1
A Mathematical Framework for
Evolutionary Economics
Jean-Pierre Aubin1 .
draft of February 21, 2017
Acknowledgments
The author dedicates this “economic program” to Pierre-Marie Larnac, to whom he owes
so much. Sadly, Pierre-Marie will not know it and the author is deprived of his critics and
comments. He thanks warmly Jean Cartelier and Jean-Philippe Terreaux for their relevant
comments, as well as Valérie Angeon, Pierre-Cyril Aubin-Frankowski, Alain Bensoussan,
Anya Désilles, Luxi Chen, Philippe Cibois, Olivier Dordan, Marie-Hélène Durand, Hélène
Frankowska, Georges Haddad, Charles-Albert Lehalle, Vladimir Lozève, Sophie Martin, Harry
Ozier-Lafontaine, Terry Rockafellar, Patrick Saint-Pierre, Roger Wets and other colleagues
of the ANR project GAIA-TROP, for friendly disputations about the viability approach
advocated in this preliminary study.
Abstract
We suggest a mathematical description of economic “variables”, such as commodities, actors,
values (quantities of numéraire), prices, durations, etc., as “evolutions” depending on time
and ranging over vector spaces. The states of these variables (stocks) are “related” at each
instant by “inert relations” and, together with their time derivatives (flows), by “kinetic
relations”. When inert and kinematic relations are given independently of each other, it may
happen than some, all or none evolutions are “viable” in the sense that they satisfy both inert
and kinematic relations. Hence the task is to reduce the inert and kinematic relations for
restoring the viability requirement: the reduced inert relation is called its “viability kernel”
and the restricted kinematic relation is called its “regulator”. This regulator governs the
viable economic evolutions of these variables and their derivatives. For instance, it allows
us to describe one of the variables, say, the endowment in means of payment of an economy,
as a function of all remaining variables. This choice can be interpreted as a “budgetary
rule” depending on both inert and kinetic relations. It may happen that a priori budgetary
rules designed independently of inert and kinematic relations, such as variants of the Walras
law, do not fit this budgetary rule, and thus trigger the dysviability of the economy. They
are indeed sufficient (but not necessary) conditions designed for proving the existence of an
equilibrium, a state which does not evolve, not for governing viable evolutions.
1
[email protected], http://vimades.com/aubin,
Société VIMADES (Viabilité, Marchés, Automatique et Décision), http://vimades.com
2
In a Nutshell
We suggest a mathematical description of economic “variables”, such as commodities, actors,
numéraire, prices, durations, etc. So, interpreting the word “variable” literally, they vary. If
they vary in function of time, they are “evolutions’ ’: they emerge and die at the two edges
of a temporal window, like the actors who appropriate them. This simple remark triggered
the suggestion to regard them not as instantaneous states or stocks, but as evolving ones.
Knowing evolutions, we know also their time derivative, or flow.
For instance, the derivative of a good describes its production if it is positive, a consumption if negative. If an economic actor is regarded as an “actor” of a given good, its
“(owner) share” evolves in time and its derivative describes an investment if it is positive, a
divestment if negative. A “value”, denoting a quantity of numéraire, can be regarded as an
endowment in means of payments (or a wealth), and its derivative as a credit if its positive,
a dept if negative. Prices of commodities evolve too, and their derivatives can be regarded
as price fluctuations. The sign of the price of the numéraire describes an inflation if it is
positive, a deflation if negative.
Various products of the components of these variables and of their derivatives describe
at each instant a particular economic qualitative behavior. For instance, if the product of
the derivatives of two goods is negative, an increase of quantity of the first good triggers the
decrease of the second good, but an increase if the product is positive.
These variables are “related”. We agree to use the adjective “inert” to denote the properties of an evolution, “kinematic” the ones involving also its derivative. We thus distinguish
two types of relations: at each instant,
1. “inert relation” involving only the states of the evolution, for instance, requiring that
both goods must abide by eudemonistic and scarcity constraints, examples of “viability
constraints”;
2. “kinematic relation” involving also both the variables and their derivatives, for describing various “processes” or engines of evolutions, emerging at a given state, or dying at
another one, or linking both an emerging and dying state (geodesic).
Do these evolutions satisfy both the inert and kinematic relations? If so, they are called
viable. Answering this “viability problem” is a preliminary task that must be completed
before asking other questions.
When inert and kinematic relations are given independently of each other, it may happen
than some of some, all or none evolutions satisfy both of the inert and kinematic relations.
Hence the task that we need to accomplish is to restrict the least possible the inert
and kinematic relations for restoring this viability requirement. In this case, the restricted
3
inert relation is called its “viability kernel” and the restricted kinematic relation is called its
“regulator”.
Knowing the regulator providing the relations between all the variables and their velocities, we can derive as many (set-valued) maps as partitions in two groups of the related
variables, the first one regarded as the input, the second as the output. For example, we may
assume that a “central banker” governs the creation of the endowment of means of payment,
which is the derivative of the value. Hence the regulator can be interpreted as a “budgetary
regulator” when we choose for output the endowment and for input the remaining variables
and their derivatives.
There is no reason why simplistic budgetary rules figure among the ones provided by
the budgetary regulator associated with the inert and kinematic constraints. Imposing these
arbitrary budgetary rules independently of inert and kinematic relations against all odds
may lead to pauperization of a some actors when their shares vanishes to the advantage
of the others. Such situation could be called “humanruptcy”, which happened much more
frequently than bankruptcy of banks, private of even central, as economic history shows. The
lessons of history are often and easily forgotten while, paradoxically, forecasts are made all
the time. History2 is much more important than mathematics for understanding economic
evolution. However, mathematics teach at least that budgetary rules designed only by
arbitrary or ill-conceived “quantitative” methods are doomed to fail. They also suggest
that qualitative methods should be used at least for working out relevant, coherent and
sound concepts.
Meanwhile, the adjective “political” should be reestablished in front of “economy” for
taking into account also the lives of the actors and the social relations they induce.
2
As well as ethology, sociology, cognitive and behavioral sciences, etc., economics requires polymathy (in
French, “polymathie” is defined in the Diderot encyclopedia by “ connoissance de plusieurs arts & sciences,
grande & vaste étendue de connoissances différentes.
4
1
From Qualitative Catallaxy to Quantitative Economy
Ludwig von Mises introduced the word catallaxy 3 , derived from the Greek verb katalatto,
“to exchange”, for denoting the science of exchange of “things”. Such process started when
sexual reproduction introduced the exchange of gametes, forcing the organisms to meet at
least once for extending their lineage after death4 .
Human beings are in some sense “digital living beings” possessing the unique mathematical capability of counting and playing with numbers. So that they isolated the “things”
endowed with a unit of measure allowing them to measure their quantity with numbers.
These “quantifiable things” are called goods, victims of the pantometric drug denounced by
Friedrich Hayek 5 .
Here, we define “economy” 6 as the restriction of catallaxy to the exchange of goods,
the quantitative “things”, letting aside the qualitative things. Using numbers, the analysis
aspired to became a “science” by dropping the adjective “political”, describing the actors
who trigger economic activity7 , including production and consumption of goods, as well as
their exchange. Commodities are sets (baskets) of quantities of goods, known in mathematics
as vectors, ranging over finite dimensional vector spaces, the dimension being the number of
goods used in the basket. Hence,
the states of economic commodities live in vector spaces.
This allows us to use linear algebra and differential calculus, familiar and rich mathematical
structures, allowing us to add commodities by adding their quantities of goods, to multiply
them by a number by multiplying the quantities of the goods by this number and to construct
evolutionary systems, evolutionary engines governing their evolution. This is the origin of
3
In his book Human Action, [52, von Mises], in which he attributed this term to Richard Whately in his
Introductory Lectures on Political Economy published in 1831, in which he wrote “the name I should have
preferred as the most descriptive, and on the whole least objectionable, is that of catallactics, or the ”Science
of Exchanges”.
4
Catallaxy was also studied by ethologists. Recent work by [55, Pelé & Sueur], [63, Sueur], etc., analyzed
in various species of macaque collective decision-making mechanisms (coordination of movements) more
sophisticated than those of sheep, perhaps simpler than those of speculators. Their study could shed some
light on mimetic learning mechanisms that take into account the size of coalitions of individuals proponents
of collective behavior and their “negotiations” during exchanges. See also La valeur n’existe pas. À moins
que [...], [14, Aubin], for the emergence of exchanges among children.
5
In his book The Fatal Conceit, [36, Hayek]. See also Section 1.2, p. 10, of Time and Money. How Long
and How Much Money is Needed to Regulate a Viable Economy, [13, Aubin].
6
Economics is the field of the relevance of the “nomos”, the management, of the “oiko”, the house (which
is becoming the whole world).
7
Beyond utility or ophelimity functions, for taking into their viability in the literal meaning of the word,
the problem is to avoid “humanruptcy”, or “persuccy, from Italian “persona uccisa”, instead of “banca
rotta”.
5
the intrusion of mathematics to provide mathematical metaphors of economic statements,
i.e., mathematical stories told to economists by uconomists8 .
1 [Uconomia] Once translated in their formal language leaving no room for polysemy,
mathematicians provide statements describing how assumptions imply conclusions, relevant
or not outside mathematics. Assumptions are the price to pay for the cost of the proofs
of the conclusions. The “weaker” is the assumption, the “stronger” is the theorem stating
the conclusions. Nobody is forced to pay this price by checking everything. Mathematical
details are not necessary to understand, i.e., to validate, the “economic messages” translating that assumptions imply conclusions through mathematical metaphors. The task of
mathematicians is to ensure that they are logically true, the one of economists is to check
that conclusions and assumptions are economically meaningful.
