗ < N1∗ :
Figure 2: Productivity shock α = β = 0.5, A1 = 1, A2 goes from 1 to 2
What we can show is that the reaction, even in the long run, is less
important in our model than in the usual one where wages are equalized.
Proposition 8 (Reaction comparison). Let’s start with two identical sectors and consider a technological shock in sector 2.
Let’s note N1∗ the equilibrium in our model (as before) and N1c∗ the equilibrium in the classical model where wages are equalized.
We have the following inequalities:
dN1c∗
dN1∗
<
<0
dA2
dA2
0<
dw1c∗
dw1∗
<
dA2
dA2
Proof:
In our model, the equilibrium is given by:
(r + λ)F −1 (N1∗ ) = w1 (A2 , N1∗ , N2∗ ) − w2 (A2 , N1∗ , N2∗ )
If we differentiate we obtain:
1
∂w1
∂w1
dN1∗
∂w1 ∂w2
∂w2 dN1∗
∂w2 dN2∗
=
−
+
−
+
−
f (F −1 (N1∗ )) dA2
∂A2 ∂A2
∂N1
∂N1 dA2
∂N2
∂N2 dA2
∂w1
1
dN1∗
∂w1 ∂w2
∂w2
∂w1
∂w2 dN1∗
⇒ (r+λ)
=
−
+
−
−
+
∗
−1
f (F (N1 )) dA2
∂A2 ∂A2
∂N1
∂N1
∂N2
∂N2 dA2
(r+λ)
⇒ (r + λ)
1
f (F −1 (N1∗ ))
dN1∗
w
w
dN ∗
= −αγ
− 2 ∗ (1 − αβγ) 1
dA2
A2
N1
dA2
In the usual model, the equilibrium would be given by:
0 = w1 (A2 , N1c∗ , N2c∗ ) − w2 (A2 , N1c∗ , N2c∗ )
If we differentiate we get:
0 = −αγ
w
dN c∗
w
− 2 c∗ (1 − αβγ) 1
A2
N1
dA2
In both case the number of worker in sector 1 decreases and we see
that:
dN1c∗
dN1∗
<
<0
dA2
dA2
Because of the relationship between labor force and wages, we easily
get:
0<
dw1c∗
dw1∗
<
dA2
dA2
As announced, the reaction to a shock is less important in our model in
addition to be slower and this is arguably a better way to model reactions
to technological shocks.
3.1
Conclusion
As far as economics is concern, the inelastic properties of our labor market
model allowed us to see how the labor force evolves in response to a
shock in one sector or the other. This model is interesting in the sense
that it models dynamics toward the equilibrium and can be simulated for
quantitative applications.
For the mathematical part, the system of ordinary differential equations
involved is close to the system of PDEs that appear in the mean field
game theory, the difference being in the presence of only two states that
makes the problem easier to solve. Another difference, though not that
important, is the fact that people cannot change their choices at all times.
Apart from these differences the system is a forward/backward system and
the intuition developed in this simple case will be useful for the rest of
the dissertation.
References
[Ace98]
D. Acemoglu. Why do new technologies complement skills? directed technical change and wage inequality. Quarterly Journal
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[Ace02]
D. Acemoglu. Technical change, inequality, and the labor market. Journal of Economic Literature, 40(1), Mar. 2002.
[Hai00a] J.-O. Hairault.
Analyse macroéconomique, tome 1.
Découverte, 2000.
La
[Hai00b] J.-O. Hairault.
Analyse macroéconomique, tome 2.
Découverte, 2000.
La
[Ioo01]
G. Iooss. Méthodes locales de théorie des systèmes dynamiques
en dimension infinie (éléments). Summer Course in Oléron,
Sept. 2001.
[MP94]
D. Mortensen and C. Pissarides. Job creation and job destruction in the theory of unemployment. The Review of Economic
Studies, 61(3), 1994.
[SS84]
C. Shapiro and J. Stiglitz. Equilibrium unemployment as a
worker discipline device. The American Economic Review, 74(3),
Jun. 1984.
Part I - Chapter 2
Portfolio Management
and Competition Between Asset Managers
Portfolio Management and Competition between
Asset Managers
Abstract
We generalize the classical portfolio management theory by taking into
account some aspects of the agency problems that arise between an asset
manager and an investor. Basically, we model how the introduction of
competition between asset managers (to get more funds under management) modifies the portfolio choices.
Résumé
Nous généralisons ici la théorie classique de la gestion de portefeuille.
Notre point de vue est de considérer le problème principal-agent qui intervient entre un gérant d’actifs et un investisseur individuel. Typiquement, nous considérons le conflit d’intérêt qui existe entre ces deux types
d’agents en raison de la concurrence des gérants d’actifs entre eux, pour
accroı̂tre leur encours.
Introduction
The agency problems that arise between a fund manager or an asset manager and its investors has long been studied. Investors would like their
money to be invested so as to maximize the risk-adjusted expected return
of their portfolio. However, asset managers can have other incentives in
addition to satisfying their current clients. An asset management company can indeed want to gamble or secure gains at the end of the year
to reach a predetermined target. It can want to maximize its value by
attracting new clients so as to get more assets under management, and
hence more fees (see Chevalier-Ellison [CE97]). As a person, an asset
manager can also have incentives linked to career concerns (see ChevalierEllison [CE99b], Scharfstein-Stein [SS90]) and it’s known that this kind of
concerns can lead to herd behaviors quite similar to those presented later
in this dissertation, concerning population issues.
Here, we are going to deal with a theoretical model that marginally modifies the 2-period Markowitz framework and embeds the willingness of
asset managers to attract new customers. We know indeed from [CE97]
that there is a positive (though complex and non-linear) relationship between past performance of a fund and the present investment flow in this
fund. Moreover, it’s arguably the relative performance of a fund and not
only the absolute performance that is taken into account by investors and
should explain the flow/performance relationship. As a consequence this
effect can simply be modeled in a 2-period model by a competition effect
in the sense of a ranking of the asset managers according to their performance. This is the purpose of the model developed below.
In the first section, we are going to present the model in a general
framework. The second section will be dedicated to a resolution that
generalize the well-known case of a CARA utility function.
1
1.1
The model
The setup
We consider a continuum of asset managers. All asset managers have
the same amount of asset under management at time 0 and are going
to choose the portfolio in which they invest. The possible portfolios are
made of any convex combinations of the two available assets. We consider
indeed a market with two assets: there is a risk-free asset whose return
is denoted r (the weight on this asset in the portfolio will be 1 − θ) and
a risky asset whose return is r + ˜ where ˜ is a random variable that is
central in what follows (the weight on this asset in the portfolio will be
θ).
The criterion used to choose the portfolio has two components. First,
there is a pure Markowitz part where the asset manager is going to maximize, in expectancy, a utility function that depends on the return. Second,
since the goal of an asset manager (seen as a company) is not only to please
the current clients but also to maximize the number of clients or, quite
similarly, the total amount of asset under management, there is competition between asset managers. Therefore, each asset manager wants to
signal that she is better than her peers and she is going to maximize, in
addition to the utility criterion, her rank inside the asset managers community.
As a consequence, each asset manager is going to maximize, according to
her believes the following expression:
E[u(X) + β C̃]
where X = 1 + r + θ˜
is the wealth at date 1, where β is a constant that
models the relative importance of the competition effect, and where C̃ is
the random variable that ranks asset managers. This variable C̃ takes
values in [0, 1], 0 being associated to the worst-performing asset manager
and 1 being for the best-performing asset manager. In other words, this is
simply the cumulative distribution function (even though it is a random
variable) of the returns in the population of asset managers.
So far, asset managers were similar to one another. However, even if
they have at time 0 the same amount 1 under management, we are going
to suppose that they have different believes about the random variable
˜. We say that a given asset manager is of the -type if she believes that
˜ ∼ N (, σ 2 ) where the variance σ 2 is supposed to be the same for all asset
managers. In other words, we suppose that they agree on the volatility of
the risky asset but they have different believes on its expected return.
In what follows f will stand for the distribution of the types and it will
be supposed to be symmetric around 0 (f is even) to guarantee that there
are as many buyers as many sellers for the two assets. To simplify, we
will assume that f is the distribution of a normal variable with variance s2 .
1.2
Resolution
To solve the problem and compare the result to a well known case without
competition, we are going to consider the case of a CARA utility function,
that is u(x) = − exp(−λx).
Let’s consider an -type asset manager. We are going to derive the
first order condition that characterizes her optimal θ.
Proposition 1 (F OC ). The first order condition associated to an -type
asset manager is:
1 2 2 2
(F OC ) −λ2 σ 2 θ −
exp
−λ(1
+
r)
−
λθ
+
λ
θ
σ
+βm(θ)C() = 0
λσ 2
2
where m stands for the probability
function of the θ’s at equi distribution
librium, where C(·) = 2 N σ· − 12 is an odd function, positive on R+
and where N is the cumulative distribution function of a gaussian variable
N (0, 1).
Proof:
The asset manager maximizes:
h
i
E u(1 + r + θ˜
) + β C̃
It’s easy to see that C̃ = 1˜>0 M (θ) + 1˜≤0 (1 − M (θ)) where M stands
for the cumulative distribution function of the weights θ.
Also,
E [u(1 + r + θ˜
)] = −E [exp (−λ (1 + r + θ˜
))]
1 2 2 2
= − exp −λ (1 + r + θ) + λ θ σ
2
Hence, the optimal θ is given by the argmax of:
1
− exp −λ (1 + r + θ) + λ2 θ2 σ 2 + βE [1˜>0 M (θ) + 1˜≤0 (1 − M (θ))]
2
Let’s differentiate the above equation. We get the first order condition
for an -type asset manager:
1 2 2 2
−λ2 σ 2 θ −
exp
−λ(1
+
r)
−
λθ
+
λ
θ
σ
+βE [1˜>0 − 1˜≤0 ] m(θ) = 0
λσ 2
2
But,
1
1
P (˜
> 0)−P (˜
≤ 0) = 2 P (˜
> 0) −
−
= 2 P N (0, 1) > −
= C()
2
σ
2
Hence we get the result.
Proposition 2 (Differential equation for 7→ θ()). Let’s consider the
function 7→ θ() that gives the optimal θ for each type. If θ is C 1 then
it verifies the following differential equation:
1 2 2 2 dθ
−λ2 σ 2 θ −
λ
θ
σ
+βf ()C() = 0
exp
−λ(1
+
r)
−
λθ
+
λσ 2
2
d
Moreover, θ must verify θ(0) = 0.
Proof:
To go from the distribution f of the types to the distribution m of the
θ’s, we need a coherence equation that is simply:
m(θ)θ0 () = f ()
Now, if we take the different first order conditions F OC and multiply
by θ0 () we get the ODE we wanted to obtain.
Now, because C(0) = 0, the equation (F OC0 ) is simply
1
−λ2 σ 2 θ exp −λ(1 + r) + λ2 θ2 σ 2 = 0
2
and the unique solution of this equation is θ = 0.
To simplify the analysis, let’s highlight that the differential equation
satisfies the following property that allows us to work only with > 0:
Proposition 3. Let θ be a solution of the ODE (∗). Then, θ defined by
θ() = −θ(−) is also a solution of (∗).
Proof:
Let’s write the ODE for −:
2
−λ σ
2
1 2
2 2
θ(−) +
exp −λ(1 + r) + λθ(−) + λ θ(−) σ θ0 (−)
λσ 2
2
+βf (−)C(−) = 0
(∗)
Since f is even and C is odd, we get:
2
−λ σ
2
1 2
0
2 2
−θ() +
exp −λ(1 + r) − λθ() + λ θ() σ θ ()
λσ 2
2
−βf ()C() = 0
1 2
0
2 2
exp
−λ(1
+
r)
−
λθ()
λ
+
θ()
σ
θ ()
⇒ −λ2 σ 2 θ() −
λσ 2
2
+βf ()C() = 0
Now, let’s go back to the economic analysis. First of all, we want θ
to be an increasing function of (this is simply because being optimistic
about the return of the risky asset should lead to buy more of this asset
than being pessimistic about it). Consequently, for > 0, the ODE implies
that
θ() >
λσ 2
Therefore, if we note 7→ θ0 () = λσ 2 the usual solution of the Markowitz
problem without ranking, then we should have ∀ > 0, θ() > θ0 () > 0
and symmetrically ∀ < 0, θ() < θ0 () < 0. This means that the ranking
effect induces a higher risk exposure.
Now, we are ready to write the exact problem we need to solve. We
want a function 7→ θ() defined for > 0 (the odd function associated
to this branch will be our solution) that verifies:
θ0 () =
βC()f ()
1
λ2 σ 2 exp (−λ(1 + r + θ()) + 12 λ2 σ 2 θ()2 ) θ() − θ0 ()
and the two additional conditions:
• θ() > θ0 () =
λσ 2
• lim→0 θ() = 0
Proposition 4 (Existence and Uniqueness). There exists a unique function that verifies the equation (∗) with the two additional constraints:
• θ() > θ0 () =
λσ 2
• lim→0 θ() = 0
Proof:
Let’s start with the proof of the uniqueness.
Let’s consider a solution θ of the problem and let’s introduce the function z defined by:
1
θ()2 − θ0 ()θ()
2
If we want to inverse this equation and get θ as a function of z then
we get:
z() =
θ() = θ0 () ±
p
θ0 ()2 + 2z()
but since θ() > θ0 () we clearly can inverse the equation and get:
θ() = θ0 () +
p
θ0 ()2 + 2z() := Θ(, z())
Now, if we differentiate the equation that defines z we have:
z 0 () = θ0 ()θ() − θ0 ()θ0 () −
⇒ z 0 () =
⇒ z 0 () =
1
1
θ() = θ0 () (θ() − θ0 ()) −
θ()
λσ 2
λσ 2
βC()f ()
1
−
θ()
λσ 2
λ2 σ 2 exp (−λ(1 + r + θ()) + 12 λ2 σ 2 θ()2 )
βC()f ()
1
Θ(, z())
−
λ2 σ 2 exp (−λ(1 + r + Θ(, z())) + 12 λ2 σ 2 Θ(, z())2 ) λσ 2
From Cauchy-Lipschitz we know that there is a unique solution z of
this equation that verifies z(0) = 0. This solution is defined in a neighborhood V of 0. From this we know that
p locally, in the neighborhood V ,
θ is uniquely defined by θ() = θ0 () + θ0 ()2 + 2z(). Since there is no
problem outside of 0 (i.e. the Cauchy-Lipschitz theorem can be applied
directly) the uniqueness is proved.
Now, we want to prove that there exists a solution on the whole domain. For that let’s consider the following ODE:
z 0 () =
βC()f ()
1
−
Θ(, z())
λ2 σ 2 exp (−λ(1 + r + Θ(, z())) + 12 λ2 σ 2 Θ(, z())2 ) λσ 2
We know that there is a local solution z (defined on a neighborhood
V of 0) satisfying this equation with z(0) = 0.
If we define θloc on V (or more exactly on an open subset of V that
contains 0, because it is not a priori defined on V ) as:
θloc () = θ0 () +
p
θ0 ()2 + 2z()
then, we have a local solution of the equation (∗) that satisfies the two
additional conditions. Let’s consider now ˆ in V . We can apply the
Cauchy Lipschitz theorem to the equation (∗) with the Cauchy condition
θ(ˆ
) = θloc (ˆ
) on the domain {(, θ)/ > 0, θ > θ0 ()} and consider θ the
maximal solution of the problem. This maximal solution clearly satisfies
lim→0 θ() = 0. We want to show that there is in fact no upper bound
for the maximal domain.
Imagine there is such an upper bound . Since θ is increasing, we have
either:
lim θ() = +∞
→
or
lim θ() = θ0 ()
→
We are going to show that these two cases are impossible.
Imagine first that lim→ θ() = +∞. Then, we can suppose there exists
an interval (, ) such that ∀ ∈ (, ), θ() > θ0 () + 1. Hence, on (, ) we
have:
θ0 () ≤
⇒ θ0 () ≤
βC()f ()
λ2 σ 2 exp −λ(1 + r + θ()) + 12 λ2 σ 2 θ()2
βC()f ()
1 2 2
2
exp
λ(1
+
r)
+
λθ()
−
λ
σ
θ()
λ2 σ 2
2
But λθ() − 12 λ2 σ 2 θ()2 ≤
∀ ∈ (, ), θ0 () ≤
2
2σ 2
so that:
βC()f ()
2
exp
λ(1
+
r)
+
λ2 σ 2
2σ 2
Hence,
Z
∀ ∈ (, ), θ() ≤ θ() +
βC(ξ)f (ξ)
ξ2
exp
λ(1
+
r)
+
dξ
λ2 σ 2
2σ 2
This implies that we cannot have lim→ θ() = +∞.
Now, let’s consider the remaining possibility that is lim→ θ() =
θ0 (). The intuitive reason why this case is also impossible is that the
slope when θ crosses the line associated to the solution θ0 should be infinite and this is absurd. To see that more precisely let’s consider the
following ODE:
0 (θ) =
λ2 σ 2 exp (−λ(1 + r + θ(θ)) + 12 λ2 σ 2 θ2 )
(θ − θ0 ((θ)))
βC((θ))f ((θ))
Let’s apply the Cauchy-Lipschitz theorem to the above equation on
the domain (R+∗ )2 with the Cauchy condition (θ0 ()) = . We have a
local solution defined on a small interval [θ0 () − η, θ0 () + η] and this solution exhibits a local minimum at θ0 (). However, we can build another
solution of the above Cauchy problem since the inverse of the maximal solution θ satisfies the equation and can be prolonged to satisfy the Cauchy
condition. Therefore, because of the local minimum, the two solutions are
different and this is absurd.
The conclusion is that the maximal interval has no upper bound.
Now, by symmetry the solution is defined on R.
One thing remains to be done. In fact, if we have found a function θ()
that verifies the differential equation and hence a distribution m coherent
with the first order condition, we still need to check that the second order
condition is verified to be sure that we characterized a maximum of the
optimization criterion. This is the purpose of the following proposition:
Proposition 5 (Second order condition). Let’s introduce
1 2 2 2
Γ(, θ) = −λ2 σ 2 θ −
exp
−λ(1
+
r)
−
λθ
+
λ
θ
σ
+βm(θ)C()
λσ 2
2
Let’s consider the unique function θ(), given by the preceding proposition,
that satisfies ∀, Γ(, θ()) = 0 and the conditions of the above proposition.
We have:
∂θ Γ(, θ()) < 0
Proof:
First, let’s differentiate the first order condition Γ(, θ()) = 0 with
respect to . We get:
∂ Γ(, θ()) + θ0 ()∂θ Γ(, θ()) = 0
Thus, the sign of ∂θ Γ(, θ()) is the sign of −∂ Γ(, θ()) and we need to
prove that ∂ Γ(, θ()) > 0.
But:
1
1 + λ2 σ 2 θ θ −
+βm(θ)C 0 ()
∂ Γ(, θ) = λ exp −λ(1 + r) − λθ + λ2 θ2 σ 2
2
λσ 2
This expression is positive for θ = θ() since θ() ≥
λσ 2
This proposition shows that everything was in fact well defined and
that we can try to compute numerically a solution. To know the shape
of the curve, it’s indeed important to compute the function θ() for an
example and to compare it to the linear function θ0 () we usually obtain
in the non-competitive case. This is what we are doing now.
1.3
Analysis on an example
To illustrate the portfolio choices, let’s consider the case where r = 2%,
σ = 20%, λ = 1, s = 1% and the coefficient β is really small: β = 5×10−5 .
Using a finite elements methodology with a dichotomy process to find the
only solution that satisfy θ(0) = 0 we get the following results:
Figure 1: β = 5 × 10−5 , r = 2%, σ = 20%, λ = 1, s = 1%
One can see that, even for a small value of β, the structure of the
Markowitz model is completely modified. In economic terms, it means
that the incentives, even if there are small, can completely change the
portfolios structure.
Conclusion
We have generalized the usual portfolio management theory by introducing a competition effect between asset managers. Asset managers compete
indeed to increase their amount of assets under management and this implies a higher risk exposure in the portfolios, on both the long and the
short side. Interestingly, the portfolio choices are really sensitive to the
presence of a ranking or competition effect, proving that the asset managers’ incentives can induce portfolios that are far from those optimal for
the investors.
References
[Aum64]
R. Aumann. Markets with a continuum of traders. Econometrica, 32(1/2), 1964.
[CE97]
J. Chevalier and G. Ellison. Risk taking by mutual funds as
a response to incentives. The Journal of Political Economy,
105(6), Dec 1997.
[CE99a]
J. Chevalier and G. Ellison. Are some mutual fund managers
better than others? cross-sectional patterns in behavior and
performance. Journal of Finance, 54(3), 1999.
[CE99b]
J. Chevalier and G. Ellison. Career concerns of mutual fund
managers. Quarterly Journal of Economics, 114(2), May 1999.
[Hud99]
S. Huddart. Reputation and performance fee effects on portfolio choice by investment advisers. Journal of Financial Markets, 2(3), Aug. 1999.
[LR81]
E. Lazear and S. Rosen. Rank-order tournaments as optimum
labor contracts. The Journal of Political Economy, 89(5), 1981.
[LSTV91] J. Lakonishok, A. Shleifer, R. Thaler, and R. Vishny. Window
dressing by pension fund managers. The American Economic
Review, 81(2), May 1991.
[Mar52]
H. Markowitz. Portfolio selection. The Journal of Finance,
7(1), Mar. 1952.
[Mer90]
R. Merton. Continuous-time Finance. Blackwell Publishing,
1990.
[SS90]
D. Scharfstein and J. Stein. Herd behavior and investment.
The American Economic Review, 80(3), Jun. 1990.
Part I - Chapter 3
Applications of Mean Field Games
to Population Issues
Applications of Mean Field Games to Population
Issues
Abstract
This chapter is the only one that doesn’t directly deal with economics
since it is an application of mean field games to individuals that want to
form a community1 . We basically explicit solutions for individuals that
want to be distributed as close as possible from one another but cannot form a Dirac distribution in presence of noise. We focus on explicit
solutions and exhibit several notions of stability for the solutions, eductive stability being a powerful viewpoint when it comes to find solutions
numerically.
Résumé
Ce chapitre est le seul qui ne soit pas directement appliqué à la science
économique puisqu’il s’agit de modéliser la formation de communautés.
Il constitue toutefois un chapitre central du point de vue méthodologique
et peut être appliqué à de nombreux sujets économiques comme celui du
choix des ménages en matière de technologie (de chauffage, ...).
Nous considérons des individus qui souhaitent être ensemble mais ne le
peuvent que partiellement, de par la présence d’un bruit. Nous insistons d’abord sur des solutions explicites puis l’on se concentre sur leur
stabilité en introduisant plusieurs notions de stabilité. L’une d’elle, la
stabilité “éductive”, se révèle être très utile pour la résolution numérique.
Introduction
Mean field games are a natural framework to deal with population issues
in which individuals optimize their position to satisfy their willingness to
be with or without their peers in addition to be at a given location for
instance. Many setups can be imagined going from people who all want
to live near a given location but do not want to live near their peers to
people who just want to live with their peers anywhere as soon as they
are gathered as a community.
On a “periodic” compact set, that is to say on a circle in dimension 1
or on a torus in higher dimensions, many results are known and the case
1 Many applications are possible though, concerning for example technological choices by
households.
where agents don’t want to live with their peers is completely solved (at
least in the no discount case) with the uniform distribution being the only
stationary solution on a torus (see [LL07a]).
Here, we are going to take the opposite point of view and consider people
that want to live with their peers and the sets are going to be R or Rn to
obtain explicit gaussian solutions.
The text is organized as follows: in a first part we present the general
mean field games framework to deal with population issues and we focus
in the second part on the special case where costs are quadratic. This
second part gives the intuition for the explicit resolution of the third part
in a special case (with logarithmic utility). The fourth part is dedicated
to the stability of our solutions. The last part is an introduction to applications with several populations. Finally, an appendix is dedicated to
numerical recipes to deal numerically with mean field games similar to
those described in this part. In particular, we focus in this appendix on
the importance of the eductive viewpoint to find numerical solutions, both
for the stationary and the dynamical problem.
1 The general framework with one population
In what follows, we consider a continuum of individuals (hereafter a population) that have preferences about living with their peers. This type of
problem is typically of the mean field game sort where individuals pay a
price to move from one point to another and have a utility flow that is a
function of the overall distribution of individuals in the population. We
are going to model it as follows:
• Each agent lives in an n-dimensional space.
• Each agent has a “utility” function v that can be decomposed in
two parts: a pure preference part g : (t, x) 7→ g(m(t, x)) (where g is
increasing to model the willingness to be together) that represents
what she gets from being in position x at time t (m is the distribution
function of the population) and a pure cost part h : α 7→ h(α) that
corresponds to the price to pay to make a move of size α (h is
typically supposed to be increasing, strictly convex and such that
h(0) = 0).
• Each agent discounts the time at rate ρ.
• Each agent is moved by a brownian motion in dimension n (specific
to herself).
The problem we are dealing with can therefore be written as a control
problem:
T
Z
u(t, x) = M ax(αs )s>t ,Xt =x E
t
(g(m(s, Xs )) − h(|α(s, Xs )|)) e−ρ(s−t) ds
with dXt = α(t, Xt )dt + σdWt 2 .
As for any mean field game we use [LL06a, LL06b] to write the associated system of partial differential equations:
Proposition 1 (Mean Field Games PDEs). The control problem is equivalent to the following system of PDEs
(Hamilton − Jacobi)
(Kolmogorov)
∂t u +
σ2
∆u + H(∇u) − ρu = −g(m)
2
∂t m + ∇ · (mH 0 (∇u)) =
σ2
∆m
2
where H(p) = M axa (ap − h(a))
Additional conditions are: m(0, ·) given, u(T, ·) = 0 and ∀t, m(t, ·) is a
probability distribution function.
Our goal here is to find stationary solutions of this problem in several
special cases where we always suppose that T is replaced by +∞ 3 :
(Hamilton − Jacobi)
(Kolmogorov)
σ2
∆u + H(∇u) − ρu = −g(m)
2
∇ · (mH 0 (∇u)) =
σ2
∆m
2
with m a probability distribution function.
2
2.1
The quadratic cost framework
Presentation
One of the simplest framework to deal with mean field games is to consider
the special case of quadratic cost: h(a) = 12 a2 . This case is indeed simpler
since it allows to replace the system of coupled PDEs by a single PDE,
√
either on u or on m (the good variable is actually ψ = m as we will
see later on). Consequently, we focus extensively on quadratic costs even
though more complex models can be used to deal with problems involving
congestion for instance.
The quadratic cost framework is characterized by a simple Hamiltonian
(H(p) = 12 p2 ) and therefore the system to solve is simplified:
Proposition 2 (Mean Field Games PDEs with quadratic costs). With
quadratic costs, the system can be written as:
(Hamilton − Jacobi)
2 We
∂t u +
σ2
1
∆u + |∇u|2 − ρu = −g(m)
2
2
will generalize to more complex structures for the brownian motion and its volatility
transversality condition that appears in the case of infinite horizon is not relevant in
the stationary case.
3 The
(Kolmogorov)
∂t m + ∇ · (m∇u) =
σ2
∆m
2
In its stationary form, the system is simply:
(Hamilton − Jacobi)
(Kolmogorov)
2.2
1
σ2
∆u + |∇u|2 − ρu = −g(m)
2
2
∇ · (m∇u) =
σ2
∆m
2
From two coupled PDEs to one
We are going to enounce two propositions that show the interest of the
quadratic costs.
Proposition 3 (One PDE in u). Let’s consider a couple (K, u) where
K is a scalar. If (K, u) is a solution of the equations (1) and (10 ) then
(u, K exp( σ2u2 )) is a solution of our initial stationary problem.
2u(x)
σ2
1
∆u(x) + |∇u(x)|2 − ρu(x) = −g x, K exp( 2 )
2
2
σ
Z
2u(x)
K exp( 2 ) = 1
(10 )
σ
Proof:
(1)
The only thing to prove is that m(x) = K exp( 2u(x)
) is a solution of the
σ2
Kolmogorov equation. Taking logs and deriving we have ∇m = 2∇u
m.
σ2
Hence, if we apply the divergence operator to each side we obtain the
Kolmogorov equation.
Another way to look at the problem is to consider an equation in m
or more exactly an equation in ψ where ψ is defined as the square root of
m.
√
Proposition 4 (One PDE in ψ = m). Let’s consider a couple (K, ψ)
where K is a scalar. If (K, ψ) is a solution of the equations (2) and (20 )
ψ
are solutions of our initial stationary
then, m = ψ 2 and u = σ 2 ln K
problem.
ψ(x)
σ 4 ∆ψ(x)
= ρσ 2 ln(
) − g(ψ 2 (x))
2 ψ(x)
K
Z
ψ(x)2 dx = 1
(20 )
(2)
x
Proof:
Let’s consider (K, ψ) solution of
the preceding equations and let’s inψ
.
troduce m = ψ 2 and u = σ 2 ln K
We have the following derivatives:
∇m
∇ψ
=2
m
ψ
∇ψ
σ 2 ∇m
=
ψ
2 m
Hence, (u, m) verifies the Kolmogorov equation.
∇u = σ 2
Now,
∆u = σ 2
⇒
|∇ψ|2
∆ψ
−
ψ
ψ2
= σ2
1
∆ψ
− 2 |∇u|2
ψ
σ
ψ(x)
σ2
1
σ 4 ∆ψ(x)
∆u(x) + |∇u(x)|2 =
= ρσ 2 ln(
) − g(ψ 2 (x))
2
2
2 ψ(x)
K
σ2
1
∆u(x) + |∇u(x)|2 − ρu(x) = −g(m(x))
2
2
Hence, (u, m) verifies the Hamilton-Jacobi equation.
⇒
The partial differential equation in ψ invites us to consider the case
where (t, x) 7→ g(m(t, x)) is the logarithm function ln(m(t, x)) as an example of our population problem that may be solved easily and explicitly.
This is our next application.
3 Application to the logarithmic utility
function
3.1
3.1.1
The basic framework
Presentation
We are going to build a very precise and explicit example that goes into
the quadratic cost framework. We consider one population that lives in
Rn and we suppose that all people in the population have the same preference function which is simply g(m(t, x)) = ln(m(t, x)).
These preferences mean that inside the population, people want to
live together. However, they are prevented to do so by the noise and our
problem is to find the optimal behavior of individuals in such a context.
To sum up, we want to find stationary solutions to the problem:
Z ∞ |α(s, Xs )|2
−ρ(s−t)
u(t, x) = M ax(αs )s>t ,Xt =x E
e
ds
ln(m(s, Xs )) −
2
t
with dXt = α(t, Xt )dt + σdWt
In other words, we want to find a solution of the following system of
PDEs:
σ2
1
∆u + |∇u|2 − ρu = − ln(m)
2
2
(Hamilton − Jacobi)
∇ · (m∇u) =
(Kolmogorov)
3.1.2
σ2
∆m
2
Resolution
Proposition 5 (Gaussian solutions). Suppose that ρ < σ22 .
There exist three constants, s2 > 0, η > 0 and ω such that ∀µ ∈ Rn , if
m is the probability distribution function associated to a gaussian variable
N (µ, s2 In ) and u(x) = −η|x − µ|2 + ω, then (u, m) is a solution of our
problem.
These three constants are given by:
σ4
4−2ρσ 2
• s2 =
2
σ
− ρ2 = 4s
2
2
1
• ω = − ρ ηnσ − n2 ln
• η=
1
σ2
2η
πσ 2
Proof:
We are going to use Proposition 3 and the PDE in u.
We are looking for a solution for u of the form:
u(x) = −η|x − µ|2 + ω
If we put this form in the Hamilton-Jacobi equation of Proposition 3
we get:
2η 2 |x − µ|2 + ρη|x − µ|2 − ρω − ηnσ 2 = − ln(K) +
2η|x − µ|2
2ω
− 2
σ2
σ
A first condition for this to be true is:
2η 2 + ρη =
2η
σ2
1
ρ
−
σ2
2
A second condition, to find ω, is related to the fact that m is a probability distribution function (equation (10 )). This clearly requires η to
be positive but this is guaranteed by the hypothesis ρσ 2 < 2. This also
implies:
⇐⇒ η =
K exp
2ω
σ2
Z
exp
Rn
−2η
|x − µ|2
σ2
⇒ ρω + ηnσ 2 =
= K exp
n
ln
2
2η
πσ 2
2ω
σ2
πσ 2
2η
n2
=1
and this last equation gives ω.
From this solution for u we can find a solution for m. We indeed know
that m is a probability distribution function and that m is given by
m(x) = K exp(
2u(x)
)
σ2
As a consequence, m is the probability distribution function of an ndimensional gaussian random variable with variance equal to s2 In where
2
σ4
s2 = σ4η i.e. s2 = 4−2ρσ
2.
Interestingly, we can come back to the control parameter α. This
control parameter describes the move each agent wants to make given her
position. We have the following result:
Proposition 6 (Optimal control). In the framework of the preceding
proposition, the optimal control parameter α is given by α(x) = −2η(x −
µ). This means that for any agent, her position Xt follows an OrnsteinUhlenbeck process that mean-reverts around µ:
dXt = −2η(Xt − µ)dt + σdWt
Proof:
Because H(p) = 21 p2 , the optimal control is α(x) = H 0 (∇u(x)) =
∇u(x).
The preceding proposition gives the result.
3.2 Generalization to a more complex brownian
motion
In this part, we present an extension of the preceding derivations in the
case of a more general noise setting. In the above text, we indeed supposed that the noise was the same in each direction. This hypothesis was
assumed for the sake of simplicity but can easily be relaxed.
With the same notations as above, we can write
dXt = α(t, Xt )dt + dBt
where B is a generalized brownian motion with a variance matrix Ω where
we assume Ω belongs to Sn++ .
To use what we have done before, we write this process as
dXt = α(t, Xt )dt + ΓdWt
where Γ ∈ Sn++ verifies Γ2 = Ω and where W is an n-dimensional standard
brownian motion.
The trick is then to consider Yt = Γ−1 Xt because of its simpler dynamics:
dYt = β(t, Yt )dt + dWt
with β(t, Yt ) = Γ−1 α(t, ΓYt )
Our control problem can be written in terms of y’s:
Z ∞ |Γβ(s, Ys )|2
v(t, y) = u(t, Γy) = M ax(βs )s>t ,Yt =y E
ln(m(ΓYt )) −
e−ρ(s−t) ds
2
t
If we introduce n the probability distribution function of y related to m
by the simple equation n(y) = m(Γy) det(Γ), then this problem can be
translated in the PDE framework where we focus on stationary solutions:
Proposition 7 (Generalization of the PDEs). The two stationary equations can be rewritten using v:
(∇v)0 Ω−1 (∇v)
1
−ρv+ ∆v = − ln(n(y))−ln(det(Γ))
2
2
1
(Kolmogorov)
div(Ω−1 (∇v)n) = ∆n
2
−1
The optimal control b is given by b = Ω (∇v).
(Hamilton−Jacobi)
Proof:
First, we can write the stationary Hamilton-Jacobi PDE associated to
the optimal control problem:
|Γb|2
1
− ρv + ∆v = − ln(m(Γy))
M axb ∇v · b −
2
2
2
We see that the Hamiltonian is H(p) = M axb p · b − |Γb|
. It’s now
2
easy to see that the optimal control is given by b = Ω−1 p and hence
0 −1
H(p) = p Ω2 p which is what we want to prove as far as the HJB equation is concerned (the right hand side comes from the relation between m
and n).
Now, since the optimal control is b = Ω−1 (∇v), the Kolmogorov equation for n is given by:
div(Ω−1 (∇v)n) =
1
∆n
2
Now we can use the intuition from the simpler case to solve the problem:
Proposition 8 (Gaussian solutions). Suppose that Sp(Ω) ∈]0; ρ2 [.
There exist a constant C and two matrices Σ, A ∈ Sn++ so that ∀µ ∈ Rn , if
n is the probability distribution function associated to a gaussian variable
N (µ, Σ) and v(y) = −(y − µ)0 A(y − µ) + C, then (v, n) is a solution of
our problem.
A and Σ are given by:
Σ = (4In − 2ρΩ)−1 Ω =
A = In −
∞
X
ρn n+1
Ω
2n+2
n=0
1
ρΩ
2
Proof:
As we did earlier, we will try to find solutions of the equations of the
following form:
v(y) = −(y − µ)0 A(y − µ) + C
(y − µ)0 Σ−1 (y − µ)
1
n(y) =
exp
−
n
2
(2π det(Σ)) 2
where µ can be any point in Rn and where we need to find the solution
matrices Σ, A ∈ Sn++ .
The Kolmogorov equation can be written as:
1
div Ω−1 (∇v)n − ∇n = 0
2
1
div −2Ω−1 A(y − µ)n + Σ−1 (y − µ)n = 0
2
Hence, our first equation that guarantees (v, n) to be a solution of the
Kolmogorov equation is:
Σ−1 = 4Ω−1 A
Now, if we focus on the Hamilton Jacobi equation we must have:
(∇v)0 Ω−1 (∇v)
1
− ρv + ∆v = − ln(n) + ln(det(Γ))
2
2
Notice now that D = 12 ∆v is actually a constant since v is a polynomial
function of degree 2. Hence:
2(y − µ)0 A0 Ω−1 A(y − µ) + ρ(y − µ)0 A(y − µ) − ρC + D
(y − µ)0 Σ−1 (y − µ)
1
n
ln(det(Ω)) + ln((2π det(Σ)) +
2
2
2
Therefore, our second equation4 that guarantees (v, n) to be a solution
of the Hamilton-Jacobi equation is:
=
4A0 Ω−1 A + 2ρA = Σ−1
Since Σ−1 = 4Ω−1 A, this second equation can be simplified and we
get A = In − 21 ρΩ.
Now that we have A, we can deduce that Σ = (4In − 2ρΩ)−1 Ω.
Now, we can very simply translate these results in terms of (u, m).
4 There
is a third equation for C but there is no issue regarding this one.
Proposition 9 (Gaussian solutions). Suppose that Sp(Ω) ∈]0; ρ2 [.
There exist a constant C̃ and two matrices Σ̃, Ã ∈ Sn++ so that ∀µ ∈ Rn , if
m is the probability distribution function associated to a gaussian variable
N (µ, Σ̃) and u(x) = −(x − µ)0 Ã(x − µ) + C̃, then (u, m) is a solution of
our problem.
Σ̃ and Ã are given by:
Σ̃ =
∞
X
ρn n+2
Ω
n+2
2
n=0
Ã = Ω−1 −
3.3
1
ρIn
2
Comments on the preceding examples
The preceding examples are interesting in the fact that we have been able
to exhibit explicit solutions. One caveat is that these solution are specific
to the logarithmic case.
Another problem with our setting is that we only describe possible stationary solutions and the possible path from an initial distribution to a
stationary solution is not dealt with.
This comment leads to the third issue in these examples which is the infinite number of solutions. This problem can be dealt with in a very simple
way. It’s indeed possible to say that, in addition to their willingness to
be together, agents in the population love a certain location µ∗ . In that
case, the stochastic control problem (we took the first setting, not the
generalized one) can be replaced by:
∞
Z
M ax(αs )s>t ,Xt =x E
ln(m(s, Xs )) − δ|Xs − µ∗ |2 −
t
|α(s, Xs )|2
2
e−ρ(s−t) ds
with
dXt = α(t, Xt )dt + σdWt
With this quadratic form, one can generalize the preceding computations and we get the following localization result.
Proposition 10 (Localization). In this new problem any gaussian solu2
tion has to be centered in µ∗ . The variance coefficient is s2 = σ4η where η
is now the unique positive solution of:
2
2η 2 − η
−
ρ
=δ
σ2
4
Stability in the logarithmic case
Let’s consider the logarithmic case of the last section and let’s work for
simplicity in dimension 15 . We have found stationary solutions of the
5 The results we will obtain can be generalized really easily in higher dimension using
Hermite Polynomials in higher dimension.
problem and up to a translation we can consider that µ = 0 so that the
stationary solution we consider is:
u∗ (x) = −ηx2 + ω
1
x2
m∗ (x) = √
exp − 2
2s
2πs2
An interesting question is the stability of this stationary solution.
We are going to consider two notions of stability. The first notion of
stability is the classical physical notion of local stability. If an equilibrium
is given, it will be said locally stable in the classical sense if, after a small
perturbation, the system goes back (perhaps asymptotically) to the initial
equilibrium. A second notion of stability is inspired from the eductive
viewpoint in economic theory (see [Gue92]). Typically, the equilibrium
will be said to be locally stable in the eductive sense if, the common
knowledge that the equilibrium is in a given neighborhood allows agents
to find, by a mental process6 (i.e. without any time-dependent learning)
the actual equilibrium.
4.1
Local physical stability
To work on the local stability in the classical sense, we consider the PDEs
of Proposition 3 and we introduce perturbations on the solutions (for
µ = 0). These perturbations can be written as:
m(0, x) = m∗ (x)(1 + εψ(0, x))
u(T, x) = u∗ (x) + εφ(T, x)
φ(T, ·) = φ(·) and ψ(0, ·) = ψ(·) are given and represent respectively
the relative perturbation on m∗ and the absolution perturbation on u∗7 .
We are going to study the dynamics of the functions φ and ψ where
we consider the linearized PDEs.
Proposition 11 (Linearized PDEs). The linearized PDEs around (u∗ , m∗ )
are:
σ 2 00
φ − 2ηxφ0 − ρφ = −ψ
2
σ 2 00
x
ψ̇ −
ψ + 2ηxψ 0 = −φ00 + 2 φ0
2
s
(Hamilton − Jacobi)
(Kolmogorov)
φ̇ +
Proof:
6 In the seminal articles on eductive stability, the mental process was linked to the notion
of rationalizable solutions, see [Gue92] for more details.
7 It’s in fact really important to consider the relative variation in the case of the probability
distribution function m.
A Taylor expansion of the ln is the only thing needed to obtain the
HJB equation.
For the Kolmogorov equation, the linearized PDE first appears as:
σ2
(ψm∗ )00 + (−2ηxψm∗ )0 = −(φ0 m∗ )0
2
2
Since (m∗ )0 = − sx2 m∗ and (m∗ )00 = xs4 − s12 m∗ , we obtain:
ψ̇m∗ −
ψ̇−
σ2
2
ψ 00 − 2
x 0
ψ +
s2
x2
1
− 2
s4
s
Using now the fact that s2 =
x
x2
ψ −2ηψ−2ηxψ 0 +2η 2 ψ = −φ00 + 2 φ0
s
s
σ2
,
4η
we obtain the result.
A more convenient way to see these linearized PDEs is to introduce
2
the L operator: f 7→ Lf = − σ2 f 00 + 2ηxf 0
Proposition 12 (Linearized PDEs). The above equations can be written
as:
(Hamilton − Jacobi)
φ̇ = Lφ + ρφ − ψ
2
(Kolmogorov)
ψ̇ = −Lψ + 2 Lφ
σ
Proof: The only thing to recall is that s2 =
σ2
.
4η
Now, we are going to use the properties of the operator L we have just
introduced. To do that we need to use some properties of the Hermite
polynomials associated to the space L2 (m∗ (x)dx).
Definition 1 (Hermite polynomials). We defined the nth Hermite polynomial of L2 (m∗ (x)dx) by:
1
x2
x2 dn
Hn (x) = sn √ (−1)n exp( 2 ) n exp(− 2 )
2s dx
2s
n!
Proposition 13 (Hermite polynomials as a basis). The polynomials (Hn )n
form an orthonormal basis of the Hilbert space L2 (m∗ (x)dx).
Proposition 14 (Hermite polynomials as eigenvectors of L). The Hermite polynomials Hn are eigenvectors of L and:
LHn = 2ηnHn
Now that we have recalled some basics about the Hermite polynomials
we can use them to solve the linearized PDEs of Proposition 12. Let’s start
first with the matrices (An )n that are going to be involved to solve the
problem:
ρ + 2ηn
−1
An =
n
−2ηn
s2
Lemma 1 (Eigenvalues of An ). Let’s consider n ≥ 2.
The eigenvalues of An are of opposite signs, λ1n < 0 < λ2n with:
λ1,2
n =
i
p
1h
ρ ± ρ2 + 16η 2 n(n − 1)
2
Proof:
The eigenvalues are the roots of the polynomials X 2 − ρX − 2ηn(ρ +
2ηn) + sn2 . We can compute ∆:
2
∆ = ρ2 + 8ηn ρ − 2 + 2ηn
σ
Hence, using the relations between η and ρ we get:
∆ = ρ2 + 16η 2 n(n − 1)
Since n ≥ 2 we have ∆ > ρ2 and therefore the two roots are real, one is
positive and the other is negative.
It’s interesting to notice that for a system of two linear PDEs like
this we are working on, one equation being forward and the other being
backward, the stability result will arise from the opposite signs of the
eigenvalues.
Proposition 15. Let’s suppose that the perturbations ψ and φ are in the
Hilbert space H = L2 (m∗ (x)dx).
φn
Let’s consider for n ≥ 2 the functions
that verify:
ψn
φ˙n
φn
=
A
n
ψn
ψ˙n
with φn (T ) equal to φn and ψn (0) equal to ψ n .
We have:
ψ
1
2
φn (t) = On ( n eλn t ) + On (φn e−λn (T −t) )
4ηn
1
2
ψn (t) = On (ψ n eλn t ) + On (φn e−λn (T −t) )
In particular,
∀t ∈ (0, T ), ∀k ∈ N, (nk φn (t))n ∈ l1 (⊂ l2 ), (nk ψn (t))n ∈ l1 (⊂ l2 )
Proof:
If we use the preceding lemma, we see that we can write:
1
2
φn (t)
1
1
1
2
= Cn,T
eλn t
+ Cn,T
eλn t
1
2
ψn (t)
vn
vn
where the v’s are found using eigenvectors of the matrix An :
vn1 = ρ + 2ηn − λ1n ,
vn2 = ρ + 2ηn − λ2n
Now, to find the two constants we need to use the conditions on φn (T )
and ψn (0):
(
1
2
1
2
φn (T ) = φn = Cn,T
eλn T + Cn,T
eλn T
1
2
ψn (0) = ψ n = Cn,T
vn1 + Cn,T
vn2
Hence:


1

 Cn,T
=

2

 Cn,T
=
2
2
φn −eλn T ψ
vn
n
2
2 eλ1
n T −v 1 eλn T
vn
n
1
1
vn
φn −eλn T ψ
n
1
1 eλ2
n T −v 2 eλn T
vn
n
Using the fact that vn1 ∼ 4ηn and vn2 ∼ ρ2 + η we can deduce the
1,2
asymptotic behavior8 of Cn,T
as n goes to infinity (with T fixed).
1
Cn,T
∼n→∞
ψn
4ηn
2
2
Cn,T
∼n→∞ φn e−λn T
,
Hence:
φn (t) = On (
ψn
4ηn
1
2
eλn t ) + On (φn e−λn (T −t) )
1
2
ψn (t) = On (ψ n eλn t ) + On (φn e−λn (T −t) )
These two estimations prove the results.
The estimates we established in the preceding proposition are the basis
of the regularization property we will obtain in the following proposition.
What we will show is indeed that whatever the regularity of the perturbations in the Hilbert space H = L2 (m∗ (x)dx), the solutions are going to
be in C ∞ on (0, T ) × R.
Proposition 16 (Resolution of the PDEs). Suppose that:
• The perturbations ψ and φ are in the Hilbert space H = L2 (m∗ (x)dx).
R
• ψ(x)m∗ (x)dx = 0 (mass preservation condition)
R
• xψ(x)m∗ (x)dx = 0 (mean preservation condition)
R
• xφ(x)m∗ (x)dx = 0 (this is guaranteed if the perturbation is even)
Let’s define (φn )n and (ψn )n by:
• φ0 (t) = φe−ρ(T −t) and ψ0 (t) = 0.
• φ1 (t) = ψ1 (t) = 0.
• ∀n ≥ 2, φn and ψn defined as in the preceding proposition.
8 Here we assume that ψ 6= 0 and φ 6= 0. If one of these coefficients is equal to 0, the
n
n
estimates of the proposition are still true and can even be improved.
P∞
P
Then φ(t, x) = ∞
n=0 ψn (t)Hn (x) are
n=0 φn (t)Hn (x) and ψ(t, x) =
∞
well defined in H, are in C and are solutions of the PDEs with the
boundary conditions associated to φ and ψ.
Proof:
First of all, the preceding proposition ensure that the two functions φ
and ψ are well defined, in C ∞ , and that we can differentiate formally the
expressions. Then, the first three conditions can be translated as ψ 0 = 0,
ψ 1 = 0 and φ1 = 0 and so the conditions at time 0 and time T are verified.
The fact that the PDEs are verified is due to the definition of φn and ψn
and also to the fact that we can differentiate under the sum sign because
of the estimates of the preceding proposition.
Now what we want to demonstrate is a stability result. We want
to show that, as T goes to infinity (the initial and final perturbations
remaining unchanged), the influence of the perturbation vanish. This is
the purpose of the following proposition:
Proposition 17 (Stability I). Suppose that:
• The perturbations ψ and φ are in the Hilbert space H = L2 (m∗ (x)dx).
R
• ψ(x)m∗ (x)dx = 0 (mass preservation condition)
R
• xψ(x)m∗ (x)dx = 0 (mean preservation condition)
R
• xφ(x)m∗ (x)dx = 0 (this is guaranteed if the perturbation is even)
Then, ∀n, ∀α ∈ (0, 12 ):
lim ||φn ||L∞ ([αT,(1−α)T ]) = 0,
lim ||ψn ||L∞ ([αT,(1−α)T ]) = 0
T →∞
T →∞
Proof:
The result is obvious for n = 0 and n = 1. For n ≥ 2, we need to go
back to the expressions of φn (t) and ψn (t).
First of all, let’s go back to the two constants:

2
v 2 φ −eλn T ψ


n
 C1 = n n
n,T
1
2
2 eλn T −v 1 eλn T
vn
n
1

2

 Cn,T
=
1
vn
φn −eλn T ψ
2
n
1
1 eλn T −v 2 eλn T
vn
n
Then9 ,
1
lim Cn,T
=
T →∞
ψn
vn1
,
2
2
Cn,T
∼T →∞ φn e−λn T
9 Here we assume that φ 6= 0. If this coefficient is equal to 0, the result is still true but the
n
2
estimate for Cn,T
cannot be written this way and is in fact better than the estimate presented
below.
Using now the expressions for the functions,
2
1
2
1
eλn t
φn (t) = Cn,T
eλn t + Cn,T
2
1
2
1
vn2 eλn t
ψn (t) = Cn,T
vn1 eλn t + Cn,T
we get:
1
2
1
2
||φn ||L∞ ([αT,(1−α)T ]) ≤ |Cn,T
|eλn αT + |Cn,T
|eλn (1−α)T
1
2
2
1
vn2 |eλn (1−α)T
||ψn ||L∞ ([αT,(1−α)T ]) ≤ |Cn,T
vn1 |eλn αT + |Cn,T
and this leads to the result really easily.
It’s noticeable that the three conditions on the perturbations are natural to obtain a stability result. First of all, the mass preservation is
natural since the total measure must remain the same. Then, the two
other conditions are necessary because of the invariance by translation of
the problem.
The result we have just obtained is a weak form of stability but
stronger stability results can be obtained by using more precise estimations. An example of such an improvement is:
Proposition 18 (Stability II). Suppose that:
• The perturbations ψ and φ are in the Hilbert space H = L2 (m∗ (x)dx).
R
• ψ(x)m∗ (x)dx (mass preservation condition)
R
• xψ(x)m∗ (x)dx (mean preservation condition)
R
• xφ(x)m∗ (x)dx (this is guaranteed if the perturbation is even)
Then:
lim supt∈[αT,(1−α)T ] ||φ(t, ·)||L2 (m∗ (x)dx) = 0
T →∞
lim supt∈[αT,(1−α)T ] ||ψ(t, ·)||L2 (m∗ (x)dx) = 0
T →∞
Proof: It’s a simple application of the Lebesgue’s dominated convergence
theorem.
4.2
Local eductive stability
Now, we are going to consider another notion of stability that has more
to do with the justification of rational expectation hypothesis or with the
process through which agents will mentally understand what will be the
stationary equilibrium.
The goal in the next paragraphs is in fact to consider an initial guess for
the stationary equilibrium (in the neighborhood of the actual equilibrium)
and to exhibit a “mental process” (this process is actually a continuous
process based on two PDEs involving what we call virtual time) that goes
from the initial guess to the true equilibrium.
Let’s consider the two equations of Proposition 3:
σ 2 00 1 02
u + u − ρu + ln(m) = 0
2
2
σ 2 00
m − (mu0 )0 = 0
2
We are going to introduce a variable θ called virtual time and consider,
given an initial guess (u(θ = 0, x), m(θ = 0, x)) for the equilibrium, the
mental process associated with the following system of PDEs:
σ 2 00 1 02
u + u − ρu + ln(m)
2
2
σ 2 00
m − (mu0 )0
∂θ m =
2
Since we only want to consider a local eductive stability, we are going
to work with the linearized version of these equations that is given by the
following proposition:
∂θ u =
Proposition 19 (Linearized mental process). The linearized mental process around (u∗ , m∗ ) is given by:
σ 2 00
φ − 2ηxφ0 − ρφ + ψ
2
σ 2 00
x
∂θ ψ =
ψ + 2ηxψ 0 − φ00 + 2 φ0
2
s
where φ and ψ are defined as before and where φ(0, ·) and ψ(0, ·) are
given.
∂θ φ =
Proof: The proof is identical to the proof of Proposition 11.
We can write these equations using the L operator introduced earlier:
Proposition 20. The above equations can be written as:
∂θ φ = −Lφ − ρφ + ψ
2
Lφ
σ2
To solve these equations, we need to introduce the matrices (Bn )n :
−(ρ + 2ηn)
1
Bn =
n
−2ηn
s2
∂θ ψ = −Lψ +
Lemma 2 (Eigenvalues of Bn ). Let’s consider n ≥ 2.
The eigenvalues ξn1 < ξn2 of Bn are both negative with:
"
#
r
1
4n
1,2
ξn =
−ρ − 4ηn ± ρ2 + 2
2
s
Proof:
The eigenvalues are the roots of the polynomials X 2 + (ρ + 4ηn)X +
2ηn(ρ + 2ηn) − sn2 . We can compute ∆:
∆ = ρ2 +
4n
>0
s2
Hence, the eigenvalues are real and are of the form given in the proposition.
Since tr(Bn ) < 0 and det(Bn ) = 2ηn(ρ + 2ηn) − 4ηn
= 4η 2 n(n − 1) > 0,
σ2
the two eigenvalues are negative.
Proposition 21. Let’s suppose that the initial conditions φ(0, ·) and
ψ(0, ·) are in the Hilbert space H = L2 (m∗ (x)dx).
φn
Let’s consider for n ≥ 2 the functions
that verify:
ψn
φn
∂θ φn
= Bn
ψn
∂θ ψn
with φn (0) equal to φ(0, ·)n and ψn (0) equal to ψ(0, ·)n .
We have:
2
φn (θ) = On (|φn (0)|eξn θ )
√
2
ψn (θ) = On ( n|φn (0)|eξn θ )
In particular,
∀θ > 0, ∀k ∈ N, (nk φn (θ))n ∈ l1 (⊂ l2 ), (nk ψn (θ))n ∈ l1 (⊂ l2 )
Proof:
The proof is similar to the proof of Proposition 15.
1
2
φn (θ)
1
1
= An eξn θ
+ Bn eξn θ
ψn (θ)
an
bn
where:
an = ρ + 2ηn + ξn1 ,
bn = ρ + 2ηn + ξn2
Now, to find the two constants we need to use the conditions on φn (0)
and ψn (0):
φn (0) = An + Bn
ψn (0) = an An + bn Bn
Hence:
(
An =
Bn =
bn φn (0)−ψn (0)
bn −an
an φn (0)−ψn (0)
an −bn
√ √
√ √
η
η
Using the fact that an ∼ − σ n and bn ∼ σ n we can deduce the
asymptotic behavior of the constants as n goes to infinity.
An ∼n→∞
φn (0)
,
2
Bn ∼n→∞
φn (0)
2
Hence, since ξn1 < ξn2 ,
2
φn (θ) = On (|φn (0)|eξn θ )
√
2
ψn (θ) = On ( n|φn (0)|eξn θ )
These two estimations prove the results.
As before these estimations show that the solutions will be far more
regular than the initial conditions.
Proposition 22 (Resolution of the PDEs associated to the mental process). Suppose that:
• The initial conditions φ(0, ·) and ψ(0, ·) are in the Hilbert space H =
L2 (m∗ (x)dx).
R
• ψ(0, x)m∗ (x)dx = 0 (this is guaranteed if the initial guess for m is
a probability distribution function)
R
• xφ(0, x)m∗ (x)dx = 0 (this is guaranteed if the initial guess is even)
R
• xψ(0, x)m∗ (x)dx = 0 (this is guaranteed if the initial guess is even)
Let’s define (φn )n and (ψn )n by:
• φ0 (θ) = φ0 (0)e−ρθ and ψ0 (θ) = 0.
• φ1 (θ) = ψ1 (θ) = 0.
• ∀n ≥ 2, φn and ψn defined as in the preceding proposition.
P∞
P∞
Then φ(θ, x) =
n=0 φn (θ)Hn (x) and ψ(θ, x) =
n=0 ψn (θ)Hn (x)
∞
are well defined in H, are in C , are solutions of the PDEs and verify
the initial conditions.
Proof:
First of all, the preceding proposition ensure that the two functions
φ and ψ are well defined, in C ∞ , and that we can differentiate formally
the expressions. Then, the first three conditions can be translated as
ψ0 (0, ·) = 0, φ1 (0, ·) = 0 and ψ1 (0, ·) = 0 and so the conditions at time 0
is verified.
The fact that the PDEs are verified is due to the definition of φn and ψn
and also to the fact that we can differentiate under the sum sign because
of the estimates of the preceding proposition.
Proposition 23 (Local eductive stability). Suppose that:
• The initial guesses φ(0, ·) and ψ(0, ·) are in the Hilbert space H =
L2 (m∗ (x)dx).
R
• ψ(0, x)m∗ (x)dx = 0 (this is guaranteed if the initial guess for m is
a probability distribution function)
•
R
xφ(0, x)m∗ (x)dx = 0 (this is guaranteed if the initial guess is even)
•
R
xψ(0, x)m∗ (x)dx = 0 (this is guaranteed if the initial guess is even)
Then the solution (φ, ψ) of the mental process converges in the sense
that:
lim ||φ(θ, ·)||L2 (m∗ (x)dx) = 0
θ→∞
lim ||ψ(θ, ·)||L2 (m∗ (x)dx) = 0
θ→∞
Proof:
We basically want to show that:
+∞
X
n=0
|φn (θ)|2 →θ→+∞ 0,
+∞
X
|ψn (θ)|2 →θ→+∞ 0
n=0
This is actually a pure consequence of the estimates proved in Proposition 21 and of the Lebesgue’s dominated convergence theorem.
This proposition proves that given an initial guess in the neighborhood
of a stationary solution, if the initial guess is symmetric around the stationary solution, then, the mental process associated to the PDEs allows
agents to find the solution. This is what we called local eductive stability.
4.3 Remarks on the conditions to have stability
results
In both the proof of the physical stability and the proof of the eductive stability, there was a need to impose symmetry conditions on the
perturbations or on the initial guesses. These conditions were necessary
to ensure stability because both A1 and B1 were singular. If one wants
to have stability results for more general initial perturbations or initial
guesses, the intuitive idea is to break the translation invariance of the
problem.
Interestingly, we have done that before in the paragraphs dedicated to
localization. This localization idea can be used once again, to have more
general stability results. If we center the problem around 0 as before, we
know that the only relevant difference between the original problem and
the problem with an additional term −δx2 , that localizes the problem
around 0, is the positive constant η that depends on δ according to the
equation:
2
2η 2 − η
−
ρ
=δ
σ2
Now, in this context we can prove that the eigenvalues of An are of
opposite signs for n ≥ 1 and that the eigenvalues of Bn are both negative
for n ≥ 1 (remember that we needed n to be larger than 2 to have these
properties in the case where δ = 0).
Lemma 3 (Eigenvalues of An and Bn for δ > 0). Suppose that δ > 0 and
n ≥ 1.
ρ + 2ηn
−1
Then, the eigenvalues λ1,2
are of opposite
n of An =
n
−2ηn
s2
signs.
−(ρ + 2ηn)
1
Similarly, the eigenvalues ξn1,2 of Bn =
are both
n
−2ηn
s2
negative.
Proof :
2
n
λ1,2
n are the two roots of the polynomial X − ρX − 2ηn(ρ + 2ηn) + s2 .
The associated ∆ is given by
2
∆ = ρ2 + 8ηn ρ − 2 + 2ηn
σ
∆ = ρ2 + 16η 2 n(n − 1) + 8nδ
√ 1,2
Hence, the eigenvalues λn
= 12 ρ ± ∆ are of opposite signs for
n ≥ 1 since ∆ > ρ2 .
Now, ξn1,2 are the two roots of the polynomial X 2 + (ρ + 4ηn)X +
2ηn(ρ + 2ηn) − sn2 . The associated ∆ is given by
n
∆ = ρ2 + 4 2
s
q
h
i
Hence, ξn1,2 = 12 −ρ − 4ηn ± ρ2 + 4n
. These two eigenvalues are
2
s
negative if and only if:
r
4n
ρ + 4ηn > ρ2 + 2
s
⇐⇒ 8ρηn + 16η 2 n2 >
⇐⇒ 2ηn >
16ηn
σ2
2
−ρ
σ2
and this is true for n ≥ 1.
This lemma can be used to prove general stability results when δ > 0.
It is indeed straightforward that all our stability results can be rewritten
exactly the same if one replaces the conditions
R
xψ(x)m∗ (x)dx = 0
R
by δ > 0 (physical stability)
xφ(x)m∗ (x)dx = 0
or
R
∗
R xψ(0, x)m∗ (x)dx = 0
xφ(0, x)m (x)dx = 0
by
δ>0
(eductive stability)
These ideas will be used extensively to study stability in multi-population
frameworks.
4.4 Concluding remarks on the two stability notions
Even though the two kinds of stability look like each other, the two notions
of stability we used are completely orthogonal. The physical stability is
indeed linked to a perturbation of the system. The system is physically
stable because, after an initial perturbation of m∗ and a final perturbation of u∗ , under some conditions, the solution of the game is stable in the
sense that agents go back to the equilibrium. Hence, the physical stability
involves forward/backward reasoning. This is not the case of the eductive
stability because the mental process is purely forward (in virtual time).
We start from a guess not too far from an equilibrium (the equilibrium
being a priori unknown) and the mental process converges toward this
equilibrium.
The fact that our solutions are stable for both the physical stability and
the eductive stability backs up the mean field game approach to find relevant solutions.
5
Applications with many populations
We will present several cases involving many populations. Our goal is
to show how the gaussian solutions found earlier can be used into more
complex cases. One of the thing we need to set quickly is the structure of
the utility functions since we have seen that the gaussian solutions were a
consequence of the logarithmic setting. Another thing we will focus on, is
the importance of some symmetries in the structure of the problem we are
going to deal with. It’s indeed predictable that problems without enough
symmetries cannot lead to gaussian distributions that exhibit some symmetries.
5.1 The basic framework and the Log-Cobb-Douglas
utility setting
In what follows we are going to be very general in the sense that we will
design a framework to be used in all the cases that will follow in the paper.
First, let’s note I the set of populations. The only constraint we put on
this set is that it must be countable.
Each i in I represents a continuum of agents like those described earlier:
all agents in a given population have the same utility function, the same
discount rate and are subject to the same type of brownian motion (to be
clear, the brownian motion is not common to all agents but the characteristics of individual brownian motions in a given population are the same:
same variance to say it simply). However, people from two different populations can have completely different utility functions, different discount
rates and even “suffer” from a totally different brownian motion.
We need to specify now the structure of the preferences. We will
continue to assume that the utility function is of a logarithmic form but
we need to take into account not only the will to be with people from the
same population but also the influence of the position of other individuals
from other populations. To do that, we introduce a set of coefficients (or
a matrix)
Θ = (θij )(i,j)∈I×I
and replace for each agent in population i the preceding term ln(m(t, x))
by
X
θij ln(mj (t, x))
j∈I
Obviously, if I is not a finite set, there is a convergence issue but we will
focus on that later since this framework must remain formal for now. To
generalize the preceding cases we also assume that ∀i, θii > 0.
Now we can write for a given population i the stochastic control problem:
"Z
∞
ui (t, x) = M ax(αs ),Xt =x E
t
"
#
|α(s, Xs )|2
θij ln(mj (s, Xs )) −
2
j∈I
X
!
#
e
with
dXt = α(t, Xt )dt + σi dWt
From this control problem we can deduce the Hamilton-Jacobi and
Kolmogorov equations but since we will use the same hint as before
(Proposition 3), we can directly focus on the following stationary equations for all populations:
∀i ∈ I, mi (x) = Ki exp(
2ui (x)
)
σi2
where Ki is a constant, and:
∀i ∈ I,
X
X
|∇ui |2
2uj (x)
σ2
− ρi ui + i ∆ui = −
θij ln(Kj ) −
θij
2
2
σj2
j∈I
j∈I
These are the basic equations we need to solve.
5.2
Conditions to have gaussian solutions
As we said before, some symmetries are needed to have gaussian distributions for solutions of that kind of problems. To find out the exact
condition, let’s assume that, to generalize our calculations for one population, we have:
∀i, ui (x) = −
−ρi (s−t)
σi2
σ2
|x − µi |2 + i ξi
2
4si
2
Proposition 24. For the above functions to be solutions of the problem,
the parameters need to solve the three following equations:
ds
1. ∀i ∈ I,
2. ∀i ∈ I,
σi4
8s4
i
P
3. ∀i ∈ I, −
σ2
+ ρi 4si2 =
i
θij
j∈I s2
j
σi2
2
θij
j∈I 2s2
j
P
(µi − µj ) = 0
σ4
ρi ξi − 4si2 n =
i
P
j∈I
θij
n
2
P
ln(2π) + n ln(sj ) + j∈I θij 2s12 |µi −
j
µj |2
Proof:
Let’s introduce the expressions for the ui ’s and the mi ’s in the HamiltonJacobi equation:
∀i ∈ I,
=
X
σi2
σi4
σi2
σi4
2
2
|x
−
µ
|
+
ρ
|x
−
µ
|
−
ρ
ξ
+
n
i
i
i
i
i
8s4i
4s2i
2
4s2i
θij
j∈I
X
1
ln(2π) + n ln(sj ) +
θij 2 |x − µj |2
2
2s
j
j∈I
n
σ2
σ2
σ4
σi4
|x − µi |2 + ρi i2 |x − µi |2 − i ρi ξi + i2 n
8s4i
4si
2
4si
n
X θ
X
ij
θij
=
ln(2π) + n ln(sj ) +
|x − µi |2 + |µi − µj |2 + 2(µi − µj ) · (x − µi )
2
2
2sj
j∈I
j∈I
⇐⇒ ∀i ∈ I,
If we identify terms of degree 0, 1 and 2 we obtain the three conditions
above.
It’s worth noticing that the third condition is not really complicated to
satisfy since it is sufficient to set ξi to the appropriate value. However, we
have admitted so far that the sum were well defined (i.e. convergent) but
this is not guaranteed per se. Consequently, the necessary and sufficient
conditions for a solution to exist actually consist in conditions 1 and 2
and in a third one ensuring the convergence of the sums involved.
5.3
Stability for multi-population frameworks
Now, we are going to study the stability of some equilibria with n populations. Because of the huge variety of equilibria, we are going to focus
on equilibria centered around 0 in dimension 1. Typically, we consider N
populations on the real line and a preference matrix Θ where the sum of
the coefficients on each line is equal to 1. To ensure equilibria around 0
we introduce a localization function −δx2 and we assume that the parameters δ, ρ and σ 2 are uniform across populations.
We know from the above calculations (the parameter δ has the same influence as before) that an equilibrium for the populations is given by the
usual one-population equilibrium, that is m1 = . . . = mN gaussian dis2
tribution functions with mean 0 and variance s2 = σ4η where η satisfies
2η 2 − η σ22 − ρ = δ.
The stability of this equilibrium will obviously depend on the preferences.
If populations are attracted by one another, it seems natural to have a
stability result. This is arguably not the case if, for instance, one population really hates the (N − 1) others and is rejected by them, in which case
the particular population can want to break the equilibrium situation.
Surprisingly perhaps, this stability question can be answered quite generally. To do that, let’s consider eductive stability and let’s write the
linearized system of PDEs.
Proposition 25 (Linearized PDEs). The linearized PDEs around the
∗
∗
∗
∗
equilibrium (u∗ = (u(1) , . . . , u(N ) )T , m∗ = (m(1) , . . . , m(N ) )T ) are
given by:
∂θ φ = −Lφ − ρφ + Θψ
2
∂θ ψ = −Lψ + 2 Lφ
σ
where φ = (φ(1) , . . . , φ(N ) )T and ψ = (ψ (1) , . . . , ψ (N ) )T are defined as
small deviations (absolute deviation for φ and relative deviation for ψ)
from the equilibrium (as before) and where φ(0, ·) and ψ(0, ·) are given.
Proof:
The proof is the same as in Proposition 19 and 20.
To solve these PDEs, we need to introduce the 2N × 2N matrices
(Cn )n defined as:
−(ρ + 2ηn)IN
Θ
Cn =
n
I
−2ηnIN
s2 N
We can now find the eigenvalues of Cn .
Proposition 26 (Eigenvalues of Cn ). The set of Cn ’s eigenvalues is:

 

s


1
4
θ̃n
−ρ − 4ηn ± ρ2 + 2  , θ̃ eigenvalue of Θ 10

2
s
Proof:
By definition, the eigenvalues are the roots of the polynomial defined
by:
det(Cn − XI2N ) = det
−(ρ + 2ηn + X)IN
n
I
s2 N
Θ
−(2ηn + X)IN
=0
This equation can be simplified11 to:
n det (ρ + 2ηn + X)(2ηn + X)IN − 2 Θ = 0
s
s2
⇐⇒ det Θ − (ρ + 2ηn + X)(2ηn + X)IN = 0
n
10 The
11 This
square root must
be understood
asa square root in C in general.
aIN
0
aIN
Θ
IN
hint is due to
=
c
cIN − a Θ + bIN
cIN bIN
0
1
Θ
a
IN
2
Therefore, γ is an eigenvalue of the matrix Cn if and only if sn (ρ +
2ηn + γ)(2ηn + γ) = θ̃ where θ̃ is any eigenvalue of Θ.
If we solve this equation in γ, we see that the eigenvalues of Cn are given
by:


s
1
4
θ̃n
γ=
−ρ − 4ηn ± ρ2 + 2 
2
s
Proposition 27 (Local eductive stability). Suppose that:
• The initial guesses φ(0, ·) and ψ(0, ·) are in the Hilbert space H =
L2 (m∗ (x)dx).
R
• ψ(0, x)m∗ (x)dx = 0 (this is guaranteed if the initial guess for m is
a probability distribution function)
• Sp(Θ) ⊂] − ∞, 1 + 2δs2 [ (we suppose that the eigenvalues are real12 )
• δ > 013
Then the solution (φ, ψ) of the PDEs converges toward 0 in the sense
that:
lim ||φ(t, ·)||L2 (m∗ (x)dx) = 0
t→∞
lim ||ψ(t, ·)||L2 (m∗ (x)dx) = 0
t→∞
Proof:
We know from previous calculations that a sufficient condition for the
convergence to be true is that all the eigenvalues of Cn for all n ≥ 1 are
negative
real parts (remember that for n = 0 we have
or have negative
−ρIN Θ
(i)
C0 =
and ψ(0, ·)0 = 0, ∀1 ≤ i ≤ N ).
0
0
Consequently, we want that for any eigenvalue θ̃ of the matrix Θ,
2
ρ2 + 4sθ̃n
2 < (ρ + 4ηn) (n 6= 0). This condition is equivalent to:
4θ̃n
< 16η 2 n2 + 8ρηn
s2
2θ̃
i.e. ∀n ≥ 1, ∀θ̃ ∈ Sp(Θ), 2 < 2ηn + ρ
σ
2θ̃
i.e. ∀θ̃ ∈ Sp(Θ), 2 < 2η + ρ
σ
∀n ≥ 1, ∀θ̃ ∈ Sp(Θ),
12 The problem can be completely solved in the general case where some eigenvalues are
complex and in that case Sp(Θ) must be a subset of the following set:




s
2 2


1 2
4nx
4nx
4ny
2
 < (ρ + 4ηn)
z = x + iy ∈ C|∀n ≥ 1,
ρ2 + 2
+
ρ + 2
+
2


2
s
s
s
13 Since
1 is always in Sp(Θ), the proposition would be irrelevant otherwise.
i.e.
∀θ̃ ∈ Sp(Θ),
i.e.
2η(θ̃ − 1)
2
2
<
2η
+
η
ρ
−
=δ
σ2
σ2
∀θ̃ ∈ Sp(Θ), θ̃ < 1 + 2s2 δ
and this is guaranteed by the hypothesis on Sp(Θ).
It’s interesting to notice that the upper bound on Sp(Θ) is the best
one in the sense that the solutions of the PDEs may not converge to 0 if
Sp(Θ) ∩ [1 + 2s2 δ, +∞[6= ∅.
As an application,
one canconsider 2 populations and the preference
1−θ
θ
matrix Θ =
. This matrix means that the preferences
θ
1−θ
are the same for the two populations. Also, if θ > 0 they like one another
and if θ < 0, they hate one another.
From the preceding proposition, we can imagine that there exists a
(negative) threshold θ̂ such that the equilibrium is stable for θ > θ̂ and
unstable otherwise.
This can be verified easily since Sp(Θ) = {1, 1 − 2θ}. The threshold is in
fact given by 1 − 2θ̂ = 1 + 2s2 δ, i.e. θ̂ = −s2 δ.
5.4 Some other examples with several populations
In this part we are going to develop special concrete cases of the preceding
theory where explicit solutions are going to be gaussian.
5.4.1
A case with three populations
We consider a set of three populations whose preferences are represented
by the following 3 × 3 matrix.