Uconomia, the “u” (not in Greek) being the same one that appears in utopia and uchronia,
for example, could describe better “mathematical economics”, since mathematicians motivated by economics are tempted to take their theorems for reality.
Mathematics are not only “quantitative”, as often required by its “clients”, but also “qualitative”. The number of variables is too important for grasping a reality, even though the
computers can treat “big data”. However, these qualitative mathematics can tell economic
stories (which may be fairy tells, but this is up to the reader to “believe” them). Even though
theorems and economic statements have no value (by lack of units of measure), theorems
compose at least a “free lunch” cooked by mathematicians for economists if they are willing
to make them “exist” in economics.
Our motivations are economics, the end results are mathematics. Are they applicable?
Only economists can answer this question.
For dealing mathematically with all other goods deprived of units of measure, we have
to abandon the linear structure, and loose the usual notion of velocity, and all the results
of differential calculus. However, it is possible mathematically to replace vector spaces and
linear algebra by mutational spaces 9 for adapting in a near future the linear properties to
8
Uconomia has been introduced in Section 5.2 of [12, La mort du devin, l’émergence du démiurge]. See
also Section 1.6, p. 23, of Time and Money, [13, Aubin] for more developments.
9
See for instance Mutational and Morphological Analysis: Tools for Shape Regulation and Morphogenesis,
[11, Aubin], and, above all, Mutational Analysis, [48, Lorenz], by Thomas Lorenz, exposing mutational and
morphological structures. Morphological equations are mutational equations restricted to the hyperspace of
subsets of a vector space. They govern the evolution of subsets in the same way that differential equations
govern the evolution of vectors. Algebraic structures of hyperspaces have been extensively studied since their
introduction by George Boole, above all when the spaces underlying hyperspaces are vector spaces (see [1,
Akian, Quadrat J.-P. & Viot], [2, Akian, Gaubert & Kolokoltsov], [3, Akian, Bapat & Gaubert], [4, Akian,
David & Gaubert], [21, Aubin & Dordan], etc.).
6
hyperspaces, the families of subsets of a space, including the space itself, the largest one, and
the empty set, the smallest one. Their study has been neglected, yet they enjoy Boolean
operations (intersection, complement, etc.). Instead of adding commodities and checking
they are smaller than a scarce one, one can study the union of sets of qualitative things and
check whether it is contained in a set describing scarcity. “Morphological equations govern
the evolution of subsets, and thus, the co-viability of evolutions of vectors and sets to which
they belong. Numbers are no longer used to measure the commodities, but most results of
differential calculus (and the mathematical stories they tell) can, in principle, be adapted
to qualitative “things” and not only for the quantitative commodities. Since the number of
relevant variables is immense and since no functional whatsoever can faithfully summarize
them, even though more powerful computer will produce “bigger and bigger data”, they
will not be grasped by human brains for a while and, in the best of cases, remain pointless
and dangerous when taken as truths to which every one should trust blindly. Quantitative
economy should leave room to qualitative catallaxy. Fortunately, mathematics produce and
will produce new concepts and methods, and some mathematicians take and will take the
risk to try the adventure if the academic system is wise enough to provide them the duration
and the means to do so without publishing too fast and to much and be punished by the
most stupid quantified system of evaluation that mankind has ever invented. There is no
unit of measure of knowledge 10 .
This study deals with the evolution of commodities and their processes (production,
consumption, storing or hoarding, etc.) involving their derivatives.
The constraints on commodities incorporate different types of viability constraints, including
1. well-fare or eudemonistic 11 constraints,
2. scarcity constrains, including preservation of personal freedom, constraints or territories, etc.
The problem we investigate is: what are the processes governing the evolutions of the
“commodities” such that these viability constraints are satisfied at each instant?
Evolution requires the polysemous concept of time, meaning both dates and durations,
which allow us to define temporal windows on which the current “time” evolves. So, except
at equilibria, commodities and other economic variables do evolve, are produced, stored,
10
See Section 3.5, Le commerce des savoirs, pp. 203-239, of La mort du devin, l’émergence du démiurge.
Essai sur la contingence, la viabilité et l’inertie des systèmes, [12, Aubin].
11
From “eu” (good) and “daimon” (spirit), Eudemonia being a minor deity, a philosophic attitude, different
from hedonism (the search of pleasure), including satisfaction, which is not measured by ophelimity functions
that Vilfredo Pareto dreamed to measure by a kind of utility function. Utility functions were introduced by
Daniel Bernoulli and Gabriel Cramer, and their derivatives (marginal utilities) by Léon Walras, William
Jevons, Carl Menger, the corner stone of the rational behavior castle, which we neither assume nor exclude.
7
consumed, etc. Commodities are not static, each of them has an history, subjected to initial
production and final consumption.
Evolutions of commodities are “elaborated” by economic actors, also subjected to birth
and death, and to viability constraints meanwhile. They are sometime producers of commodities when they increase, sometime consumers when they decrease, transactors or exchangers when they remain constant.
In the economic discourses the elaboration process is called “labor”, and is implicitly
described by a one-dimensional good, playing a fundamental role after time and duration,
and is independent of exchange (which, by the way, requires commercial labor by actors)
and of the measure instruments such as numéraire and prices. It has its own value, the labor
value, which is then difficult to define by lack of a rigorous definition of labor or work.
For William Petty, “work is the father and nature the mother of wealth”, for David Hume,
“every thing in the world is purchased by labor”, for Adam Smith, the value of a commodity
“is equal to the quantity of labor which it enables him to purchase or command [...] and
labor is the real measure of the exchangeable value of all commodities”, for David Ricardo,
labor is measured on temporal window of fixed duration, for Karl Marx, both working force
and duration are involved and for Alfred Marshall, duration, supply and demand play an
important role as a component of value for small durations, whereas, at long-term, working
time in involved.
For simplicity, we may represent an actor’s elaborations by the “shares” of the actors in
the elaboration 12 process, which also evolve.
Some of the commodities, the ones which possess unit of measures, can be measured by
numéraire by using prices. They, too, evolve, not for their own sake, but for processing
information on the evolution of commodities satisfying the constraints. This may be an
opposite view to familiar approaches, the Walrasian one, for instance, which studies the
evolution of prices, obeying an a priori “budgetary law”, independent of the well-fare and
scarcity constraints. This law may be a necessary condition for proving the existence of a
Walrasian equilibrium of price (under strong assumptions), but by no means sufficient. And
equilibria, in the sense of stationary evolutions, do not exist in life sciences. Restricting the
12
The term “ elaboration ”, from the Latin elaboratus, involves the notion of labor, and being less used, is
not very polysemous. We shall use the adjective elaborative and coin the word elaborativity for describing
the first derivative with respect to time of an evolution of the share of elaboration (or marginal elaboration).
We could have used the word “labor” instead of elaboration, except that labor is not a really well defined
concept, or rather, a polysemous one. The elaboration (or work, labor, etc.) is first a transformation process
of a socio-economic environment. Mathematics can help us to restrict this polysemy by multiplying some of
its different meanings, even if introducing different neologisms may seem pedantic. An alternative could be
the use of mathematical symbols, as kinds of ideograms, independent of the language used, in the same way
that sinograms are common to all Chinese dialects.
8
Walrasian tâtonnement to the sole evolution of prices, the evolution of commodities depend
upon the evolution of measurement tools, and not the converse, hoping furthermore that
either an invisible abstract deity (the “Market”) or planners choose prices and numéraire.
Instruments of measure are useful: using thermometers provide the measure of fever for
diagnosing a disease and to cure it for the fever to disappear. You could also change the
thermometer or tamper with a thermometer to show the safe level and leave the patient with
no cure. Inventing the thermometer for measuring the temperature, a single variable with a
unit of measure, was difficult, but much simpler than measuring the million of commodities,
the billion of actors. It is much easier to change the thermometer rather than to maintain
the eudemonistic and scarcity constraints. Instead, it could be wiser to adjust the means
of payments, by trial and error, to the price of cyclic evolutions. Even though mathematics
can provide an answer to state that there exist adequate means of payment to be provided
for maintaining the viability of the economy, no childish analytical formula nor computers,
as powerful as they are, can provide the right amount of means of payments. This is the
only lesson that qualitative mathematics could deliver: get as much quantified information
as you can, but, please, do not break the thermometer by forgetting the qualitative variables
(confidence of actors, for instance) and by forbidding the creation of means of payments
for the actors to live, instead of believing to simplistic and often wrong budgetary rules,
actually, to the assumptions under which they operate13 . Freed, and thinking, human brains
will do better than the most artificially intelligent programs they provide to computers. We
assume that the coalitions of commodity actors are visible: they govern the co-evolution of
the commodities and their measurement tools when they exist.
For taking into account these evolutionary concepts, we may appeal to Maupertuis, the
inventor of the “principle of least action”, reconciliating Fermat, Leibniz and Newton, opening the way to Euler, Lagrange, Hamilton et Jacobi, ..., Richard Feynman, etc., to describe
the evolution of mechanical systems by introducing the time derivatives of their components,
not only mechanical, but biologic as well as psychologic14 .
This accumulated knowledge amassed by physicists for studying the simple evolution
of inert matter could by using what they call a variational approach can serve as starting
directions 15 for creating mathematical metaphors of “bio-socio-economic” evolutions, in the
13
See Box 2, p. 39.
The laws of movement and of rest deduced from this principle being precisely the same as those observed
in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative
growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander,
so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely
established, suffice for all movements. He even introduced it in the definition of happiness in his Essay de
philosophie morale, [50, Maupertuis], defined as “the product of the intensity of pleasure by the duration”.
15
Starting directions, but not final results, i.e., by introducing temporal windows described by dates and
durations, and the first aand second time derivatives of the components of economic commodities and the
actors of their elaboration. However, we start from the definitions of tangents by Fermat and bypass the
14
9
wake of Maupertuis, d’Holbach, and so many scientists who adhered to Horace’s motto Sapere
aude that Kant used as a slogan of the Enlightenment.
Among so many economist’s proposals, we shall try to answer one of the questions raised
by Max Weber in the introduction of his The Protestant Ethic and the Spirit of Capitalism,
[66, Weber]: Where capitalistic acquisition is rationally pursued, the corresponding action
is adjusted to calculations in terms of capital. This means that the action is adapted to a
systematic utilization of goods or personal services as means of acquisition in such a way
that, at the close of a business period, the balance of the enterprise in money assets (or, in
the case of a continuous enterprise, the periodically estimated money value of assets) exceeds
the capital, i.e. the estimated value of the material means of production used for acquisition
in exchange. It makes no difference whether it involves a quantity of goods entrusted in
natura to a travelling merchant, the proceeds of which may consist in other goods in natura
acquired by trade, or whether it involves a manufacturing enterprise, the assets of which
consist of buildings, machinery, cash, raw materials, partly and wholly manufactured goods,
which are balanced against liabilities. The important fact is always that a calculation of
capital in terms of money is made, whether by modern book-keeping methods or in any other
way, however primitive and crude. Everything is done in terms of balances: at the beginning
of the enterprise an initial balance, before every individual decision a calculation to ascertain
its probable profitableness, and at the end a final balance to ascertain how much profit has
been made. For instance, the initial balance of a commenda3 transaction would determine an
agreed money value of the assets put into it (so far as they were not in money form already),
and a final balance would form the estimate on which to base the distribution of profit and
loss at the end. For him, the role of economics is to provide solutions to the social problems
of his time. This objective is even more urgent in our time.
Here, we shall translate the polysemous word “capital”, nowadays meant as patrimonial
capital, by the more neutral term “endowment of means of payments” to lubricate the
economic system in order to preserve the viability of the welfare of the economic actors and
the scarcity constraints. Prices and numéraire, the thermometers of economic activity, are
determined for this purpose, an objective underlying the above Max Weber ’s statement of
this problem.
standard differential calculus by introducing sets of interrelated evolutions and their tangents, the topics of
set-valued and mutational analysis.
10
2
Evolutionary Commodities
We describe the mathematical notations which are used in this study.
1. We denote by [T − ∆, T ] a temporal window of duration ∆ ≥ 0 ending at actualization
date T ∈ R and starting from investment date T − ∆. Durations are equipped with
a unit (the second), and with an origin, equal to 0. Dates (or instants, chronological
times, etc.) t ∈ R are temporal windows t := [t, t] with duration 0.
2. The ` economic goods (or services they provide) are labelled by h and are equipped
with units of measure eh . They constitute the canonical basis of the commodity vector
space X := R` . Commodities 16 denote baskets of quantities xh of ` goods h denoted
by vectors x ∈ R` defined by
x := (xh )1,...,` =
`
X
xh e h ∈ X
(1)
h=1
Hence a good is itself a one-dimensional commodity reduced to itself.
The polysemy of the words in economics, accounting, finance, etc., may be dimmed by
introducing the following definitions.
Definition 2.1 [Evolutionary Commodities] An inert evolutionary commodity
(T, ∆, x(·)) denotes evolution defined on a temporal window [T − ∆, T ] by
t ∈ [T − ∆, T ] 7→ (t, x(t)) ∈ R × X
(2)
An kinematic evolutionary commodity (T, ∆, x(·), x0 (·)) denotes the joint evolution of the
commodity and of its time derivative
t ∈ [T − ∆, T ] 7→ (t, x(t), x0 (t)) ∈ R × X × X
At date t, the state of a good h is a

 production if x0h (t) > 0
stockpile
if x0h (t) = 0

consumption if x0h (t) < 0
16
(3)
(4)
Commodity involves many different meaning, commodity price, commodity exchange, commodity production, commodity stocks, commodity tax, commodity economy, etc.
11
Commodities evolve on a temporal window except at equilibrium where it remains constant for all times. Unfortunately, there is no equilibrium in life sciences, so that all the
wonderful mathematical theories on optima and equilibria are economically pointless.
We use stockpile instead of stock for denoting a good which is stored or hoarded during
a temporal window for avoiding confusion with the accounting stock. Indeed, since its
beginnings, accounting and finance use the terminology stocks for denoting the quantity of a
good at a given instant and flow 17 for grasping differences of the stocks at the two extremities
of a temporal window. Irving Fisher acclimated the stock-flow terminology in economics.
Hence, we may use flow as synonym of time derivative of an evolution in the continuous
time version as velocity is used in physics for the same purpose.
Both the word derivative, sacralized by the mathematicians for nearly half a millennium,
and the word stock have been hijacked by speculators to designate financial products on
stock-markets! Hence kinematic evolutions (T, ∆, x(·), x0 (·)) can be regarded as “stock-flow
phases18 ”.
17
The “price/cash flow ” is the ratio used to compare a company’s market value to its cash flow.
By analogy with physics, when Ludwig Boltzmann, Henri Poincaré, and Willard Gibbs among other
investigated state-velocity pairs, here, stock-flow evolutions under the name d phases. Their graphs along
an evolution phase diagrams, are the graphs of maps, often set-valued map, which make all their charm.
See for instance [53, Nolte]. When the commodity is a one-dimensional good x(t), it is tempting to divide
x0 (t)
by two the dimension of the phase (x(t), x0 (t)) ∈ R2 by replacing it by the rate
∈ R, an avaricious
x(t)
perversion that physicists avoid. Indeed, the use of rates instead of velocities induces an “exponential curse”:
if bounded velocities induce linear growth, which is the case of the production of commodities facing scarcity
constraints, bounded rates induce exponential growth, which is the case of money lent with interest rates,
making the debts irredeemable. However, rates are useful when one-dimensional variables are the products
of a finite differentiable functions, since the rate of a product is the sum of the rates, but this is not a real
excuse. See Section 1.3, p. 13, of Time and Money,[13, Aubin].
18
12
quantity
u
Evolutionary Commodity
scarcity threshold
P
E
production
C
stockpile
consumption
D
B
(0, 0) T − ∆ S − ∆
F
[temporal window of duration ∆ > 0]
S
T
date
Figure 2.2 [Symbolic Diagram of an Evolutionary Commodity] This diagram represents the graph of an evolutionary commodity made of two goods, the dates being represented in abscissa, the quantities of units of good in ordinate. The first evolutionary good
is
1. at investment date T − ∆, instantaneously or impulsively invested at B (impulse
x01 (T − ∆) = +∞);
2. then produced from B to P until it reaches the scarcity constraint u (x01 (t) > 0);
3. stored from P to C at the constraint level u: (x01 (t) = 0);
4. consumed from C to D until actualization date (x01 (t) < 0);
5. and, at actualisation date T , instantaneously or impulsively consumed at D (impulse
x01 (T ) = −∞).
The second good x2 (·) evolves from investment date S − ∆ to actualization date S alternating consumption, production and a consumption stages from E to F . Other combinations
are possible: evolutionary goods alternating production, consumption, storage, production,
storage, consumption phases during their life span, continuously and impulsively, for instance.
The flows are the “slopes” of the tangents, positive during production phases, zero at stored
phases and negative during consumption.
The first evolution good is not differentiable in the classical and official sense because
1. at the investment and actualization dates T − ∆ and T at B and D are impulses, so
13
that x0h (T − ∆) = +∞ and x0h (T ) = −∞ are symbolic notations which have to defined
(double arrows);
2. when the scarcity threshold is reached and left, the derivatives of the evolutions are
discontinuous, so that we have to introduce derivatives from the left (retrospective)
and from the right (prospective).
For the time, we use the classical notations for the derivatives, even when they do not
obey the classical definitions of derivatives frozen by Cauchy, but which can be adapted19
to any set-valued map, and thus, to these officially pathological cases.
It is not enough to use the derivatives of the individual goods to detect whether they are
in a production, storing or consumption state. Interactions between the derivatives of the
different commodities must also be taken into consideration for capturing the nature of the
process at each time.
If for instance the product x0h (t)x0k (t) < 0 of the flows of two goods is negative and
if x0h (t) < 0, then the consumption of the good h implies the production of the good k at
time t. This means that the good h is the input for the production of the output k in the
process governing the evolution of the commodity. This economic process among goods of
a commodity is thus “hidden” in the description of the commodity and evolves with it. In
other words, it is implicitly described by evolutionary commodity.