+ + +
Θ= + + − 
+ − +
This means that population 1 is attractive to populations 2 and 3 and
is not reluctant to live with them. However, populations 2 and 3 don’t
want to live together.
Also, we suppose for the sake of simplicity that structural parameters do
not depend upon i, that is ρ and σ are the same for everybody. P
Also, we
suppose that preferences are normalized in the sense that θ := 3j=1 θij
doesn’t depend upon i.
In that case, it’s easy to see that the variances will not depend on i
and that they are all equal to
s2 =
σ4
4θ − 2ρσ 2
where we implicitly assumed that ρσ 2 < 2θ.
Then to find the means, we may assume that µ1 = 0 and (µ2 , µ3 ) has to
solve14 :
θ12 µ2 + θ13 µ3 = 0
θ21 µ2 + θ23 (µ2 − µ3 ) = 0
θ31 µ3 + θ32 (µ3 − µ2 ) = 0
Here, we can see that we need what I called “symmetries”. We actually
need the preferences to be such that:
−
θ12
θ21
θ31
=1+
=1+
<0
θ13
θ23
θ32
Along with the conditions on the sum, these leave 5 degrees of (partial15 )
freedom in the matrix Θ. An example of such a matrix is:


1
1
1
2
−1 
Θ= 2
2 −1
2
5.4.2
The lattice example
The last example with three populations can be generalized to more complex cases but those would have limited interest once we have seen the
3-population case.
A good example, that is slightly different from these possible cases, is the
case where I is infinite and where the populations’s centers are located
on a lattice.
First of all, let’s say that we keep the uniformity hypothesis upon ρ
and σ and the same assumption for the sum of the θ’s. In that case we
already know that variances are the same and that:
s2 =
σ4
4θ − 2ρσ 2
where θ is defined as above (and must also be finite).
Secondly, let’s consider n linearly independent vectors in the n-dimensional
space: (u1 , . . . , un ). I is simply Zn and we will try to find a solution with
µi := µ(i1 ,...,in ) = i1 u1 + . . . + in un
Now, we need to verify that condition 2 is verified that is:
X
θij (µi − µj ) = 0
j∈I
⇐⇒
X
j∈I
14 We
θij
n
X
(ik − jk )uk = 0
k=1
focus on solutions different from (0, 0) for (µ2 , µ3 ).
some inequalities have to be satisfied.
15 because
⇐⇒
X
l∈I
θi,i+l
n
X
lk u k = 0
k=1
Therefore, if θi,j is an even function of x = j − i, let’s say θ(x) = γ |x| ,
then the condition is satisfied because of the symmetry in the lattice.
A particular and perhaps easier case is when n = 1 since in that case we
can for instance set the populations’ centers on Z and write the “matrix”
Θ as:
Θ = (θij )ij = (γ |j−i| )ij
1+γ
The additional condition in that case is 2 1−γ
> ρσ 2 .
Appendix: Numerical recipes
This appendix is dedicated to numerical methods to solve the mean field
games presented in the above text in more general cases where explicit solutions cannot be exhibited. The approach we present is inspired from the
eductive stability notion because eductive stability is based on a purely
forward reasoning. The forward/backward structure of mean field games
is indeed quite problematic when it comes to find numerical solutions.
The introduction of a virtual time in the eductive stability helps a lot to
circumvent this issue.
First, we are going to present methods to find stationary solutions and
then, we will use the same ideas to find dynamical solutions, an issue that
is not really dealt with in the above text, except locally, around a stationary equilibrium.
We must keep in mind that the functions we are looking for are, in practice, approximated by finite Fourier series.
Stationary equilibrium
First, let’s recall the two equations that characterize a stationary equilibrium:
(Hamilton − Jacobi)
1
σ2
∆u + |∇u|2 − ρu = −g(x, m)
2
2
σ2
∆m
2
The Hamilton-Jacobi
equation can be simplified using the change of
variable β = exp σu2 and we obtain:
(Kolmogorov)
(Hamilton − Jacobi)0
(Kolmogorov)0
∇ · (m∇u) =
σ2
1
∆β = β ρ ln(β) − 2 g(x, m)
2
σ
∇β
σ2
∇ · σ 2 (m
) =
∆m
β
2
The two equations (Hamilton − Jacobi)0 and (Kolmogorov)0 can be
written in a more practical way for numerical resolutions by “inverting”
the ∆ operators. This can be done in the Kolmogorov equation by restricting the Laplace operator to probability distribution functions (since
in practice we restrict ourselves to Fourier series with only a finite number
of harmonics) and we obtain:
(Kolmogorov)0
−m+
σ2
∆
2
−1 ∇β
σ 2 ∇ · (m
) =0
β
This cannot be done in the case of the Hamilton Jacobi equation but
2
we can invert an operator like σ2 ∆ − Id for any > 0. This gives:
(Hamilton−Jacobi)
0
2
−1 σ
1
−β+
∆ − Id
β ρ ln(β) − 2 g(x, m) − =0
2
σ
Using these equations we can consider the ideas of eductive stability
and try to obtain solutions by solving the following equations where we
introduce the virtual time θ:

h
i−1  ∂θ m = −m + σ2 ∆
σ 2 ∇ · (m ∇β
)
2
β
h
i
 ∂ β = −β + σ2 ∆ − Id −1 β ρ ln(β) −
θ
2
1
g(x, m)
σ2
−
Numerically these equations are quite easy to solve using Fourier meth√
ods. An example is shown below where g(x, m) = m−δx2 with σ 2 = 0.4,
ρ = 0.4, δ = 0.5 on the domain [−1, 1] (we took = ρ3 ).
Figure 1: Initial guess: N (0, 0.3). Solution after 8000 iterations with dθ ' 0.01
(an iteration is drawn every 40 iterations). Only 15 harmonics are used.
Dynamical problems
Similar ideas can be used to solve dynamical problems on a time interval [0, T ]. If we write the two equations that characterize the dynamics
of a mean field game we get, using the same change of variable as above:
0
(Hamilton − Jacobi)
(Kolmogorov)0
σ2
1
∂t β +
∆β = β ρ ln(β) − 2 g(x, m)
2
σ
σ2
∇β
) =
∆m
∂t m + ∇ · σ 2 (m
β
2
Two constraints must be added that are m(0, ·) = m and β(T, ·) = β.
These two equations can be written using a forward and a backward
heat operator:
σ2
1
F− β = ∂t β +
∆β = β ρ ln(β) − 2 g(x, m)
2
σ
∇β
σ2
∆m = −∇ · σ 2 (m
)
F+ m = ∂t m −
2
β
The F− operator is invertible if we restrict the operator to functions16
β’s where β(T, ·) = β. Similarly, the F+ operator is invertible if we
restrict it to functions m’s where m(0, ·) = m. Therefore, we can write
the eductive equations as follows:
(
−1
∂θ m = −m + F+
−σ 2 ∇ · (m ∇β
)
β
−1
∂θ β = −β + F−
β ρ ln(β) − σ12 g(x, m)
Solutions to these equations can be found using Fourier methods.
Hereafter, we take the case of a population initially normally distributed
around −0.3 (with a standard deviation of 0.2) and agents have a utility
√
function of the form g(x, m) = m − δx2 . Parameters value are σ 2 = 0.4,
ρ = 0.4, δ = 0.5 and the domain is [−1, 1].
16 Functions
are Fourier series with a finite number of harmonics.
Figure 2: Initial condition: N (−0.3, 0.2). t goes from 0 to T = 40 with dt ' 0.1.
u(T ) = 0. 800 iterations are considered before drawing (with dθ ' 0.01). Only
15 harmonics are used.
References
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M. Abramowitz and I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover,
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[Aum64] R. Aumann. Markets with a continuum of traders. Econometrica, 32(1/2), 1964.
[Eva68]
L. Evans. Partial Differential Equations. American Mathematical Society, Jun. 1968.
[Gue92] Roger Guesnerie. An exploration of the eductive justifications of
the rational-expectations hypothesis. The American Economic
Review, 82(5), Dec. 1992.
[LL06a] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen i. le cas
stationnaire. C. R. Acad. Sci. Paris, 343(9), 2006.
[LL06b] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen ii. horizon
fini et contrôle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
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of Mathematics, 2(1), Mar. 2007.
[LL07b] J.-M. Lasry and P.-L. Lions. Mean field games. Cahiers de la
Chaire Finance et Développement Durable, (2), 2007.
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D. Scharfstein and J. Stein. Herd behavior and investment. The
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N. Touzi. Stochastic control and application to finance. Scuola
Normale Superiore, Pisa. Special Research Semester on Financial Mathematics, 2002.
Part I - Chapter 4
Application of Mean Field Games to Growth Theory
Application of Mean Field Games to Growth
Theory
Abstract
This chapter discusses the interaction between economic growth in
the sense of human capital accumulation and the dynamics of inequalities. We use a mean-field game framework in which individuals improve
their human capital both to improve their wages and to avoid potential
competition with less skilled individuals.
Our contribution is twofold. First, we exhibit a mechanism in which
competition between a continuum of people regarding human capital accumulation lead to growth. Second, our model highlights the importance
of Pareto distributions to describe inequalities since power laws appear
naturally as explicit solutions of our problem.
Résumé
Ce chapitre traite du lien entre croissance économique et inégalités
dans un modèle de capital humain. Nous utilisons un jeu à champ moyen
au sein duquel les agents accroissent leur capital humain pour améliorer
leur salaire mais aussi pour lutter contre la concurrence potentielle d’autres
individus. Nous mettons en place un mécanisme dans lequel la concurrence entre agents concernant l’accumulation du capital humain aboutit
à un processus de croissance. Aussi, nous insitons sur le rôle particulier
joué par les distributions de Pareto dans la théorie des inégalités dans
la mesure où des solutions explicites apparaissent naturellement pour ce
type de distribution.
Introduction
Recent theories of economic growth, following the Schumpeterian model
developed essentially by P. Aghion and P. Howitt ([AH92]), mainly focus
on research and industrial innovation as the only way to generate a non
decreasing growth process.
Here, we are back to former ideas to explain growth with human capital
accumulation only. However, our framework is quite new since we use
the recent theory designed by J.-M. Lasry and P.-L. Lions on mean-field
games ([LL06a, LL06b, LL07a]). This new framework allows us to model
in a simple way the interaction between people and growth will be a byproduct of the interaction and competition between people to improve
their welfare.
We basically model a continuum of individuals whose wages depend not
only on their own human capital but also on the whole distribution of human capital. This distribution dependency is important in two different
ways. First, we take into account a competition effect in the labor market. Second we model the easiness or difficulty to improve human capital
depending on the proximity to the technological frontier.
Like in the recent paper by Aghion et al (2001) ([AHHV01]) or as in
Aghion and Howitt ([AH]), growth is fostered by an escape competition
effect. However, in our setting, it is the threat of competition that forces
people to improve their human capital and not competition by itself: because individuals less skilled than a given person represent a threat for
this person, she is forced to accumulate human capital. That leads eventually to economic growth.
In addition to the growth process, the mean-field game framework allows
us to deal with the distribution of wages across people. The distribution
of wages is indeed known to be quite well described by a Pareto distribution (at least for the tail) and we show that Pareto distributions for
human capital and wages are indeed stable in our setting.
Our contribution is therefore to shed light on a growth process which is a
consequence of a competition effect and which is compatible with power
law distributions for inequalities.
In the first section we will present the setup of our model and derive
a solution using classical methods (Euler Lagrange). In a second part,
we exhibit comparative statics and then, in a third part, we go deeply
into an analysis of the mechanisms that are involved in our model. In the
fourth section, we use the mean field game partial differential equations
to solve the problem in a different way and we generalize the solution to
a stochastic framework. The first three parts are aimed at readers with a
purely economic background whereas the fourth part (and the appendix
that contains proofs omitted in the first parts) are targeted at readers
with a slightly higher technical and mathematical background.
1
1.1
The model
Introduction
We assume that there is initially a working population of size 1 with a
given distribution of human capital. Human capital will be denoted q and
the distribution, at time t, will be referred to as m(t, q).
For a given worker, wage per hour will be a function not only depending
on the individual human capital but also on the scarcity of her specific
human capital1 . In other words, the salary of a worker with human capital
q is, at time t, given by:
1 One can think this is a very restrictive way to model competition since it imply a very low
substitution between people. Another setting that leads to similar results could be to replace
the density function by the tail function.
w(t, q) = G(q, m(t, q)),
G(|{z}
· , |{z}
· )
−
+
The main interest of this equation is to model a competition effect in
the labor market: unskilled people have a small salary because they are
unskilled and also because most of the time they are so numerous that
they can be replaced by other similar unskilled people2 .
Individuals - who live forever - can improve their human capital with
a cost depending on two factors. First, the cost, in monetary terms, is a
function of human capital change and, second, it is also a function of the
position of initial human capital in the distribution. More precisely, we
will assume that the cost (in monetary terms) at time t is given by:
C(t, q,
dq
dq
) = H( , F (t, q)),
dt
dt
H(|{z}
· , |{z}
· )
+
−
R∞
where F (t, q) = q m(t, u)du is the number of people in the population
with a human capital greater than q. That is to say it is more costly for
skilled workers to improve their human capital than for unskilled workers. This hypothesis is relevant since it is often more difficult to improve
human capital for an individual in the right tail of the distribution since
she is near the technological frontier.
1.2
The optimization problem
As in the classical Mincerian approch to human capital accumulation,
we are going to suppose that people improve their human capital all life
long. However, we do not focus on schooling choices in the sense that we
consider a given working population. Human capital accumulation must
therefore be seen as the consequence of on-the-job training.
Each individual chooses her effort continuously to maximize her utility.
Her intertemporal utility is classically given by an expression of the form:
Z ∞
u(ct )e−ρt dt
0
) − ct . This gives
with a wealth constraint ṡt ≤ rst + w(t, q) − C(t, q, dq
dt
a unique intertemporal constraint that is:
Z ∞
Z ∞
dq
−rt
ct e dt ≤ s0 +
w(t, q) − C(t, q, ) e−rt dt
dt
0
0
Therefore, if we assume that r is exogenous the only thing the agent is
going to maximize is the right hand side of the constraint which is her
intertemporal wealth.
Basically, an individual with human capital q has the following program:
2 We
assume here, as it will be the case in what follows, that m(t, ·) is a decreasing function.
∞
Z
M ax(qs ),q0 =q
G(qs , m(s, qs )) − H(a(s, qs ), F (s, qs )) e−rs ds
0
where a(·, ·) is defined by dqs = a(s, qs )ds.
1.3
1.3.1
Resolution
A specific setup
To solve the problem we need to specify the two functions G and H. Our
specification is the following:
To derive the wage function we are going to start with a discrete
setting. Imagine that there are n types of workers with human capital
q1 , . . . , q n .
A standard production function for a representative firm would be3
Y =A
n
X
qiα L1−β
i
α > 0, β ∈ (0; 1]
i=1
The wage associated to a worker of type i, wi , is then proportional to
qiα
β
Li
.
Therefore, if we go from this discrete setting to a continuous one, we
can assume that:
G(q, m(t, q)) =
(
qα
C m(t,q)
β,
0
if q is in the support of m(t, ·)
otherwise
For the cost, we use the simple specification that follows:
H(
dq ϕ
E dt
dq
, F (t, q)) =
,
dt
ϕ F (t, q)δ
∀q in the support of m(t, ·)
where C and E are two constants and where α, β, δ and ϕ are four positive parameters subject to technical constraints that are: α + β = ϕ ,
β = δ and we want typically ϕ to be strictly greater than 1 .
This specification can be considered ad hoc but in fact it must be regarded
as quite general since we have two degrees of freedom to choose the parameters. An easier specification without any degree of freedom would have
been to set α = β = δ = 1 and ϕ = 2 but we want to remain quite general.
Now, we assume that the initial distribution of human capital is a
Pareto distribution4 . We can use a normalization and assume that the
3 The
limit case where β = 1 is a logarithmic case.
distribution is usually used in the literature on economic inequalities (see Piketty’s
papers for example, or Atkinson ([Atk05]))
4 This
minimal point of the initial distribution is 1. The Pareto coefficient5 of
the initial distribution is denoted k, so that:
1
1q≥1
q k+1
This Pareto distribution is central in the study of economic inequalities and will be stable in our model in the sense that our solution involves
Pareto distributions of human capital at all time.
m(0, q) = k
1.3.2
Explicit resolution
To start the explicit resolution of our problem, let’s begin with the Euler
Lagrange equation associated to it.
Proposition 1 (Euler-Lagrange’s equation). Let’s note G̃(t, q) = G(q, m(t, q))
and H̃(t, q, q̇) = H(q̇, F (t, q)).
The optimal path has to satisfy the following Euler-Lagrange equation:
i
d h
∂q G̃(t, q) − ∂q H̃(t, q, q̇) = −
∂q̇ H̃(t, q, q̇) + r∂q̇ H̃(t, q, q̇)
dt
Proof:
This is a pure application of the Euler-Lagrange’s principle with a discount rate r.
This equation can be solved easily and this is the result of the following
proposition:
Proposition 2 (Growth rate). If ϕ(ϕ − 1) < βk then, there is a unique γ
so that the solution of the preceding equation is characterized by a constant
growth γ:
• qt = q0 exp(γt)
1q≥exp(γt)
• m(t, q) = k exp(γkt)
q k+1
Moreover, γ is implicitly given by:
ϕ(ϕ − 1) − βk ϕ
C(ϕ + βk)
γ = rγ ϕ−1 −
ϕ
Ekβ
Proof: See appendix.
(∗)
Before going into the comparative statics and the analysis of the model,
we need to verify that the above solution satisfy the transversality condition. In other words we need to verify that the integral in the criterion
remains finite.
5 It is usually a measure of inequality. For example, it is related to the Gini coefficient by
1
the formula: G = 2k−1
.
Proposition 3 (Transversality condition). For the solution exhibited in
Proposition 2 to be an actual solution of the optimization problem, we need
to have γ < ϕr
Proof: See appendix.
These propositions are central in our discussion. We have indeed
proved that a constant growth rate was a possible outcome of our model
if the parameters satisfied some constraints. Moreover, the distribution of
human capital and hence the distribution of wealth is always of the Pareto
type and this is interesting in the light of the usual theories of inequalities.
2
Comparative statics
In that part, we analyze the growth rate formula derived previously (equation (∗)). Basically, γ can be seen as a function of three meaningful parameters: r, E and k.
• r has to be seen as a parameter linked to impatience. We expect the
growth rate to be a decreasing function of r
• E is a measure of the cost to improve human capital. A small E
indicates efficient on-the-job training in our model and we expect
growth to decrease with E.
• k is a measure of the initial homogeneity in the distribution of human
capital. If we consider that the initial human capital distribution is
a result of the basic educational system then a high k means a very
equalitarian educational system whereas a smaller k represents a
more free educational system that leads to more heterogeneity. The
sign of dγ
will be interesting to evaluate the link between growth
dk
and social homogeneity.
Growth as a decreasing function of r
As expected, γ is a decreasing function of r. This is straightforward if
we consider equation (∗) or the following graph on which we plotted the
left hand side and the right side of (∗).
Figure 1: The impact of an increase in r
Growth as a decreasing function of E
As expected, γ is a also a decreasing function of E. The same argument as before applies.
Figure 2: The impact of an increase in E
Growth can be fostered by heterogeneity
The last dependency we analyze is on k. The result is in general
ambiguous but we can say the following:
Proposition 4 (Dependence on k6 ). Suppose as before that ϕ(ϕ−1) < βk
• For β = 1, the function k 7→ γ(k) is decreasing.
• For β < 1 and as k goes to infinity, γ(k) tends to zero as k
β
−ϕ
.
Proof: see appendix.
3
Analysis of our model
Our model generates a constant growth rate for human capital, both for
the entire society and for each single individual. In what follows we discuss
the underlying source of growth and relate our finding to recent papers in
the economic literature.
To begin with, the basic reason why people change their human capital is
due to two effects. First, there is a pure wage effect since, ceteris paribus,
wage increases with human capital. However, this effect cannot explain by
itself the continuous improvement of human capital at a constant growth
rate. The effect needed to ensure a convincing explanation is a competition effect, or to say it as in Aghion and Howitt ([AH]), even though
the comparison is not entirely relevant, an escape competition effect. A
given individual taken at random in the population is threaten by people
who have less human capital than he has (say q̃). Indeed, if part of those
people were to improve there human capital so that they end up with
a human capital q̃ they will compete with our individual on the labor
market, reducing her wage. This effect is the origin of continuous growth
in our model. Contrary to Aghion et al., we have here a continuum of
agents and therefore, for any given individual, there is always a threat.
We think therefore that the Schumpeterian effect which basically assumes
that people won’t improve their human capital if the gains are two small
is reduced to nothing because there is always a potential competitor and
that’s why a Darwinian effect (competition effect) dominates. Let’s indeed highlight how tough is the threat effect. Each agent knows that every
one is threaten by every one, and that fear will induce behaviors that will
make the frightening event happen and be more important. This model
shows that the growth process is not only due to those who innovate, that
is to say “researchers” near the technological frontier, but is in fact a process that involves the whole population and is fostered by those who are
far from the technological frontier and threaten the leaders by improving their human capital. Also, our model gives a striking example of the
fact that the Darwinian competitive pressure can be much more intense
between agents with rational expectations than between myopic agents.
Myopic agents would fear other agents moves, while agents with rational
expectations fear also the competitive moves of other agents induced by
their own competitive behavior. In other words, Darwinian competition,
6 To relate this result to better known measures of heterogeneity, just notice for instance
1
that the Gini coefficient in the case of a Pareto distribution is simply given by G = 2k−1
.
Therefore, we basically show that γ is increasing as a function of the Gini coefficient for the
human capital distribution at least if this Gini coefficient is small enough.
as a general concept, when extended to competition between agents with
rational expectations, leads to an extremely tough competitive scheme.
One of the characteristics of our model is also related to the structure
of economic inequalities. Starting with a given Pareto distribution with
parameter k, the solution exhibited above, is always a Pareto distribution
of order k (with a support that depends on time obviously). Recalling
that the Gini coefficient is only determined by k (it is straightforward to
1
get the formula G = 2k−1
), we have also the interesting property that the
Gini coefficient is constant and cannot therefore be modified by the kind
of human capital accumulation process we model.
Is this consistent with reality? The answer depends on the country.
To deal with this issue we
have to consider wages instead of human capital.
qα
Because w(t, q) = C m(t,q)
β , the wages on the optimal path are given by:
w(t, q) =
C
exp(−βkγt)q βk+ϕ
kβ
k
Distribution of wages is therefore a Pareto distribution of order p = βk+ϕ
for all t. It means that wage inequalities are very stable over time. This
is difficult to test but great work has been done in Piketty ([Pik03]) and
Piketty and Saez ([PS03]) for wages, respectively in France and in the
US7 . Even if there are fluctuations in wage inequalities, these inequalities seem (perhaps surprisingly) to have been stable over the twentieth
century in France. However, this is not the case for the US where wages
are less and less equally distributed (Today, the ”working rich” celebrated
by Forbes magazine seem to have overtaken the ”coupon-clippers” - see
[PS03]).
Also, in a book printed recently ([AP07]), Atkinson and Piketty showed
that this stability is true for most of non english-speaking developed countries whereas inequalities are less stable in the US, the UK, Ireland, Australia, New Zealand ...
The conclusion is that our model is quite consistent with the continental
Europe experience in the long run as far as wages inequalities are concerned.
We discussed earlier the impact of initial inequalities represented ei1
on the growth rate (notice here that the growth
ther by k or by G = 2k−1
rate of wages is simply given by ϕγ: that can be derived easily from
the above equation for w(t, q)). Our finding is that an education system
which leads to a very homogeneous population could be responsible for
a small growth rate. This has natural policy implications and supports
liberalization of the schooling system and therefore less uniform schools.
However, one must not forget that in our model, on-the-job training is
always feasible and that’s why the main implication of our model is certainly better expressed by: economic inequalities can be good for growth
as soon as there is no segregation i.e. large access for everybody to the
human capital accumulation process.
7 Another
article by Atkinson deals with the UK but concerns revenues and not wages
4 Mathematical complements and generalization to a stochastic framework
4.1 The mean field game partial differential equations
In the first part, we found a solution to our problem using a EulerLagrange methodology. This is not in fact the most relevant mathematical
method to deal with the problem. The problem involves indeed the probability distribution function and the tail function of the human capital
across the population and the mean field games partial differential equations are in a way far more relevant to solve the problem.
Let’s first introduce the Bellman function of the problem:
Z ∞
J(t, q) = M ax(qs ),qt =q
G(qs , m(s, qs )) − H(a(s, qs ), F̄ (s, qs )) e−r(s−t) ds
t
We can translate this optimization problem into the two mean field
games partial differential equations:
Proposition 5 (The mean field games partial differential equations). Our
optimization problem can be represented by the two following PDEs:
(HJB)
G(q, m(t, q)) + ∂t J + M axa a∂q J − H(a, F̄ (t, q)) − rJ = 0
(Kolmogorov)
∂t m(t, q) + ∂q (a(t, q)m(t, q)) = 0
a∂q J − H(a, F̄ (t, q)) is the optimal control
where a(t, q) = ArgM axa
function.
In the special case we solved, the two equations can be written as:
C
β
ϕ
qα
ϕ−1 1
+
F (t, q) ϕ−1 (∂q J) ϕ−1 + ∂t J − rJ = 0
1
m(t, q)β
ϕ E ϕ−1
!
1
ϕ−1
F (t, q)β
m(t, q) = 0
∂t m(t, q) + ∂q
∂q J(t, q)
E
and the optimal control is given by:
a(t, q) =
1
ϕ−1
F (t, q)β
∂q J(t, q)
E
Proof:
The optimal control function is given by a(t, q) = Argmaxa a∂q J −
.
1
β
ϕ−1
aϕ
Hence, a(t, q) = F (t,q)
∂
J(t,
q)
and M axa a∂q J − E
can be
q
E
ϕ F (t,q)β
aϕ
E
ϕ F (t,q)β
replaced by
ϕ−1
ϕ
1
1
β
ϕ
F (t, q) ϕ−1 (∂q J) ϕ−1 in the HJB equation.
E ϕ−1
These PDEs can be solved easily if we add the constraint a(t, q) = γq
that corresponds to a uniform and constant growth rate.
Proposition 6 (Resolution of the PDEs). If ϕ(ϕ − 1) < βk, there is
a unique triple (J, m, γ) that satisfies both the PDEs and the additional
equation on the optimal control function: a(t, q) = γq.
Solutions are of the following form:
m(t, q) = k
exp(γkt)
1q≥exp(γt)
q k+1
J(t, q) = B exp(−βkγt)q βk+ϕ 1q≥exp(γt)
1
where γ and B are related by γ = B
(βk + ϕ) ϕ−1
E
Proof:
First of all, the additional condition is equivalent to a constant growth
rate for qt and therefore, we obtain the Pareto distribution m(t, ·) stated
above.
Therefore, we have the following equation for ∂q J(t, q) if q ≥ exp(γt):
∂q J(t, q) = E(γq)ϕ−1 F (t, q)−β = E(γq)ϕ−1 e−βkγt q βk
Hence (the constant being nought),
J(t, q) =
E
γ ϕ−1 e−βkγt q βk+ϕ
βk + ϕ
If we plug this expression into the Hamilton-Jacobi equation we get:
C βk+ϕ −βkγt ϕ − 1
q
e
+
Eγ ϕ q βk+ϕ e−βkγt
kβ
ϕ
E
E
−βkγ
γ ϕ−1 e−βkγt q βk+ϕ − r
γ ϕ−1 e−βkγt q βk+ϕ = rD
βk + ϕ
βk + ϕ
From this we get:
C
ϕ−1
E
E
+
Eγ ϕ − βk
γϕ − r
γ ϕ−1 = 0
kβ
ϕ
βk + ϕ
βk + ϕ
This is exactly the equation (∗) of Proposition 2 and therefore γ is
unique.
This PDE approach allows us to generalize the model. We are indeed
going to add a common noise in the model and show how this noise affects
the results.
4.2
The model with common noise
So far, our model was completely deterministic whereas most applications
of mean field games are in a random setting. Here, it is not natural to
introduce a specific noise for each individual if we want to keep explicit
solutions with m being a Pareto distribution (because of the threshold).
However, it’s possible to introduce randomness through a common noise
on the evolution of the human capital.
More precisely, we can replace the dynamics for q by a stochastic one:
dqt = a(t, qt )dt + σqt dWt
where W is a noise common to all agents. This specification seems complicated since all the functions J, m and F are now random variables.
However, the intuitions developed above can be applied mutatis mutandis
and our specification is robust and deep enough to be generalized to a
complex stochastic framework.
First, consider the Bellman function J. From Proposition 6, we can
see that the expression for J was in fact a function of q and of the lower
bound q m of the q’s so that it’s more natural in general to define here
J = J(t, q, q m ) as:
∞
Z
M ax(qs )s>t ,qt =q,qtm =qm E
C
t
qα
E a(t, q)ϕ −r(s−t)
−
e
ds|F
t
m(t, q)β
ϕ F (t, q)β
Proposition 7 (Partial differential equations with common noise). The
Hamilton Jacobi equation corresponding to the above optimization problem
can be written in the following differential form:
M axa C
qα
E aϕ
−
− rJ
β
m(t, q)
ϕ F (t, q)β
σ 2 m2 2
σ2 2 2
2
mJ = 0
q ∂qq J + a0 ∂qm J +
q ∂qm qm J + σ 2 qq m ∂qq
2
2
where a0 is a(t, qtm )8 .
+∂t J + a∂q J +
The optimal control function is given by the same expression as in the
deterministic case:
1
ϕ−1
F (t, q)β
a(t, q) =
∂q J(t, q)
E
Lemma 1. If a(t, q) = γq, then the probability distribution function of
the q’s is m(t, q) = k
(qtm )k
1 m.
q k+1 q≥qt
Proof:
Assuming a(t, q) = γq we get:
qt = q0 exp ((γ −
σ2
)t + σWt ) = q0 qtm
2
2
exp (k(γ − σ2 )t + σkWt )
(qtm )k
1q≥exp((γ− σ2 )t+σW ) = k k+1
1q≥qtm
⇒ m(t, q) = k
k+1
t
q
q
2
8 This
is exogenous in the optimization because individuals are atomized
Proposition 8 (Resolution of the PDEs). If ϕ(ϕ − 1) < βk and r >
σ2
ϕ(ϕ − 1), then, there is a unique growth rate γ compatible with the
2
problem and J is of the form:
J(q, q m ) = Bq βk+ϕ (q m )−βk 1q≥qm
where γ and B are related by γ =
Moreover, γ is given by (∗0 ):
B
(βk
E
+ ϕ)
1
ϕ−1
C(ϕ + βk)
ϕ(ϕ − 1) − βk ϕ
σ2
γ = (r − ϕ(ϕ − 1) )γ ϕ−1 −
ϕ
2
Ekβ
(∗0 )
Proof:
First, if a(t, q) = γq then,
∂q J(t, q, q m ) = E(γq)ϕ−1 F (t, q)−β = Eγ ϕ−1 q βk+ϕ−1 (qtm )−βk
From this we deduce that the solution is of the stated form with
E
B = βk+ϕ
γ ϕ−1 .
If we want to find B or γ we need to plug the expression for J in the
Hamilton Jacobi equation. This gives:
C
E
q βk+ϕ−1 (q m )−βk β − γ ϕ − rB + γ(βk + ϕ)B − βkγB
k
ϕ
σ2
+ B ((βk + ϕ)(βk + ϕ − 1) + (−βk)(−βk − 1) + 2(βk + ϕ)(−βk)) = 0
2
C
E
σ2
− γ ϕ + γϕB − (r − ϕ(ϕ − 1) )B = 0
β
k
ϕ
2
C(βk + ϕ)
βk + ϕ ϕ
σ2
−
γ + ϕγ ϕ − (r − ϕ(ϕ − 1) )γ ϕ−1 = 0
β
Ek
ϕ
2
ϕ(ϕ − 1) − βk ϕ
C(ϕ + βk)
σ2
γ = (r − ϕ(ϕ − 1) )γ ϕ−1 −
ϕ
2
Ekβ
As for (∗), it’s clear that, given our hypotheses, this equation has a unique
solution.
The main conclusion is that the introduction of a common noise leads
to an increase in γ (as it can be seen on the following graph).
To conclude this part, let’s just note that the transversality condition
is modified:
Proposition 9 (Transversality condition). For the solution exhibited in
the above proposition to be an actual solution of the optimization problem,
2
we need to have γ < ϕr − (ϕ − 1) σ2 .
Proof:
The expression in the integral that defines the criterion is:
Figure 3: The introduction of a noise
C kβ+ϕ m ϕ E ϕ kβ+ϕ m ϕ
(qt )
q
(qt ) − γ q0
kβ 0
ϕ
Hence, for the solution to be well defined, the function t 7→ E [(qtm )ϕ ] e−rt
has to be integrable.
But:
σ2
σ2
σ2
E [(qtm )ϕ ] = E exp(ϕ(γ −
)t + ϕσWt ) = exp(ϕ(γ −
)t + ϕ2 t)
2
2
2
We therefore need to have r > ϕ(γ −
wanted to prove.
σ2
)
2
2
+ ϕ2 σ2 and this is what we
Conclusion
Using a mean field game framework, this paper presents a growth model
where growth is fostered by the fear of individuals about the possible
competition of their peers. This model can either be solved by classical
Euler-Lagrange methods or using the partial differential equations of the
mean field games theory. This second approach is a good way to show the
robustness of the model when it comes to the introduction of randomness.
Appendix
Proof of Proposition 2:
Let’s consider a solution of the form qt = q0 exp(γt). If this is true for
every single individual, then, the probability distribution function m(t, ·)
has to be of the Pareto form m(t, q) = k exp(γkt)
1q≥exp(γt) . This expression
q k+1
for m(t, ·) leads to F (t, q) = exp(γkt)
1q≥exp(γt) .
qk
Therefore, for q ≥ exp(γt) we have:
• G̃(t, q) = G(q, m(t, q)) =
C
kβ
• H̃(t, q, q̇) = H(q̇, F (t, q)) =
q α+β(k+1) e−γβkt
E ϕ
q̇
ϕ
exp(−γδkt)q δk
Hence, if we use the preceding proposition, we must have:
C(α + β(k + 1)) α+β(k+1)−1 −γβkt Eδk ϕ
q
e
−
q̇ exp(−γδkt)q δk−1
kβ
ϕ
i
d h ϕ−1
=−
E q̇
exp(−γδkt)q δk + rE q̇ ϕ−1 exp(−γδkt)q δk
dt
Since q̇ = γq we obtain:
C(α + β(k + 1)) α+β(k+1)−1 −γβkt Eδk ϕ
γ exp(−γδkt)q δk−1+ϕ
q
e
−
kβ
ϕ
i
d h ϕ−1
=−
Eγ
exp(−γδkt)q δk+ϕ−1 + rEγ ϕ−1 exp(−γδkt)q δk+ϕ−1
dt
⇒
C(α + β(k + 1)) α+β(k+1)−1 −γβkt Eδk ϕ
q
e
−
γ exp(−γδkt)q δk−1+ϕ
kβ
ϕ
= γδkEγ ϕ−1 exp(−γδkt)q δk+ϕ−1 −E(δk+ϕ−1)γ ϕ−1 exp(−γδkt)γq δk+ϕ−1
+rEγ ϕ−1 exp(−γδkt)q δk+ϕ−1
Hence, using the various assumptions on the parameters, we get:
C(ϕ + βk)
Eβk ϕ
−
γ = βkEγ ϕ − E(βk + ϕ − 1)γ ϕ + rEγ ϕ−1
kβ
ϕ
ϕ(ϕ − 1) − βk ϕ
C(ϕ + βk)
γ = rγ ϕ−1 −
ϕ
Ekβ
This equation in γ has a unique solution if, as it is supposed here,
ϕ(ϕ − 1) < βk.
⇒
Figure 4: The solution for γ
Proof of Proposition 3:
The integrand in the criterion is:
C βk+ϕ
E ϕ βk+ϕ
γ
q
exp((βk
+
ϕ)γt
−
kγβt)
e−rt
q
exp((βk
+
ϕ)γt
−
kγβt)
−
0
0
kβ
ϕ
Hence, we must have:
(βk + ϕ)γ − kγβ − r < 0
γ<
r
ϕ
Proof of Proposition 4:
First, let’s differentiate the equation (∗) with respect to k:
(ϕ(ϕ−1)−βk)
⇒
dγ ϕ−1 β ϕ
dγ
βCϕ −β−1 βC(1 − β) −β
γ
− γ = r(ϕ−1) γ ϕ−2 +
k
−
k
dk
φ
dk
E
E
β
dγ βCϕ −β−1 βC(1 − β) −β
(ϕ(ϕ − 1) − βk)γ ϕ−1 − r(ϕ − 1)γ ϕ−2 = γ ϕ +
k
−
k
dk
φ
E
E
If β = 1, it’s then obvious that k 7→ γ(k) is decreasing.
Otherwise, if β < 1, we can see from (∗) that the only limit point for γ is
β
0 and then, ϕ
kγ ϕ ∼ Cβ
k1−β . This leads to the result.
E
References
[AAZ06]
D. Acemoglu, P. Aghion, and F. Zilibotti. Distance to frontier, selection, and economic growth. Journal of the European
Economic Association, 4(1), Mar. 2006.
[ABB+ 05] P. Aghion, N. Bloom, R. Blundell, R. Griffith, and P. Howitt.
Competition and innovation: An inverted-u relationship.
Quarterly Journal of Economics, 120(2), May. 2005.
[AH]
P. Aghion and P. Howitt. Forthcoming book.
[AH92]
P. Aghion and P. Howitt. A model of growth through creative
destruction. Econometrica, 60(2), Mar. 1992.
[AHHV01] P. Aghion, C. Harris, P. Howitt, and J. Vickers. Competition, imitation and growth with step-by-step innovation. The
Review of Economic Studies, 68(3), Jul. 2001.
[AP07]
A. Atkinson and T. Piketty. Top Incomes over the Twentieth
Century: A Contrast between European and English-Speaking
Countries. Oxford University Press, 2007.
[Atk05]
A. Atkinson. Top incomes in the united kingdom over the 20th
century. Journal of the Royal Statistical Society, 168, 2005.
[Aum64]
R. Aumann. Markets with a continuum of traders. Econometrica, 32(1/2), 1964.
[Hai00a]
J.-O. Hairault. Analyse macroéconomique, tome 1.
Découverte, 2000.
La
[Hai00b]
J.-O. Hairault. Analyse macroéconomique, tome 2.
Découverte, 2000.
La
[Jon95]
C. Jones. R&d-based models of economic growth. The Journal
of Political Economy, 103(4), Aug. 1995.
[LL06a]
J.-M. Lasry and P.-L. Lions. Jeux à champ moyen i. le cas
stationnaire. C. R. Acad. Sci. Paris, 343(9), 2006.
[LL06b]
J.-M. Lasry and P.-L. Lions. Jeux à champ moyen ii. horizon
fini et contrôle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
[LL07a]
J.-M. Lasry and P.-L. Lions. Mean field games. Japanese
Journal of Mathematics, 2(1), Mar. 2007.
[LL07b]
J.-M. Lasry and P.-L. Lions. Mean field games. Cahiers de la
Chaire Finance et Développement Durable, (2), 2007.
[Min58]
J. Mincer. Investment in human capital and personal income
distribution. The Journal of Political Economy, 66(4), Dec.
1958.
[Pik03]
T. Piketty. Income inequality in france, 1901-1998. Journal
of Political Economy, 111(5), 2003.
[PS03]
T. Piketty and E. Saez. Income inequality in the united states,
1913-1998. Quarterly Journal of Economics, 118(1), Feb. 2003.
[Rom86]
P. Romer. Increasing returns and long-run growth. The Journal of Political Economy, 94(5), Oct. 1986.
[Rom90]
P. Romer. Endogenous technological change. Journal of Political Economy, 98(5), Oct. 1990.
[Tou02]
N. Touzi. Stochastic control and application to finance. Scuola
Normale Superiore, Pisa. Special Research Semester on Financial Mathematics, 2002.
Part I - Chapter 5
Another (though similar) Growth Model
Another (though similar) Growth Model
Abstract
As in the preceding part, we present a mean field game model of
growth. If the preceding one was a growth model involving human capital,
this one is embedded into an imperfect competition framework. Firms are
going to improve their productivity to increase their profit and to avoid
market power losses due to other firms that improve their productivity.
It’s noticeable that in spite of some marginal differences the underlying
mechanisms in the two models are arguably similar.
Résumé
Comme précédemment, nous présentons un modèle de croissance sous
la forme d’un jeu à champ moyen. Contrairement au précédent modèle
qui était un modèle de croissance par accroissement du capital humain,
le présent modèle est un modèle de croissance par accroissement de la
productivité de firmes dans un cadre de type concurrence imparfaite. Les
firmes tâchent d’améliorer leur productivité pour accroı̂tre leur profit et
éviter de perdre des parts de marché. Malgré des différences liées au sujet
traité, ce modèle et le précédent sont similaires concernant les mécanismes
sous-jacents.
Introduction
In a monopolistic competition framework, the profit of a firm depends
on its price and on a price index which can be seen as a function of the
whole distribution of the prices across firms. Therefore, a monopolistic competition model can be seen as a 1-period mean field game where
the players and the strategies are respectively the firms and the prices
at which they sell their products. If we consider a simple monopolistic
competition framework with linear costs, the prices are proportional to
the unit costs or, equivalently, inversely proportional to the productivities. Consequently, if firms can pay to improve their productivity, they
just play a dynamical mean field game in which their intertemporal profit
depends on the dynamics of the productivity distribution (or equivalently
on the unit costs distribution). This mean field game is similar to the
one embedded in the preceding model where the economic mechanisms
for human capital can be kind of translated in terms of productivity. This
is the framework of our model in which firms are going to pay to reduce
their costs.
In the first section we will present our model and derive the solution
in a specific case using the Euler-Lagrange methodology. In the second
section, we will present some comparative statics and discuss the underlying mechanisms.
1
The model
1.1
Introduction
In what follows, we consider an industrial sector and we assume that there
is a continuum of firms of size 1 in this sector. A representative agent consumes a continuum of types of goods from this sector, each type of good
being produced by a different firm.
Demand side
We assume that the goods are differentiated goods with an elasticity
1
of substitution σ = 1−α
> 1, that is to say the utility function of the
representative agent is given by:
Z 1
u=
x(i)α di
0
where x(i) is the quantity of a good of type i.
With these hypotheses, we can derive the demand function:
Proposition 1 (Demand function). The optimal demand function for the
representative agent is:
1
x(i) = R R 1
0
p(i) α−1
α
p(i) α−1 di
R1
where R = 0 p(i)x(i)di is the total amount of money dedicated to all the
varieties of goods.
Proof:
The representative agent maximizes:
Z 1
Z
x(i)α di
s.t.
0
1
p(i)x(i)di ≤ R
0
Introducing a lagrangian multiplier, we see that there exists a constant
λ so that ∀i, x(i)α−1 = λp(i).
R1
1
1
1
α
Consequently, ∀i, x(i) = λ α−1 p(i) α−1 and hence R = λ α−1 0 p(j) α−1 dj.
Replacing λ by its expression, we get:
1
x(i) = R R 1
0
p(i) α−1
α
p(j) α−1 dj
Supply side
Each type of good is produced by a specific firm with constant return
to scale, using labor only. However, firms are heterogeneous in the sense
that producing one unit of good i will require a(i) units of labor1 (one unit
costs the wage w) where the a’s are distributed according to the cumulative distribution function F (or equivalently, according to the probability
distribution function m).
From the result on the demand function, we can easily deduce the
price set by each firm as a function of its productivity:
Proposition 2 (Equilibrium price). The price p(i) set by the profitmaximizing firm i is given by:
p(i) =
wa(i)
α
Proof:
The firm i maximizes its profit:
π(i) = p(i)x(i) − wa(i)x(i)
The first order condition is:
0=
dx(i)
x(i)
dπ(i)
1
= x(i) + (p(i) − wa(i))
= x(i) −
(p(i) − wa(i))
dp(i)
dp(i)
1−α
p(i)
Hence,
p(i) =
wa(i)
α
We see that the price is indeed proportional to the unit cost and inversely proportional to the productivity. From this expression we can
deduce easily the expression for the profit of each firm.
Proposition 3 (Equilibrium profit). The equilibrium profit of the profitmaximizing firm i is given by:
α
(1 − α)R R ∞
0
1A
measure of productivity is therefore
a(i) α−1
α
a α−1 m(a)da
1
.
a(i)
Proof:
From the preceding equations we get:
π(i) =
1−α
wa(i)x(i)
α
α
= (1 − α)R R 1
0
a(i) α−1
α
a(j) α−1 dj
α
= (1 − α)R R ∞
0
1.2
a(i) α−1
a
α
α−1
m(a)da
The optimization problem
Each firm continuously chooses the extent to which it improves its productivity to maximize its intertemporal profit. Basically, a firm has the
following program:
Z ∞
M ax
[π(t, at ) − H(γ(t, at ), F (t, at ))] e−rt dt
0
where γ(·, ·) is defined by dat = −γ(t, at )at dt, π(·, ·) is the instantaneous
profit of the firm and H(·, ·) is the cost of the improvement γ (in percentage), this cost depending on the proximity to technological frontier.
This optimization problem is associated to a Euler-Lagrange equation:
equation). Let’s introduce Φ(t, a, ȧ) =
Proposition 4ȧ (Euler-Lagrange
π(t, a) − H(− a , F (t, a)) .
The Euler-Lagrange equation with a discount r is:
∂a Φ =
d
∂ȧ Φ − r∂ȧ Φ
dt
Proof:
It’s a simple application of the Euler-Lagrange equation with a discount rate r.
This equation is not the only one to be solved since the evolution
of the distribution must be in accordance with the choices of the firms:
this is the purpose of the Kolmogorov equation in the mean field games
theory. Here however, in the special case we are going to consider, this
“coherence equation” will not be an issue because there will be a constant
growth rate.
1.3
Resolution
To solve the problem we need to specify the function H. Our specification
is simple and resembles the specification of the model presented in the
previous part:
ȧ ϕ
ȧ
E −a
H(− , F (t, a)) =
, ∀a in the support of m(t, ·)
a
ϕ F (t, a)β
where E is a constant2 and where ϕ and β are two parameters (with
ϕ > 1).
This specification is quite natural. Basically, it’s more expensive to
reduce one’s costs if the percentage of change is higher and if costs are
already relatively small compared to competitors’ costs (this is due to the
proximity to the technological frontier).
Here, since we do not use the PDE approach but the Euler-Lagrange
methodology, we can allow for different parameter values across firms.
More precisely, we will use E as a free parameter depending on firm characteristics.
Now, we assume that the initial distribution of the a’s is a power distribution: F (0, a) = ak 1a∈(0,1) 3 .
We can now exhibit a solution:
α
Proposition 5 (Resolution). Assume first that k > 1−α
.
Assume then that the constant E(i) associated to the firm i is set at time
0 once for all for all i’s and given by4 :
E(i) = ξa0 (i)
α
kβ− (1−α)
Then, there exists a unique positive constant γ so that a solution of
the problem is given by:
at (i) = a0 (i) exp(−γt)
The equation that implicitly gives γ is the following:
αR
k
α
k+ α−1
= ξ[γ ϕ
kβ
+ rγ ϕ−1 ]
ϕ
(∗)
Proof:
Let’s consider that at (i) = a0 (i) exp(−γt). In that case the distribution evolution is such that:
F (t, a) = ak exp (γkt)1a∈(0,exp (−γt))
m(t, a) = kak−1 exp (γkt)1a∈(0,exp (−γt))
2E
will soon depend on each firm’s characteristics.
ensure convergence of the integral in the definition of profit we need to have k >
and we will assume this for the rest of this text.
4 This is quite ad hoc but since β is free, one can basically choose the exponent.
3 To
α
1−α
Now, we just need to find γ compatible with the Euler-Lagrange equation. To do that, we need to go step by step:
First, let’s compute ∂a Φ:
1
at (i) α−1
∂a Φ(t, at (i), ȧt (i)) = −αR R ∞
α
x α−1 m(t, x)dx
ϕ
ȧt (i)
ȧt (i)
−
−
at (i)
at (i)
βE(i)
1
+E(i)
+
m(t, a)
ai (t) F (t, at (i))β
ϕ F (t, at (i))β+1
1
αR
= − k a0 (i) α−1 exp(γt) + E(i)γ ϕ a0 (i)−(kβ+1) exp(γt)
0
ϕ
α
k+ α−1
+
kβE(i) ϕ
γ a0 (i)−(kβ+1) exp(γt)
ϕ