So, the matrix of the products of derivatives of the components of the commodity play
the dynamic version20 of the connection matrix of vector random variables for describing the
dynamic correlations between the components of the commodity:
Definition 2.3 [Differential Connection Tensors of An Evolutionary Commodity] Let us consider the derivatives from the left x0h (t) and from the right x0k (t) of two goods
→
−
of an evolutionary commodity at time t. The differential connection tensors ( D ⊗ D)x(t) is
the matrix of entries x0h (t)x0k (t) describing the dynamic correlations between the components
of the commodity.
19
See [5, Aubin], [8, Aubin], for precise mathematical definitions.
It has been introduces and studied in [19, Aubin, Chen Luxi & Dordan] to which we refer to for more
details
20
14
We may use the tools of qualitative physics 21 for observing qualitative behaviors of the
commodity.
Definition 2.4 [Qualitative Behavior] We introduce the sign set
Q := {−1, 0, +1}` of sign vectors q := (qh )h=1,...,` where qh ∈ {−1, 0, +1}
(5)
→
−
At each time t, we define the qualitative classes Q(Dx(t), q) and Q(( D ⊗ D)x(t), q) defined
by
∀ q ∈ {−1, 0, +1}` , Q(Dx(t), q) := {h ∈ {1, . . . , `} such that sign(x0h (t)) = qh }
→
−
2
∀ q ∈ {−1, 0, +1}` , Q(( D ⊗ D)x(t), q) := (h, k) such that sign(x0h (t)x0k (t)) = q(h,k)
(6)
allow us to track the evolution of these qualitative indicators along time.
The time spans of these qualitative states are measured by
∀ q ∈ Q, meas ({t ∈ [T − ∆, T ] such that sign(x0 (t)) = q})
(7)
∀ q ∈ Q2 , meas({t ∈ [T − ∆, T ] such that sign(x0h (t)x0k (t)) = q})
Then, we may study the order of visits of these qualitative classes when time evolve.
21
See for instance Analyse qualitative, [28, Dordan], Chapter 9, p. 160, of Neural Networks and Qualitative
Physics: a Viability Approach, [9, Aubin] and Section 8.8, p. 302, of Viability Theory. New Directions, [15,
Aubin, Bayen & Saint-Pierre]. Actually, qualitative physics started with “qualitative economics” when
Paul Samuelson introduced in the beginning of the 1970’s matrices of signs (see Foundations of Economic
Analysis, [59, Samuelson], and developed by many authors, among whom Kelvin Lancaster, [44, Lancaster]).
15
3
The Ultimate and Yet Forgotten Goods: Time and
Durations
There are many examples of goods, among which the durations ∆ ≥ 0, equipped with the
unit of measure, the second, and which has a origin22 , 0. Hence, duration is a commodity
allowing to define temporal windows [T − ∆, T ], parameterized by two-dimensional vectors
1
of a duration is
(T, ∆) ∈ R × R+ , describing the “dual nature of time”. The inverse
∆
regarded as a liquidity 23 . If v(t) ∈ R is an economic value and v(T ) − v(T − ∆) a profit,
v(T ) − v(T − ∆)
then
can be interpreted as an enrichment, inversely proportional to the
∆
investment duration and proportional to the profit. This ratio could be the basis for a
“Shareholder Value Tax” instead of the “Added Value Tax”, as the Tobin tax on financial
transactions24 . Instants T := [T − 0, T ] ∈ R are temporal windows of duration ∆ = 0,
and thus, parameterized by one-dimensional scalars, easier to deal with. They are regarded
as a an evolving present T , separating evolving past from evolving future. They should
rather be called an “absent”, since instants do not physically exist yet25 ! Instants or dates
t := [t, t] ∈ [T − ∆, T ] are also called “current dates”, or, unfortunately, “times”. The
chronological time was measured by the ephemerides through gnomons and sundials, and
now, by clocks, whereas the duration was obtained by clepsydra, and, since the xth century,
by hourglasses, or by the difference between two chronological times.
With any temporal window of duration ∆ ≥ 0 and any current dates t ∈ [T − ∆, T ],we
associate the
1. forward duration t ∈ [T − ∆, T ] 7→ t − (T − ∆) vanishing at date T − ∆ with a constant
velocity equal to 1, measuring the duration from the beginning T − ∆ of the temporal
1
is its (instantaneous) liquidity;
window. For any t ∈]T − ∆, T ], the inverse
t − (T − ∆
2. backward duration t ∈ [T − ∆, T ] 7→ T − t vanishing at T with constant velocity equal
to −1, measuring the duration up to the end T of the temporal window. For any
1
is called the (instantaneous) liquidity.
t ∈ [T − ∆, T [, the inverse
T −t
The smallest duration measured so far is the yoctoseconde (10−24 seconds). Therefore, for the time,
instants do not exist yet physically, and remain the privilege of mathematicians who do not hesitate to let
durations converge to 0 in average flows (difference quotients) for obtaining derivatives of all kinds, orders
and obediences.
23
see [27, Chen Luxi], and, in a financial context, [46, Lehalle & Laruelle], [45, Lehalle], etc.
24
See Section 1.4, p. 18, of Time and Money,[13, Aubin].
25
See Didier Nordon in his book Scientaisies. Chroniques narquoises d’un mathématicien, [54, Nordon],
p. 87, quoting the novel Deux heures moins dix by Mikhail Shishkin.
22
16
0
duration ∆
duration
duration ∆
duration
duration
∆
∆
time
T −∆
T
T +∆
time
Figure 3.1 [Forward and Backward Durations]
[Left]. Forward and Backward Standard Durations
Right. Forward and Backward Durations on Temporal Windows
1. Forward duration t 7→ t−(T −∆) defined on retrospective temporal window [T −∆, T ]
from T − ∆ up to T ;
2. Backward duration t 7→ T + ∆ − t defined on prospective temporal window [T, T + ∆]
from T up to terminal time T + ∆.
If durations have an origin, there is no physical origin of time, except than arbitrary and
conventional origin26 O and yet, consensual for being real, in the literal sense and as a real
number. The chronological time t 7→ t − O ∈ R is thus a “pseudo evolution” in the extent
that O is a pseudo-origin of time. Another convention leads us to fix O = 0, so that time
t ∈ R is identified with a real number27 t ∈ R.
In some sense, chronological time plays the role of a “numéraire of evolutions” 28 for
comparing evolutions between them by comparing each of them with the duration t 7→
α(t) := t − O ∈ R where O, with constant velocity equal to 1. It is a conventional good, as
long as we agree on an origin of time, usually chosen as O = 0, so that α(t) := t is called
the standard time standard time (or standard duration, to be precise).
26
“What was God doing before He created the Heavens and the Earth?” asked Augustine of Hippo in his
confessions. What was the universe behaving before the Big Bang, ask some physicists? Introducing the
concepts of temporal windows and duration function circumvents the question of origin of time.
27
By doing so, only its total order structure should be used, by stating that an instant s is anterior or
posterior to another instant t if the duration s − t is non negative. This allows us to use the concepts of
infimum and supremum. The max-plus algebra structure on R, in particular, hyperspaces (of subsets of a
given space) an another example, could provide an adequate structure for taking into account only its order
relation to study evolutionary problems.
28
See Section 8.2, p. 601, of [12, La mort du devin, l’émergence du démiurge].
17
Average velocities of both retrospective and prospective durations are equal to 1. However, it is perceived that time, i.e., duration, flies29 , and that their flow, called fluidity, is no
longer constant. This suggest to introduce also
Definition 3.2 [Durations with Variable Fluidities]
−
1. forward(kinematic) duration t ∈ [T − ∆, T ] 7→ →
τ (t) vanishing at date T − ∆ with a
→
−
τ (T ) − →
− (T − ∆)
−
variable positive fluidity →
τ (t) > 0 and a constant average fluidity
∆
1
equal to 1. For any t ∈]T − ∆, T ], the inverse →
is called the (instantaneous)
−
τ (t)
→
−
τ 0 (t)
is called the (forward) duration rate, equal to the
liquidity and the growth rate →
−
τ (t)
product of the fluidity and the liquidity;
−(t) vanishing at T a variable
2. backward (kinematic) duration t ∈ [T − ∆, T ] 7→ ←
ω
←
−
←
−
−(t) < 0 and a constant average fluidity ω (T ) − ω (T − ∆) equal
negative fluidity ←
ω
∆
1
to −1. For any t ∈ [T −∆, T [, the inverse ←
−(t) is called the (instantaneous) liquidity
ω
←
−0 (t)
ω
and the growth rate ←
−(t) is called the (backward) duration rate.
ω
29
Tempus fugit, as Virgil notices in his Georgics: Sed fugit interea fugit irreparabile tempus, singula dum
capti circumvectamur amore. Indeed, the more the years go by, the faster time flies, the more precious it
becomes! The velocity of such a duration is no longer constant, and could be called the fluidity of time.
18
impulse
(∆, ∆)
(∆, ∆)
∆
impulse
impulse
∆
0
impulse
∆ time
0
∆ time
Figure 3.3 [Piecewise Linear Durations and Impulses] The figure displays example
of piecewise linear durations with two and three fluidities and, in the figure from the left,
impulse durations with jumps : when ∆ converges to T , the duration function τT (·) is the
impulse duration such that τT (t) = 0 if t ∈ [T − ∆, T [ with a zero fluidity and τT (T ) = ∆
with infinite fluidity and the duration function τ(T −∆) (·) is the impulse duration such that
τ(T −∆) (T − ∆) = ∆ with infinite fluidity and τ(T −∆) (t) = ∆ if t ∈]T − ∆, T ] with a zero
fluidity.
Other examples are depicted below:
19
∆
duration
ϕ=0
ϕ=1
0
∆
time
Figure 3.4 [Forward
Kinematic
Standard Durations] This figure displays some forward kinematic durations
with zero acceleration (in blue) and with
constant opposite accelerations (in green
and orange). Other kinematic durations
function have variable accelerations.
Perennial duration with zero fluidity is
defined on the temporal window ] − ∞, T ]
of infinite duration.
Forward and backward durations are crucial variables, which should be introduced explicitly among the goods of any commodity. Not only evolutionary commodities depend
−
upon the “standard” time α(t) := t, but also upon forward durations →
τ (t) := t − (T − ∆)
→
−
←
−
or backward duration ω (t) := T − t or ω (t):
t ∈ [T − ∆, T ] 7→ (t, t − (T − ∆), T − t, x(t), etc.(t))
(8)
t ∈ [T − ∆, T ] 7→ (t, t − (T − ∆), T − t, x(t), etc., x0 (t), etc.0 (t))
For simplifying the notations, we do not introduce them explicitly, assuming that they are
components of the evolutionary commodities: only standard instants t are mentioned in
the description of evolutionary commodities x(·), with which we shall add later many other
specific variables summarized by etc.(t) and their derivative etc.0 (t):
t ∈ [T − ∆, T ] 7→ (t, x(t), etc.(t))
(9)
t ∈ [T − ∆, T ] 7→ (t, x(t), etc., x0 (t), etc.0 (t))
Remark: Time and Money Another important good will be distinguished, the
numéraire: it is a specific good (concrete, and, by now, abstract) used as the unit of measure
for defining the concept of value (capital, money, etc.), which is a quantity of units of
numéraire. By definition, the numéraire is used by economic actors (commodity actors,
as we shall define them) for fluidifying their exchanges of commodities, by comparing the
value of each of their goods with the value of the numéraire. It plays the role of an enzyme
which catalyses a metabolic reaction, accelerating its velocity without being consumed by
the reaction. Unfortunately, the activity of this economic enzyme does not always remain
constant, so that economic alchemists of all kinds hoard quantities of units of numéraire for
20
transmuting means of payment into wealth, the real power that lenders exert on borrowers.
The social existence of a numéraire is a matter of trust: it depends on the consensus to
use it as means of payment. Contrary to durations, trust has no unit of measure, it is only
catallactic. Time is not really money: the first one physically exists, the second one exists
only socially, for the convenience of “human computers”. Money could be regarded as the
value of duration, which enjoys a unit of measure. This is developed in Section 8, p. 33. 21
4
Inert and Kinematic Relations: The Viability Question
We face two types of evolutionary constraints on evolutionary commodities described by
relations linking time, commodities and their derivatives:
Definition 4.1 [Inert and Kinematic Relations] Evolutionary commodities and their
derivatives are required to satisfy inert and kinematic constraints described respectively by
independent relations:
1. a inert relation K ⊂ R × X describing commodity constraints
∀ t ∈ [T − ∆, T ], (t, x(t)) ∈ K
(10)
2. by a kinematic relation M ⊂ R × X × X linking both an evolutionary commodity and
its flow (T, ∆, x(·), x0 (t)) to abide by kinematic constraints
∀ t ∈ [T − ∆, T ], (t, x(t), x0 (t)) ∈ M
(11)
We denote by (K, M) the inert-kinematic or stock-flow relation.
Remark:
For the time, the inert and kinematic relations30 are assumed to be constant subsets. But they also can evolve themselves together with their elements. Indeed,
mutational and morphological analysis31 provide the mathematical tools for governing the
evolution of sets as well as its elements. The same qualitative stories can be adapted to the
case of evolving commodity and kinematic relations (at a higher mathematical price). 30
At this stage, they could be written in the more familiar form
x(t) ∈ K(t) and x0 (t) ∈ M (t, x(t))
(12)
but the introduction of at least forward and backward durations, actors, numéraire and prices, etc., does
not allow us to decide at this stage which variables depend on others, since they are interrelated. Up to a
permutation of the labels of the variables and their derivatives, the relation can be regarded as the graph
of an input-output set-valued map. All such set-valued maps share the same graphical properties than the
ones generated by the relation. This is why we shall use only relations instead of maps. See Figure ??, p. ??
and Section 8.2; of Traffic Networks as Information Systems. A viability Approach, [20, Aubin & Désilles],
for more details.
31
See for instance Mutational and Morphological Analysis: Tools for Shape Regulation and Morphogenesis,
[11, Aubin], and Mutational Analysis, [48, Lorenz], by Thomas Lorenz.
22
Since the inert and kinematic relations are independent, it is not always possible to
process evolutionary commodities which satisfy the commodity constraints. The consistency
between commodity and kinematic relations constitutes the first question to answer:
Definition 4.2 [Viability of Inert Relations under Kinematic Relations] We
shall say that a inert relation K is viable under a kinematic relation M if, for any
(T, ∆, x) ∈ K, there exists an evolutionary stock-flow evolution (T, ∆, x(·), x0 (·)) ∈ X × X
satisfying
1. the actualization condition x(T ) = x at actualization time T ;
2. both commodity and constraint relations:

(t, x(t))
∈ K ⊂
R×X

and
∀ t ∈ [T − ∆, T ],

(t, x(t), x0 (t)) ∈ M ⊂ R × X × X
(13)
The question we are facing is simple to enunciate: take our sculptor chisel to carve in
the inert relation K and the kinematic relation M viable ones:
Definition 4.3 [The Viability Solution to the Confrontation between Commodity and Kinematic Relations] The viability solution to the confrontation between the
inert and kinematic relations (K, M) is defined by the viability kernel and its regulator
(G, K)
1. made of the largest commodity sub-relation G ∈ K;
2. viable under its regulator R ∈ M.
This viability kernel G and its regulator R can be constructed under elaborate assumptions
and share the properties of viability kernels which need to be translated in this situation.
This definition does not exclude the trivial case when the viability kernel G is empty. It
is a sad information, but still an information.
Naturally, this is one part of the task. We may add also “investment” conditions at
investment date T − ∆:
Definition 4.4 [Investment Relation] An investment relation is a set-valued map as-
23
sociating with any investment date T − ∆ a sub-relation D ⊂ K of investment commodities
(T − ∆, 0, c) ∈ K. The investment kernel is the pair of sub-relation (G[D], R) ⊂ (K, M) of
(T, ∆, x) ∈ K such that there exists a stock-flow evolution (T, ∆, x(·), x0 (·)) satisfying

 x(T ) = x and (T − ∆, x(T − ∆)) ∈ D
∀ t ∈ [T − ∆, T ], (t, x(t)) ∈ K
(14)

0
∀ t ∈]T − ∆, T [, (t, x(t), x (t)) ∈ R
Note that the regulation relation R depends only on K and is independent of the investment
relations.
They are “economic geodesics” linking an investment x(T − ∆) at investment date T − ∆
to an actualization x at actualization date T .
This mathematical framework is the simplest one. However, many other variables may be
added: second derivatives of commodities, commodity actors, numéraire and prices and their
velocities, etc. These variables and their derivatives are assumed to be linked by relations
describing constraints on the variables and constraints between them and their derivatives.
The issue is the viability of their evolutions. We begin by stating our statements in this simple
case and proceed crescendo by introducing actors, numéraire, prices, durations among many
other variables.
To grasp the mathematical details necessary to find the viability solution to the consistency between commodity and commodity flow relations is not necessary to understand the
“economic message” sent:
1. are inert and kinematic relations consistent?
2. if not, describe the viability kernel (G, R) or (G[D], R) when investment conditions are
required.
What are missing are the assumptions. This is a mathematical task which is not presented
in this study, since the developments would occupy the size of a book. They are provided
by “viability theory”, a nickname for the set of mathematical tools (set-valued analysis32 ,
differential inclusions, mutational and morphological inclusions33 ), designed for answering
this type of interrogations dealing with the consistency between inert and kinematic constraints and the construction of regulators providing viable evolutions of commodities, prices
and values in numéraire among many other variables. This task is left to uconomists. Only
32
See Set-Valued Analysis, [22, Aubin & Frankowska], Variational Analysis, [58, Rockafellar & Wets], etc.
See Viability Theory. New Directions, [15, Aubin, Bayen & Saint-Pierre], Mutational and Morphological
Analysis: Tools for Shape Regulation and Morphogenesis, [11, Aubin], Mutational Analysis, [48, Lorenz], etc.
33
24
the mathematical question, here, the construction of viability kernels and their regulators
relevant to economists, need to be explained.
We further introduce
1. elaborative energy function depending on dates, durations, elaboration and elaborativity
2. elaborative action on a temporal window, equal the integral of the elaborative energy
function on an elaboration evolution on this temporal window.
25
5
The Energy
26
6
Economic Actors
Evolutionary commodity actors are the first variables to be added after time and duration,
before all the variables which are the components of commodities and services. For simplicity, the elaboration of economic actors in the transformation process of the economic
environment are described by one dimensional variables, called “elaboration shares” of the
evolving commodities for producing them (positive derivatives of commodities), consuming
them (negative derivatives) or storing them (null derivatives). In other words, they are evolutionary actors. We have to understand how they interact with the above processes for
sharing at each instant an evolving commodity.