= exp(γt) −
αR
k
α
k+ α−1
a0 (i)
1
α−1
ϕ
+ E(i)γ a0 (i)
−(kβ+1)

kβ 
(1 +
)
ϕ
Second, let’s compute ∂ȧ Φ:
∂ȧ Φ(t, at (i), ȧt (i)) = E(i)
ϕ
− ȧatt (i)
(i)
1
ȧi (t) F (t, at (i))β
= E(i)γ ϕ−1 a0 (i)−(kβ+1) exp(γt)
Therefore:

exp(γt) −
αR
k
α
k+ α−1
a0 (i)
1
α−1
ϕ
+ E(i)γ a0 (i)
−(kβ+1)

kβ 
(1 +
)
ϕ
= E(i)γ ϕ−1 a0 (i)−(kβ+1) exp(γt)(γ − r)
1
kβ
αR
+ rγ ϕ−1 ]
a0 (i) α−1 = a0 (i)−(kβ+1) E(i)[γ ϕ
⇐⇒
k
ϕ
k+ α
α−1
Now, we are nearly done with the problem since we just need to find
γ solution of:
αR
k
α
k+ α−1
= ξ[γ ϕ
kβ
+ rγ ϕ−1 ]
ϕ
The left-hand side is a positive constant and the right-hand side is
increasing and goes from 0 to +∞. It’s then easy to see that there is a
unique solution γ to our problem.
This proposition leads to several comments. First of all, the solution
we exhibited is characterized by a constant growth rate in the sense that
the unit costs decrease at constant rate γ. Therefore, in our partial equilibrium setup, the prices decrease at rate γ and the quantities sold increase
at rate γ and this is arguably growth (even though not in the usual sense
of national accountability). Second, the solution we found is valid for
any value of the parameters, because the transversality condition is never
violated for r > 0 (it is straightforward to see that the integral is always
well defined). The last remark concerns the hypotheses. If the hypothesis
α
kβ− (1−α)
on k is only technical, the second one (E(i) = ξa0 (i)
) can seem
weird at first sight since the date 0 has no particular meaning and hence
there is no reason why this equation should be true for any value of the
parameters. If one want to get rid of this equation, another possibility to
α
keep the same solution is to impose that β = k(1−α)
. Whatever the option
we consider, it is a real issue to deal with comparative statics because the
parameters are related to one another in a way that doesn’t seem to be
meaningful. However, with the hypothesis of the proposition, we can supα
kβ− (1−α)
pose that the exponent in E(i) = ξa0 (i)
doesn’t remain constant
α
and move with the parameters (i.e. we do not set β so that kβ − 1−α
is equal to some predetermined constant) so that comparative statics can
be computed. This is one of the topics of the next section.
2 Comparative statics and analysis of the
model
2.1
Comparative statics
To deal with comparative statics, we need to suppose that the exponent
α
kβ− (1−α)
in the expression E(i) = ξa0 (i)
varies with the parameters (i.e.
is not fixed). In that case, it’s possible to use the equation (∗) that defines
γ to find the signs of the derivatives with respect to the four parameters
of interest: ξ, R, σ, r.
Growth as a decreasing function of ξ
An increase in ξ means that it’s more expensive to reduce costs and
hence γ should be a decreasing function of ξ. This is indeed the case:
Proposition 6.
dγ
<0
dξ
Proof:
We log-differentiate the equation (∗).
d[γ ϕ kβ
+ rγ ϕ−1 ]
dξ
ϕ
+
=0
ξ
γ ϕ kβ
+ rγ ϕ−1
ϕ
kβγ ϕ−1 + r(ϕ − 1)γ ϕ−2
dξ
= −dγ
ξ
γ ϕ kβ
+ rγ ϕ−1
ϕ
⇒
dγ
<0
dξ
Growth as a increasing function of the total wealth R of the
sector
Proposition 7. The growth rate is an increasing function of the total
wealth R of the sector:
dγ
>0
dR
Proof:
We log-differentiate the equation (∗).
kβγ ϕ−1 + r(ϕ − 1)γ ϕ−2
dR
dγ
= dγ
>0
⇒
ϕ−1
R
dR
+
rγ
γ ϕ kβ
ϕ
The interpretation here is twofold. First, it can be considered a pure
wealth effect meaning that richer sectors or richer countries should grow
faster. Second, it can be seen as a size effect meaning that larger sectors
or larger countries should grow faster. This size effect is present in several
growth models such the famous article by Romer [Rom86]. It’s unfortunately known to be inconsistent with reality ([Jon95]).
Growth as a increasing function of the elasticity of substitution σ
The elasticity of substitution σ is a good proxy of the competition
intensity: the more substitute the goods are, the more competitive the
sector is. Therefore, if our growth process is, as we think it is, due to a
competitive or Darwinian effect, then the growth rate should be an increasing function of σ.
This is indeed the case:
Proposition 8.
dγ
>0
dσ
Proof:
d α k+
Hence,
k>
dγ
dα
α
.
1−α
α
α−1
R
= ξ[kβγ ϕ−1 + r(ϕ − 1)γ ϕ−2 ]dγ
k
α(2−α)
.
(1−α)2
dγ
hence dσ >
has the same sign as k −
Therefore,
dγ
dα
> 0 and
But this is positive because
0 (since
Growth as a decreasing function of r
dα
dσ
> 0).
The interest rate r is a parameter that can be seen as impatience.
Therefore, γ should be decreasing as a function as a function of r.
Proposition 9. The growth rate is a decreasing function of the interest
rate:
dγ
<0
dr
Proof:
0 = kβγ φ−1 dγ + γ φ−1 dr + r(φ − 1)γ φ−2 dγ ⇒
dγ
<0
dr
Now that we have presented the comparative statics, we can present
the underlying mechanisms of the growth process.
2.2
Economic analysis of the model
The growth process we exhibited resembles the one presented in the preceding part. The mechanisms are indeed really similar if we consider that
human capital is replaced by productivity.
The reason of the growth is twofold as before. First, there is an incentive
to reduce unit costs in order to reduce prices and to be more competitive.
However, this first effect is not the only one. There is a second effect
(the Darwinian effect) that makes firms reduce their costs because their
competitors with higher costs can do so and steal some of their market
share by reducing their costs.
If the first effect is due to a forward-looking reasoning (firms that want
to compete firms ahead of them in term of technology), the second effect
is more backward-looking since firms are improving their productivity to
avoid a more intense competition with less technically gifted firms.
It’s therefore interesting to notice that the growth process is not only due
to the leaders who innovates but also due to the less productive firms that
push the process forward. This is arguably the main idea of the two models we presented and it can be thought of as a consequence of an escape
competition effect as in Aghion [AHHV01].
Conclusion
Using mean field games ideas, this part deals with a growth model similar to the previous one where growth comes from the combination of a
Schumpeterian effect and a Darwinian effect. We have shown that the
growth process was fostered not only by leaders that innovate but also
by the least productive firms that push the process forward by adopting,
for cheap, new technologies that reduce their costs and allow them to
compete more easily.
References
[AAZ06]
D. Acemoglu, P. Aghion, and F. Zilibotti. Distance to frontier, selection, and economic growth. Journal of the European
Economic Association, 4(1), Mar. 2006.
[ABB+ 05] P. Aghion, N. Bloom, R. Blundell, R. Griffith, and P. Howitt.
Competition and innovation: An inverted-u relationship.
Quarterly Journal of Economics, 120(2), May. 2005.
[AH]
P. Aghion and P. Howitt. Forthcoming book.
[AH92]
P. Aghion and P. Howitt. A model of growth through creative
destruction. Econometrica, 60(2), Mar. 1992.
[AHHV01] P. Aghion, C. Harris, P. Howitt, and J. Vickers. Competition, imitation and growth with step-by-step innovation. The
Review of Economic Studies, 68(3), Jul. 2001.
[Aum64]
R. Aumann. Markets with a continuum of traders. Econometrica, 32(1/2), 1964.
[HMY04]
E. Helpman, M. Melitz, and S. Yeaple. Export versus fdi with
heterogeneous firms. The American Economic Review, 94(1),
Mar. 2004.
[Jon95]
C. Jones. R&d-based models of economic growth. The Journal
of Political Economy, 103(4), Aug. 1995.
[LL06a]
J.-M. Lasry and P.-L. Lions. Jeux à champ moyen i. le cas
stationnaire. C. R. Acad. Sci. Paris, 343(9), 2006.
[LL06b]
J.-M. Lasry and P.-L. Lions. Jeux à champ moyen ii. horizon
fini et contrôle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
[LL07a]
J.-M. Lasry and P.-L. Lions. Mean field games. Japanese
Journal of Mathematics, 2(1), Mar. 2007.
[LL07b]
J.-M. Lasry and P.-L. Lions. Mean field games. Cahiers de la
Chaire Finance et Développement Durable, (2), 2007.
[Mel03]
M. Melitz. The impact of trade on intra-industry reallocations
and aggregate industry productivity. Econometrica, 71(6),
Nov. 2003.
[Rom86]
P. Romer. Increasing returns and long-run growth. The Journal of Political Economy, 94(5), Oct. 1986.
[Rom90]
P. Romer. Endogenous technological change. Journal of Political Economy, 98(5), Oct. 1990.
[Tou02]
N. Touzi. Stochastic control and application to finance. Scuola
Normale Superiore, Pisa. Special Research Semester on Financial Mathematics, 2002.
Part I
Appendix
Appendix 1
This appendix, based on Pierre-Louis Lions courses at the Collège de France, is
dedicated to the foundations of the mean field hypothesis. Basically, the different hypotheses made in what we call mean field games are the following:
• There are infinitely many agents modelled with a continuum.
• Agents are anonymous in the sense that they are identical and can be
replaced by one another without any global change.
• If the influence of other agents on a given individual depends on p variables
Z1 , . . . , Zp , then, it only depends on the distribution of these variables in
the population.
If the first two hypotheses are quite natural, the third one seems to be specific
to mean field games. However, we are going to explain why the first two hypotheses can easily lead to the third one. In fact, the purpose of this appendix is
to prove that symmetrical functions of N variables can be considered functions
of measures as N goes to +∞ if the functions weakly depend on the variables
(the exact sense will be explained later).
Let’s consider in what follows a compact set K ⊂ Rp and let’s denote by P(K)
the set of probability measures on K. The most useful property of P(K) is a
compactness property.
Proposition 1 (Properties of P(K)). P(K) is a weakly compact set (in the
sense of the weak convergence of measures) whose topology can be represented
with a metric.
The weak convergence of measure can be represented by metrics. Among them,
two kinds of metrics are useful in our context.
Definition 1 (Levy-Prokhorov metric). Let’s consider two probability measures
µ1 , µ2 ∈ P(K). The Levy-Prokhorov distance between these two measures is:
dLP (µ1 , µ2 ) = inf { > 0|∃(X1 , X2 ) r.v.0 s, X1 ∼ µ1 , X2 ∼ µ2 , P(|X1 − X2 | > ) < }
where Xi ∼ µi means that µi is the probability measure associated to the random
variable Xi .
Definition 2 (Monge-Kantorovich metric). Let’s consider two probability measures µ1 , µ2 ∈ P(K). The p-Monge-Kantorovich (1 ≤ p < +∞) distance between these two measures is:
o
n
1
dpM G (µ1 , µ2 ) = inf E[|X1 − X2 |p ] p |X1 ∼ µ1 , X2 ∼ µ2
where Xi ∼ µi means that µi is the probability measure associated to the random
variable Xi .
Proposition 2 (Comparison). Let’s consider two probability measures µ1 , µ2 ∈
P(K). If 1 ≤ p < +∞ then:
1
dpM G (µ1 , µ2 ) ≤ [dLP (µ1 , µ2 )p + |K|p dLP (µ1 , µ2 )] p
Proposition 3 (Weak convergence). Let’s consider a sequence (µn )n of distribution measures in P(K). ∀1 ≤ p < +∞:
µn * µ ⇐⇒ dpM G (µn , µ) → 0 ⇐⇒ dLP (µn , µ) → 0
Now we can
to understand these metrics using simple probability measures
Ptry
N
such as N1 i=1 δxi where the xi ’s are points in K.
Proposition 4. Let’s consider x = (x1 , . . . xN ), y = (y1 , . . . yN ) ∈ K N and let’s
PN
PN
1
1
N
note mN
x = N
i=1 δxi and my = N
i=1 δyi . Then:


N
1 X
N
dpM G (x, y) := dpM G (mN
,
m
)
=
min
|xi − yσ(i) |p
x
y
 N
i=1
! p1
|σ ∈ SN



Similarly,
dLP (x, y) :=
N
dLP (mN
x , my )
#{i/ |xi − yσ(i) | > }
<
= min > 0|∃σ ∈ SN ,
N
We see from these examples that the Levy-Prokhorov distance is a good candidate to define functions that weakly depends on variables as N goes to +∞
since two sets of points x = (x1 , . . . xN ), y = (y1 , . . . yN ) ∈ K N are close in K N
(or in fact in K N /SN ) if, up to a permutation, only a few coordinates are far
from each other.
Definition 3 (Weakly dependence on variables). Let’s consider a sequence of
symmetrical functions (uN ) with uN : K N → Rd . We say that the uN ’s uniformly weakly depend on variables if there exists a modulus of continuity1 ω such
that:
∀N, ∀x, y ∈ K N , |uN (x) − uN (y)| ≤ ω(dLP (x, y))
Now, we have everything needed to enounce the main proposition of this text:
Proposition 5. Let’s consider a sequence of symmetrical functions (uN ) with
uN : K N → K 0 where the uN ’s uniformly weakly depend on variables and where
K’ is a compact set of Rd .
Up to a subsequence, there exists a function U ∈ C 0 (P(K)) (where P(K) is the
metric space associated with the metric dLP ) such that:
lim
sup |uN (x) − U (mN
x )| = 0
N →+∞ x∈K N
1 We
suppose that ω is continuous, ω(0) = 0, ω is increasing and ω is sub-additive.
Proof:
Let’s define uN on P(K) by inf-convolution:
uN (µ) = inf uN (y) + ω(dLP (µ, mN
y ))
y∈K N
The definition for uN is clearly compatible with the earlier definition (if we
identify x and mnx ) and we have the same uniform modulus of continuity:
∀N, ∀µ1 , µ2 ∈ P(K), |uN (µ1 ) − uN (µ2 )| ≤ ω(dLP (µ1 , µ2 ))
Now, since (P(K), dLP ) is a compact metric space we can apply Ascoli’s theorem and the result is proved.
Now, it’s interesting to understand what it means for a symmetrical function to
be weakly dependent on the variables. Because of Proposition 2 we know that a
uniform modulus of continuity with a Monge-Kantorovich metric implies weak
dependence. A good criterion that uses d2M K or dqM K is the following:
Proposition 6 (Sufficient condition for weak dependence). Let’s consider a
sequence of symmetrical functions (uN ) with uN : K N → Rp . A sufficient
condition to have weak dependence is the following:
∃C > 0, ∀N,
N
X
|∇i uN |2 ≤
i=1
C
N
or more generally for q > 1:
∃C > 0, ∀N,
N
X
i=1
|∇i uN |q ≤
C
N q−1
What is proved in this appendix is that the third hypothesis that clearly seemed
to be strong and to limit the intuitions to a small class of games is in fact quite
general: most of the games that verify the first two hypotheses are in fact mean
field games!
Part II
Discount Rates
Part II - Chapter 1
Economic Calculus and Sustainable Development
Economic Calculus and Sustainable Development∗
Abstract
This part discusses the discount rate to be used in projects that aimed
at improving the environment. The model is quite stylized and involves
two different goods, one is the usual consumption good whose production
increases exponentially, the other is an environmental good whose quality
remains limited.
We define an ecological discount rate and we exhibit different relations
that link this discount rate with the usual interest rate and the growth
rate. We also discuss, mainly in the long run, the sensibility of this ecological discount rate to the parameters.
Résumé
Ce chapitre a pour objet l’étude des taux d’escompte à utiliser dans
le cadre de projets environnementaux. Le modèle utilisé est un modèle
stylisé à deux biens : un bien de consommation dont la production croı̂t exponentiellement et un bien dit environmental dont l’offre est par définition
limitée. Nous définissons un taux d’escompte écologique et nous étudions
ses liens avec les notions économiques habituelles : taux d’intérêt et taux
de croissance. Nous discutons la sensibilité sur le long terme de cette
notion aux différents paramètres d’intérêt.
Introduction
Determining the discount rate to use to evaluate projects which aim
at improving the quality of environment is a real issue. However, it seems
that this question has remained unanswered in the economic literature.
If one uses the traditional figures of economic calculus in the context of
environmental projects, the resulting consideration for the future would
be so weak in the long run that it would be incompatible with sustainable
development in its very definition: answer present needs without jeopardizing the ability of future generations to answer theirs.
The goal here is to reconcile economic calculus and sustainable development. Economists cannot indeed continue to consider environmental
issues as if they were equivalent to their usual economic ones. This has
in fact a really negative impact on the very economic theory that appears
∗ This text slightly differs from a text written with Roger Guesnerie and Jean-Michel Lasry
for an article to be published. Any error in this text is mine.
as myopic if not only selfish.
Here we develop a theory for the discount rates to be applied in the context of sustainable development and our main conclusion is to use, as in
the Stein report, very low discount rates when it comes to projects with
environment improving goals.
1
Model and preliminary insights
1.1
Goods and Preferences
We are considering a world with two goods. Each of them has to be
viewed as an aggregate. The first one is the standard aggregate private
consumption of growth models. The second one is called the environmental good. Its “quantity” provides an aggregate measure of “environmental
quality” at a given time. It may be viewed as reflecting biodiversity, the
quality of landscapes, nature and recreational spaces, the quality of climate. Later, however, for the sake of interpretation, we will view the
index, as integrating, in a broader way, many non-markets dimensions of
welfare.
We call xt the quantity of private goods available at period t, and yt
the level of environmental quality at the same period. Generation t, that
lives at period t only, has ordinal preferences, represented by a concave,
homogenous of degree one utility function:
σ−1
σ
σ−1
σ−1
v(xt , yt ) = xt σ + yt σ
However, the measurement of cardinal utility involves an iso-elastic
function.
0
1
v(xt , yt )1−σ
1 − σ0
The above modelling calls for the following comments that concern
respectively v and V.
V (xt , yt ) =
• Concerning v, we stress the standard properties of a CES utility
function, where σ is the elasticity of substitution, by referring to
the consumer’s choice if he were faced with a price both for the
environmental good and the private good. (Here the price of the
environmental good has to be viewed as an implicit price)
– When the ratio (implicit) price of the environmental good over
price of the private good decreases by one per cent, then the
ratio quantity (here quality) of the environmental good over
quantity of the private good increases by σ per cent. Equivalently, when the ratio environmental quantity (here quality) over
private good quantity decreases by one per cent the marginal willingness to pay for the environmental good increases by ( 1/σ) per
cent
It follows that if, as we often suppose in the following, environmental quality is constant and equals y, and the private good
consumption increases at the rate g, then the marginal willingness to pay for the environmental good increases at the rate
(g/σ), which is greater, (resp. smaller) than g, if σ is smaller,
(resp. greater) than one.
– The reader has noted that both xt and yt appear with the same
coefficient in the function v. However this is without loss of
generality as soon as we keep control of the freedom in the
measurement of yt . Indeed, as we shall see later, one can define
at each period a “green GDP”, (the product of the implicit price
of the environmental good by its quantity) and the standard
GDP, (the product of the quantity of private goods by its price).
The ratio of green GDP over the standard GDP is indeed, see
1
below, ( xytt )1− σ , and we may calibrate the model, i.e. choose the
units of measurement of the environmental good by assessing
the relative value of green GDP at the first period.
• Let us come to V . The marginal utility of a “util” of v, takes the
0
form v −σ : when v increases by one per cent, marginal cardinal
utility decreases by σ 0 per cent. This is the standard coefficient
linked to intertemporal elasticity of substitution ( σ10 ), relative risk
aversion (σ 0 ) or intertemporal resistance to substitution.
1.2
Social welfare
Social welfare is evaluated as the sum of generational utilities. In line
with the argument of Koopmans, we adopt the standard utilitarian criterion1 :
+∞
0
1 X −δt
e v(xt , yt )1−σ
0
1 − σ t=0
Two comments can be made:
• The coefficient δ is a pure rate of time preference. Within the normative viewpoint which we mainly stress here, the fact that this coefficient is positive has been criticized, for example by Ramsey who
claims that this is “ethically indefensible and arises merely from the
weakness of the imagination” or Harrod (1948) who views that as
a “polite expression for rapacity and the conquest of reason by passion”. To reconcile these feelings with Koopmans’ argument2 , we say
that “ethical considerations become preponderant” when δ tends to
zero. We may view the number as the probability of survival of the
planet3 .
• We may view the coefficient σ 0 , as a purely descriptive one, reflecting
intertemporal and risk behavior, or as a partly normative coefficient,
1 Notice that the elasticity σ 0 can be thought of as an ethical viewpoint of an external
planner instead of the above interpretation.
2 “Overtaking” would be another, different, way to proceed.
3 This is more satisfactory when we model adequately the uncertainty of the problem,
than within a deterministic framework where a higher δ may sometimes be a proxy for the
inappropriate treatment of uncertainty.
reflecting the desirability of income redistribution across generations.
In the following we will vary the interpretation according to the
viewpoint.
2 Preliminaries: Investigation around a
simple reference trajectory
2.1
The reference trajectory and first insights
In order to give some intuition on the discount rates question, we
shall consider a reference trajectory of the economy where environmental
quality is fixed at the level y and where the sequence of private goods
consumption denoted x∗t is also given (we often assume that the growth
rate g of consumption is itself fixed)4 . Note that the opposition between a
finite level of environmental good and an increasing level of consumption
good reflects an essential feature of the ecological question: sites, species
are finite and up to now we only have a planet where modern optimism
leads to believe that consumption goods may be multiplied without limit.
Note that our formulation, at least at this stage, does not assume either
“limits to growth” due to the finite ecological resources nor even deterioration of the ecological production due to growth. But it leaves open
the degree of substitutability between standard consumption good and
environment. We will argue later that this is one key uncertainty of the
problem, although at this stage, we leave the elasticity of substitution σ
as fixed.
The question we examine is: What are the discount rates, standard
interest rate for private goods, i.e. the return to private capital rt , and the
ecological rate for environmental goods implicit to the fixed trajectory?
At this stage, one may question the long run validity of the plausible
short run hypothesis that marginal willingness or environmental amenities
grows faster than private wealth, i.e. that σ < 1. At this stage, we will not
decide about the value of σ, but will take for granted that it is temporally
stable. We shall come back on this assumption5 .
2.2
Implicit discount rates
We shall first investigate the implicit discount factors at the margin of
our reference trajectory,with fixed environmental quality and exponential
growth. We sometimes refer to this approach as the reform viewpoint.
In the next section, we shall then take the optimization viewpoint and
show conditions under which our reference trajectory is a first best social
optimum.
The reference trajectory ∗ has consumption growing at the rate gt∗ (by
∗
definition x∗t+1 = egt x∗t ) and the environmental quality equal to y.
4 These sequences can be thought of as resulting from an optimization problem with exogenous progress. However, models à la Ramsey-Solow do not necessarily lead to a stationary
equilibrium (for a review on these issues, see Guesnerie-Woodford (1992))
5 It’s arguably difficult to guess the right value for σ because it is not only the elasticity of
substitution between the two goods but also a complete characterization of preferences .
We want to compute the implicit discount rates that sustain this trajectory, that is the discount rates that make it locally optimal.
Definition 1. The implicit discount rate for private good between periods
t and t + 1, is rt∗ such that
∗
e−rt = e−δ
∂1 V (xt+1 , y)
∂1 V (xt , y)
i 1−σσ0
h σ−1
σ−1 ( σ−1 )
1
x− σ .
where ∂1 V (x, y) = x σ + y σ
As in finance, it is possible to define the discount rate between periods
0 and T with the sequence of short-term rates:
R∗ (T ) =
T −1
1 X ∗
rt
T t=0
Now, we can define by analogy our new notion.
Definition 2. The ecological implicit discount rate between two consecutive periods is βt∗ defined by:
∗
e−βt = e−δ
∂2 V (xt+1 , y)
∂2 V (xt , y)
The discount rate between periods 0 and T is:
B ∗ (T ) =
T −1
1 X ∗
βt
T t=0
∗
Proposition 1. Along our reference trajectory, x∗0 , . . . , x∗t+1 = egt x∗t , . . .
the implicit private discount rate for the private good between periods t
and t + 1 can be equivalently written as,
either:
0
1+ρ∗
rt∗ = δ + gt∗ σ 0 + 1−σσ
ln 1+ρ∗t
σ−1
t+1
or
−1
0
1+ρ∗
t
rt∗ = δ + gt∗ /σ + 1−σσ
ln
∗
−1
σ−1
1+ρt+1
1 −1
σ
yt ∂2 V
is the ratio of Green GDP over standard
where ρt = xt ∂1 V = xytt
∗
GDP, and ρt is its value along the trajectory.
Proposition 2. The ecological discount rate between periods t and t + 1
is:
βt∗ = rt∗ − gt∗ /σ
Proof of Proposition 1:
The implicit discount rate rt∗ for private goods between periods t and
t + 1 is uniquely defined by:
−rt∗
e
∂1 V (x∗t+1 , y)
= e−δ
= e−δ
∂1 V (x∗t , y)
x∗t+1
x∗t
− 1
x∗t+1
σ
x∗t
σ−1
σ
+y
σ−1
σ
+y
+y
σ−1
σ
σ−1
σ
0
! 1−σσ
σ−1
σ−1
σ
Taking logarithms, this gives:
rt∗
=δ+
gt∗ /σ
1 − σσ 0
ln
−
σ−1
rt∗ = δ + gt∗ /σ −
1 − σσ 0
σ−1
rt∗ = δ + gt∗ /σ +
1 − σσ 0
σ−1
x∗t+1
σ−1
σ
σ−1
!
σ−1
x∗t σ + y σ
1 + ρ∗t+1 −1
ln
1 + ρ∗t −1
1 + ρ∗t −1
ln
1 + ρ∗t+1 −1
This is the second formula of Proposition 1. The first formula can be
obtained by the same reasoning if we go back to:
!
σ−1
σ−1
x∗t+1 σ + y σ
1 − σσ 0
∗
∗
rt = δ + gt /σ −
ln
σ−1
σ−1
σ−1
x∗ σ + y σ
t

σ−1
σ
y
∗ σ−1 1 +
0
σ

xt+1
x
1 − σσ
t+1
∗
∗
ln 
rt = δ + gt /σ −
σ−1
 x∗
σ−1
σ
t
y
1 + xt+1

1 + ρ∗t+1
1 + ρ∗t
rt∗ = δ + gt∗ /σ −
1 − σσ 0 σ − 1 ∗ 1 − σσ 0
gt −
ln
σ−1
σ
σ−1
rt∗ = δ + gt∗ σ 0 +
1 − σσ 0
ln
σ−1
1 + ρ∗t
1 + ρ∗t+1



This formulation will be useful when σ > 1 whereas the other one will
be useful for σ < 1.
Proof of Proposition 2:
We remind that
Hence, βt∗ =
=
1/σ
x
y
.
(∂ V )
(∂ V )
xt+1 1/σ
= e−δ (∂2 2 Vt+1
= e−δ (∂1 1 Vt+1
)t
)t
xt
rt∗ − gt∗ /σ and this proves Proposition 2.
−βt∗
Therefore, e
∂2 V
∂1 V
∗
∗
= e−rt egt /σ
2.3 Long run discount rates: the reform viewpoint
Now we stress the long run behavior of the discount rates, under
the assumption that the average growth rate of consumption converges:
1
T
PT −1
t=0
gt∗ → g ∗ .
We focus our attention on the long run discount rate for private good,
i.e.Pthe limit of the discount rate between periods 0 and T , R∗ (T ) =
T −1 ∗
1
t=0 rt , when T becomes high. Similarly, the long run ecological
T
discount
is the limit, when T increases indefinitely of B ∗ (T ) =
PT −1 rate
∗
1
t=0 βt
T
At this stage, one should give some intuition on the qualitative differences between the cases σ < 1 and σ > 1 that will appear strikingly in
the results.
We have:
v(xt , yt ) = xt [1 + (
σ
yt ( σ−1
) σ ) ]( σ−1 )
xt
First, let us consider σ > 16 . Now, v grows as xt whenever xytt tends to
zero and social marginal utility of consumption will decrease at σ 0 times
the growth rate of v.
On the contrary, in the case where σ < 17 it is useful to write:
v(xt , yt ) = yt [1 + (
σ
yt ( 1−σ
) σ ) ]( σ−1 )
xt
In that case v does not grow any longer indefinitely with xt , but tends
to y. Then, social marginal utility of consumption is approximately equal
0
to (y)−σ and tends to zero at a speed independent of σ 0 .
These differences are reflected in a particularly spectacular way in the
behavior of long run discount rates.
Proposition 3. At the margin of the reference situation, when T tends
to +∞,
- When σ is greater than one,
R∗ (T ) → δ + g ∗ σ 0
B ∗ (T ) → δ + g ∗ (σ 0 − 1/σ)
- When σ is smaller than one
R∗ (T ) → δ + g ∗ /σ
B ∗ (T ) → δ
6 For
example, with σ = 2
v(xt , yt ) = xt [1 + (
7 For
yt (1/2) 2
)
]
xt
σ = 1/2,
v(xt , yt ) = yt [1 + (
yt (−1)
)]
xt
Proof:
We have the formula of Proposition 1:
rt∗ = δ + gt∗ σ 0 +
1 − σσ 0
ln
σ−1
1 + ρ∗t
1 + ρ∗t+1
We note that when σ is greater than one, as soon as gt∗ has a lower
bound strictly greater than zero, ρt tends to zero. In this case, it is
straightforward to see that:
rt∗ → δ + gt∗ σ 0
It’s now easy to conclude that R∗ (T ) → δ + g ∗ σ 0 using Césàro’s theorem.
For the long run ecological discount rate, we use Proposition 2 to
conclude that B ∗ (T ) − R∗ (T ) → g ∗ /σ and this leads to the result:
B ∗ (T ) → δ + g ∗ (σ 0 − 1/σ)
In the σ < 1 case we come back to the other part of Proposition 1:
1 − σσ 0
1 + ρ∗t −1
rt∗ = δ + gt∗ /σ +
ln
σ−1
1 + ρ∗t+1 −1
Here, however, in the long run, ρ∗t → +∞ so that we have directly:
rt∗ → δ + g ∗ /σ
Using Césàro’s theorem we get R∗ (T ) → δ + g ∗ /σ and with the help
of Proposition 2 we have the result on B that is B ∗ (T ) → δ
This proposition is really central in our paper and need some comments:
• The cases σ < 1 and σ > 1 are really different in the sense that
asymptotic results are discontinuous at σ = 1. The first case (σ >
1) can be thought of as a traditional case since the two goods are
good substitutes. The second case (σ < 1) is characterized by a
low substituability between the private good and the environmental
good and corresponds to a case where environmental issues become
preponderant in the long run.
• In the σ < 1 case the asymptotic ecological discount rate only depends on δ and not on the growth path. This is one of the most
important properties in this part since it may be an incitation to
take very low discount rate for the environmental good.
The σ = 1 case
For the sake of completeness, we have to consider the σ = 1 case. For
this case the utility function is not defined but we can use a very common
√
trick that is: limσ→1 1σ v(x, y) = xy.
2 σ−1
Consequently, we can use the same definitions and reasonings as those
we used but with a Cobb-Douglas function. The results are the following:
Proposition 4. If σ = 1, at the margin of the reference situation, when
T tends to +∞,
R∗ (T ) → δ + 21 g ∗ (σ 0 + 1)
B ∗ (T ) → δ + 21 g ∗ (σ 0 − 1)
The case of environmental good exhaustion
We can also generalize our result by making an other hypothesis on
yt∗ . We can indeed consider an exhaustion of the environmental good at
a positive rate g 0 . This means that the condition yt∗ = y is replaced by
0
yt∗ = ye−g t .
With this setting, the above propositions are slightly modified but we
can use the same methodology to get the asymptotic results on the two
interest rates:
Proposition 5. If g 0 is not too large, at the margin of the reference situation, when T tends to +∞,
- When σ is greater than one,
R∗ (T ) → δ + g ∗ σ 0
0
B ∗ (T ) → δ + g ∗ (σ 0 − 1/σ) − gσ
- When σ is smaller than one
0
R∗ (T ) → δ + g ∗ /σ + g 0 1−σσ
σ
B ∗ (T ) → δ − σ 0 g 0
Robustness
We said before that the parameter σ was hard to calibrate. One can
therefore ask whether the result is still true in the case where σ is not
constant but depends on t8 .
Proposition 6. If limt→+∞ σt = σ 6= 1 then, at the margin of the reference situation, when T tends to +∞,
- When σ is greater than one,
R∗ (T ) → δ + g ∗ σ 0
B ∗ (T ) → δ + g ∗ (σ 0 − 1/σ)
- When σ is smaller than one
R∗ (T ) → δ + g ∗ /σ
B ∗ (T ) → δ
The case where σt → 1 is undetermined and depends, among other
things, on the convergence speed.
8 By
this we mean that σt is a deterministic function (the random case being hard to tackle.)
3 Discount rates along an optimized growth
path
The above results hold at the margin of any trajectory, whether it is
non-optimal, or optimal either in a first best sense or in a second best
sense. Optimal solutions of growth models, either first best or second best
have the limit properties that we attributed to our reference solution and
hence the results of Proposition 2 apply in a variety of worlds (naturally
in a second best world the implicit discount rate for private good may
differ from market prices). Hence in a sense, the above analysis exhausts,
at least in a large class of contexts, the qualitative properties of long run
ecological discount rates.
However, we will now turn to a fully defined optimization framework
in order to say more on the whole trajectory of ecological discount rates.
The model will also allow us to provide a better assessment of an a priori
worrying discontinuity of our results as a function of σ.
We consider a model à la Keynes-Ramsey where the interest rate r
is exogenous. This interest rate can come from an AK model or from a
market that has no link with the model: a good example of this is given
by r extracted from a research arbitrage equation as in Aghion Howitt.
3.1 The model and characterization of the first
best social optimum
We consider a representative agent, living for ever, who maximizes the
following intertemporal utility function:
∞
X
exp(−δt)V (xt , yt )
t=0
where we assume, partly for convergence reasons, that (1 − σ 0 )r < δ < r9 .
The representative agent, sometimes denoted below as the planner,
has economic and environmental constraints:
Economic constraints: at+1 = exp(r)at + wt − xt , where at stands for
the wealth at date t and wt is an exogenous production flow,
Environmental constraints: The available quality for the environmental good is limited to y that is: yt ≤ y.
To solve the optimization problem we introduce the lagrangian associated to the problem and this gives:
L=
∞
X
exp(−δt)[V (xt , yt ) + λt (exp(r)at + wt − xt − at+1 ) + µt (y − yt )]
t=0
The first order conditions are the following:
9 It’s
interesting to note that if σ 0 > 1, one can take negative values for the discount rate δ.