Each actor is thus defined by its elaboration share of each component of a commodity at
each instant:
Definition 6.1 [Economic Actors] We introduce the set {1, . . . , n} of n actors labelled
by i. Each actor i associates with any good h = 1, . . . , ` its elaboration share ai,h ∈ [0, 1] of
good h. We denote by ai := (ai,h )h=1,...,` the elaboration vector of actor i and by A ⊂ Rn`
the elaboration simplex, which is the set of elaboration matrices a := (ai,h )i=1,...,n;h=1,...,`
required to satisfy
∀ i = 1, . . . n, ∀ h = 1, . . . , `, ai,h ∈ [0, 1] and, ∀ h = 1, . . . , `,
n
X
ai,h ≤ 1
(15)
i=1
If x := (xh )h=1,...,` is a commodity, its elaboration matrix a ◦ x is defined by
a ◦ x := (ai,h xh )i=1,...,n;h=1,...,`
(16)
The inert evolutionary elaboration (T, ∆, a(·)) is defined by
t ∈ [T − ∆, T ] 7→ a(t) := := (ai,h (t))i=1,...,n;h=1,...,` ∈ A ⊂ Rn`
(17)
and (T, ∆, (a ◦ x)(·)) denotes the inert evolution of elaborated commodities.
The derivative (T, ∆, a0 (·)) is called the elaboration flow. For each h = 1, . . . , `, the actor
i is a

 investor if a0i,h (t) ∈ ]0, +∞]
holder if
a0i,h (t) = 0
(18)

disvestor if a0i,h (t) ∈ [−∞, 0[
→
−
We introduce also the differential elaboration tensors ( D ⊗ D)a(t) of entries a0i,h (t)a0j,k (t)
→
−
and the tensor ( D a ◦ x)(t)(Da ◦ x)(t) of entries a0i,h (t)xh (t)a0j,k xk (t)(t) describing the dynamic correlations between the elaborated goods of the ` commodities by the n actors.
27
We summarize in the table below the interaction between the investment of the actors
and the nature of the processes of commodities:
consumption
production
x0h (t) < 0
x0h (t) > 0
disvestor a0i,h (t) < 0 “down-fare” disvestment
investor a0i,h (t) > 0
“up-fare”
investment
(19)
We observe that the flow of the elaborated commodities is equal to
da(t) ◦ x(t)
= a0 (t) ◦ x(t) + a(t) ◦ x0 (t) =
dt
(a0i,h xh + ai,h x0h ) i=1,...,n;h=1,...,` ⊂ X
(20)
so that, for any actor i and any good h, the growth rates of the elaborated goods satisfy
a0i,h (t) x0h (t)
(ai,h (t)xh (t))0
=
+
ai,h (t)xh (t)
ai,h (t) xh (t)
Z
a00 (t) · a(t)dt
Elaborative Work
Elaboration
a(t)
Effort a00 (t)
Elaborarivity
Regulator
Z
a
bor
Elaborativity
a0 (t)
(21)
Po
tive
wer
0 (t)dt
a
·
00 t)
a (
Effort
Regulator
Ela
Figure 6.2 [Elaboration, Elaborativity, Effort] This diagram displays the evolutions
of the elaboration, the elaboritivity (its first derivative) and the effort (its second derivative)
respectively.
Dotted lines symbolize the elaborative work between elaboration and elaboritivity and the
elaborative power between effort and and elaborativity.
We adapt to this new situation the concepts of elaborated inert and kinematic relations
introduced in Definition 4.1, p. 21:
28
Definition 6.3 [Elaborative Inert and Kinematic Relations] Evolutionary commodities elaborated by actors and their derivatives are required to satisfy independent inert
and kinematic constraints described respectively by relations:
b ⊂ R × X × A requiring that the evolution
1. inert elaborated relation described by K
(T, ∆, x(·), a(t)) satisfies
b
∀ t ∈ [T − ∆, T ], (t, x(t), a(t)) ∈ K
(22)
links or relates time, commodities and their elaboration by actors.
c ⊂ R × X × A × X × Rn`
2. kinematic elaborated relation on flows described by M
requiring that the kinematic evolution (T, ∆, x(·), a(t), x0 (t), a0 (t)) satisfies
c
∀ t ∈ [T − ∆, T ], (t, x(t), a(t), x0 (t), a0 (t)) ∈ M
(23)
Our basic viability problem can be formulated in this case:
Definition 6.4 [The Viability Solution to the Elaborative Inert and Kinematic
Relations] The viability solution to the confrontation between the inert and kinematic
b M)
c is defined by the viability kernel and its regulator (G,
b K)
b made of the two
relations (K,
largest consistent
b
1. inert sub-relation Gb ∈ K;
b∈M
c
2. regulator R
This viability kernel can be constructed under elaborate assumptions and shares the
properties of viability kernels which need to be translated in this situation.
In classical economics, although both labor and its value are not clearly defined, the labor
value (without any qualifying adjective) refers to the quantity of labor necessary to produce
a unit of commodity. If we identify the quantity of labor with the elaboration, then the
product λ(t) := a(t) · x(t) of the elaborative share and a commodity, that we may call the
elaborative value, may provide one aspect of labor value. Then, we call elaborativity value
the first derivative λ0 (t) = a0 (t) · x(t) + a(t) · x0 (t).
29
Elaboration
Credit
Commodity
Value
Elaborativity Value
λ0 (t) = a(t) · x0 (t) + a0 (t) · x(t)
a
x
·
·
x0
0
a
Flow
Elaborativity
a0 (t)
Stock
Elaboration
x(t)
Kinematic
commodity x0 (t)
a · x
Commodity x(t)
a · x
Value
Elaborative Value
λ(t) = a(t) · x(t)
Figure 6.5 [Elaboration, Elaborativity, Commodities and Kinetic Commodies]
This diagram describes the f
1. elaboration a(t) (stocks) and derivative a0 (t) (elaborativity);
2. commodity x(t) and its derivative x0 (t) (kinematic commodity);
3. elaborative value λ(t) = a(t) · x(t) and its elaborativity value λ0 (t); = a(t) · x0 (t) + a0 (t) ·
x(t).
30
7
Money : A Desktime Story Told by Children
How did human beings invent “money” and its “valometer”? Deprived of the testimony of
the history of the emergence of economic systems, it would suffice, on the basis of Ernst
Haeckel ’s biological motto, ontogeny recapitulates phylogeny 34 to observe the evolution of
relations between children facing the exchange of their goods and services as they grow up.
These testimonies might suggest an idea of ??what prevailed at the dawn of humanity. Even
if this hypothesis is not validated, this recapitulation is instructive:
1. Young children begin by resolving their differences with claws, and it is only after the
speech has been mastered that they realize that its use can lead to the same result
at a lower cost. Diplomacy is the pursuit of war by other means, in the reverse order
advocated by Clausewitz, when war becomes necessary after the failure of diplomacy.
These two phases succeed one another, when they do not become entangled:
2. If the attack of the strongest is successful, the case is settled. If the forces are balanced, the dispute can be transmuted into negotiation or bargaining. Animals dispute
territories, sexual partners and prey, but do not negotiate;
3. Negotiations and bargaining involve awareness of time and confidence in promises that
bind the future. These negotiations take many forms, from the smile of the baby in
recognition of the food to the affectation of the little girls against the sweets of the
boys proud to make them a gift. It is the phase of donation both to satisfy the power
and the munificence of the donor to satisfy the desire of the beneficiary;
4. The gift then gives way to bartering when children begin to know how to enumerate:
the number is used to ”lubricate” this draft trade. It is involved in comparing toys
and services between them. The division of desires and the division of talents into
rendering services intervene in order to agree on a transaction;
5. Advancing in age, children move from the enumeration phase to that of the concept of
cardinal numbers and are initiated into the pantometric drug. Small objects (marbles,
if I take the example of the boys of my generation), easy to transport and count, serve
not only to play, but also as ”means of payment” to exchange objects and services;
6. The marbles are first exchanged between them, marbles against marbles, in games
where address and practice (examples of talents and work) determine the exchange.
Competition between talent and work is emerging. Then, the marbles are negotiated
against different objects, agreeing on the number of marbles that ”is worth the object”:
34
In 1874, after observing that the evolution of the fetuses during the embryogenesis ”recapitulates” that
of the species, Ernst Haeckel proposed this lapidary formula. This hypothesis is not to be taken literally, as
it suffers from exceptions, detected by Stephen J. Gould in his famous book Phylogenesis and Ontogenesis,
cite [Gould] Gould77 published in 1977.
31
7. The marbles thus become units of account (of cash). The choice of cash is variable
from one playground to the next. Once accepted somewhere, this choice is diffused
in other schools, without the help, at this level, of advertisements or other marketing
manipulations;
8. Mimetism and conformism helping, every object desirable by others becomes attractive
for each and takes on value, that of envy and jealousy. Through mechanisms that are
still mysterious to me, the ascendancy of a child with a certain charisma on others
inspires them to imitate it and creates a demand for marbles. This aspiring opinion
leader35 , interested in marbles, both to use them in his own games and to exchange
them, gives them an intrinsic value, upstream of the value of other objects and services.
The value he instilled in his comrades exceeded that which the marbles had as the only
means of exchange. Children will want marbles for their own value, and not just for
the services they render in trade. The marbles become also a wealth;
9. Some children, whether they are less prone to objects and services, or that they use
them in moderation in their games, hoard these marbles, reserve them and appropriate
them for future exchanges. The perception of time becomes indispensable: the marbles
buy time (actually, duration);
10. In order to continue their exchanges, children without marbles contact those who have
hoarded them to obtain new marbles. Against what ? Against promises of repayment,
in objects, services or marbles. This step requires at least the trust, empathy and /
or altruism of the lenders. The savers then advance these marbles: for example 10
marbles today against 13 in two weeks, to satisfy the immediate desire of an aspiring
young borrower, with a gain of three marbles for the time of deprivation of 10 marbles
for two weeks;
11. The interest, the price of deprivation, is then invented. It allows lenders to pocket
more and more wealth. Lenders have gained power over borrowers, that of denying
loans, or that of increasing interest.
12. Exchanges begin with friends. Then, the system becomes generalized, to others, between unknowns6, to whom these friends have trusted in their turn. The marbles pass
from hand to hand along a chain of exchanges without the lender knowing, in fine, to
whom his marbles will arrive.
13. The chain of exchanges is very imperfect since some cheat or do not keep their promises,
preferring plundering to negotiations whenever confidence gives way to defiance.
35
Concept introduced by Paul Lazarsfeld, founder of the two-step flow of communication (see On Social
Research and Its Language, [?, Lazarsfeld]).
32
14. Once deprived of available wealth, borrowers can no longer carry out their transactions
and put an end to them. Lenders then find themselves with bags of marbles that no
longer serve them for lack of borrowers. The marbles no longer have any value. The
system got stuck. Negotiations may again give way to violence.
15. However, these children have parents who care about their happiness. They give them
marbles to feed their games, and, consequently, exchanges. With their new bags, the
children resume their games and continue their exchanges.
16. But the more generous the parents, the more the lenders accumulate marbles, the
easier it is to lend them as long as the cost of marble deprivation remains the same,
the more they are insured against a shortage of marbles. Their power is strengthened
as their bags of balls swell, feeding in others a perception of the inequality and the
frustrations that it causes. Then comes the time of revolts.
The system would not have worked without parents, ”donors of last resort” of means of
payment, on the one hand, and it would be packed if the supply of marbles had been too
abundant, on the other hand.
These observations, which each parent has made, correspond in their main lines to the
reconstructions of economic history. Parents instinctively play the role of sovereigns calculating the amount of pocket money of their teenagers according to the demand and resources
of their child. They too are victims of mimetism by taking inspiration from what the other
parents give to theirs.
Naturally, many factors have Been neglected in this short nostalgic recapitulation, first
step towards abstraction, that I will continue by passing from marbles to credit cards, from
desktime story to mathematical metaphors (see Time and Money. How Long and How Much
Money is Needed to Regulate a Viable Economy, [13, Aubin].
Exchanging assets for cash units requires ”accountancy” skills. In addition to computation, confidence, that involved in the reciprocity of futures exchanges, is still unavoidable to
trust and measure this intertemporal trade. Money is a cultural regulon for the economic
actors who trust it, and as long as they trust it. Conversely, sub-societies can create ”local”
currencies, streamlining black markets, from casino chips to the bits of digital computers,
into their most recent avatar.
33
8
Measures of Commodities: Numéraire and Prices
Up to now, the preceding section dealt only with exchange (barter) mechanisms. Whatever
their reasons or purposes, as soon as human brains discovered numbers and how to manipulate them, they used instruments of measure for “valorising” goods endowed with a unit of
measure: they are made of a numéraire (monetary unit) and of prices36 .
For exchanges measured by prices to take place, amounts of numéraire are used for
catalyzing the exchanges: they are means of payment in numéraire, entering the process
without being consumed. At least, in theory. Unfortunately, our ancestors already designed
“creative finance”: they also hoarded amounts of numéraire, disrupting this catalytic process
by using parts of the means of payments as wealth. Means of payment fuel the exchanges,
wealth fuels greed, by increasing the accumulation of not spent value:
1. providing means of payment has an eudemonistic role, for insuring the viability constraint;
2. hoarding wealth has a hedonist function, for accumulating wealth providing the actor
at least
(a) a private insurance against the uncertainty of the future;
(b) the power of lending or not, at their conditions, other actors who are deprived of
means of payments and left with no other choice than to borrow them.
Hence, price and numéraire, when they exist, play different roles. We begin by their
role as instrument of measure by unit of numéraire used as means of payment and not the
hoarding of wealth.
Namely, we introduce
1. a singular commodity, the numéraire, labelled by h = 0. The amount v := v0 ∈ R in
units of numéraire is called a value (or a patrimonial values, or a capital according to
Irving Fisher, etc.).
2. The value e?h of the unit of good eh in units of numéraire is called its “ unit price”,
?
or “price of the commodity”. They form the canonical basis of the dual X ? := R` of
prices (price vectors or systems) denoted by
h
p := (p )1,...,` =
`
X
h=1
36
See, for instance, [24, 25, Cartelier].
ph e?h ∈ X
(24)
34
Prices p are linear value functions
x ∈ X 7→ hp, xi :=
`
X
p h xh ∈ R
(25)
h=1
associating with commodities x ∈ X their valuehp, xi in units of numéraire. The numéraire
being a good, it has also a price π := p0 , which should be equal to 1, but actually may
evolve, and a monetary value πv in means of payments.
Being linear, prices are different from nonlinear value functions associating with a commodity a value in numéraire, such as cost or value functions derived from specific valuation
or valorization problems.
Prices evolve, too, since they are produced by human beings, who are price makers or
price takers, knowing how to count, units of numéraire, in particular. Hence their derivatives
express how fast they evolve:
Definition 8.1 [Price Fluctuations and Credits]
1. The derivatives v 0 (t) = hp(t), x0 (t)i + hp0 (t), x(t)i of the value is called an credit of
means of payment (also called a cash flow). If it is created by an endowment provider,
it is a credit if v 0 (t) ≥ 0 and a debt if v 0 (t) ≤ 0;
2. Inflation means (only) the impact of the derivative of prices π 0 of the unit of
numéraire. The derivatives p0 (·) of other goods are called price fluctuations for lack
of an existing name.
Furthermore,
1. the value hp(t), x0 (t)i, called the kinematic value, is known under several names, such
as transaction value in finance whenever x0 (t) is regarded as a transaction of shares
of assets;
2. the value hp0 (t), x(t)i can be regarded as impacts of price fluctuations on commodities.
In finance, the self-financed assumption requires that this impact is equal to 0.
3. the sum hp(t), x0 (t)i+hp0 (t), x(t)i of these two values is called here an credit in means
of payment.
35
We summarize in the table below the interaction between flow and price fluctuation:
pricef luctuation
price def lation
price inf lation
consumption
production
0
xh (t) < 0
x0h (t) > 0
p0h (t) < 0 improvement depreciation
p0h (t) > 0 pauperization appreciation
Commodity
(26)
Price
Numéraire
Credit v 0 (t)
Credit
0
,x
hp
Flow
Flow x0 (t)
Stock
Commodity
x(t)
hp 0
,x
i
i
Fluctuation
p0 (t)
hp, xi
Price p(t)
hp, xi
Value
Value v(t)
Figure 8.2 [Commodities, Prices, Flows and Fluctuations] This diagram displays
the four finite dimensional vector spaces isomorphic to R` on which evolve the evolutions of
the
1. commodities x(t) (stocks) and their derivatives x0 (t) (flows);
2. prices p(t) and their derivatives p0 (t) (fluctuations);
3. values (capitals) v(t) = hp(t), x(t)i and their derivatives v 0 (t); = hp(t), x0 (t)i +
hp0 (t), x(t)i (credits) symbolized by orange and violet double arrows.
36
9
Joint Viability of Commodities, Actors, Prices and
Values
We now introduce elaboration matrices a := (ai,h )i=1,...,n;h=1,...,` of the n actors i = 1, . . . , n
and ai,0 shares of the value. They are required to satisfy
∀ h = 1, . . . n, ai,h ∈ [0, 1] and
n
X
ai,h ∈ [0, 1]
(27)
i=1
For the numéraire h = 0, we may take into account the debts by assuming that the elaboration of the value may be negative. In this case, we still require that
ai,0 ≤ 1 and
n
X
ai,0 ∈ [0, 1]
(28)
i=1
The value of the good h elaborated by actor i is equal to ai,h ph xh and the value of an
`
X
elaborated commodity x := (xh )h=1,...,` elaborated by an actor i is thus equal to
ai,h ph xh .
h=1
For any coalition S ⊂ N := {1, . . . , n} of actors,
X
i∈S
ai,0 vh −
`
X
h=1
!
h
ai,h p xh
is called a
saving if it is non negative
of coalition S.
debt
if it is negative
(29)
We observe that growth of the value of the good h by actor i is the sum
a0i,h (t) x0h (t) ph0 (t)
(ai,h (t)ph (t)xh (t))0
=
+
+ h
ai,h (t)ph (t)xh (t)
ai,h (t) xh (t)
p (t)
of the three growth rates of the elaboration, the commodity and the prices.