∗
∗
 ∂xt L = 0 ⇐⇒ ∂x V (xt , yt ) = λt
∂at+1 L = 0 ⇐⇒ λt+1 exp(r − δ) = λt

∂yt L = 0 ⇐⇒ ∂y V (x∗t , yt∗ ) = µt
From these first order conditions we can derive the asymptotic economic growth rate:
Proposition 7. The asymptotic growth rate for the private good x∗t depends on σ and is given by the following formulae:
∗
- If σ < 1 then g∞
= σ(r − δ) and there is a real environmental issue.
∗
- If σ > 1 then g∞
= r−δ
and this is the traditional result without any
σ0
consideration of environmental goods.
∗
- If σ = 1 then g∞
=
2(r−δ)
1+σ 0
and it is a specific case.
Proof:
The first thing to note is that yt∗ = y.
Then, since we supposed that r is greater than δ we have exp(r−δ) > 1
so that λt and ∂x V (x∗t , y) are both decreasing and tend to zero. The natural consequence is that the consumption of the private good x∗t grows
and tends to +∞.
The growth path x∗t is then characterized by:
∂x V (x∗t , y) = λt =
λ0
exp((r − δ)t)
Therefore,
x∗t
1
−σ
h
x∗t
σ−1
σ
+y
σ−1
σ
0
i 1−σσ
σ−1
=
λ0
exp((r − δ)t)
As we did in the preceding parts we are going to consider two cases
depending on σ being larger or smaller than 1.
• The σ > 1 case:
x∗t
−σ 0
∼∞
λ0
exp((r − δ)t)
x∗t+1
r−δ
) ∼∞
x∗t
σ0
Hence, the asymptotic growth rate is the same as if there were no
consideration of the environmental good:
ln(
∗
g∞
=
r−δ
σ0
• The σ < 1 case:
x∗t
1
−σ
∼∞
λ0
1
0
y σ −σ exp((r − δ)t)
x∗t+1
) ∼∞ σ(r − δ)
x∗t
Hence, the growth rate in that case is given by:
ln(
∗
g∞
= σ(r − δ)
As before, it is also possible to consider σ = 1 by taking a Cobb1−σ 0
Douglas function for V : V (xt , yt ) =
∗
g∞
=
(xt yt ) 2
1−σ 0
and we eventually obtain
2(r−ρ)
1+σ 0
In this framework we really see the difference between the two cases
σ > 1 and σ < 1. In the former case, the asymptotic growth is not affected by the presence of the environmental good whereas in the latter
case, the growth rate is different from the usual one and decreases when
σ gets lower.
Now, we can apply our finding to the ecological interest rates B ∗ . We
have the following results:
∗
Proposition 8. The asymptotic ecological discount rate B∞
= limT →+∞ B ∗ (T )
is given by the following formulae:
∗
- If σ < 1 then B∞
= δ.
∗
- If σ > 1 then B∞
= (1 −
∗
- If σ = 1 then B∞
=δ−
1
)r
σσ 0
1−σ 0
(r
1+σ 0
+
1
δ.
σσ 0
− δ).
Proof:
We just need to combine Proposition 2 and 7 and to apply Césàro’s
theorem to have the result. The result of Proposition 2 is indeed valid in
this setting.
3.2 More on the apparent discontinuity of the results with σ
Both the general setting and the optimization setting we are now dealing with suffer from what can be thought of as a real caveat. The ecological
interest rate seems indeed to be a discontinuous function of σ.
If we plot the asymptotic ecological interest rate as a function of σ we
obtain the following results:
∗
Figure 1: Dependence on σ of the variable B∞
when σ 0 = 2
In what follows we show that this discontinuity is specific to the asymptotic case and that B ∗ (T ) is a continuous function of σ when T is fixed
(and finite).
Proposition 9. ∀T < ∞, σ 7→ B ∗ (T ; σ) is continuous.
Proof: (This proof can be omitted at first reading)
The first thing to do is to write the result of Proposition 2 and to
deduce a useful expression for B ∗ (T ).
We have:
βt = r −
gt∗
σ
T −1
1 1 X ∗
gt
σ T t=0
∗
1 1
xT
⇒ B ∗ (T ) = r −
ln
σT
x∗0
⇒ B ∗ (T ) = r −
Therefore, the only thing to prove is that ∀t, x∗t is a continuous function
of σ. But we know that the growth path is defined by the first order
λ0
10
condition ∂x V (x∗t ; σ) = exp((r−δ)t)
where we omitted the reference to y
here since we focus on σ. Then it is easy to see that the only two things
we need to prove are:
• The Lagrange multiplier λ0 is a continuous function of σ.
10 It’s
1
2
very important here to consider v(x, y) =
h
1 σ−1
x σ
2
+ 12 y
σ−1
σ
i
σ
σ−1
with the weights
to extend the function properly. Obviously, it doesn’t change anything to our preceding
results since it is only a multiplicative scalar adjustment
• The function g(ξ, σ) implicitly defined by ∂1 V (g(ξ, σ); σ) = ξ is continuous.
The second point is easy. Notice first that the function (x, σ) 7→
V (x; σ) can be extended to a C 2 function (the proof is easy). Then, by
2
the implicit function theorem, g(ξ, σ) is a C 1 function ((ξ, σ) ∈ R+∗ ).
Therefore, the only thing to prove is that the first Lagrange multiplier
λ0 is a continuous function of σ. Let us recall that λ0 is defined by the
resources constraint:
∞
X
x∗t e−rt = a0 +
t=0
∞
X
∞
X
wt e−rt (:= Λ∞ 11 )
t=0
g(λ0 exp((δ − r)t), σ)e−rt = Λ∞
t=0
Here, we cannot apply directly the implicit function theorem to the left
hand side. However, if we consider the restricted optimization problem
with a fixed time horizon T 12 then the associated Lagrange multiplier
(λT0 ) is implicitly defined by
T
X
g(λT0 exp((δ − r)t), σ)e−rt = a0 +
t=0
and the implicit function theorem applies:
T
X
wt e−rt (:= ΛT )
t=0
λT0
is a C 1 function of σ.
Now, we can approximate λ0 by λT0 and this gives:
|λ0 (σ) − λ0 (σ̃)| ≤ |λ0 (σ) − λT0 (σ)| + |λT0 (σ) − λT0 (σ̃)| + |λT0 (σ̃) − λ0 (σ̃)|.
Hence, we see that the only thing to prove is a pointwise convergence in
the sense that, for σ fixed, we have a convergence of λT0 (σ) towards λ0 (σ)
as T → ∞.
P
To prove that let’sPintroduce FT : z 7→ Tt=0 g(z exp((δ − r)t), σ)e−rt and
∞
similarly F : z 7→ t=0 g(z exp((δ − r)t), σ)e−rt . These two functions are
positive and decreasing because g is a positive and decreasing function
of ξ. Moreover, FT is continuous and there is a pointwise convergence
of FT towards F . By monotony, FT converges towards F uniformly on
every compact set and therefore, F is a continuous function and so is the
inverse of the function F .
By the second Dini’s theorem then, the inverse of the function FT converges uniformly on every compact set towards the inverse of the function
F.
But λT0 − λ0 = FT−1 (ΛT ) − F −1 (Λ∞ ) and hence, since ΛT → Λ∞ , we are
done with the proof.
This Proposition 9 is important in the sense that the discontinuity in
Proposition 3 is proven to be specific to the asymptotic case. Therefore,
this discontinuity is not really a problem in our model.
11 This
12 M ax
quantity is supposed finite for the problem to have a solution.
PT
t=0 exp(−ρt)u(xt , y) s.t. at+1 = exp(r)at + wt − xt
3.3
The dynamics of ecological discount rates
The Ramsey-Keynes framework is also interesting to deal with the ecological discount rates in finite horizon and to see the evolution of ecological
discount rates with time. In other words, we are going to be interested in
what can be called yield curves for ecological discount rates B ∗ (T ).
P −1 ∗
Since B ∗ (T ) = δ − σ1 T1 Tt=0
gt , we see that the dynamics of the ecological discount rate is linked to the dynamics of growth.
Therefore, we are going to focus on the way gt∗ converges toward its
limit.
Proposition 10. gt∗ converges monotonically toward its limit according
to the following rules:
- If σ < 1 and σσ 0 > 1 then gt∗ is increasing.
- If σ > 1 and σσ 0 > 1 then gt∗ is decreasing.
And less importantly:
- If σ < 1 and σσ 0 < 1 then gt∗ is decreasing.
- If σ > 1 and σσ 0 < 1 then gt∗ is increasing.
- If σ = 1 or σσ 0 = 1 the optimal growth rate is constant.
Proof:
Let us go back the the first order conditions that define the growth
path.
We have:
∗
∂x V (x∗t , y) = er−δ ∂x V (x∗t egt , y)
Therefore, g, as a function of x is defined implicitly by (we now omit
the y terms):
V 0 (x) = er−δ V 0 xeg(x)
If σ = 1, we are dealing with Cobb-Douglas functions and then the
growth rate is clearly independent of x and the result is proved.
Otherwise, since x∗t is an increasing sequence, the variation properties
of gt∗ are given by the sign of g 0 (x) that is going to be computed now.
Taking logs and deriving we get:
V 00 (x)
V 00 (xeg(x) ) g(x)
=
e
(1 + g 0 (x)x)
V 0 (x)
V 0 (xeg(x) )
Hence, the sign of g 0 (x) is the sign of V 0 (x)V 00 (xeg(x) )eg(x) −V 0 (xeg(x) )V 00 (x).
0
g
d V (xe )
This sign is simply the sign of dx
where g is now an indepenV 0 (x)
dent variable.
The latter expression can be written as:
−g/σ
e
d
dx
"
σ
y + (xeg ) σ−1
y+x
# 1−σσ0
σ−1
σ
σ−1
The sign of this derivative is the sign of:
1 − σσ 0 g σ−1
1 − σσ 0 σ − 1 g σ−1
e σ −1 =
e σ −1
σ−1
σ
σ
Since g > 0 in our context, this expression has the same sign as the
product (1 − σσ 0 )(σ − 1) and this proves our result.
The result of Property 10 is in fact the superposition of two effects.
First, there is a substitution effect between the two goods. If this
substitution is very low, we are incited not to postpone consumption of the
private good and therefore the growth is naturally decreasing (respectively
increasing if σ is high).
Second, there is an intertemporal substitution effect. If the substitution between two periods is low (if σ 0 is high), the growth is supposed to
be decreasing (respectively increasing if σ 0 is low).
Now using Proposition 2, we can deduce the shape of the yield curve
for ecological discount rate:
Proposition 11. The shape of the yield curve is the following:
- If σ < 1 and σσ 0 > 1 then T 7→ B ∗ (T ) is decreasing.
- If σ > 1 and σσ 0 > 1 then T 7→ B ∗ (T ) is increasing.
And less importantly:
- If σ < 1 and σσ 0 < 1 then T 7→ B ∗ (T ) is increasing.
- If σ > 1 and σσ 0 < 1 then T 7→ B ∗ (T ) is decreasing.
- If σ = 1 or σσ 0 = 1 then T 7→ B ∗ (T ) is constant.
To illustrate our proposition, we drew yield curves using a simulation
of the growth path13 . Two examples are given below where the x-axis
represents years and the y-axis the value of the ecological discount rate.
The first case corresponds to σ < 1 and σσ 0 > 1 in which case the
yield curve is decreasing and converges towards δ.
The second case corresponds to σ > 1 and σσ 0 > 1 in which case the
.
yield curve is increasing and converges towards r − r−δ
σσ 0
13 We
simply used a Newton-Raphson methodology to find the growth path
Figure 2: Yield curve example (σ = 0.8, σ 0 = 1.5, r = 2%, δ = 0.1%)
Figure 3: Yield curve example (σ = 1.2, σ 0 = 1.5, r = 2%, δ = 0.1%)
An interesting fact is that ecological discount rates converge slowly to
their asymptotic value.
4 Uncertainty about the elasticity of substitution σ
We have seen that the two cases, σ < 1 and σ > 1, were really different. An interesting issue is therefore to understand what happens if we
do not know in which case we are. To better understand the issue, we are
going to consider two possible values for σ. More precisely, we consider
that σ can be equal to σl < 1 with probability p or it can be equal to
σh > 1 with probability 1 − p, p being in the interval (0; 1). To simplify
the approach, we also consider that σ 0 is large enough so that σl σ 0 > 1.
∗
We are going to show that, independently of p > 0, B∞
= δ. This is in
accordance with the intuition that, asymptotically at least, the rate to be
taken into account is the smallest.
To prove that, we are going to start again with the optimization problem:
∞
X
exp(−δt)[pV (σl ; xt , y) + (1 − p)V (σh ; xt , y)]
t=0
s.t. at+1 = exp(r)at + wt − xt
Let’s start with a result on the growth rate:
Proposition 12. The asymptotic growth rate for the private good x∗t does
∗
not depend on p > 0 and is equal to g∞
= σl (r − δ)
Proof:
Using the same notations as in Proposition 7 we know that the growth
path x∗t is characterized by:
p∂x V (σl ; x∗t , y) + (1 − p)∂x V (σh ; x∗t , y) =
λ0
exp((r − δ)t)
Therefore,
1
−
px∗t σl
σl −1
x∗t σl
+y
σl −1
σl
1−σl σ0
σl −1
1
−
+(1−p)x∗t σh
σh −1
x∗t σh
+y
σh −1
σh
1−σh σ0
σh −1
Using the asymptotic results derived earlier in Proposition 7, we get
that:
px∗t
− σ1
l
− 1
−σ 0
−σ 0
+ o x∗t σl + (1 − p)x∗t
+ o x∗t
=
But, since we supposed that σl σ 0 > 1, we have:
λ0
exp((r − δ)t)
=
λ0
exp((r − δ)t)
λ0
exp((r − δ)t)
Hence, as in Proposition 7, we get the asymptotic growth rate which
∗
is here g∞
= σl (r − δ).
px∗t
− σ1
l
∼∞
This proposition means that, in terms of growth, everything is asymptotically as if p were equal to 1: the environmental issues dominate.
∗
Now we can find the asymptotic ecological discount rate B∞
in this
random context:
∗
Proposition 13. The asymptotic ecological discount rate B∞
does not
∗
depend on p > 0 and is equal to B∞
=δ
Proof:
Let’s recall first the definition of B ∗ (T ) in this context:
B ∗ (T ) = δ −
p∂y V (σl ; xT , y) + (1 − p)∂y V (σh ; xT , y)
1
ln
T
p∂y V (σl ; x0 , y) + (1 − p)∂y V (σh ; x0 , y)
To prove our result, it is sufficient to prove that the expression in the
logarithm remains bounded as T increases. Hence, we are going to prove
that the following expression is bounded:
py
− σ1
l
x∗T
σl −1
σl
+y
σl −1
σl
1−σl σ0
σl −1
+ (1 − p)y
− σ1
h
σ −1
1−σh σ0
σh −1
σh −1
∗ h
xT σh + y σh
0
The first part of the expression converges toward py −σ and is therefore bounded.
σh −1
σh −1
For the second part of the expression, x∗T σh + y σh → ∞ so that,
0
hσ
< 0, the second part of the expression tends toward 0 and
since 1−σ
σh −1
this proves the result.
An important conclusion is that the very possibility that the environmental issues dominate implies that, in the long run, everything happens
as if the environmental issues do dominate. Obviously, this result is only
asymptotic and based on the hypothesis that we never learn σ, an hypothesis that is quite restrictive. However, since we do not know σ and since
we do not know when we would know it, this hypothesis seems to be an
acceptable approximation of reality and the case σ < 1 is really relevant
for that very reason.
5 Application to the expected environmental return on investment
Investing now to get benefits in the future is one of the central issues in
finance. Here we develop a notion equivalent to the time value of money
in an environmental framework. In other words, we investigate the fair
increase in environmental good at time T to compensate, in utility terms,
an investment at time 0. As before we are going to reason at the margin
of an equilibrium trajectory.
Definition 3. We denote by ωt∗ the marginal increase in environmental
good at time t + 1 to compensate a marginal investment at time t. This
value is defined by:
∗
e−ωt = e−δ
∂2 V (x∗t+1 , y)
∂1 V (x∗t , y)
The associated discount rate between periods 0 and T is defined by14 :
∗
e−Ω
(T )T
= e−δT
∂2 V (x∗T , y)
∂1 V (x∗0 , y)
Now we can relate Ω∗ (T ) to the ecological discount rate and deduce
asymptotic results for Ω∗ (T ).
Proposition 14. Ω∗ (T ) is related to B ∗ (T ) by the following equation (for
σ 6= 1):
1
ln(ρ∗0 )
Ω∗ (T ) = B ∗ (T ) +
(σ − 1)T
Proof:
By definition:
e−Ω
∗
(T )T
= e−δT
∗
∗
∂2 V (xT , y)
∂2 V (x0 , y)
= e−B (T )T
= e−B (T )T
∂1 V (x∗0 , y)
∂1 V (x∗0 , y)
y
x∗0
− 1
σ
To conclude, we just need to take logarithms of both sides.
It’s now easy to deduce the asymptotic properties of Ω∗ (T ).
Proposition 15. If we define Ω∗∞ by limT →∞ Ω∗ (T ) then:
∗
Ω∗∞ = B∞
This result must be noticed since it means, at least in the case where
∗
σ < 1, that there is no link between the long run interest rate R∞
and
∗
the long run return for investments in environmental projects, Ω∞ .
As before we can also construct yield curves T 7→ Ω∗ (T ). From Proposition 14, we see that yield curves for the ecological discount rate differ
from yield curves for Ω by a purely economic term that vanishes in the
long run. However, this term is important since it embeds the wealth
of the whole economy. Indeed, this term ( σ−1
ln(ρ∗0 )) is decreasing in
σ2 T
x∗0 (which is a good proxy for the total wealth of the economy without
liquidity constraint) and turns out to be negative for large values of x∗0 .
14 Here
we cannot just sum the ω’s because the two goods involved are different
Therefore, the shape of the Ω yield curve factors in the total wealth of
the economy, even though this dimension vanishes asymptotically.
The results of our simulations are presented below and we can see an
hump-shaped yield curve that appears because of the complex combination of the wealth effect with the two traditional effects discussed above
(depending on σ and σ 0 ).
Figure 4: Yield Curve for Ω (σ = 0.8, σ 0 = 1.5, r = 2%, δ = 0.1%, y x∗0 )
Figure 5: Yield Curve for Ω (σ = 0.8, σ 0 = 1.5, r = 2%, δ = 0.1%, y ' x∗0 )
Figure 6: Yield Curve for Ω (σ = 0.8, σ 0 = 1.5, r = 2%, δ = 0.1%, y x∗0 )
Without the wealth effect, the curves would be increasing. However,
the wealth effect can be of great importance in the short run and we see
that a poor country is characterized by large rates in the short run (and
therefore a decreasing curve at least for the first years) whereas for a rich
country the rates can be negative during dozens of years.
Conclusion
We have developed a notion of discount rate for environmental goods.
This notion allowed us to transpose financial yield curves into a sustainable development framework. Properties of these yield curves, both
asymptotic and for finite horizons, have been derived.
If we believe in a low substituability between environment and consumption then, we have also shown the disconnection between classical
interest rates (or growth) and ecological discount rates: in the long run,
one need to use very low discount rates to evaluate projects or investments
that aim at improving the environment. Hence, we also proved that the
classical argument used against sustainable development, that it is nonsense to invest for generations that are going to be richer, is not relevant.
Finally, our application to environmental return on investments exhibited the differences between poor and rich countries when it comes to
invest in environmental projects. In the short or medium run, our results
plead for transfer from North to South.
References
[AH92]
P. Aghion and P. Howitt. A model of growth through creative
destruction. Econometrica, 60(2), Mar. 1992.
[GLZ07] O. Guéant, J.-M. Lasry, and D. Zerbib. Autour des taux
d’intérêt écologiques.
Cahiers de la Chaire Finance et
Développement Durable, (3), 2007.
[Gue04] R. Guesnerie. Calcul économique et developpement durable.
Revue Economique, 55, 2004.
[Hai00a] J.-O. Hairault.
Analyse macroéconomique, tome 1.
Découverte, 2000.
La
[Hai00b] J.-O. Hairault.
Analyse macroéconomique, tome 2.
Découverte, 2000.
La
[Hot31]
H. Hotelling. The economics of exhaustible resources. The Journal of Political Economy, 39(2), Apr. 1931.
[Ste06]
N. Stern. Stern review on the economics of climate change. Oct.
2006.
[Wei01] M. Weitzman. Gamma discounting. The American Economic
Review, 91(1), Mar. 2001.
Part II - Chapter 2
Negative Discount Rates
Non Positive Discount Rate
Abstract
This chapter must be seen more as an appendix than as a proper
chapter since most of the results are often assumed to be known in many
articles.
It discusses the possibility to consider non positive discount rate in optimization problems such as those of growth models. If a positive discount
rate is often assumed, in some circumstances a discount rate equal to zero
or even negative is certainly relevant. Examples of such circumstances
can easily be found in the field of sustainable development where we could
want to give equal weight to the present and the future. Our goal is to
show how randomness can lead to the possibility to consider small or even
negative discount rates.
Introduction
Most optimization problems seem to require for convergence reasons (and
therefore for bad reasons from an economist viewpoint) a positive discount
rate. If one consider the Ramsey problem where an agent has to maximize
(under wealth constraints) her intertemporal utility given by:
Z ∞
u(c(t))e−ρt dt
0
we often require ρ to be positive and therefore different from 0 whereas a
discount rate equal to nought could make sense for sustainable development issues.
In what follows, we are going to show that the introduction of a noise
in the Ramsey optimization problem can allow for lower discount rates,
sometimes even strictly negative, depending on the intensity of the noise
and on the concavity of the utility function. This idea is not new and the
introduction of a Poisson process is the best example to allow for smaller
purely psychological discount rates. Here however, we want to focus on
the usual introduction of a brownian motion in optimization problems
and show that the more randomness we introduce, the smaller or the
more negative discount rates can be. This is particularly interesting for
sustainable development problems since they are characterized by both
randomness and the willingness to set small or negative discount rates.
In the first section we are going to recall the framework of the Ramsey
model. A second section will be dedicated to its rigorous resolution. The
third section will generalize the computations to a random framework.
1
The optimization problem without noise
The problem we are dealing with is the traditional optimization problem
of an agent with a given wealth who has to decide whether to save or
consume so as to maximize her intertemporal utility.
Due to the introduction later on of a noise, we do prefer to set up the
problem in continuous time and therefore the optimization problem can
be written as:
R∞
supc(·) 0 u(c(t))e−ρt dt
s.t. da(t) = ra(t)dt − c(t)dt
a(0) given
This problem is a priori not well defined since the integral value can
be infinite and therefore we must consider first the problem with a finite
horizon.
For any given T , we will define cT (t) as the solution of the following
optimization problem:
RT
supcT (·) 0 u(cT (t))e−ρt dt
s.t. da(t) = ra(t)dt − cT (t)dt
a(0) given
Our goal is to know the possible values for ρ such that the sequence of
functions cT (·) converges (uniformly on all compacts) towards a function
c(·) which defines a solution to the first optimization problem1 .
To make the problem tractable, we are going to consider a family of
utility functions that is:
c1−θ
,
θ ∈ (0, 1)
1−θ
First we are going to demonstrate the classical requirement for ρ and
show that, for the problem to have a solution, one need to have ρ > (1−θ)r
and therefore ρ positive.
Then we will introduce noise in the wealth constraint and show how the
introduction of randomness is useful to allow lower values of the discount
factor ρ.
u(c) =
2
Resolution
To start with, we define the Bellman function JT associated with the finite
horizon T .
RT
JT (t, a) = sup(cT (s))s≥t t u(cT (s))e−ρ(s−t) ds,
s.t. da(s) = ra(s)ds − cT (s)ds
a(t) = a
1 The idea here is that we “avoid” the often confusing issue (at least ex ante) of the
transversality condition to define properly the first problem.
Proposition 1. ∀T > 0, JT is solution of the following PDE:
(
1
θ
(∂a JT )1− θ − ρJT + ∂t JT + ra∂a JT = 0
1−θ
s.t. JT (T, a) = 0, ∀a ≥ 0
Also, we have
cT (t)−θ = ∂a JT (t, a(t))
Proof:
The function JT satisfies the following Hamilton-Jacobi Bellman PDE:
supc u(c) − ρJT + ∂t JT + ∂a JT (ra − c) = 0
⇐⇒ [supc u(c) − ∂a JT c] − ρJT + ∂t JT + ra∂a JT = 0
with the condition JT (T, a) = 0, ∀a ≥ 0
Since we specified the utility function u, the control parameter c can
easily be deduced from that equation and we have:
u0 (c) = c−θ = ∂a JT (t, a)
So that the equation can be rewritten in the form presented above.
Proposition 2. Assume that ρ > (1 − θ)r.
Then, ∀T > 0, JT can be written in the following form (for t ∈ [0, T ]):
JT (t, a) = K [1 − exp(−ζ(T − t))]θ a1−θ
where K and ζ are two constants.
Importantly,
ρ − (1 − θ)r
ζ=
θ
Proof:
First of all the terminal condition for JT is verified.
Let’s define for convenience φ(t) = 1−exp(−ζ(T −t)) so that JT (t, a) =
Kφ(t)θ a1−θ .
Now, just replace this form in the PDE. This gives:
1− 1
θ θ
K(1 − θ)φ(t)θ a−θ
− ρKφ(t)θ a1−θ + Kθφ0 (t)φ(t)θ−1 a1−θ
1−θ
+rK(1 − θ)φ(t)θ a1−θ = 0
1
1
θK 1− θ (1 − θ)− θ − ρKφ(t) + Kθφ0 (t) + rK(1 − θ)φ(t) = 0
But since φ0 (t) = ζ(φ(t) − 1) we will find the two constants using the
two equations:
1
1
θK 1− θ (1 − θ)− θ = Kθζ
−ρK + ζKθ + rK(1 − θ) = 0
−1
1
K θ (1 − θ)− θ = ζ
⇐⇒
−ρ + ζθ + r(1 − θ) = 0
⇐⇒
K(1 − θ) = ζ −θ
ζ = ρ−(1−θ)r
θ
Since ζ > 0, K is well defined and we have indeed exhibited a solution
of the HJB equation.
Proposition 3. Assume that ρ > (1 − θ)r.
Then, ∀T > 0, cT can be written in the following form (for t ∈ [0, T )):
cT (t) = a(0)
Z t
ζ
ζ
exp rt −
ds
1 − exp(−ζ(T − t))
0 1 − exp(−ζ(T − s))
Proof:
We know that cT satisfies:
cT (t)−θ = ∂a JT (t, a(t))
Therefore:
cT (t)−θ = K(1 − θ) [1 − exp(−ζ(T − t))]θ a(t)−θ
⇒ cT (t) = ζ [1 − exp(−ζ(T − t))]−1 a(t)
As a consequence, going back to the wealth constraint:
da(t) = ra(t)dt − ζ [1 − exp(−ζ(T − t))]−1 a(t)dt
Z t
ζ
⇒ a(t) = a(0) exp rt −
ds
0 1 − exp(−ζ(T − s))
This gives the result:
cT (t) = a(0)
Z t
ζ
ζ
exp rt −
ds
1 − exp(−ζ(T − t))
0 1 − exp(−ζ(T − s))
It’s now very easy to derive the final result in the deterministic case.
Proposition 4 (Local Uniform Convergence in the Deterministic
Case). Assume that ρ > (1 − θ)r.
Let’s consider a given T1 > 0. cT (·) converges uniformly on [0, T1 ], as
T → +∞, towards a function c(·) defined (independently of T1 ) by:
r − ρ c(t) = a(0)ζ exp ((r − ζ)t) = a(0)ζ exp
t
θ
Proof:
Let’s introduce R(t, T ) =
ζ
.
1−exp(−ζ(T −t))
∀t ∈ [0, T1 ], lim
T →+∞
The key point is that:
ζ
=ζ
1 − exp(−ζ(T − t))
Moreover,
∀t ∈ [0, T1 ], ζ
ζ
− ζ ≤
−ζ
1 − exp(−ζ(T − t))
1 − exp(−ζ(T − T1 ))
The uniform convergence on any set [0, T1 ] can now be deduced easily.
This proposition shows that for ρ > (1 − θ)r the problem is well defined. However, if θ ∈ (0, 1) is fixed, it’s impossible to consider very low
discount rates and a fortiori a non positive ρ.
This is the reason why we are going to introduce noise in the next paragraphs to allow for lower discount rates.
Surprisingly perhaps, all the methods developed above will work mutatis
mutandis.
3
The problem with noise
The introduction of noise will be done with a brownian motion in the
wealth constraint. For each horizon T , the agent problem will be the
following:
(
supcT (·) E
hR
T
0
i
u(cT (t))e−ρt dt
s.t. da(t) = ra(t)dt − cT (t)dt + σa(t)dW (t)
a(0) given
As before, we continue to suppose that:
u(c) =
c1−θ
,
1−θ
θ ∈ (0, 1)
We define the Bellman function JT associated with the finite horizon
T.
(
JT (t, a) = sup(cT (s))s≥t Et
hR
T
t
i
u(cT (s))e−ρ(s−t) ds ,
a(t) = a
s.t. da(s) = ra(s)ds − cT (s)ds + σa(s)dW (s)
Proposition 5. ∀T > 0, JT is solution of the following PDE:
(
1
2
θ
(∂a JT )1− θ − ρJT + ∂t JT + ra∂a JT + 12 σ 2 a2 ∂aa
JT = 0
1−θ
s.t. JT (T, a) = 0, ∀a ≥ 0
Also, we have as in the deterministic case
cT (t)−θ = ∂a JT (t, a(t))
Proof:
The function JT satisfies the following Hamilton-Jacobi Bellman PDE:
supc u(c) − ρJT + ∂t JT + ∂a JT (ra − c) +
1 2 2 2
σ a ∂aa JT = 0
2
⇐⇒ [supc u(c) − ∂a JT c] − ρJT + ∂t JT + ra∂a JT +
1 2 2 2
σ a ∂aa JT = 0
2
with the condition JT (T, a) = 0, ∀a ≥ 0
Therefore, the reasoning is the same as in the deterministic case with
the convexity term in addition in the PDE and the proposition is proved
by the same arguments.
Proposition 6. Assume that ρ > (1 − θ)(r − 12 σ 2 ).
Then, ∀T > 0, JT can be written in the following form (for t ∈ [0, T ]):
JT (t, a) = K [1 − exp(−η(T − t))]θ a1−θ
where K and η are two constants.
Importantly,
ρ − (1 − θ)(r − 12 σ 2 )
η=
θ
Proof:
First of all the terminal condition for JT is verified.
Let’s define for convenience φ(t) = 1 − exp(−η(T − t)) as before and
replace the expression for JT in the PDE. This gives:
1− 1
θ θ
K(1 − θ)φ(t)θ a−θ
− ρKφ(t)θ a1−θ
1−θ
1
+Kθφ0 (t)φ(t)θ−1 a1−θ + rK(1 − θ)φ(t)θ a1−θ − σ 2 Kθ(1 − θ)φ(t)θ a1−θ = 0
2
1
1
1
θK 1− θ (1−θ)− θ −ρKφ(t)+Kθφ0 (t)+rK(1−θ)φ(t)− σ 2 Kθ(1−θ)φ(t) = 0
2
But, since φ0 (t) = η(φ(t) − 1) we will find the two constants using the
two equations:
1
1
θK 1− θ (1 − θ)− θ = Kθη
−ρK + ηKθ + rK(1 − θ) − 12 σ 2 Kθ(1 − θ) = 0
−1
1
K θ (1 − θ)− θ = η
⇐⇒
−ρ + ηθ + r(1 − θ) − 12 σ 2 θ(1 − θ) = 0
(
K(1 − θ) = η −θ
⇐⇒
ρ+ 1
σ 2 (1−θ)−(1−θ)r
2
η=
θ
Since η > 0, K is well defined and we have indeed exhibited a solution
of the HJB equation.
We need to notice that the set of admissible ρ’s is larger now due to
the diffusion parameter.
Proposition 7. Assume that ρ > (1 − θ)(r − 12 σ 2 ).
Then, ∀T > 0, cT can be written in the following form (for t ∈ [0, T )):
Z t
η
1 2
η
cT (t) = a(0)
exp (r − σ )t −
ds + σW (t)
1 − exp(−η(T − t))
2
0 1 − exp(−η(T − s))
Proof:
We know that cT satisfies:
cT (t)−θ = ∂a JT (t, a(t))
Therefore:
cT (t)−θ = K(1 − θ) [1 − exp(−η(T − t))]θ a(t)−θ
⇒ cT (t) = η [1 − exp(−η(T − t))]−1 a(t)
As a consequence, going back to the wealth constraint:
da(t) = ra(t)dt − η [1 − exp(−η(T − t))]−1 a(t)dt + σa(t)dW (t)
Using Ito’s lemma we have:
Z t
1 2
η
⇒ a(t) = a(0) exp (r − σ )t −
ds + σW (t)
2
0 1 − exp(−η(T − s))
This gives the result.
It’s now very easy to derive the final result in the stochastic case.
Proposition 8 (Local Uniform Convergence in the Stochastic
Case). Assume ρ > (1 − θ)(r − 12 σ 2 ).
Then, ∀T1 > 0, cT (·) converges in L∞ ([0, T1 ]), as T → +∞, towards a
function c(·) defined (independently of T1 ) by:
1
c(t) = a(0)η exp (r − σ 2 − η)t + σW (t)
2
!
r − ρ − 21 σ 2
= a(0)η exp
t + σW (t)
θ
Proof:
The problem is exactly the same as before. The only thing to be
careful with is exp (σW (t)) but esssupt∈[0,T1 ] exp (σW (t)) < +∞ so that
there isn’t any problem.
We have seen that the initial problem is well defined if the inequality
ρ > (1 − θ)(r − 12 σ 2 ) is satisfied. Consequently, if the noise is important
enough, that is if r < 12 σ 2 , the set of admissible ρ’s contains non positive
numbers.
Conclusion
Without noise, it’s often impossible to consider very low discount rates or
even non positive values for ρ.
The introduction of randomness has been shown to allow for lower values
of ρ. Interestingly, the more random the environment is, the lower the
discount rate can be. This is particularly relevant for sustainable development issues since it’s the archetypal situation where the uncertainty (the
noise) is important and where we could want to consider a very low ρ,
typically equal to 0 or even negative.
Conclusion
In this dissertation we presented mean field games through applications. We
tried to present applications in various fields but we also tried to present various types of mean field games. The first chapter deals indeed with a dynamical
mean field game with two states and the structure of the model is typically the
structure of a mean field game with a discrete state space (system of ODEs).
This type of game is particularly relevant to deal with club theory, as far as dynamical aspects are concerned. The second chapter is a static mean field game
and this game is not of the forward/backward type. Chapter 3 is the archetype
of a mean field game with a continuum of states in a random environment.
Partial differential equations associated to this game are those of the seminal
articles by J.-M. Lasry et P.-L. Lions. The last two chapters of the first part
use dynamical mean field games with a continuous state space but the games
are embedded in deterministic models of growth.
Hence, we have tried to present the different ways mean field games could be
used in applications.
Also, even though this seems to be only in the appendix of chapter 3, one of
our contributions to the mean field game theory concerns numerical methods
to solve the PDEs. Inspired by the notion of eductive stability, we exhibited a
general method to solve forward/backward equations.
New applications are now in progress, concerning for instance the international
market of oil. We believe that mean field game theory and the forward/backward
approach will help to solve numerous problems in economics.
As far as the secondary topic is concerned, an article is to be submitted in collaboration with J.-M. Lasry and R. Guesnerie. We do think that the concept
of ecological rate is relevant for economic decision making linked with environmental issues.
Conclusion
Dans cette thèse nous avons présenté les jeux à champ moyen dans un contexte
appliqué. Nous nous sommes efforcés de présenter des problèmes non seulement dans divers domaines économiques mais aussi dans divers domaines de la
théorie des jeux à champ moyen elle-même. Ainsi, le premier chapitre a pour
support un jeu à champ moyen dynamique dont la structure est typique des
jeux à champ moyen à espace d’états discret (système d’EDO). Ce type de jeu
apparaı̂t naturellement dans d’autres domaines comme dans la théorie des clubs,
théorie pour laquelle les jeux à champ moyen sont très utiles pour étudier les
problèmes dynamiques. Le second chapitre est un jeu à champ moyen statique,
l’aspect forward étant remplacé par une équation de bouclage. Le chapitre 3 est
l’archétype d’un jeu à champ moyen dynamique à espace d’états continu avec un
aspect stochastique. Les équations aux dérivées partielles de ce chapitre sont
celles des articles initiaux de J.-M. Lasry et P.-L. Lions. Les deux chapitres
suivants sont encore des jeux à champ moyen dynamiques à espace d’états continu mais sont, eux, déterministes.
Ainsi, nous avons présentés les divers aspects sous lesquels les jeux à champ
moyen pouvaient apparaı̂tre ou être utilisés.
Aussi, même si cela n’apparaı̂t que dans l’annexe au chapitre 3, l’une des contributions de ce texte, au-delà même des exemples d’application, est l’aspect
“méthodes numériques”. Si celles-ci ne constituent pas le cœur de notre sujet,
nous avons avancé sur ce sujet délicat de la résolution numérique de jeux à
champ moyen de manière drastique en nous inspirant du concept de stabilité
éductive et en l’utilisant à des fins algorithmiques, tant pour la résolution de
problèmes stationnaires que pour des problèmes dynamiques complexes.
De nouvelles applications des jeux à champ moyen sont aujourd’hui en cours
d’écriture, notamment dans le domaine des marchés pétroliers et il faut croire
que la théorie des jeux à champ moyen a de beaux jours devant elle.
Concernant maintenant le second sujet de la thèse, un article est en cours
d’écriture avec R. Guesnerie et J.-M. Lasry. Nous pensons que la notion de
taux écologique est pertinente dans le cadre de la gestion de projets environnementaux ainsi que pour justifier certains transferts Nord-Sud en matière technologique.
Vu : le Président
Vu : les suffragants
M._____________
M.______________
Vu et permis d’imprimer :
Le Vice-Président du Conseil Scientifique Chargé de la Recherche de l’Université Paris
Dauphine
Résumé :
Introduite par J.-M. Lasry et P.-L. Lions, la théorie des jeux à champ moyen
simplifie les interactions entre agents économiques selon une approche inspirée
des théories physiques. Des applications économiques sont présentées concernant le marché du travail, la gestion d’actifs, les problèmes de répartition de
population(s), ainsi que la théorie de la croissance. Les modèles présentés
utilisent la théorie des jeux à champ moyen sous des formes diverses, parfois statiques, souvent dynamiques, à espace d’états discret ou continu et dans
un environnement déterministe ou stochastique. Diverses notions de stabilité
sont discutées dont la notion de stabilité éductive, qui a inspiré des méthodes
numériques de résolution. Nous présentons en effet des méthodes numériques
qui permettent d’obtenir des solutions, tant aux problèmes stationnaires qu’aux
problèmes dynamiques, en s’abstrayant de la structure forward/backward, a priori problématique d’un point de vue numérique. En marge de la théorie des jeux
à champ moyen, la problématique des taux d’escompte idoines pour traiter des
problèmes de développement durable est abordée. Nous discutons de la notion
de taux écologique introduite par R. Guesnerie et apportons des propriétés non
asymptotiques nouvelles, de continuité notamment.
Mots clés : Equations aux dérivées partielles - Théorie des jeux - Contrôle
optimal - Croissance économique - Développement durable
Summary:
Introduced by J.-M. Lasry and P.-L. Lions, mean field games simplify models
of economic interactions, using an approach inspired from physics. Economic
applications are presented concerning labor market economics, portfolio management, population(s) distribution issues and growth theory. The models use
different types of mean field games, either static or dynamic, with either a discrete or a continuous state space, either in a deterministic or in a stochastic
setting. Several stability notions are discussed and eductive stability plays an
important part since numerical methods are inspired from this stability notion.
We indeed present numerical methods to solve mean field games for both stationary and dynamic problems and eductive stability allows us to circumvent
the difficulty linked to the forward/backward structure. After the chapters on
mean field games, we deal with the issue regarding the right discount rate to be
used for sustainable development projects. We discuss the notion of ecological
discount rate introduced by R. Guesnerie and exhibit new continuity properties
for the non-asymptotic rates.
Keywords: Partial differential equations - Game theory - Optimal control Economic growth - Sustainable development

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