Let us consider viability and kinematic relations

 z}|{
K ⊂ R × (X × A × X ? × R)
z}|{
 M ⊂ R × (X × A × X ? × R × R) × (X × A × X ? × R × R)
(30)
(31)
1. The evolution of commodities, actors, price and value are required to obey the following
constraints
z}|{
t ∈ [T − Ω, T ] 7→ (t, x(t), a(t), p(t), v(t), π(t)) ∈ K
(32)
37
2. and their derivatives
z}|{
∀ t ∈ [T − Ω, T ], (t, x(t), a(t), p(t), v(t), π(t), x0 (t), a0 (t), p0 (t), v 0 (t), π 0 (t)) ∈ M
(33)
Elaboration
Credit
v 0 (t)
Remuneration
Numéraire
Elaborative Credit
= hs(t), a0 (t)i + hp0 (t), x(t)i
hs 0
0i
,a
a
,
i
hs
Flow
Elaborativity
a0 (t)
Stock
Elaboration
x(t)
Remunerative
Fluctuation
s0 (t)
hs, ai
Remuneration
s(t)
hs, ai
Value
Elaborative Value
v(t) = hs(t), a(t)i
Figure 9.1 [Elaboration, Elaborativity, Remuneration et Remunerative Fluctuations] This diagram describes the four vector spaces isomorphic to R` in which evolve
the
1. elaboration a(t) (stocks) and derivative s0 (t) (elaborativity);
2. remuneration s(t) and its derivative s0 (t) (remunerative fluctuation);
3. remunerative value ) v(t) = hs(t), s(t)i and its credit v 0 (t); = hs(t), s0 (t)i+hs0 (t), a(t)i.
38
z}|{ z}|{
Hence, the question is to know whether ( K , M ) are consistent, and, if not, to carve in
z}|{
z}|{
the viability relation K its viability kernel and its regulator in M :
Definition 9.2 [The Viability Solution to the Consistency between CommodityActor-Price Relations and their Kinematic Relations] The viability solution to the
z}|{ z}|{
confrontation between the inert and kinematic relations ( K , M ) is defined by the viability
z}|{ z}|{
kernel and its regulator ( G , K ) made of the two largest consistent consistent
z}|{ z}|{
1. commodity sub-relation G ∈ K ;
z}|{ z}|{
2. regulator R ∈ M
10
Are Arbitrary Budgetary Laws Violating Viability
of Elaborative Commodities?
z}|{
In particular, the regulator R relates
∀ t ∈ [T − Ω, T ], (t, x(t), a(t), p(t), v(t), π(t), x0 (t), a0 (t), p0 (t), v 0 (t), π 0 (t)) ∈
z}|{
R
(34)
Among these constraints, some should be “hard”, the ones dealing with commodities and
actors:
b ⊂ K
b
(t, x(t), a(t), x0 (t), a0 (t)) ∈ R
(35)
since they guarantee the viability of the elaborated commodities described by
(t, x(t), a(t)) ∈ Gb
(36)
The constraints on p(t), v(t), π(t) involving numéraire and prices are “soft” in the sense that
there are rules imposed by human brains which have also evolved by trial and errors, actually,
too many deadly errors. Their inputs, (natural) intelligence, thought, reflection, imagination, knowledge, discovery, dissent , wisdom, trust, etc., escape the present mathematical
idealization involving vector spaces, and thus, unit of measures. Knowledge economy advocated by Peter Drucker in Chapter 12 of The Age of Discontinuity, [29, Drucker], should be
called knowledge catallaxy, since it cannot be measured by numbers, such as the “Knowledge
Economy Index” (KEI) suggested by the World Bank Institute using simple averages of non
quantifiable variables such as education, innovation, etc.
Using these useful monetary instruments of measure should comfort or improve the rez}|{
lation Gb in the sense that the projection of G onto the space R × X × A should contain
b
the sub relation G.
39
This unfortunately is not the case whenever ill designed constraints on the instrument of
measure p(t), v(t), π(t), p0 (t), v 0 (t), π 0 (t) amount to shrink Gb too much so that its projection
b sometime, up to pauperization. This is the case when static or dynamic budgetary
slits G,
constraints (of Walras type, for instance) are imposed. They are indeed sufficient condition
for finding viable equilibria or viable evolutions, but, by no means, necessary37 .
2 [Warning: a priori Budgetary Law and Economic Dysviability] The regulation
map links at each instant many variables: commodities and their velocities, elaboration matrices and their velocities, value and credit, prices and their fluctuations, inflation. If we
choose to single out an arbitrary budgetary rule for governing the endowment, it may happen
that it is not consistent with the regulator which takes into account the inert and kinematic
constraints. In this case, we face the alternative of guaranteeing the viability of the elaborated
commodities and violating the arbitrary budgetary rule, of abide by this budgetary rule and
violate the viability property described by the inert constraints. In other words,
1. maintain the viability of the allocation of elaborated commodities and correct the measure instruments by acting on prices, or providing endowments by adequate credits or
debts;
2. maintain the budgetary rule and trigger the dysviability of the commodities.
This is a political or ideological problem. We personally choose the first solution and to both
abide by the regulation law and maintain the viability of allocation of commodities at the
price of violating ill-designed budgetary rules.
Even though mathematics hints at the possibility of conciliating a monetary policy with
the viability requirements, an operational corollary can be presented. If a monetary policy
does not work, at it has been shown in many historical example, should not stick to it against
all odds.
We illustrate in a symbolic way the regulator
∀ t ∈ [T − Ω, T ], (t, x(t), a(t), p(t), v(t), π(t), x0 (t), a0 (t), p0 (t), v 0 (t), π 0 (t)) ∈
z}|{
R
(37)
37
The book Time and Money. How Long and How Much Money is Needed to Regulate a Viable Economy,
[13, Aubin], deals with this issue for “computing” the endowment that should be provided to an economy for
guaranteeing its viability. It is summarized and completed in [16, Aubin & Chen Luxi], where “averagers”
are used for providing a mathematical definition of the “velocity of money”, which is not a velocity in the
sense of physical sciences, but just the ratio of the product of averages over the average of the products of
two evolutions on a temporal window. For necessary static and dynamic Walras type budgetary laws, see
Mathematical Methods of Game and Economic Theory, [6, Aubin], Dynamic Economic Theory: a Viability
Approach, [10, Aubin].
40
as a budgetary rule
z}|{
∀ t ∈ [T − Ω, T ], (v 0 (t), t, x(t), a(t), p(t), v(t), π(t), x0 (t), a0 (t), p0 (t), π 0 (t)) ∈ Rσ
(38)
under the permutation σ : (t, x, a, p, v, π, x0 , a0 , p0 , v 0 , π 0 ) 7→ (v 0 , t, x, a, p, v, π, x0 , a0 , p0 , π 0 ) expressing the credit e := v 0 as a function of the remaining variables. Since we can represent
graphically only ternary relations, we sacrifice several variables to extract two ternary relations
∀ t ∈ [T − Ω, T ],
(v 0 (t), v(t), p0 (t)) relating credit, value and price fluctuation
(v 0 (t), x(t), p(t)) relating credit, good and price
(39)
For each of those two relations, we shall extract the graphs of two equivalent set-valued maps
obtained by permutation of the variables:
Credit-Value-Price
Fluctuation Relation
Credit-Good-Price Relation
good x
value v
dit
u
val
credit e
Credit
Map
Fluctuator
fluctuation p0
goo
e,
d{
x}
/
dit
cre
credit e
Credit
Regulator
good/price{x, p}
value/fluctuation {v, p0 }
re
e/c
e}
{v ,
Pricer
price p
Figure 10.1 [Ternary Relations and Examples of Two Set-Valued Maps] Since
the credit plays such an important role, we may restrict our attention to the sub relations of
the regulator to three variables by letting aside all the other variables for graphical symbolic
purpose. The two examples involve the credit e := v 0 :
[Left]. The relation links credit e := v 0 , value v and price fluctuation p0 . We may express
the credit as a function of value and price fluctuation (the credit map), or, alternatively,
express price fluctuation in terms of value and credit (the price fluctuator).
41
[Right]. The relation links credit e := v 0 , good x and price p. We may express credit as a
function of good and price (the credit regulator), or the price as a function of good and credit
(the pricer).
Up to a permutation, the graph of the credit map and the fluctuator are the same, so that
they share their “graphical properties” of the credit-value-price fluctuation relation. In the
same way, the graphs of the credit regulator and the pricer are the same and equal to the
relation.
42
Table of Notations
State
commodity
x(t)
price
p(t)
value
v(t) =
p(t)x(t)
Lagrangien
State
elaboration
remuneration
elaborative
value
Lagrangien
V elocity
kinetic
commodity
price
f luctuation
credit
action
V elocity
a(t)
elaborativity
s(t)
remuneration
f lucturation
λ(t) =
elaborativity
s(t)a(t)
value
action
x0 (t)
p0 (t)
v 0 (t) =
+ p0 (t)x(t)
Acceleration
dynamic commodity x00 (t)
???
p00 (t)
???
p(t)x0 (t)
a0 (t)
s0 (t)
λ0 (t) =
+ s(t)a0 (t)
s0 (t)a(t)
regulateur
Acceleration
ef f ort
???
???
???
regulateur
a00 (t)
s00 (t)
43
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44
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45
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Contents
1 From Qualitative Catallaxy to Quantitative Economy
4
2 Evolutionary Commodities
10
3 The Ultimate and Yet Forgotten Goods: Time and Durations
15
4 Inert and Kinematic Relations: The Viability Question
21
5 The Energy
25
6 Economic Actors
26
7 Money : A Desktime Story Told by Children
30
8 Measures of Commodities: Numéraire and Prices
33
9 Joint Viability of Commodities, Actors, Prices and Values
36
10 Are Arbitrary Budgetary Laws Violating Viability of Elaborative Commodities?
38